Mathematical Studies of Pattern Formation and Cooperative Organization in Bacterial Colonies

Thesis Submitted for the Degree ”Doctor of Philosophy”

Inon Cohen

Submitted to the Senate of Tel-Aviv University

2006

The research work for this thesis has been carried out at Tel-Aviv University under the supervision of Professor Zeev Schuss and Professor Eshel Ben-Jacob

This work was carried out under the supervision of Professor Zeev Schuss and Professor Eshel Ben-Jacob.

Acknowledgments

This work is dedicated to the memory of my mother, Ruthy Cohen, how did not live to see its completion.

Foremost, I would like to thank my supervisor and mentor, Eshel Ben-Jacob, for his guidance, patience and support. I have beneﬁted immensely from his profound physical insight, and his inexhaustible curiosity. I am especially grateful for his ability to direct me to interesting problems and their not less interesting solutions.

I would like to thank my supervisor Zeev Schuss for his patience and instructive guidance in advanced mathematics. His vast knowledge was of great help for me.

This dissertation was submitted in a delay of several years due to personal reasons. I would like to apologize to my supervisors for this ineptness, and thank them for not giving up on me.

I had the fortune to learn from Herbert Levine during our collaboration.

I am thankful to David Gutnick for aiding me in the world of micro biology. His vast biological knowledge and experience was of great help for me.

Most importantly, I thank my wife Galit for her ceaseless encouragement and support.

Abstract

The focus of this dissertation is the study of bacterial colonies. In the past, bacterial colonies were usually grown in the laboratory on soft, semi-solid substrates (intermediate agar concentration) with high levels of nutrient. In such ‘friendly’ conditions, bacterial colonies grow in simple compact patterns. Such patterns ﬁt well into the view that bacterial colonies are collections of independent unicellular organisms (‘non-interacting particles’, in physicists’ notation). In nature, bacteria must often cope with hostile environmental conditions. To do so they have developed sophisticated cooperative behavior and intricate communication capabilities, organized within the bacterial colonies. Utilizing such capabilities, the colonies develop complex spatio-temporal patterns in response to adverse growth conditions. When hostile conditions are created in petri-dishes in the laboratory, by using a very low level of nutrients, a hard surface (or both), very complex colonial patterns are often observed (see ﬁgure 1.1 in chapter Introduction).

In this work I construct mathematical description of such complex biological systems and use it to advance our knowledge and understanding of these biological systems, and the principals governing their development. I endorse the approach that the main theoretical tool we have in such study is the formulation of appropriate mathematical models (‘generic models’). By this approach, the models should conform to the known biological facts and capture the essence of the biological system, yet they should not be overloaded with obscure details irrelevant to the biological phenomena under consideration. ii

Once such a model is formulated, and veriﬁed to describe the biological phenomena, it can supply us with understanding of the biological system in two ways. First, the mere ability to distinguish between relevant and irrelevant biological facts is of great importance, because understanding the biological interpretation of the model’s mathematical details show us the biological features essential for the formation of the biological phenomena. Sec- ondly, analysis of the model can be further used to produce predictions about the biological system, such as unknown biological features included in the model, the system’s behavior in untested conditions, or the effect of new elements on the system.

Bacterial colonies present a wealth of beautiful patterns formed during colonial development for different environmental conditions. Invoking ideas from pattern formation in non-living systems and using ‘generic’ modeling we are able to reveal novel survival strategies which account for the salient features of the evolved patterns. Using the models, I demonstrate how communication leads to self-organization of the colonies via cooperative behavior of the cells.

Three types of colonial patterns are studied in the research I present here. Each one of them is formed by a different type of bacteria, and each one of them deserves a unique mathematical treatment in order to explain the development of the bacterial colonies. The ﬁrst type of patterns is tip-splitting branching patterns (see Fig. 1.1(a) ). These patterns are formed by Paenibacillus dendritiformis var. dendron bacteria [213], referred to as T . The colonial patterns are branched, and new branches are formed during development as wide, old branches split into two. The bacteria in the colony perform two-dimensional random walk within the boundaries of the colony.

The second type of patterns is chiral branching patterns (see Fig. 1.1(b) ). These Abstract iii patterns are formed by Paenibacillus dendritiformis var. chiralis [213] bacteria, referred to as C . The branches are narrow, elongated, and twist sideways as they grow (always in the same direction, in all parts of the colony). New branches are side branches of old ones.

The bacteria in the colony perform quasi one-dimensional random walk in the branches, as the branches are too narrow for the bacteria to rotate in.

The third type of patterns is vortex-led branching patterns (see Fig. 1.1(c) ). These patterns are formed by Paenibacillus vortex bacteria [212], referred to as V . They are distinguished by the movement of the bacteria: at the tip of each growing branch (and sometimes also within it), there is group of bacteria rotating in a synchronized motion, like a vortex.

I study each type of bacteria with at least two models. Double-checking with two models can give us more conﬁdence in the biological interpretation of our mathematical claims. The models can be grouped into two wide categories. (1) Discrete models, where the bacteria are represented by discrete entities which perform realizations of a stochastic processes.

Such entities can consume nutrients and produce chemicals, reproduce, perform random or biased movement, and respond to chemicals. The time evolution of the chemicals is described by reaction-diffusion equations. (2) Continuum reaction-diffusion models, with multiple coupled diffusive fields. In these models the microorganisms are represented via their 2D density, and the time evolution of this density is described by reaction-diffusion partial differential equation. The equation of bacterial density is coupled to other reaction- diffusion equations for the chemical fields (e.g. nutrient). In some models the bacterial movement is described also by additional fields, such as velocity field or orientation field.

For the T bacteria, I find that formation of some kind of boundary to the colony is iv essential for the tip-splitting branching patterns, as could be expected. Using a continuum model I show that the boundary could not be described by a phase shift in the reaction term (food consumption, reproduction and death), but rather it is related to lubrication material produced by the bacteria to facilitate their movement on the hard surface. The effect of the lubrication material on the bacterial movement is described by reaction-diffusion equation for the bacterial density field, where the diffusion is non-linear and it is density dependent. The colonial boundary in this case is a singularity line of a weak solution to the resulting system of PDEs. Additional aspects of the patterns of the colonies are derived from chemotaxis of the bacteria (Chemotaxis is a response of the bacteria to gradients of external chemical field, where the bacteria bias their movement and move up or down the gradient). Using a discrete model I show that the governing form of chemotaxis is not food chemotaxis, the most common form of bacterial chemotaxis, but rather a signaling chemotaxis. The bacteria respond to chemicals emitted by other bacteria. Chemical communication between the bacteria leads to cooperative behavior which affects the colony structure.

The T bacteria that build the tip-splitting branching patterns are closely related to the C bacteria that build the chiral branching patterns. Both the continuum and the discrete models show that the crucial different between the two types is the length of the bacteria.

The bacteria that build the chiral branching patterns are much longer then their relatives. Their length restrict their rotation in narrow places. As new, small, branches form, the bacterial rotation is limited. The branches become more elongated and narrower and the bacteria are conﬁned to back-and-forth motion. The chirality of the branches is related to the propeller-like action of the bacterial engine, and its interaction with the surface of the substrate. As with their tip-splitting relatives T , results show that the colonial development Abstract v is affected by chemotaxis. The models also give an additional support to the claim that the T bacteria respond the signaling chemotaxis. In some conditions where nutrient is scarce, the T bacteria become somewhat more elongate than they usually are. The branches of the colony become narrower and somewhat twisted to the side. This phenomenon is referred to as weak chirality. The models for the C bacteria, adapted for elongated T bacteria, show that the chirality is due to bacterial response to signaling chemotaxis.

The V bacteria that form the vortices have a different mode of motion than T or C bacteria. They do not perform a random walk, but rather a forward notion. Models show the vortices are formed due to contact interactions between the bacteria. The interactions couple the movement of neighboring bacteria. In large parts of the parameters space, there are two steady-state solutions to the bacterial motion equations. One is uniform forward motion where all the bacteria move as a group. The other solution is synchronized rotation.

In the last chapter of this work, I show how the models can be used for further research. I deal with the phenomena of bursting of sectors in growing colonies. I show how it can be explained by a small change in the bacteria in the sector, represented by a change of a single parameter. Different representations of the bacteria, by different models, are utilized to describe different types of bursts.

To conclude, the variety of patterns presented by bacterial colonies seems almost end- less due to the response of different bacteria to different environmental factors. Mathe- matical study, like the one presented here, can capture the essential biological features of the biological systems. Construction of appropriate generic models is an important step towards understanding the bacteria. We can be even more conﬁdent in our claims about biology if we double-check it by using several dissimilar models, like the continuum models vi and the discrete models. Contents

1 Introduction 1

1.1 Models Formation vs. Models Study ...... 1

1.2 Motivation to study bacteria ...... 3

1.2.1 Complex patterns of bacterial colonies ...... 6

1.3 Classes of models for bacterial colonies ...... 9

1.4 Outline ...... 12

2 Observations and Biological Background 15

2.1 Introducing Bacteria ...... 15

2.2 Complex bacterial patterns ...... 16

2.2.1 Branching growth of bacterial colonies ...... 18

2.2.2 Chiral patterns ...... 28

2.2.3 Collective migration and formation of vortices ...... 32

2.3 Biological Background ...... 36 2.3.1 Bacterial movement and chemotaxis ...... 36

2.3.2 Bacterial metabolism and growth ...... 46

3 Mathematical Background 49

3.1 Propagating fronts and diffusive instabilities ...... 49

3.1.1 The Fisher-Kolmogorov equation ...... 50

3.1.2 Propagation in two-dimensional space ...... 54

3.1.3 Linear stability analysis of two ﬁelds systems – the DFK model . . 57

3.1.4 Meta-stable node and diffusive instability ...... 61

3.1.5 Non-constant diffusion Coefﬁcient ...... 65

3.1.6 Pattern selection ...... 71

3.2 Discrete Models ...... 76

4 Reaction-Diffusion Models for Branching Patterns 79

4.1 Comparison between models ...... 79

4.1.1 Linear diffusion – effective meta-stable reaction term ...... 82

4.1.2 Nonlinear diffusion ...... 92

4.2 Chemotaxis and colonial self-organization ...... 101

4.2.1 Food chemotaxis and chemotactic signaling ...... 101

4.2.2 Weak chirality ...... 105 5 Analysis of Models with Non-Linear Diffusion 111

5.1 Existence and uniqueness of sharp-front traveling wave solutions ...... 112

5.1.1 Local analysis ...... 116

5.1.2 Existence of a continuum of smooth front solutions ...... 117

5.1.3 Uniqueness of sharp type solution ...... 121

5.1.4 Existence of sharp type solution ...... 124

6 Mean Orientation Field Model for Chiral Patterns 129

6.1 Modeling chiral growth ...... 132

6.1.1 Orientation Dimension ...... 132

6.1.2 Mean Orientation Field ...... 136

6.1.3 Chiral branching patterns ...... 138

6.2 Chemotaxis and weak chirality ...... 141

7 Atomistic Models for Branching Patterns 147

7.1 The Communicating Walkers model ...... 150

7.1.1 The walkers, the boundary and local interaction ...... 150

7.1.2 Food consumption, internal energy, reproduction and sporulation . . 152

7.1.3 Results of numerical simulations ...... 153

7.1.4 Chemotaxis-based adaptive self-organization ...... 156

7.2 Orientation ﬁeld and the Communicating Spinors Model ...... 165 7.2.1 Oriented particles and orientation ﬁeld ...... 166

7.2.2 Chemotaxis in the Communicating Spinors model ...... 173

7.2.3 Weak chirality in the Communicating Spinors model ...... 176

8 Vortex Formation 179

8.1 Atomistic model ...... 182

8.2 continuum model ...... 189

9 Mutants Spread in Colonies 193

9.1 Transitions and sectors formation in bacterial colonies ...... 193

9.1.1 Observations of colonial development ...... 196

9.2 Modeling mutants and mutations ...... 199

9.2.1 Single cell densities in the Non-Linear Diffusion model ...... 201

9.2.2 Variable group size in the Communication Walkers model . . . . . 205

9.3 Modeling formation of sectors ...... 210

9.3.1 Mutation during compact growth ...... 210

9.3.2 Mutation during Branching growth ...... 211

9.3.3 The effect of chemotaxis ...... 212

9.4 Mutants Spread in Colonies – Conclusions ...... 214

10 Discussion 217 Appendices 225

A Biological Units and Dimensionless Equations 225

A.1 The Fisher-Kolmogorov equations ...... 225

A.1.1 Deriving the equations ...... 225

A.1.2 Evaluation of the parameters ...... 227

A.2 Reaction-diffusion models for branching patterns ...... 229

A.2.1 A cutoff in the reaction term ...... 229

A.2.2 The model of Mimura at al...... 229

A.3 Atomistic models for branching patterns ...... 230

Bibliography 233

Chapter 1

Introduction

The study presented in this dissertation was done within the department for applied mathematics. The mathematics is applied here is as a tool for the study of biological systems, an approach that is sometimes referred to as mathematical biology. In effect, the subject of this dissertation is the construction of mathematical models for biological systems (bacterial colonies, in this case), and the biological conclusions that can be drawn from the models. The mathematical analysis of a model, as interesting as it may be, is irrelevant to the subject if the model does not ﬁt the biological system, or if no conclusions (preferably predictions) about the biological system can not be drawn from it.

1.1 Models Formation vs. Models Study

In the field of mathematical biology (not unlike the field of theoretical physics), mathematicians tend to work in one of two typical modes. Some mathematicians prefer to work in a ”specialization” approach: they let the experts of the field (biology, in our case) formolate a suitable mathematical model, and than they study and analyze the model as a pure mathematical entity. Other mathematicians prefer to work in ”generalization” approach: they

1 2 study the ﬁeld themselves, formulate mathematical models, study them, and change them as the research progresses. (Of cause the actual mode of work is always some mixture of the two approaches, and different members of the same research team may approach the research in different ways).

The pros of the specialization approach are that each expert can deepen is mastering in her ﬁeld. At its best, this approach leads to a cross-disciplinary teamwork. The cons of the specialization approach are the detachment of each expert from the other’s discipline. The models that a biology expert can construct cannot be more complex than her knowledge of mathematics, and both the mathematician and the biologist might miss the biological implications of the mathematical results.

The cons of the generalization approach are that it requires the mathematician to practically become an expert in biology (or at least in her subject of interest), with deep understanding both in mathematics and biology. The pros of this approach are that the feedback between mathematics and biology is immediate and profound. Any biological detail or relation can trigger a related change in a model, or can even trigger a switch to a new class of models. Any biological implication of the model can be a base for model rejection or for a new biological interest.

My work lends towards the generalization approach. I ﬁnd it imperative for the model constructor to have immediate grasp of the possible mathematical models, and for the mathematician studying the model to have a through understanding of the biological interpretation of all the parts of the model and all the outcomes of the model’s analysis.

This dissertation presents an effort to study bacterial colonies using mathematical models. In this context, mathematical understanding of the models is not a purpose of its own; CHAPTER 1 – INTRODUCTION 3 its serves the goal to study bacterial colonies. In fact, I devote here more space and more elaboration in the development of useful proper models than to their analysis. Proper models are models which their details agree with the known biological facts (or at least do not contradict them), and the models can reproduce many of the global features of the colony, and the models can be studies analytically or numerically. Useful models are models which contains as few details as possible (to ease the understanding and study of the models), and if possible, can produce some previously unknown result – a testable prediction. A model which does not produce previously unknown result is useful if it includes the claim that it is a minimal model – it implies the claim that all the biological details incorporated into the model are required to produce the observed phenomenon. This later claim is also a testable prediction.

1.2 Motivation to study bacteria

This dissertation considers mathematical problems arising in pattern formation in certain biological systems. It provides theoretical support to the experimental investigations of bacterial colonies carried out in the laboratory of Prof. Eshel Ben-Jacob. The research motivation is to use patterns and bacterial dynamics as hints to the principles of organization and cooperation in bacterial colonies. Mathematical modeling of the colonial patterns observed in the laboratory has been initiated in [35, 34, 60], continued in my M.Sc. thesis and in many publications, and this work is a continuation of this effort.

To clarify the issue at hand, Fig. 1.1 shows three representative patterns of bacterial colonies grown in Petri dishes (see chapter 2 for details). The three patterns were created by three different strains of bacteria, and they are manifestation of the different properties and 4 behavior of the different bacteria. The driving forces of the colonial pattern formation are the propagation of the bacteria on the surface, and the limitation on bacterial reproduction imposed by diffusion of nutrients.

Many systems display the emergence of interfacial patterns during diffusive growth, ranging from the growth of snowflakes to the aggregation of a soot particle, from oil re- covery by fluid injection to solidification of metals and the formation of a coral reef. Other patterns emerge in the bulk of systems driven out of equilibrium [67, 71], such as spiral waves in Belusov-Zhabotinsky reactions, Liesegang rings in reaction-diffusion systems, and fluid rolls convection cells in Rayleigh-Benard convection. Some of the colonial patterns are reminiscent of branching patterns observed in many such physical systems, yet there is no a-priori reason to assume that similar mechanisms generate the patterns in the so different systems, and there is no a-priori reason to assume that the same mathematical description fit the different systems. The microscopic description of the mentioned non-living systems gets down to atoms, while the “elementary particles” of the colonies are bacteria. Many properties distinguish bacteria as particles from atoms, not the least of which is self-propulsion of the bacteria, bacterial reproduction and mortality, and changes in the environment actively induced by the bacteria.

Among non-equilibrium dynamical systems, living organisms are the most challenging ones that scientists can study. A biological system constantly exchanges material, energy and information with the environment as it regulates its growth and survival. The energy and chemical balances at the cellular level involve an intricate interplay between the microscopic dynamics and the macroscopic environment, through which life at the intermediate mesoscopic scale is maintained [3]. The development of a multicellular structure requires non-equilibrium dynamics, as microscopic imbalances are translated into the macroscopic CHAPTER 1 – INTRODUCTION 5

(a) (b)

(c)

Figure 1.1: Three characteristic branching patterns of bacterial colonies discussed in this work: (a) Simple branching (tip-splitting) exhibited by T morphotype (Paenibacillus dendritiformis var. dendron). The colonial patterns are branched, and new branches are formed as wide, old branches split into two. (b) Chiral branching pattern (chiral dendritic branching) exhibited by C morphotype (Paenibacillus dendritiformis var. chiralis). The branches are narrow, elongated, and twist sideways as they grow (in the same direction in all parts of the colony). New branches are side branches of old ones. (c) Branching pattern created by rotating vortices of V morphotype (Paenibacillus vortex). At the tip of every growing branch (and sometimes also within it), there is group of bacteria rotating in a synchronized motion, like a vortex. New branches are side branches of old ones, formed as new vortices break through the boundaries of old branches. The colonies are grown in Petri dishes with (a) 0.01 g/l peptone and 1.75% agar concentration (b) 1.4 g/l peptone and 0.75% agar concentration, and (c) 7.5 g/l peptone and 2.25% agar concentration 6 gradients that control collective action and growth [182].

A tremendous amount of effort by many researchers is devoted to the search for basic principles of organization (growth, communication, regulation and control) on the cellular and multicellular levels [189, 122, 197, 96, 176, 201, 119, 230]. Under the advise of my supervisors, my approach is to use the successful conceptual framework for pattern formation in non-living systems as a tool to unravel their significantly more complex biological counterparts. Of critical importance is the choice of starting point, i.e. the choice of which phenomena to study: it has to be simple enough to allow progress, but also well motivated by the significance of the results. Cooperative organization is bacterial colonies is well suited for these requirements, as I explain below. The bacterial colonies are described in detail in chapter 2. Of course I hope that my findings will be generalized to other systems, but I believe that specific and detailed work must precede general insight in order to avoid meaningless general statements.

1.2.1 Complex patterns of bacterial colonies

The bacteria I study propel themselves by rotating flagella – long helical filaments that extend outside of the bacterial cell and are rotated by molecular engines [76]. Some of the bacteria (like T and C morphotype ) use the flagella [145] to perform random-walk-like motion (swimming): straight runs interrupted by short periods of random rotation called tumbling [41]. Note that unlike Brownian motion, the duration and length of the steps are determined by the bacteria and not the by environment. In the experiments of Ben-Jacob’s group, the bacteria swim on the surface of a semi-solid substrate within a lubrication fluid

[104], which they produce themselves [156]. Microscopic observations reveal that the lubrication has a wetting-like appearance (finite angle of contact with the surface) and CHAPTER 1 – INTRODUCTION 7 it confines the bacterial movement. The bacterial motion can be biased so as to follow gradients of chemical fields and reach locations of higher (or lower) concentrations [1]. Bacteria consume nutrients from the media in order to reproduce [5]. Reproduction in bacteria has a maximal rate and it is done by division of the cell into two identical daughter cells [105, 88]. The experiments show that a single bacterium can reproduce into a fully- grown colony. This indicates that in a continuum description, the growth process is unstable near zero density. When food is deficient and the bacteria are starved for long time, they shift to immobile stationary phase (spores) that enables them to survive for longer time [105].

Traditionally, bacterial colonies are grown on substrates with a high nutrient level and intermediate agar concentration [105]. Such “friendly” conditions yield colonies of simple compact patterns, which ﬁt well the traditional view of bacterial colonies as a collec- tion of independent unicellular organisms (non-interacting “particles”). However, bacterial colonies in nature must regularly cope with hostile environmental conditions [197, 210]. When hostile conditions are created in a Petri dish by using a very low level of nutrients, a hard surface (high concentration of agar), or both, very complex patterns are observed (Figs. 1.1, 2.2).

Drawing on the analogy with diffusive patterning in non-living systems [117, 138, 28, 17, 18], one can argue that complex patterns are to be expected. The cellular reproduction rate that determines the growth rate of the colony is limited by the level of nutrients available for the cells. The latter is limited by the diffusion of nutrients towards the colony (for low nutrient substrate). Hence colony growth under certain conditions should be similar to diffusion limited growth in non-living systems [28, 17, 18]. The study of diffusive patterning in non-living systems teaches us that the diffusion ﬁeld drives the system towards 8 decorated (on many length scales) irregular fractal shapes [150, 149, 193, 222, 80]. In- deed, bacterial colonies can develop patterns reminiscent of those observed during growth in non-living systems [85, 151, 86, 153, 33, 30, 37, 32, 155, 157].

But, this is certainly not the end of the story. In fact, the colonies exhibit far richer behavior. This is ultimately a reﬂection of the additional levels of complexity involved

[30, 37, 35, 34, 26, 27, 68, 22]. The building blocks of the colonies are themselves living systems, each having its own autonomous self-interest and internal degrees of freedom. Yet, efﬁcient adaptation of the colony to adverse growth conditions requires self- organization on all levels, self-organization that can only be achieved via cooperative behavior of these individual cells. Thus, pattern formation at the colony level may be viewed as the outcome of a dynamical interplay [30, 37, 32] between the micro-level (the individual cell) and the macro-level (the colony). For this interplay to work, the effects of changes at the micro-level must make themselves felt at the macro-level.

To achieve the required level of cooperation, bacteria have developed various communication capabilities, such as: (A) direct cell-cell physical and chemical interactions [161, 72], (B) indirect physical and chemical interactions, e.g., production of extracellular “wetting” ﬂuid [156, 98], (C) long range chemical signaling, such as quorum sensing [87, 140],and (D) chemotactic response to chemical agents which are emitted by the cells [53, 44, 54]. The communication capabilities enable each cell to be both an “actor” and a

“spectator” (using Bohr’s expressions) during the complex patterning.

Another important point to keep in mind is that cells can change their behavior both by direct biochemical means and by regulating the expression of their genes; that is, by preferentially turning on and off certain genes in response to external signals. And gene CHAPTER 1 – INTRODUCTION 9 products themselves are often arranged as a sort of computational network allowing the expression of one gene to control the expression of other interconnected ones [143]. For researchers in the pattern formation ﬁeld, the above communication, regulation and control mechanisms open a new class of tantalizing complex models exhibiting a much richer spectrum of patterns than the models for non-living systems. And, knowing when we have the correct explanation for an experimental fact in a very complex system as opposed to merely a possible explanation will remain an exciting but difﬁcult challenge.

1.3 Classes of models for bacterial colonies

In the study of 2D patterns generated by propagating fronts in non-living systems, several modeling approaches to handle global morphology have been proposed. Stefan-like models [28, 17] include an explicit boundary separating two regimes of diffusive fields. Phase-field-like models (Landau-Ginzburg models) [131, 17] use only continuous fields to describe the system. The front in such models connects a stable phase to a meta-stable one, i.e. the growth term is bi-stable. In the limit of vanishing front width, the front can be replaced by an explicit boundary, and the model is reduced to a Stefan-like one [131]. Atomistic models, such as DLA [231, 193], use particles moving stochastically (random walk) to describe molecules in a solution. In this approach solid matter is represented by stationary particles. DLA stands for diffusion limited aggregation, as particles from the solution aggregate to form the solid.

Preliminary attempts to model bacterial colonies were done by Ben-Jacob et. al. [30,

37, 33] and Matsushita et. al. [151, 86, 153]. While Matsushita et. al. [153] attempted to measure growth parameters using Fisher-Kolmogorov equations, Ben-Jacob et. al. [33] 10 showed the limitation of this model. All the terms in this equation (diffusion term and a growth term which is unstable at zero density) agree with the microscopic bacterial details, but such models cannot produce the macroscopic branching patterns. Matsushita and Fu- jikawa [151, 86] suggested a DLA model to describe the colony. This model can reproduce the global structure of the colony. However, the model has moving particles outside the aggregate, unlike the colony where the moving bacteria are inside the colony. Ben-Jacob et. al. [33] suggested that a phase-ﬁeld-like model, with bi-stable growth term, can reproduce the global branching pattern. Bi-stable growth term does not agree with the details of the bacterial reproduction process, nor does it agree with the experiments (the growth process should be unstable near zero density, see above).

It is well worth emphasizing the beneficial aspects of having a connection between a discrete model and a related continuum model. It is usually difficult to do much beyond simulation for a discrete model; so, having a continuum analog allows for analysis that helps guide the simulations and vice versa. For the DLA class of models, the relationship between these automata and the continuous approach to crystal growth as captured in the phase field model (and the free surface reduction thereof) has proven invaluable. Once the basics are understood, of course, one can modify the simulation to encompass more details of the actual system and thereby obtain more reliable results. The few paragraphs from the beginning of this section demonstrate that the construction of the ”right” model (as defined is section 1.1) is usually the most significant step in a research effort.

There is also a literature on using discrete models for other, more complex reaction- diffusion processes; see for example the work by Kapral et al. [148] on simulations of 3D knotted labyrinths. In real reaction-diffusion processes, it is almost always the case that one is using the discrete model as a simpler stand-in for the true continuum dynamics; after CHAPTER 1 – INTRODUCTION 11 all, one cannot hope to match the actual number of molecules (in the order of Avogadro number, 1023) by discrete simulational entities, and the number of particles is enough such that a continuum description is valid (recall, though, the cutoff effect for type I systems, see Sec. 3.1.3). Using a small number of particles in the simulation as a stand-in introduces extra noise into the simulation, and this is the price one pays for a more flexible and more efficiently-coded numerical scheme. Similar remarks hold for lattice-gas automata [45], in which one uses discrete objects to model systems with fluid flow.

In biological multicellular systems, computational convenience is not the only reason why one can make good use of discrete entities. First, the numbers match more closely. The number of bacteria in a typical experiment is 109; one can almost approach these numbers computationally and therefore one is not plagued by the extra noise issue. Perhaps more importantly, cells contain large numbers of internal degrees of freedom which modulate their response to external signals form other cells. Hence, describing a population of cells with something as non-informative as a density field is usually insufficient. At the very least, one would have to introduce either new variables (which advect with the cell velocity as these are tied to the cells, see Sec. 8.2 and the Mean Orientation Field model in Sec. 6.1) or even new coordinates (see a model in Ref. [78], where the cell’s age is taken as a relevant coordinate for the density field, or the Orientation Dimension model in Sec. 6.1 where the cells’ orientation is taken as a relevant coordinate for the density field); this makes for “ugly” continuum equations. Tracking cells as individual objects makes it easy to add internal degrees of freedom; one just attach extra labels to the cell and postulate transition rules as to how these labels change in time. This flexibility is, I feel, quite useful and hence many of the models to be discussed keep cells discrete. At the same time, though, continuum analysis is used to shed light on the simulations, and it forms an indispensable 12 part of an integrated effort to understand microbiological pattern formation.

1.4 Outline

The outline of this dissertation is as follows: due the interdisciplinary nature of the subject of the dissertation, I decided to break the introduction of the dissertation into three different chapters. This chapter covers general topics of the introduction. In chapter 2 I present the problem to be studied, namely bacteria and bacterial colonies. I present known characteristic of bacteria, bacterial colonies and the behavior of bacteria in bacterial colonies. Since the material is too vast too be covered in this scope, I focus on aspects relevant to this dissertation. The material is presented mainly from biological perspective, with some deviation to mathematical description of facts presented. In chapter 3 I present a review of mathematics of pattern formation in general, with emphasize on aspects relevant to this dissertation.

Chapter 4 concentrates on reaction-diffusion models for branching bacterial colonies. In this chapter I examine several models presented in the literature as models for simple branching bacterial colonies (as of the patterns of Paenibacillus dendritiformis var. dendron). I do not examine all the models capable of reconstructing branching patterns, only models that were introduced for the study bacterial colonies. The models are compared in terms of their ability to reproduce the observations, the validity of their biological interpretation and the simplicity of the mathematical description. An important test for the models is their ability to incorporate chemotaxis and chemotactic signaling into the description of the colonies. Chemotaxis is a response of the bacteria to gradients of external chemical ﬁeld, where the bacteria bias their movement and move up or down the gradient. Chemo- CHAPTER 1 – INTRODUCTION 13 tactic signaling is a chemotactic response of the bacteria to a chemical ﬁeld created by other bacteria, and it is an important form of communication in bacterial colonies.

Chapter 5 does not present any new model. In this chapter I try to analyze several properties of models presented in chapter 4. Specifically I prove the existence and uniqueness of weak solution of propagating front for reaction-diffusion system with density-dependent diffusion coefficient. I prove that this weak solution can be approximated by numerical schemes that were developed for finding smooth solutions of reaction-diffusion systems.

Chapter 6 presents a continuum model for chiral branching bacterial colonies. The colonies, created by Paenibacillus dendritiformis var. chiralis, exhibit curved dendritic branches. The model is reaction-diffusion system with density-dependent diffusion coefﬁ- cient. The diffusion is anisotropic in two dimensions, where the anisotropy is determined locally by the dynamics of the model. An additional complex ﬁeld, that represents the mean local orientation of the bacteria, determines the anisotropy. The orientation interactions between the bacteria seem to facilitate the chiral dendritic growth, which enables easy propagation of the colony’s branches and coverage of the surface with low mean density of bacteria. These traits of the colony are favorable in the conditions imposed on the bacteria, conditions of hard growth surface and low nutrient concentration.

Chapter 7 presents atomistic models for branching bacterial colonies. Its ﬁrst section present the Communicating Walkers model for tip-splitting branching patterns (like the colonies of P. dendritiformis var. dendron ). This model was presented in many publications, including my M.Sc. thesis, and it is presented here only as a foundation for other studies presented in this dissertation. Section 7.2 presents one such study. It presents an atomistic model for chiral branching colonies patterns (like the colonies of P.dendritiformis 14 var. chiralis ). Unlike the Communicating Walkers model and unlike the continuum models, this model includes realistic description of the movement of the bacteria. With the aid this model I try to understand the forces acting on a bacteria in a chiral branching colony and the interactions between the bacteria that facilitate orientation-mediated cooperation in the formation of chiral dendritic colonies.

Chapter 8 models different kind of colonial growth, the colonies of Paenibacillus vortex. The main feature of these bacteria is their ability to coordinate their movement into coherent vortices. Such vortices can act like circular saw and pave the way for other bacteria. I present both atomistic and continuum models that show how intermediate-range attraction between bacteria can lead to the formation of such vortices. Unlike previous models that have been presented in the literature, the models presented in chapter 8 do not assume restricting boundary conditions or revolving movement terms.

Chapter 9 presents a study in the expression of mutations in bacterial colonies. It concentrates on the geometric manifestation of different kind of mutations in different colonies.

In a sense it is an application of the research in earlier chapters to a new biological problem. The study employs both the continuum reaction-diffusion model and the atomistic walkers’ model as complementary research tools. Both models are modiﬁed to accommodate for a mutation of a single bacterium. The study demonstrates that in stressful conditions, where corporation between bacteria is needed, mutations can be maintained and expressed more easily than in favorable conditions.

Chapter 10 is a discussion. It summarizes the entire research, put it in a broader context of cooperation and feedback loop in bacterial colonies, and suggests possible directions for future research. Chapter 2

Observations and Biological Background

In this chapter we elaborate on the phenomena of colonial pattern formation, and we present a brief summary of the existing biological knowledge relevant to the colonial patterning and its modeling. The main interest of this thesis is colonies of bacteria of strains Paenibacil- lus dendritiformis and Paenibacillus vortex, which can produce beautiful and interesting branching patterns (see Fig. 1.1 in the introduction). Some of the material presented here has been published in Refs. [34, 35, 60, 20, 21, 25, 24, 62].

2.1 Introducing Bacteria

Bacteria, or prokaryotes, are one of the five kingdoms of living organism, distinct from eukaryotes: animals, plants, fungi and protoctists. They are the first living organisms to appear on earth, and for more than three billion years they were the only existing living organisms. Even today life on earth could not have existed without the support of bacteria. The other name for bacteria – prokaryotes – reflects the lack of internal nucleus encapsu-

15 16 lating the DNA, a nucleus present in eukaryotic cells. This lack of nucleus is a symptom to the lack of internal spatial organization in the bacterial cell. Usually bacteria are considered more “primitive” then eukaryotes, which have sub-cellular spatial organization, as well as multi-cellular spatial and functional organization. While the ﬁrst of these characteristics clearly distinguish between the two groups, the second one is less clear. Most types of bacteria can live as single-celled organisms, but many can create multi-cellular functional and spatial organizations [189, 109, 123, 53, 132, 58, 30, 37, 200, 22, 63, 66, 4].

One of major divisions of the kingdom of bacteria is according to the shape of the cells. The P. dendritiformis bacteria belong to a group of rod-shaped bacteria – Bacilli. The size of a P. dendritiformis bacterium is about 1µm in diameter and 2−10µm long (Fig.

2.1). About 5-10 filaments called flagella emerge from random locations at the sides of the body. The flagella extend into the surrounding media for 5 − 10µm. They are rotated by a molecular engines, and their function is to propel the bacterium (see more details in the

Sec. 2.3.1).

2.2 Complex bacterial patterns

In 1989, a Japanese group [85, 151, 86] reported for the ﬁrst time that bacterial colonies can grow elaborate branching patterns of the type familiar from the study of fractal formation in the process of diffusion limited aggregation (DLA). This work was done with Bacillus subtilis, but was subsequently extended to other bacterial species such as Escherichia coli and Salmonella typhimurium. They showed explicitly that nutrient diffusion was the relevant dynamics responsible for the instability; later, we will see how models which couple nutrient diffusion to bacterial density can naturally account for these structures. CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 17

Figure 2.1: Electron Microscope pictures of a P. dendritiformis bacterium with its ﬂagella (the long ﬁlaments surrounding the bacterium).

Motivated by these observations, Ben-Jacob et al. [30, 37] conducted new experiments to see how adaptive the bacterial colonies could be in the presence of external “pressure”, here in the form of a limited nutrient supply and hard surface. The endeavor started with

B. subtilis 168 (which is non-motile on a solid agar surface), from which a new species of bacteria – Paenibacillus dendritiformis – has been isolated [30, 37, 213]. This new species is motile on the hard surface and its colonies exhibit branching patterns (Fig. 1.1). This new mode of tip-splitting growth was found to be inheritable and transferable by a single cell [30, 37], hence it is referred to as a distinctive morphotype, and, to indicate the tip-splitting character of the growth, it was denoted T morphotype.

In ﬁgure 1.1 we also show patterns developed by two additional distinct morphotypes.

One is characterized by a strong twist (of a specific handedness) of the branches of its colonies [37, 27]. We refer to this property as strong chirality. The bacteria was identified to 18 belong to the same species, P. dendritiformis [213], and we name it C (chiral) morphotype. The most noticeable character of another morphotype is its ability to form vortices, hence we refer to it as V (Vortex) morphotype. The V bacteria was identified as a new species, Paenibacillus vortex [212].

Each of the morphotypes exhibits its own profusion of patterns as the growth conditions are varied. These beautiful complex shapes reﬂect sophisticated strategies employed by the bacteria for cooperative self-organization as they cope with unfavorable growth conditions.

As we will describe in this thesis, lengthy and close inspection of the evolved patterns, combined with understanding gained from the study of patterning in azoic systems and the use of the generic modeling approach, has led to signiﬁcant progress in unraveling these novel biological patterns.

2.2.1 Branching growth of bacterial colonies

2.2.1.1 Macroscopic Observations

Some examples of the patterns exhibited by colonies of the T morphotype are shown in Fig. 2.2. All the colonies in this example grow at intermediate agar concentrations (about 1.5% – 1.5g in 100ml). At very high peptone levels (above 10g/l) the patterns are compact. At somewhat lower but still high peptone levels (about 5 − 10g/l) the patterns are reminiscent of viscous fingering patterns in Hele-Shaw devices [38, 106, 117]; they exhibit quite pronounced radial symmetry and may be characterized as dense fingers, each finger being much wider than the distance between fingers. For intermediate peptone levels, branching patterns with lower fractal dimension (reminiscent of electro-chemical deposition) are observed. The patterns are “bushy”, with branch width smaller than the distance between CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 19 branches. As the peptone level is lowered, the patterns become more ramified and fractal– like. Surprisingly, at even lower peptone levels (below 0.25g/l for 2% agar concentration) the colonies revert to organized structures: fine branches forming a well-defined global envelope. We characterize these patterns as fine radial branches. For extremely low peptone levels (below 0.1g/l), the colonies lose the fine radial structure and again exhibit fractal patterns. For high agar concentration the branches are very thin.

Figure 2.2: Examples of patterns exhibited by bacterial colonies of T morphotype on the surface of a semi-solid agar. The figures (a)-(d) are for decreasing amounts of food in the substrate. (a) Shows dense fingers at high concentrations of food, (b) and (c) show fractal-like branching patterns for intermediate concentrations, and (d) shows fine radial branches.

At high agar concentration and very high peptone levels the colonies display a structure of concentric rings (Fig. 2.4). At high agar concentrations the branches also exhibit a global twist with the same handedness, as shown in Fig. 2.3. Similar observations during 20 growth of Bacillus subtilis been reported by Matsuyama et al. [157, 155]. We refer to such growth patterns as having weak chirality, as opposed to the strong chirality exhibited by the C morphotype.

Figure 2.3: Weak (global) chirality exhibited by the T morphotype during growth on 4g/l peptone and 2.5% agar concentration.

A closer look at an individual branch (Fig. 2.5) reveals a phenomenon of density variations within the branches. These 3-dimensional structures arise from accumulation of cells in layers. The aggregates can form spots and ridges which are either scattered randomly, ordered in rows, or organized in a leaf-veins-like structure. The aggregates are not frozen; the cells in them are motile and the aggregates are dynamically maintained.

2.2.1.2 Microscopic Observations

Looking through the microscope at colonies of T morphotype, one can see cells performing a random-walk-like movement in a fluid. We assume that this lubrication fluid is excreted by the cells and/or drawn by the cells from the agar [35, 34]. The cellular movement is confined to this fluid; isolated cells spotted on the agar surface do not move. The boundary CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 21

Figure 2.4: A structure of concentric rings “superimposed” on a branched colony. The rings structure is expressed in the entire colony, regardless of branches and clustering of branches.

of the fluid thus defines a local boundary for the branch (Fig. 2.6). Whenever the cells are active, the boundary propagates slowly as a result of the cellular movement pushing the envelope forward and production of additional wetting fluid. Electron microscope observations reveal that these bacteria have flagella for swimming.

The observations reveal also that the cells are active at the outer parts of the colony, while closer to the center the cells are stationary and some of them sporulate (form spores – see Fig. 2.7). It is known that certain bacteria respond to adverse growth conditions by entering a spore stage until more favorable growth conditions return. Such spores are metabolically inert and exhibit a marked resistance to the lethal effects of heat, drying, freezing, deleterious chemicals, and radiation.

At very low agar concentrations (below 0.5%) the bacteria swim inside the agar and not on its surface. Between 0.5% and 1% agar concentration some of the bacteria move on the surface and some inside the agar. 22

Figure 2.5: A structure of ordered aggregates within branches. The picture shows variation in the height of the branches. The more bacteria are in a unit area, the more layers the bacteria are in, and the higher the area seems. CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 23

Figure 2.6: Closer look on the tip of one branch of a T colony. (a) The branch as deﬁned by the ﬂuid layer. (b) The bacteria within the branch.

Figure 2.7: Electron microscope observation of T bacteria. Round or oval shapes with a bright center are spores. Elongated shapes are living cells. The cells engulﬁng oval shapes are pre-spores. 24

2.2.1.3 Morphology Selection, Morphology Diagram and Velocity-Pattern Correla- tions

The emerging understanding of pattern determination in non-living systems includes the concepts of morphology diagram, morphology selection, morphology velocity correlations and morphology transitions [29, 28, 17, 206, 207, 205, 129]. In short, the patterns formed in many non-living evolving systems may often be grouped into a small number of “essential shapes” or morphologies, each representing a dominance of a different underlying effect. If each morphology is observed over a range of growth conditions, a morphology diagram may exist. The existence of a morphology diagram implies the existence of a morphology selection principle and vice versa. Ben-Jacob et al. [29, 28] proposed the existence of a new morphology selection principle: the principle of the fastest growing morphology, a principle which should be applicable for a wide range of growth conditions. In general, if more than one morphology is a possible solution, only the fastest growing one is nonlinearly stable and will be observed, i.e. selected.

The selection principle implies that the average velocity is an appropriate response function for describing the growth processes and hence should be correlated with the geometrical character of the growth. In other words, for each regime (essential shape) in the morphology diagram, there is a characteristic functional dependence of the velocity on the growth parameters. At the boundaries between the regimes there is either discontinuity in the velocity (ﬁrst order-like transition) or in its slope (second order-like transition).

At present, there are many examples which support the existence of the selection principle in non-living systems. Yet there are also possible counter-examples [14]. The new principle might also be valid for pattern determination during colonial development in bac- CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 25 teria [37, 18]. The bacterial patterns may be grouped into a small number of “essential shapes”, each observed over a range of growth conditions [85, 86, 30, 37, 164]. To prove this hypothesis, the next step would be to demonstrate the velocity-pattern correlation during colonial growth.

Ideally, the measurements of the growth velocity should be performed under constant growth conditions. However, in practice, due to humidity variations, finite size and edge effects, conditions do vary during growth [30, 37]. This introduces some uncertainty in the mapping of the morphology diagram. In the experiments presented in Ref. [37] the morphology is identified and its growth velocity is measured when the colony fills about half of the petri dish.

In Fig. 2.8 we show a plot of the growth velocity as a function of nutrient level for

1.5% agar concentration. For the presented range of peptone levels it was found that the velocity shows three distinct regimes of response, each corresponding also to a distinct morphology (the fine radial branches, branching patterns and dense fingers, see Fig. 2.2, pictures (d), (c) and (a) respectively), as was predicted for non-living systems. The change in velocity suggests that the switching between morphologies is indeed a real morphology transition and not a simple cross-over (see Ref. [17]). The transition at low peptone level (between fine radial branches and branching structure) might be a first order morphology transition, i.e. a transition characterized by a jump in the velocity and hysteresis. The transition at the higher peptone level (from branching to dense fingering) seems to be a second order morphology transition. These observations of velocity-pattern correlations strongly support the existence of a morphology selection principle which determines the selected colonial morphology for a given morphotype. The work of Wakano et al. [226] lends some theoretical support for this graph, although their model does not exactly match 26 the observed patterns (at infinite time all the gaps between braches close).

Figure 2.8: Colonial growth velocity of T colonies as a function of nutrient level. Each data point is an average over ﬁve plates. The agar concentration in all plates is 1.5%.

In non-evolving (equilibrium) systems there is a phenomenon of critical ﬂuctuations when the system is kept at the transition point between two phases. At that point the system consists of a mixture of the two phases. In Ref. [17] it was shown that an analogous phenomenon exists in evolving non-living systems and it was explained that this fact provides additional support for the idea of morphology transitions. Fig. 2.9 shows patterns exhibited by colonies grown at “critical” peptone levels, where transitions between two morphologies occur. Similarly, for the ﬂuctuations displayed by non-living systems [17, 25], we observe a combination of the morphologies characterizing the patterns above and below the critical point. These observations provide additional support for the relevance of the concepts of morphology selection and morphology transition to colonial development. CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 27

Figure 2.9: Coexistence of two morphologies near the .The colony shows combinations of the ﬁne radial branches and bursts of faster fractal sectors.

2.2.1.4 Branching colonial patterns of other bacteria

Matsushita and co-workers [151, 85, 86, 155, 158] studied in detail the colonial branching patterns and morphology diagram of the bacteria B. subtilis OG-01. A typical morphology diagram is shown in Fig. 2.10. Note that here the horizontal axis is the inverse agar concentration and the vertical axis is the nutrient level. These bacteria are not efﬁcient in producing a lubricating ﬂuid, hence above about 0.8% agar concentration they can not move on the agar surface. At such conditions that the bacteria cannot move and low level of nutrients (below 1g/l peptone), DLA-like patterns are observed. As the level of nutrients is increased, the patterns become compact, with a cellular structure at the interface.

For low agar conditions (so that the bacteria can move) and low level of nutrients, dense branching patterns are observed. These patterns are replaced by compact growth for higher levels of nutrients. 28

Figure 2.10: Morphology diagram of B. subtilis colonies grown by Matsushita et al. [85, 151, 155]. Taken with permission from [183].

Beautiful patterns of concentric rings imposed on a dense branching growth are observed at high levels of nutrients and intermediate agar concentration (about 0.75%) (Fig.

2.11). For more details see Refs. [183].

Branching patterns of lubricating bacteria (strains which are capable of producing a wetting ﬂuid) Serratia marcescens and Salmonella anatum are reported by Matsuyama et al. [156, 157]. They also show that a strain defective in the production of a wetting ﬂuid exhibits weak chirality, similar to that exhibited by the T morphotype grown on a hard surface. We will return to this point in chapters 6 and 7.

2.2.2 Chiral patterns

Chiral asymmetry (ﬁrst discovered by Louis Pasteur) exists in a whole range of scales, from subatomic particles through human beings to galaxies, and seems to have played an CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 29

Figure 2.11: Colony of B. subtilis. Pattern of concentric rings superimposed on a branched colony. Taken with permission from [183]. important role in the evolution of living systems [101, 11]. Bacteria display various chiral properties. Mendelson et al. [161, 163, 165, 162] showed that long cells of B. subtilis can grow in helices, in which the cells form long strings that twist around each other. They have shown also that the chiral characteristics affect the structure of the colony. Ben-Jacob et al. [30, 37, 27] have found yet another chiral property – the strong chirality exhibited by the C morphotype. Here, the ﬂagella handedness acts as a microscopic perturbation which is ampliﬁed by the diffusive instability, leading to the observed macroscopic chirality (see

Sec. 7.2). This appears to be analogous to the manner in which crystalline anisotropy leads to the observed symmetry of snowﬂakes [17].

2.2.2.1 Morphology Diagram and Closer Look at the Patterns

C morphotype exhibits a wealth of different patterns according to the growth conditions (Fig. 2.12). As for T morphotype, the observations may be organized in a morphology diagram and there is a pattern-velocity correlation [37]. Also, as for T morphotype, the patterns are generally compact at high peptone levels and become ramiﬁed at low peptone 30 levels. At very high peptone levels and high agar concentration, C morphotype conceals its chiral nature and exhibits branching growth similar to that of T morphotype.

a) b)

c) d)

Figure 2.12: Patterns exhibited by the C morphotype for different growth conditions. a) Thin disordered twisted branches at 0.5g/l peptone level and 1.5% agar concentration. b) Thin branches, all twisted with the same handedness at 2g/l peptone level and 1.25% agar concentration. c) Pattern similar to (b) but on softer agar: 1.4g/l peptone level and 0.75% agar concentration. d) Four inocula on the same plate, conditions of 1g/l peptone level and 1.25% agar concentration.

Below 0.5% agar concentration the C morphotype exhibits compact growth with density variations. These patterns are almost indistinguishable from those developed by the T morphotype. In the range of 0.5%-1% agar concentration the C morphotype exhibits its most complex patterns (Fig. 2.13). Surprisingly, these patterns are composed of chiral CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 31 branches of both left and right handedness. Microscopic observations reveal that part of the growth is on top of the agar surface while in other parts the growth is in the agar. Our model of the chiral growth explains that indeed growth on top of the surface and in the agar should lead to opposite handedness.

a) b)

Figure 2.13: In agar soft enough for the bacteria to swim in, the branches lose the one- side handedness they have on harder agar. The two colonies of (a) and (b) are of 5g/l peptone level and 0.6% agar concentration. The two patterns are of two strains of the C morphotype, strains whose patterns are indistinguishable on harder agar. c) Closer look (×10 magniﬁcation) on a colony grown at 8g/l peptone level and 0.6% agar concentration.

Optical microscope observations indicate that during growth of strong chirality the cells move within a well deﬁned envelope. The cells are long relative to those of T morphotype, and the movement appears correlated in orientation (Fig. 2.14). Each branch tip maintains 32 its shape, and at the same time the tips keep twisting with speciﬁc handedness while propagating. Side branches are emitted at angles with mean of 90◦ from the main branch, but there are large variations in the emission angle. Electron microscope observations do not reveal any chiral structure on the cellular membrane [34].

2.2.3 Collective migration and formation of vortices

Many microbial genera (e.g. Proteus, Vibrio, Serratia, Chromobacterium, Clostridium) can exhibit collective migration, via gliding or via swarming (see Sec. 2.3.1.1 for clas- sification of surface translocation methods). More then half a century ago, observations of migration phenomena of Bacillus circulans on hard agar surface have been reported [83, 208, 233]. The observed phenomena include “turbulent like” collective flow, complicated eddy (vortex) dynamics, merging and splitting of vortices, rotating “bagels” and more. During studies of complex bacterial patterning, a new species which exhibits similar behavior to B. circulans have been isolated from colonies of B. subtilis [37, 33, 74]. A representative strain from this species was identified in Ref. [212] as P. vortex strain V168. We refer to this new species as V (vortex) morphotype, as their most noticeable character is their ability to form vortices.

2.2.3.1 Bacterial patterns and bacterial ﬂow dynamics

A wide variety of branching patterns are exhibited by the V morphotype as the growth conditions are varied. Some representative patterns are shown in Fig. 2.15. Each branch is produced by a leading droplet and emits side branches, each with its own leading droplet.

Microscopic observations reveals that each leading droplet consists of hundreds to mil- CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 33

a) b)

Figure 2.14: Optical microscope observations of branches of C morphotype colony. a) ×20 magnification of a colony at 1.6g/l peptone level and 0.75% agar concentration, the anti- clockwise twist of the thin branches is apparent. The curvature of the branches is almost constant throughout the growth. b) ×10 magnification of a colony at 4g/l peptone level and 0.6% agar concentration the branches are not thin, but have a feathery structure. The curvature of the branches varies, but it seems that at any given stage of growth the curvature is similar in all branches. That is, the curvature is a function of colonial growth. c) ×500 magnification of a colony at 1.6g/l peptone level and 0.75% agar concentration. Each line is a bacterium. the bacteria are long (5µm-50µm) and mostly ordered. 34

Figure 2.15: Representative patterns observed during colonial development of V morphotype. Growth conditions are: (a) 7g/l peptone level and 2.25% agar concentration. (b) 10g/l peptone level and 2% agar concentration. (c) 15g/l peptone level and 2.25% agar concentration. (d) 20g/l peptone level and 2.25% agar concentration. CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 35 lions of cells that circle a common center (hence the term vortex) at a cellular speed of up to 10µm/sec (Fig. 2.16). Both the size of a vortex and the speed of the cells can vary according to the growth conditions and the vortex location in the colony. Within a given colony, both clockwise and anti-clockwise rotating vortices are observed. The vortices in a colony may consist of either a single or multiple layers of cells. We occasionally observed vortices with an empty core, which we refer to as “bagel” shaped. After formation, the number of cells in the vortex increases, the vortex expands and it translocates as a unit. The speed of the vortices is slower than the speed of the individual cells circulating around the vortex center.

a) b)

Figure 2.16: Microscope observations of vortices. Magniﬁcation is ×16 (a) and ×50 (b). Growth conditions are 4g/l peptone level and 2.5% agar concentration.

Bacterial cells are also contained in the trails left behind the leading vortices (Fig.

2.16). Some of them are immobile while others move, swirling with complex dynamics; groups containing a few to thousands of cells move in various directions, changing 36 direction abruptly and sometimes returning to a prior direction. When two such groups pass by each other they can unite into a single group, or they might remain separate despite the close contact. The migrating groups of cells are very reminiscent of the “worm” motion of Dictyostelium (cellular slime mold) or schools of multicellular organisms. The whole intricate dynamics is conﬁned to the trail of the leading vortex, and neither a single cell nor a group of moving cells can pass the boundary of the trail. New vortices may be formed in the trail and only they can break out of the trail and create a new branch.

Microscopic observations of V morphotype also reveal that the bacterial motion is performed in a fluid on the agar surface, like the motion of T and C morphotypes [35]. The cellular movement is confined to this fluid; isolated cells spotted on the agar surface do not move. The fluid’s boundary thus defines a local boundary for the branch. We do not observe tumbling motion nor movement forward and backward. Rather, each cell’s motion is forward along the cell’s long axis, and the cell tend to move in the same direction and speed as its surrounding cells, in a synchronized group. Electron microscope observations show that the bacteria have flagella which indicates the movement to be swarming (Sec.

2.3.1.1).

2.3 Biological Background

2.3.1 Bacterial movement and chemotaxis

2.3.1.1 Classiﬁcation of Bacterial Movements

In the course of evolution, bacteria have developed ingenious ways of moving on surfaces.

The most widely studied and perhaps the most sophisticated translocation apparatus used by bacteria is the flagellum [76], but other mechanisms exist as well [104]. We describe CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 37 here briefly only some of the different types of bacterial surface translocation, which clas- sically are defined to be [104]:

• Swimming – Surface translocation produced through the action of ﬂagella. The cells move individually and at random in the same manner as ﬂagellated bacteria move

in wet mounts (i.e., nearly straight runs separated by brief tumbling). Swimming takes place only in sufficiently thick surface fluid. Microscope observations reveal no organized flow-field pattern.

• Swarming – Surface translocation produced through the action of ﬂagella, but unlike swimming, the movement is continuous and regularly follows the long axis of the

cell. The cells are predominantly aggregated in bundles during their movement, and microscope observations reveal ﬂow-ﬁeld patterns highly organized in whirls and bands.

• Gliding – Surface translocation occurring only in non-ﬂagellated bacteria and only when in contact with solid surface. In all other respects, gliding is identical to swarm-

ing.

• Twitching – Surface translocation occurring in both ﬂagellated and non-ﬂagellated

bacteria, but not through the action of ﬂagella. The movement is usually solitary (although small aggregates may occur), appears intermittent and jerky and does not regularly follow the long axis of the cell.

• Sliding – Surface translocation produced by the expansive forces in a growing culture in combination with special surface properties of the cells that reduce the friction 38

between cell and substrate. The microscopic observations reveal a uniform sheet of

closely packed cells in a single layer that moves slowly as a unit.

• Darting – Surface translocation produced by the expansive forces developed in an

aggregate of cells inside a common capsule. These forces cause ejection of cells from the aggregate. The resulting pattern is that of cells and aggregates of cells distributed at random with empty areas of substrate in between. Neither cell pairs

nor aggregates move except during the ejection which is observed as ﬂickering in the microscope.

Those types of bacterial movement can be organized in two major categories – solitary or clustered. In the case of solitary movement, no long-range correlations exist between the movement of different cells, and the resulting density can often be approximated by a diffusion equation. In contrast, in the case of clustered movement the interactions between cells have profound inﬂuence on the movement and the resulting dynamics; thus the equations describing the cellular density are not as nearly as simple.

2.3.1.2 Swimming in Bacterial Colonies

As for the movement of T morphotype, based on microscope and electron microscope observations of ﬂagella we identify the movement as swimming, i.e. whenever possible they move in straight runs interrupted by tumbling. Cells tumble about every τT ≈ 1−5sec, depending on external conditions. The speed of the bacterium during a run is very sensitive to conditions such as the liquid viscosity, temperature and pH level. Typically, it is of the order of v = 1 − 10µm/sec.

Swimming in a liquid can be approximated by a random walk with variable step size. CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 39

With isotropic and space-independent steps size, the probability density of a random walker is governed by a diffusion equation, and in the case of a T bacterium the diffusion coef-

2 −8 −5 2 ﬁcient is Db ≡ v τT = 10 − 10 cm /sec. At low cellular densities and in a uniform liquid, the same diffusion equation can describe the dynamics of the cellular density, as swimming is a solitary movement. Low cellular densities means that the mean free path between collisions lc is longer than the tumbling length lT ≡ vτT , thus collisions between the bacteria can be neglected. The mean free path (or collision length) is

− 1 lc ∝ σ 2 (2.1) where σ is the projection of the 3D bacterial density on the surface.

At high cellular densities (lc < lT ), the collisions cannot be neglected. In an attempt to approximate the dynamics in those conditions, one may want to consider the time of straight motion to be lc/v instead of τT . Hence Db depends on the bacterial density to yield

− 1 Db ∝ vσ 2 . (2.2)

This approximation is valid under the assumptions that a collision event is identical to a tumbling event (abrupt uncorrelated change in direction of motion), that a tumbling event is independent of the collisions, and that the speed between such events is not affected by their frequency.

The assumption that a collision event is like a tumbling event posses many problems. Even if the bacteria do not activate any special response to collisions, it is unrealistic to assume that collisions are elastic, or that the ﬂagella adopt a completely new orientation the moment following a collision. Instead, it is reasonable to assume strong correlation between the cell’s orientation before collision and the cell’s orientation after collision. In 40 addition, the orientation after the collision should be biased according with the average direction of motion of the surrounding bacteria, as they carry the liquid with them. The important parameter is not the collision length lc but re-orientation time τr. The re-orientation time is the time it takes a bacterium to looses memory of its initial orientation, i.e. the time span on which the ﬁnal orientation has effectively no correlation with the initial orientation.

At low densities the re-orientation time τr is equal to the tumbling time τT . As the density rises and the collisions become more frequent, τr decreases. It is quite possible that the experimental densities are high enough so as to make the velocity and even the type of motion dependent on bacterial density, thus making Eq. (2.2) (and even the whole notion of a diffusion “constant”) irrelevant. In any case, high cellular densities does mean an effective decrease in the diffusion coefﬁcient related to the bacterial movement.

When swimming in an unstirred liquid, very low cellular densities also effect the movement. The bacteria secrete various materials into the media and some of them, like enzymes and polymers, change significantly the physical properties of the liquid and make it more suitable for bacterial swimming. The secretion of these materials depend on cellular density, thus at not-too-high densities the speed of swimming rises with the cellular density. In some conditions, the bacteria cannot swim at all as individuals; on semi-solid surface the bacteria have to produce their own layer of liquid to swim in it. In such cases the bacterial speed is related to the bacterial density by a power law with a positive exponent. All these arguments together indicate that the diffusion coefficient related to the bacterial movement should be a nontrivial function of the bacterial density. Moreover, the specific functional form might depend on the specific bacterial strain.

The composition of the wetting ﬂuid produced by the T bacteria is not known, but other bacterial species can provide insights. Serratia marcescens (and other species of CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 41

Serratia) produced a wetting ﬂuid which is essential for its surface translocation and for the formation of colonial branched patterns [159, 156, 154]. The essential element of this wetting ﬂuid is a surfactant, serrawettin. Serrawettin is a lipopeptide, consists of a fatty acid and a cyclic peptide of 5 amino acids. The production of surfactants is common among bacteria [65, 70]. Bacillus subtilis produces a surfactant called surfactin [6, 127], which is similar to serrawettin both in chemical structure and physiological functions. In fact surfactin can partially replace for the role of serrawettin in colonies of S. marcescens [154].

Surfactin and serrawettin are produced under conditions of nutritional stress and their production depends on bacterial density. These materials are very potent materials, pen- etrating and punctuating bi-layered membranes of all cells; eukaryotic, prokaryotic, and membrane coated viruses [217, 203, 224]. It has been suggested that their production is part of a general stress response, aimed to neutralize external attacks. But, in B. subtilis, the membrane modifying properties of surfactin are related to competence (genetic competence is a physiological state of bacteria in which the cell tends to take in and use large chunks of foreign DNA). Indeed one of the regulating genes of competence, comS, is part of the surfactin production operon (i.e. the protein ComS is produced only when surfactin is produced) [73, 97]. This genetic coupling is a clear indication that surfactin production is not a general stress response, but a part of the intercellular communication system. It is not clear if B. subtilis have any other roles for surfactin, besides the modiﬁcation of membranes. One obvious possibility is aiding surface translocation, like one of the roles S. marcescens has for serrawettin. Another possibility is creating intercellular media which ease the transfer of DNA molecules between the cells. 42

2.3.1.3 Chemotaxis

Chemotaxis means changes in the movement of the cell in response to a gradient of certain chemical ﬁelds [2, 42, 133, 40]. The movement is biased along the gradient either in the gradient direction or in the opposite direction. Bacteria are too short to measure spatial gradients of chemicals by simply comparing concentrations at different locations on their membrane [42]. They deduce the spatial gradients by calculating temporal derivatives along their path. It is known that Escherichia coli, for example, can compare successive measurements over a time interval of 3 seconds. The actual chemotaxis in swimming bacteria is implemented by decreasing the tumbling frequency as cells swim up the gradient of an attracting agent or down the gradient of a repelling agent. Thus the straight runs are important for gradient perception and the tumbling timing is important for the response to this gradient.

Usually chemotactic response means a response to an externally produced ﬁeld like in the case of food chemotaxis. However, the chemotactic response can be also to a ﬁeld produced directly or indirectly by the bacterial cells. We will refer to this case as chemotactic signaling [35, 26, 60, 20, 22]. E. coli, for example, when exposed to oxidative stress, emit the amino acid aspartic acid that attract neighboring cells and allow the cooperative degradation of toxic materials [53, 54, 26].

Chemotaxis towards high concentration of nutrients is a well studied phenomenon in bacteria [2, 1]. When the center of a semi-solid agar plate (0.35% agar concentration) is inoculated with chemotactic cells, distinct circular bands of bacterial cells become visible after a few hours of incubation. In fact, these patterns are used as semi-quantitative indi- cators of chemotactic response [1]. Genetic experiments show that the creation of each of CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 43 those bands depends solely on the chemotactic response to a single chemical in the substrate (these chemicals are usually metabolizable, but even cells that have lost the ability to metabolize a certain chemical form bands, as long as they are attracted to it [2]). Berg et al. [41] showed that the bacteria realize chemotactic response by modulating the periods between tumbling events – the period is increased when moving in a preferred direction along the chemical gradient. This makes a bias in the random walk which result in a mean drift of the bacteria in the desired direction, a drift that can be as large as v/10.

Bacteria sense the local concentration C of a chemical via membrane receptor binding [2, 42, 133]. The cell measures the concentration by calculating the relative number of occupied receptors No , where N and N are the number of occupied and free receptors No+Nf o f respectively. The simplest assumptions about the reception mechanism are as follow: for a given chemical C, we assume that No is determined by two characteristic times: the mean time of a receptor occupation – τo , determined by internal cellular processes – and the mean time lapse when the receptor is free (τ f ). Since τ f is inversely proportional to the concentration of the chemical (with the proportion coefﬁcient determined by the receptor afﬁnity to the chemical), we get:

N τ C o = o = , (2.3) Nf + No τ f + τo K +C ¡ ¢ where K ≡ C τ f /τo is constant. It is crucial to note that when estimating gradients of chemicals, the cells actually measure changes in the receptors’ occupancy No and not Nf +No in the concentration itself. Using Eq. (2.3) and assuming that τo does not change in space, we obtain: µ ¶ ∂ No K ∂C = 2 . (2.4) ∂x No + Nf (K +C) ∂x This means that the chemical gradient times a factor K/(K +C)2 is measured. This depen- 44 dence in known as the ‘receptor law’ [171]. According to the ‘receptor law’, for very high concentration the chemotaxis response vanishes due to saturation of the receptors.

The ‘receptor law’ indicates that there is a limited range of concentrations (a range of about one order of magnitude centered around K) of high sensitivity for gradients. This does not agree with experimental data. Experiments show that bacteria can be sensitive to concentration gradients over a range of five orders of magnitudes [166, 43]. This discrepancy may be due to coupling between close receptors [91], i.e. a non-occupied receptor may send the “occupied” signal to the cell due to coupling with the neighboring receptor state. The spatial distance between receptors changes constantly – the receptors are proteins embedded in the cell’s membrane, and since they are not hooked in place, Brownian motion (at room temperature) suffices to allow them to float around the entire membrane in 10−1sec. The diffusing of receptors on the membrane aids the bacteria overcome the limitations of the ‘receptor law’. This coupling together with receptors’ repositioning can create temporary clusters which can act as “mega-receptors”, where a single chemical molecule can change the state of all the receptors in the cluster (spontaneous symmetry breaking and

“magnetization” of the receptors is prevented by the receptors’ diffusion, which raise the effective temperature above the critical value) [47]. Bigger clusters has both shorter τ f (as the effective afﬁnity of a group of identical receptors is bigger than that of a single receptor) and longer τo (as it takes more time to the change of state to spread over all the receptors in the cluster). The combined effect is smaller K for larger clusters. Dynamic adjustment of the coupling between neighboring receptors enable the bacteria to measure gradients over a very wide range of gradients.

Eventually, very high concentration does saturate the receptors. The saturation explains the bands reported by Adler [2, 1]. It can be easily shown that linear chemotactic response CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 45 to a nutrient cannot produce such bands. A non-linear response like the ‘receptor law’ must be included for the bands to form. Moreover, high concentration of the attractant represses both the strength of the chemotactic response [2] and the velocity of the expanding band

[234]. These observations are accounted for by the ‘receptor law’ (or any other expression which accounts for saturation) for chemotactic response if one assumes that the average gradient around the cells is proportional to the initial concentration of the chemical [2, 234].

In the case of high bacterial densities, collisions between bacteria can hinder both the perception of chemical gradient and the bacterial response. As the collisions prevent the bacteria from moving in straight lines between tumbling events, the measurement of spatial derivative become inaccurate. At the same time, the collisions decrease the cell’s ability to modulate the length of runs, and the effective response to chemotaxis is reduced.

2.3.1.4 Chemical-Aided Communication, Regulation and Control

Many means of chemical communication are employed by bacteria [110, and references therein]. Here we mention only the following example. Bacillus subtilis and Myxococcus xanthus, like many other species of bacteria, sporulate (see deﬁnition in Sec. 2.2.1.2) as a way to survive conditions they cannot live through. They do not do it as solitary cells, though. B. subtilis sporulate in response to extracellular differentiation factors, two signals that are sent by other cells and relay information about the cellular density and the state of the colony [94]. M. xanthus use several similar signals during the sporulation process. The A signal, for example, is used at early stages to estimate the density of the surrounding cells, and the sporulation process is not activated until critical density is achieved [132, 110].

In the above example and in many others, the colony creates ﬁelds of chemicals that 46 provide the bacterial cells with global information, i.e. information about an area much larger then the cell size, which could not have been gathered directly by a single cell.

2.3.2 Bacterial metabolism and growth

Most bacteria reproduce by ﬁssion of the cell into two daughter cells which are practically identical to the mother cell. The crucial step in the cell division is the replication of the genetic material and its sharing between the daughter cells. Hasty replication of DNA might lead to many errors – most organisms limit the rate of replication to about 1000 bases per second. Thus the reproduction has minimal reproduction time τR. This reproduction time

τR is about 25min in Bacilli.

For reproduction, as well as for movement and other metabolic processes, bacteria and all other organisms need an inﬂux of energy. Any organism which does not gets its energy directly from sunlight (by photosynthesis) needs an external supply of food. In the patterning experiments, the bacteria eat nutrient from the agar. As long as there is enough nutrient and no signiﬁcant amount of toxic materials, food is consumed (for cell replication and internal processes) at maximal rate Ωc. To estimate Ωc we assume that a bacterium −12 needs to consume an amount of food CR of about 3 × 10 g. It is 3 times its weight – one quanta for doubling body mass, one quanta used for movement and all other metabolic processes during the reproduction time τR, and one quanta is for the reduced entropy of making −15 organized cell structures out of food. Hence Ωc is about 2 f g/sec (1 f g = 10 gram).

If nutrient is deﬁcient for a long enough period of time, some stains of Bacilli (P. dendritiformis and P. vortex included) may enter a special stationary state – a state of a spore – which enables them to survive much longer without food (Fig. 2.7). The bacterial cells CHAPTER 2 – OBSERVATIONS AND BIOLOGICAL BACKGROUND 47 employ very complex mechanisms tailored for the process of sporulation. They stop normal activity – like movement – and use all their internal reserves to metamorphose from an active volatile cell to a sedentary durable ’seed’. While the spores themselves do not emit any chemicals (as they have no metabolism), the pre-spores (sporulating cells) emit a very wide range of waste materials, some of which unique to the sporulating cell. These emitted chemicals might be used by other cells as a signal carrying information about the conditions at the location of the pre-spores. Ben-Jacob et al. [35, 18] suggested that such materials repel the bacteria (’repulsive chemotactic signaling’) so as to allow them to escape from an unfavorable location.

When bacteria are grown in a petri dish, nutrients are usually provided by adding peptone, a mixture including all the amino acids and sugars as source of carbon. Bacteria which are not defective in synthesis of any amino acid can grow also on a minimal agar, where a single source of carbon and no amino acids are provided. Such growth might seem to be easier to model as the growth is limited by the diffusion of a single chemical. However, during growth on minimal agar there is usually a higher rate of waste products accumulation, introducing other complications into the model. Moreover many of our strains are auxotrophic i.e. defective in synthesis of some amino acids and need an external supply of it. Providing the bacteria with these amino acids and only a single carbon source might pose us the question as to what is the limiting factor in the growth of the bacteria. For all those reasons we prefer to use peptone as nutrient source.

We said that if there is ample supply of food, bacteria reproduce in a maximal rate of one division in τR. If the available amount of food is limited, bacteria consume the maximum amount of food they can. In the limit of low bacterial density, the available amount of food √ over the tumbling time τT is the food contained in the area τT DbDn, where Db and Dn 48 are the diffusion coefﬁcients of the bacteria and the food, respectively. Hence the rate of √ food consumption depend on the local concentration of food n and it is given by n DbDn, whether Db is constant or not.

In a continuous model, reproduction of bacteria translates to a growth term of the bacterial density σ which equals σ times the consumption rate per bacteria. In the limit of high nutrient it is σ/τR, and in the limit of low nutrient it is proportional to nσ. This brings to mind Michaelis-Menten law [171] of Knσ/(1 + γn) with K, γ constants. Many authors take only the low nutrient limit of this expression, Knσ, although it is not biologically established that the bacteria in the experiments are always limited by the availability of food and not by their maximal consumption rate. Chapter 3

Mathematical Background

In this chapter, we provide an introduction to the modeling of systems forming branched patterns. This is a vast topic and we cannot be all-inclusive as to topics, methods and (especially) references. We have therefore chosen to focus on mechanisms that have to date proven useful for the microbiological applications that are our main focus. Speciﬁcally, we will discuss in turn diffusive instabilities and the resultant pattern selection problem.

3.1 Propagating fronts and diffusive instabilities

In this section we discuss pattern formation in continuum, reaction-diffusion systems. We will not try to follow strict mathematical deﬁnitions, but rather to clarify the subject as best as possible. All functions are real functions and are C2 on the relevant interval (usually [0,1] ), unless stated otherwise. All constants whose value is not stated are positive. The solutions of the PDE are assumed to be real and C2 (or piecewise C2 for weak solutions) on their entire domain (which is one of R, R2, R × [0,∞) or R2 × [0,∞) , as can be understood from the context).

49 50

3.1.1 The Fisher-Kolmogorov equation

The patterns that are of interest in the bacterial colonies form at the interface between the occupied and unoccupied regions of the petri dish. This interface is clearly moving as the colony expands and hence the study of these structures involves the issue of how interfaces propagate under non-equilibrium conditions. We will start our discussion of this topic by studying the simplest interface model, that due to Fisher [82] and independently, Kolmogorov et al. [126]. The Fisher-Kolmogorov (FK) model involves one ﬁeld u(x,t) which obeys a reaction-diffusion equation given by ∂ u = D∇2u + f (u) (3.1) ∂t The reaction (growth) term is usually taken to be a logistic (Pearl-Verhulst) growth term

f (u) = u(1 − u) (3.2)

This equation was ﬁrst introduced by Fisher and Kolmogorov to study spread of mutants in a population, where u denotes their fraction in the population (u(x,t) ∈ [0,1] ∀(x,t) ∈ R ×[0,∞)). It also naturally emerges as the continuum description of the infection reaction

A + B → 2A with equal diffusivities for the two species [146, 147].

This equation has two homogeneous solutions; A stable solution u ≡ 1 and an unstable solution u ≡ 0. Thus, we can study the propagation of the stable state (u = 1) into the unstable one (u = 0). To study such propagation one looks for 1D traveling wave solutions

(traveling fronts between 0 and 1) when the system initial conditions are u = 0, with a small perturbation at x → −∞. To derive a front propagating at a steady velocity v, the equation is written in the moving frame z = x − vt to yield ∂2 ∂ D u + v u + f (u) = 0 (3.3) ∂z2 ∂z CHAPTER 3 – MATHEMATICAL BACKGROUND 51

This equation can be interpreted as the motion of a particle of mass D in the potential

Z u F (u) = f (w)dw (3.4) 0 in the presence of friction v (in the case of the logistic growth term, F (u) = u2/2 − u3/3).

The front consists of motion from the potential maximum at u = 1 to the minimum at u = 0. Such a representation makes it clear that there is a propagating front solution for every v > 0. For large enough v (over-damped motion) the trajectory do not cross u = 0 while for small enough v (under-damped motion) the trajectories oscillate around u = 0

(Fig. 3.1). There is a critical damping vc which separates the two classes. vc is found by kz linearizing about u = 0, setting u ' e and ﬁnding the roots k; at vc, the two resulting roots are equal. For the original growth term of the FK equation, vc = 2.

Figure 3.1: Traveling waves solutions of the Fisher-Kolmogorov equation for various values of the propagating velocity v. For v = 0.5 there is an unstable solution. The selected solution has a velocity v = 2. For v = 4 there is a stable solution which is not selected.

It is important to contrast this type of equation with formally similar equations where f (u) has three roots – one stable, one unstable and one meta-stable (Fig. 3.2). Here, a 52 propagating front solution should connect the stable and meta-stable roots. In the moving particle interpretation, this is like a particle moving from a global maximum at u = 1 and coming to a rest at a local maximum at u = 0. It is evident that only one speciﬁc value of the friction v can stop the particle at the maximum, i.e. there is a propagating front solution only for one value of the velocity v.

Figure 3.2: The inverted potential F(u); (a) for the original FK equation. (b) for the meta- stable case.

Since for FK there exists a continuous family of traveling solutions, the question of selection of front velocity arises. For the FK equation itself, Aronson and Weinberger [10] showed rigorously that any localized initial perturbation will lead to a front with v=vc= 2. Unfortunately, their proof can not be extended to more complex models. In an attempt to look for a general principle, Ben-Jacob et al. [19] showed that the v=vc front is the marginally stable one. That is, if one performs linear stability in the moving frame, all the fronts with v > vc are linearly stable and those with v < vc are linearly unstable (stability in the moving frame means that even if the perturbation grows in the stationary frame, it decreases when viewed from the moving frame). It is worthwhile to see how this analysis CHAPTER 3 – MATHEMATICAL BACKGROUND 53 works in some detail. If we analyze the equation for a nominal perturbation δueωt near positive inﬁnity, we discover that δu ∼ ekx with r v v2 k = − ± − 1 + ω (3.5) 2 4

If we had chosen a base solution with v < 2, we would discover that all positive ω ranging

v2 up to 1 − 4 would give rise to two equally decaying wave-vectors, as the square root is imaginary. Having two modes with equal decay constants means that we can satisfy any reasonable boundary condition and hence those values of ω correspond to allowed (continuum) modes. Ben-Jacob et al. demonstrated that the above marginal stability criterion for the selected velocity is valid for more complicated systems with two ﬁelds, so they postu- lated it to be a general criteria. Indeed, its validity has by now been demonstrated in many systems [219, 220].

For more general forms of f (u), the structure of the front in the vicinity of u = 0 is

Ae−q1z + Be−q2z . (3.6)

Both the decay factors q1 and q2 and the amplitudes A and B are functions of v and depends on the form of f (u). In general [134], marginal stability occurs when q1 = q2 (referred to as “case I” or “pulled front”) or when the amplitude of the faster decaying part (say

A) vanishes (referred to as “case II” or “pushed front”). Ben-Jacob et al. provided the following example: ∂ ∂2 u u = u + (a + u)(1 − u) , (3.7) ∂t ∂x2 a and looking for a front connecting the stable (or meta-stable) state u = 1 with the unstable

1 state u = 0. They showed that for 2 < a < 2 the selected solution is for q1 = q2, i.e. this 1 is a ”case I” situation and the velocity of the solution is vc = 2. However, for 0 < a < 2 54 or a > 2 the selected solution is for zero amplitude of the faster decaying part, i.e. this is a

“case II” situation and the velocity of the solution is v = (2a)1/2 + (2a)−1/2.

More recently, the validity of the marginal stability criterion has been traced to the idea of structural stability of the problem of propagation into an unstable state [178]. That is, one can look at a sequence of models in which f (u) is changed to be of the “meta-stable” type, in which there is a unique traveling front solution; a simple possibility is to take the reaction term to be identically zero if u is below some small cutoff ε (this f (u) is of a “marginally- stable” type but the mass-in-potential analog can show that it too as a unique traveling front solution). As this sequence converges to the FK equation, the selected velocity converges to the marginally stable FK velocity. Since in some loose sense, the initial conditions with compact support for u not equal to zero acts as a sort of time-dependent cutoff on how far the ﬁeld could have spread, one can turn this structural stability into a reason for dynamical selection. For more details, see Refs. [121, 51].

3.1.2 Propagation in two-dimensional space

The FK equation in 2D leads to a compact (versus branching) growth of the stable state

[33, 59]. That is, there is no instability which tends to enhance a small-scale roughness in the interface (in any physical or simulated system, these small-scale perturbations are inevitable). The proof of this is quite trivial. If we linearize around the marginally stable front velocity in the moving frame of reference, translation invariance in the y direction (perpendicular to the interface’s direction of propagation) guarantees that any mode can be taken to have the form eωt+iqy. Substitution of this expression into the equation leads to an eigenvalue problem for ω + Dq2. Hence, having non-zero q can only diminish the growth rate ω. Fig. 3.3 demonstrates the 2D stability of the FK equation (case I). CHAPTER 3 – MATHEMATICAL BACKGROUND 55

Figure 3.3: Linear stability of a flat interface of the FK equation. Shown here are two snapshots of the density field, an initial state with a curved ‘interface between dense and non-dense regions and the final state after the equation of motion is integrated. Note that the fluctuations smooth out in time, since in there is no Mullins-Sekerka (diffusive) instability. 56

Our main goal is to get interface models which can exhibit spatial patterning in the form of branched growth. To understand how this can happen, let us consider first the well-studied example of solidification from supersaturated solution. Here, one finds that the linear stability modes grow in time, a fact referred to as the Mullins-Sekerka instability [170] or diffusive instability. This result is easily understood, since the velocity of the interface is proportional to the gradient of the concentration field at the interface. Any

“bump” on the interface has at its tip a higher gradient than at the rest of the interface as it is further into the diffusion ﬁeld. Therefore, the ”bump” will grow faster than the rest of the interface. This argument is valid for any spatial mode q so all modes are unstable. Of course, linear stability merely reveals that a ﬂat (and similarly circular) interface is unstable. It cannot provide us information about the actual evolving pattern, which is determined by nonlinear effects [28, 17].

So, the lesson is that one must have a diffusing field that is coupled to the interface in order to get an instability. As the growth of a colony in a petri dish is limited by the diffusion of nutrient from the surrounding agar, one can hypothesize that coupling a nutrient field to the bacterial density field should be able to produce branching patterns. In fact, the idea of having an diffuse interface in one equation which then becomes diffusively unstable as it couples to a second field is the concept behind the successful application of phase

field approach to solidification [64, 136, 142, 84, 125, 229, 130, 129]. In the simplest case, the nutrient field is coupled to the bacterial density field only through the growth (reaction) terms. The chemotaxis towards food which was mentioned above is an additional complication that should be included in the model only if simpler models prove inadequate. We will discuss the actual modeling of branching bacterial patterns in Chapter 4. Here, we will focus our attention on the general properties of this class of models. CHAPTER 3 – MATHEMATICAL BACKGROUND 57

3.1.3 Linear stability analysis of two ﬁelds systems – the DFK model

Thus, as a ﬁrst generalization to the FK equation we will study the following system of equations, to be referred to as the “Diffusive Fisher-Kolmogorov equation” (DFK) [37, 153, 33]:

∂b = D ∇2b + n f (b) ∂t b ∂n = D ∇2n − ηn f (b) , (3.8) ∂t n where η > 0 and f (b) has an unstable node at b = 0, stable one at b = b1 > 0, and f (b) > 0 for all b ∈ (0,b1). In dimensionless variables the equations are:

∂b = ∇2b + n f (b) (3.9) ∂t ∂n = D∇2n − ηn f (b) (3.10) ∂t where D ≡ Dn and f (b) are rescaled to have f 0 (0) = 1, a stable node at b = 1, and f (b) > 0 Db for all b ∈ (0,1). We will not try to completely solve this system. We will use linear stability to check the stability of ﬂat interface. We present here only the outline of the analysis and its results.

We explain here the details of how one carries out a stability calculation for an interface in a reaction-diffusion system. First, one must ﬁnd the allowed steady-state interface pro-

files in a moving frame of reference. The analogy of a particle in a potential was useful for one component systems but it can not be generalized, and we need a methodology which will replaces it. The key idea is to check the dimension of the space of all solutions of the system which leave the fixed point at negative spatial infinity (the dimension of the unstable manifold of this fixed point) and the dimension of the space of solutions which enter the 58

fixed point at plus infinity (the dimension of the stable manifold of this fixed point), and to compare it to the number of constraints on the solution. To see this idea in use, we focus ³ ´ n∞ on the low nutrient case n∞ < η and look for a front of ~u ≡ (b,n) going from ~u = η ,0 at x = −∞ to ~u = (0,n∞) at +∞. Linearizing the equations at large x, we find that the most general solution that approaches the desired fixed point takes the form

3 (∞) (∞) λ(∞) ~ x i ~u(x) = (0,n∞) + ∑ ai Ai e (3.11) i=1 where

p p (∞) v (∞) v 1 (∞) v 1 λ = − ; λ = − − v2 − 4n∞ ; λ = − + v2 − 4n∞ (3.12) 1 D 2 2 2 3 2 2

(∞) ~ Here the ai are arbitrary and the Ai are normalized eigenvectors. There are three un- knowns, corresponding to a three dimensional attracting subspace. Similarly, at large nega- r ³ ´ (−∞) ³ ´ n∞ (−∞) (−∞) λ (−∞) n∞ ∑2 ~ x i λ v 1 2 η tive x,~u(x) = η ,0 + i=1 ai Ai e where now 1 = − 2D + 2D v + 4 D f η λ(−∞) and 2 = 0. This is a subspace of dimension two and the sum of the dimensions of the two subspace is 5. Due to translation invariance, the front solutions must necessarily form a one dimensional manifold. This means that in addition to the four matching conditions at the origin that ensures that one can smoothly connect the solution at positive x to the one at negative x (continuity of ~u and ∂x~u at x = 0), we must apply one more condition to specify a solution uniquely. The fact that the number of conditions (5) matches the number of dimensions of the two subspace (5) means that the solution can be constructed for a continuous range of possible velocities. We note in passing that for this speciﬁc case, the η (−∞) ∞ global conservation law for b + n ensures that a2 at − will turn out to be zero.

We can contrast this with what would have occurred had the reaction terms exhibited bi-stability, namely that there would have been no linear instability of the b = 0 state. CHAPTER 3 – MATHEMATICAL BACKGROUND 59

0 λ(∞) Then, f (0) would have been negative and the mode corresponding to 3 would have been disallowed. The counting argument would then have led to the need for ﬁxing one more variable, meaning that only discrete values of the velocity would allow us to form front solutions.

We next proceed to the stability calculation, which works in a related manner. We linearize the equation around the previously determined steady-state solution ~u(0) by letting

ω ~u = ~u(0) + δ~ueiqye t (3.13) and we re-do the asymptotic analysis. Near +∞, we have the eigenvalues r ³ ´ ∞ v v 2 (ω + Dq2) λ( ) = − ∓ + 1,2 2D 2D D r³ ´ (∞) v v 2 λ = − ∓ − n∞ + ω + q2 (3.14) 3,4 2 2 and near −∞ v u ³ ³ ´ ´ u³ ´ η n∞ ω 2 ∞ v t v 2 f η + + Dq λ(− ) = − ∓ + 1,2 2D 2D D r³ ´ ∞ v v 2 λ(− ) = − ∓ + ω + q2 (3.15) 3,4 2 2

Some sort of boundary conditions should be specified at large |x|; the precise form is irrelevant. If none of the growth rates (i.e. the real part of the λ’s) are degenerate, we will invariably find that there is an exponential suppression of the two modes with the fastest growth rates at +∞ and the two modes with slowest growth rates at −∞. Again, there is automatically a one parameter family of solutions, this time corresponding to the amplitude freedom in the linear system. Hence, the fact that there only four degrees of freedom requires us to fix ω as well to a set of discrete values. 60

The other possibility is that some growth rates are degenerate. The only possible way this could occur for positive real ω is for the velocity to be less than 4n∞; then all real ω less 2 2 λ(∞) λ(∞) than n∞ − v /4 − q give rise to Re 3 = Re 4 . In this case, we have an extra degree √ of freedom available and we do not need to specify ω. Hence, v = 2 n∞ is the marginally stable velocity with respect to these continuum modes. By direct calculation, it turns out that there are no unstable discrete modes (hence this is a type I system). More importantly, adding ﬁnite q always reduces the growth rate of the continuum modes, as the upper limit obeys ω = −q2; there is no diffusive instability!

So, what went wrong in our attempt to find a two-field model which exhibits diffusive growth patterns? Essentially, interfaces with type I marginal stability can never have diffusive instabilities. Physically, a diffusive instability requires that the diffusion length of the second field be much larger than the interface width of the first field. Type I fronts spread as much as possible and they really have no width per se so the above criterion is never satisfied. The DFK model is hence inadequate for our purposes. It turns out that there are two possible remedies. One approach is to have some degree of meta-stability for the b = 0 state; as we will discuss next, it appears that this meta-stability is a natural consequence of trying to account for the finite number of interacting particles (leading thereby to inevitable fluctuation effects) in a deterministic reaction-diffusion model. Subsequently, we will turn to a second possibility which assumes that diffusive motion at low density is necessarily cooperative. This implies that the diffusion “constant” should really be taken to be field dependent [59, 124, 93, 61, 128]. It turns out that this too provides a finite width to the interface and therefore allows for an instability. CHAPTER 3 – MATHEMATICAL BACKGROUND 61

3.1.4 Meta-stable node and diffusive instability

It is known that diffusive systems with bi-stable reaction term can produce branching patterns. Such a model is the phase field model, ∂φ δF (φ,T) = ∇2φ − (3.16) ∂t δφ ∂T 1 ∂φ = D ∇2T + (3.17) ∂t 2 ∂t (where F is a suitable free energy containing a gradient-squared term and a pair of stable states at φ = ±1 whose relative weights vary with the temperature T). Langer [136] and Collins and Levine [64] were the first to use the phase-field model to study pattern formation in solidification from an under-cooled melt. Both analytical investigations and large scale numerical simulations showed that such models can produce branching patterns in different systems [55, 57, 56, 81, 228, 130]. The solutions to these models all include a layer of transition from one phase to another (front), the width of which is determined by the model’s parameters. In the limit of vanishing front width, one can derive a free boundary problem; we will discuss this later in some detail.

As we mentioned earlier, the growth term of the DFK model makes the system unstable around zero density. But, the reaction-diffusion approach is only the mean-field infinite particle limit of the actual dynamics of a set of finite number of organisms. This may sound like a quibble since the number of organisms can go up to 1010, but this first impression is wrong. Type I systems are controlled by their leading edge, where the number of particles is always small (regardless of overall population size), thus one cannot ignore finite number effects. To include these effects explicitly means handling a stochastic system, which is much more difficult to solve than a deterministic PDE. A possible shortcut has emerged from work on continuum models of DLA [48] and from direct study [120] of the reaction 62

A + B → 2A as compared to a mean field theory treatment thereof. The shortcut is that the leading effect of the fluctuations is to impart a small (order 1/N) meta-stability to the b = 0 state. This has been checked by comparing the velocity of the discrete system to the velocity of a modified DFK-like model µ ¶ 1 0 = vc0 + c00 + c c Θ c − (3.18) A A A B A N µ ¶ 1 0 = vc0 + Dc00 − c c Θ c − (3.19) B B A B A N where Θ is the Heaviside step function, which gives 1 for positive argument and 0 for negative argument. Both the modified continuum model and the finite particle system give

2 vMS − v(N) ∼ 1/log N [51, 121, 180], where the marginal stability prediction at N = ∞ is p vMS = 2 cB (∞).

While the velocity shift just described is quantitatively signiﬁcant, the cutoff effect does not alter the qualitative form of the interface in a one dimensional system. In higher dimensions the effect is more pronounced. Kessler and Levine [120] concluded that in higher dimensions the modiﬁed system exhibits a diffusive instability above a critical value of D.

1 The critical value is a function of N and it varies only logarithmically with N. The loga- rithmic dependence arises because the effective size of the interface region scales as logN. Hence, one can have a transition to a pattern forming behavior at a physically realizable D even for huge numbers of particles. The calculation which shows this is a straightforward generalization of the one just presented; for details, see Ref. [120]. The ﬁnal analytic result is depicted in Fig. 3.4. A simulation showing that the discrete particle system really does have the predicted instability is shown in Fig. 3.5. Note though that the nonlinear state formed by the system does not exhibit large-scale branching, but instead is best described as “cellular”. We will return to this point shortly. CHAPTER 3 – MATHEMATICAL BACKGROUND 63

Figure 3.4: Results of stability analysis: upper - Dependence of the critical value εc of the cutoff as a function of the diffusion constant ratio D. lower - spectrum at D = 10, ε = .0014 . 64

Figure 3.5: Results of a simulation of the microscopic reaction model A + B → 2A with a reaction rate constant k. The cases shown are the cases D = 10 (upper) and D = 1 (middle), both for N = 100 and k = .001; and D = 10, N = 106 with k = 10−7 (lower). The graph shows the interfacial region for a box of width 100; the color scale corresponds to the number of A particles. Methods: Each site of a 100×200 lattice has some number of A and B particles. Initially, A particles are introduced at one edge, whereas B particles exist everywhere, with an average density of N. In each time step, particles can diffuse and same-site particles can react. The data shown is a typical snapshot of the interface once a statistical steady-state has been established. CHAPTER 3 – MATHEMATICAL BACKGROUND 65

Such fluctuation-induced instability is not the only possibility. The stability of the DFK model relies on the fact that the front never saturates at a finite scale. So, any effect that would lead to a finite front width would generally allow for the instability. As an example,

Horvath and Toth [107] recently analyzed a reaction-front where the nature of the chemical kinetics leads to a growth term that varies as b2 at small b. This is indeed sufficiently different from a DFK model and does have an instability. There is no justification to model the bacterial system with such a growth term, but there is a different possibility – the mod- ification of the diffusion term at small density. This idea is discussed in the next section.

3.1.5 Non-constant diffusion Coefﬁcient

We will argue later that the complicated dynamics of the lubrication fluid which enables bacterial motion on hard agar can in some instances be subsumed into taking the diffusion coefficient to be dependent on the bacterial density. There is a significant mathematical literature on reaction-diffusion systems with nonlinear diffusion [172, 52, 139, 177, 95, 9, 173, 174, 190, 192]. We now summarize some of the known results in this field.

The porous medium equation

∂u ³ ´ = ∇ · uk∇u k > 0 (3.20) ∂t rises in isentropic ﬂow of an ideal gas in a homogeneous porous media. In this equation u is

1 the rescaled density and k = α , where α is the power of the pressure in the state equation of the gas [172]. The same equation appears in problems like thin fluid films spreading under gravity (k = 3, Ref. [52]), radiative heat transfer by Marshak waves (k = 6, Ref. [139]) and other fields. For any k > 0 one can use the transformation v(x,t) = u(x,t)k to get the 66 equation ∂ 1 − k v = ∇ · (v∇v) + (∇v)2 (3.21) ∂t k

Oleinik, Kalashnikov and Yui-Lin [177] proved the existence and uniqueness of weak solutions of the 1D porous medium equation (although the derivative of u is not continuous, the flux is). They also proved that if the initial conditions are zero u outside a finite interval, then the solution is zero outside a finite interval at all later times for which it exist. Thus, there is a sharp boundary separating regions of zero and non-zero values of u – a front. With initial condition u(x,0) = Eδ(x) (where δ(x) is the delta distribution function whose integral is the step function), there is an exact solution for the porous medium equation (Fig. 3.6) Ã" #+!1/k −1 k x2 u(x,t) = t 2+k E − (3.22) 4 + 2k 2 t 2+k where [x]+ denotes max(0,x), the positive part of the argument. The front of this solution is propagating at a finite speed which decreases in time. The solutions approach the front ∂u with zero slope (lim = 0) for 0 < k < 1, a finite slope for k = 1, and an infinite slope u→0 ∂x for k > 1.

Gurtin and MacCamy [95] considered biological population with density-dependent diffusion and a Malthusian (linear) growth term

∂ ³ ´ u = ∇ · uk∇u + µu k > 0, µ ∈ R (3.23) ∂t

They mapped this problem to the porous media equation and deduced the properties of the former from the properties of the latter.

Aronson [9] and Newman [173] considered density-dependent diffusion with logistic CHAPTER 3 – MATHEMATICAL BACKGROUND 67

t=0.7

t=1.0

Figure 3.6: Similarity solutions for the porous medium equation (k = 1, δ function as initial conditions). As there is a compact support to the initial conditions, there is a compact support to the solution at all times.

(Pearl-Verhulst) growth term µ ¶ ∂ uk u = ∇ · ∇u + u(1 − u) k > 0 (3.24) ∂t k + 1 They showed that there are traveling wave solutions to the 1D problem. There is a critical speed v∗ under which there are no traveling wave solutions. For v > v∗ there are solutions with exponential decay (as z → ∞, where z = x − vt is the spatial coordinate moving with the solution), i.e. the solution is everywhere nonzero. For v = v∗ there is a solution with sharp front (Fig. 3.7). Starting from initial conditions u(x,0) > 0∀x < xβ, u(x,0) = 0∀x > xβ (where xβ) the sharp-edge solution is the only attainable asymptotic solution. Initial conditions which are non-zero only in a ﬁnite interval leads to separation of the asymptotic solution into two traveling wave solutions moving in opposite directions, both with sharp edge. For k = 1 and the initial condition u = 0 on the half-line, they used phase-space methods to ﬁnd an exact solution: the critical speed is v∗ = 1/2 and the solution for v = v∗ is £ ¤+ 1 − ez−zβ (3.25)

∗ where z = x − v t and zβ is some constant (reﬂecting the translational invariance of Eq. 68

(3.24) ). u

- 1

v=v* v>v*

Figure 3.7: Traveling wave solutions for the density dependent diffusion analog of Fisher- Kolmogorov equation. For the critical velocity v∗ there is a sharp front, like in the solutions of the porous medium equation.

Newman [174] considered the generalized equation µ ¶ ∂ ∂ uk ∂ ³ ´ u = u + u 1 − uk (3.26) ∂t ∂x k + 1 ∂x He used integral invariants to ﬁnd the exact traveling wave solution 1 v∗ = (3.27) k + 1 µ ¶ h i+ 1/k u(z) = 1 − ek(z−zβ) (3.28)

In all those examples an initial interface between zero and non-zero values of u develops into a sharp front which propagates at a finite speed. The quantitative shape of u as it approaches zero is determined only by k (zero slope for 0 < k < 1, finite slope for k = 1 and infinite slope for k > 1). Sanchez-Gardu´ no˜ and Maini [190, 191] proved that those properties are carried to any FK-like reaction term (stable node at u = 1, unstable one at u = 0 and f (u) > 0∀u ∈ (0,1)) and any non-negative diffusion coefficient D(u) with D(0) = 0, D0 (0) > 0 and D(u) > 0∀u ∈ (0,1] (generalization of the case k = 1). CHAPTER 3 – MATHEMATICAL BACKGROUND 69

Because of the finite support for the field, we expect on general grounds that this class of system can exhibit diffusive instabilities if we couple the density field to a nutrient field. We can investigate this possibility numerically with the simple choice D(b) = bk. Specifically, we take ³ ´ σ˙ = ~∇ · D(b)~∇b + εG(b,n) (3.29)

n˙ = ∇2n − G(b,n) (3.30) where n is the nutrient concentration, and G = nb is the growth term. For a one dimensional front, the numerical results shown in Fig. 3.8 demonstrate the formation of a sharp front, as expected. For two-dimensional simulations, some care must be taken with the singular nature of the front. Fig. 3.9 shows a typical time evolution of the numerical solution. Again, the pattern is cellular, without large scale branching.

Figure 3.8: Traveling wave solutions for the density-dependent diffusion model with σn as a growth term. The monotonically decreasing ﬁeld with the sharp front is σ, the monotonically increasing is n.

Both the nonlinear diffusivity model and the reaction-cutoff model has the feature of rapid re-organization of b once the front has passed [93, 61, 128]. Our current belief is that to get a really branching patterns in either of these models the re-organization must be 70

Figure 3.9: Time evolution of a typical 2D solution for the density-dependent diffusion model with σn as a growth term and without death term. The different lines show the support of the solution at different times, with the square at the middle as initial conditions. The different lines show the support of σ at successive times. prevented. It can be done in two ways. One possibility is to make the diffusion constant D(b) dependent on n as well as on b [115]. Since n goes to zero inside the region of non- zero b, this will lock the pattern in place. Alternatively, one can explicitly include a “death” term −µb (where µ is a small positive constant) in the equation for σ˙ [59, 124]. Such term accounts for the transition of the bacteria into pre-spore state. It causes the density to go back down to zero after the front passes. Plotting the sum of the b ﬁeld and the density of “dead” bacteria s deﬁned by

s˙ = µb , (3.31) we see a full branching structure in Fig. 3.10. We again emphasize the fact that the density- dependent diffusion is responsible for the qualitative difference in behavior between this system and the diffusive FK model. Similar results hold for a cutoff model with a death CHAPTER 3 – MATHEMATICAL BACKGROUND 71 term (see Sec. 4.1.1.1).

Figure 3.10: Result of a typical simulation of the density-dependent diffusion model with σn as a growth term and with a death term. The grey levels indicate the density of σ + S. Noise was introduced through the diffusion terms in order to reduce implicit anisotropy of the underlying lattice.

The phenomenon presented here, of 2D branching patterns driven by density-dependent diffusion, differs from the models usually used by the physicists community. Unlike phase- ﬁeld type of models, the front is a sharp one in the model as is, not in some limit. Unlike

Stefan-like models, the front is not forced from outside, but it is a singularity in the solution which spontaneously arises from the nonlinear diffusion term in the equations.

3.1.6 Pattern selection

We would like to finish our discussion of diffusive instabilities with a review of one of the most important concepts in this field, that of pattern selection via solvability [117, 137, 181, 50]. It is this idea which ultimately explains how seemingly small details of the microscopic kinetics can in fact play a controlling role in deciding upon the macroscopic structure. The most famous application of this theory is the stabilization of snowflake- like dendritic morphologies due to the small crystalline anisotropy in the surface energy 72 between solid and liquid. A bacterial “demonstration” of this anisotropy effect is shown in

Fig. 3.11.

Since there exist several exhaustive reviews of this topic, we will do with a qualitative discussion of the dendrite problem. First, we note that the phase-field model [136, 64, 55, 57, 56, 81, 228, 130] as well as the modified DFK model can lead, under the assumption that the interface thickness in the first field is much smaller than the diffusion length of the second field, to a direct equation of motion for a sharp interface. In the phase field model the dynamics satisfy this assumption for large enough D . In the non-linear diffusion model the assumption is satisfied for small enough initial levels of n as b and hence the diffusion coefficient of b are everywhere small.

Let us use the phase ﬁeld model of solidiﬁcation as our example

∂φ δF = ∇2φ − (3.32) ∂t δφ ∂T 1 ∂φ = D ∇2T + (3.33) ∂t 2 ∂t

First, one solves for the φ ﬁeld at ﬁxed T, since T varies on a much larger scale. This leads to an equation relating the value of T at the φ front to the velocity of the front and the front curvature; the curvature arises via the fact that

∂2φ ∂φ ∇2φ ' + κ (3.34) ∂r2 ∂r where r is the coordinate normal to the local interface. The equation relating the value of

T at the front, Tf , to the velocity and curvature has the form

Tf = ηvn + γκ (3.35) where vn is the normal velocity of the front, γ is a surface energy term and η is called the CHAPTER 3 – MATHEMATICAL BACKGROUND 73

Figure 3.11: “Bacterial snowflakes.” Growth of bacterial colonies in the presence of 6 − f old line anisotropy (see Ref. [36] for details). The darker areas in the background of the top figures is due to photography artifacts. Bottom left: Closer look showing dendritic tip. Bottom right: a real (ice) snowflake captured in a snowstorm (taken from Ref. [39]) 74 kinetic coefficient. If η is small, we get the local equilibrium Gibbs-Thomson boundary condition Tf = γκ [135].

Next, we consider the T equation. Since φ approaches its fixed point value on the small spatial scale, the source term can be replaced by a δ function whose magnitude reflects the integrated source strength. This magnitude is correlated with the velocity. This occurs explicitly in the phase-field model but is approximately true as well in the modified DFK equation. Note that the T field reaction term is the same (up to a factor and a sign) with the

φ ﬁeld reaction term and hence, to the extent that we neglect the curvature, it is proportional

0 00 to vnφ +φ ; and, the latter term integrates to zero. All of this serves to motivate the classic Stefan problem [135] Z ∂T = D∇2T − Lv (s)δ(~x −~x(s))ds (3.36) ∂t n with

Tf = ηvn + γκ (3.37) and T → T∞ at large distances. The integral in the Eq. (3.36) is taken over the entire boundary separating the two macroscopic phases of the reaction system.

It turns out that one can look for steady-state solutions of these equations and make tremendous analytic progress. Results to date include

1. At γ = 0, a continuous family of propagating needle crystals solve the dynamic equation [108]. These crystals are parabolas (paraboloids of revolution in 3d, where the ρ ρv tip radius can be used to form the dimensionless Peclet number 2D which must be

chosen as a speciﬁc function of T∞.

2. Adding a ﬁnite γ eliminates all solutions if the surface energy (and kinetic coefﬁcient) CHAPTER 3 – MATHEMATICAL BACKGROUND 75

are isotropic [116, 160]. If γ is made angle-dependent, a unique stable solution is

selected [16, 13, 118, 15].

3. Fluctuations near the tip (induced, say, by thermodynamic ﬂuctuations or by intrinsic particle number noise) will, at large enough anisotropy, leave the tip intact and generate side-branches [12, 49]. At small anisotropy, these ﬂuctuations can destabilize the

tip and lead to a disordered pattern. One can use this approach to obtain a qualitative picture of morphology transitions in this system [50]

The major lesson for us in our task of modeling microorganism patterns is the difﬁculty in anticipating (i.e. guessing before doing a signiﬁcant amount of work) which small details can have surprising macroscopic consequences. One guideline is that a model must always have a stabilization mechanism for large wave-vector and thus, details which affect the cutoff term are inherently critical. Similarly, altering the symmetry of the macroscopic transport driving the instability should typically lead to major changes. Also, the tips of the patterns are what control the overall morphology and so one might expect dramatic consequences if one perturbs this region.

Finally, this sensitivity opens up the possibility of signiﬁcant feedback from the macroscopic level back down to the microscopic level of organization. Suppose there are two different crystal structures for some material with concomitantly differing surface tension anisotropies, kinetic coefﬁcients etc. As a function of the macroscopic transport conditions, one or the other might lead to faster overall growth and hence be selected. This is the scenario we have already described for the transitions between the T and C morphotypes in the bacterial system. 76

3.2 Discrete Models

We can gain much insight into instability mechanisms and nonlinear states from the continuous models of biological processes. Often, though, it is more convenient to compute with discrete “automata” models, which in some sense are designed to be simulated. In fact, we will see that perhaps the most convenient approach for microbial systems seems to be a hybridization of continuum and atomistic methods. In this part, we survey discrete analogs of the models discussed so far.

Let us start with systems exhibiting diffusive instabilities. Initially, the simplest discrete analog was afforded by diffusion-limited-aggregation (DLA) [231, 232]. Here, discrete walkers move diffusively in space and attach to a growing cluster. In the limit of taking one walker at a time (i.e. of extremely slow growth) and purely irreversible attachment at any nearest-neighbor site, one obtains the classic DLA fractal [231, 232]. Subsequent workers extended these ideas, producing a variety of systems which could semi-realistically mimic the physics of crystal growth and related processes. One such example is the diffusion- transition scheme of Ben-Jacob et al. [206, 207, 205], which was used to study morphology transitions in solidiﬁcation from a supersaturated solution. This particular model served as the inspiration for the Communicating Walkers model that we will discuss in Sec. 7.1.

It is well worth emphasizing the beneficial aspects of having a connection between a discrete simulation and a related continuum model. It is usually difficult to do much beyond simulation for a discrete model; so, having a continuum analog allows for analysis that helps guide the simulations and vice versa. For the DLA class of models, the relationship between these automata and the continuous approach to crystal growth as captured in the phase field model (and the free surface reduction thereof) has proven invaluable. Once the CHAPTER 3 – MATHEMATICAL BACKGROUND 77 basics are understood, of course, one can modify the simulation to encompass more details of the actual system and thereby obtain more reliable results. But, just doing the simulations or even the analysis is insufficient; one must understand the fundamental mechanisms at the heart of an observed simulated structure as these can then be compared to the true underlying biological dynamics.

There is also a literature on using discrete models for other, more complex reaction- diffusion processes; see for example the work by Kapral et al. [148] on simulations of 3d knotted labyrinths. In physical reaction-diffusion processes, it is almost always the case that one is using the discrete model as a simpler stand-in for the true continuum dynamics; after all, 1023 particles is usually enough such that a continuum description is valid (recall, though, the cutoff effect for type I systems in Sec. 3.1.3) and also one cannot hope to match the actual number of molecules by discrete simulational entities. So, there is extra noise introduced by having a small number of particles in the simulation and this is the price one pays for a more flexible and more efficiently-coded numerical scheme. Similar remarks hold for lattice-gas automata [45], in which one uses discrete objects to model systems with fluid flow.

Aside from computational convenience, there are good reasons why modeling of biological systems can make good use of discrete entities. First, the numbers match more closely. The number of bacteria in a typical experiment is 108 − 1010; one can almost approach these numbers computationally and therefore one is not plagued by the extra noise issue. Perhaps more importantly, cells contain large numbers of internal degrees of freedom which modulate their response to external signals form other cells. Hence, describing a population of cells with something as non-informative as a density field is usually insufficient. At the very least, one would have to introduce either new variables (which advect 78 with the cell velocity as these are tied to the cells, see Sec. 8.2 and the Orientation Field model in Sec. 6.1) or even new coordinates (see the model in Ref. [78], where the cell’s age is taken as a relevant coordinate for the density field, or the initial model in Sec. 6.1 where the cells’s orientation is taken as a relevant coordinate for the density field); this makes for complicated reaction-diffusion systems. Tracking cells as individual objects makes it easy to do this; we just attach extra labels to the cell and postulate transition rules as to how these labels change in time. This flexibility is, we feel, quite useful and hence some of the models to be discussed keep cells discrete. At the same time, though, continuum analysis is used to shed light on the simulations and forms an indispensable part of an integrated effort to understand microbiological pattern formation. Chapter 4

Reaction-Diffusion Models for Branching Patterns

In this chapter we deal with continuum, reaction-diffusion models for bacterial growth. We compare between various models, both modeling wise and results wise. The models were intended to model branching colonies of either Paenibacillus dendritiformis var. dendron (T morphotype) or Bacillus subtilis. The two biological systems may differ in their underlying reasons for branching, but we will check all of the models only as appropriate models for colonies of T morphotype described in the biological background (Chapter 2). We repeat here some of the ﬁgures presented in that chapter, just as a reminder of our subject.

4.1 Comparison between models

Not all the models we compare here are construced by us. In addition to our own models, we will present models due to Kessler and Levine [120], Kitsunezaki [124], Kawasaki et al. [115] and Mimura et al. [152, 168]. The models are all two-dimensional (2D), with b(~x,t) denoting the bacterial density projected on a 2D plane and n(~x,t) is the 2D nutrient

79 80

Figure 4.1: Patterns exhibited by the T morphotype as a function of peptone level (increasing from left to right) and agar concentration (1.5% bottom row, 2% middle row, 2.5% top row). CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 81

Figure 4.2: A copy of figure 2.6: Closer look on the tip of one branch of a T colony. (a) The branch as defined by the fluid layer. (b) The bacteria within the branch. concentration. The equations for the various models will be written in dimensionless units.

In Appendix A we present the relation between the dimensionless equations and the real (biological) units.

In general, the rate of change of the bacteria density can be described by[171]:

∂b = movement + “birth” − “death” (4.1) ∂t

As discussed in Chapter 2, the movement of bacteria consists of various possible mechanisms, of which we will concentrate on swimming, so that the motion is described as diffusion (either linear or non-linear). The “birth” term in Eq. 4.1 corresponds to bacterial reproduction, which depends on the supply of nutrients. The “death” term represents the transition of bacteria into a non-motile state (this transition is not death, but a transition into pre-spore state, see Sec. 2.3.2). The density of bacteria in the non-motile state is denoted by s(~x,t). 82

4.1.1 Linear diffusion – effective meta-stable reaction term

As was said in Chapter 3, the basic reaction-diffusion model for pattern formation is the Fisher-Kolmogorov equation [82, 126]. This equation was an invaluable tool for the study of pattern formation in reaction-diffusion systems, but it cannot serve as a model to the growth of bacterial colonies as it cannot produce any 2D pattern but a uniform disk (in a circular geometry). Coupling the bacterial field to a nutrient field produces the Diffusive Fisher-Kolmogorov (DFK) model[153, 33] ∂b = D ∇2b + n f (b) (4.2) ∂t b ∂n = D ∇2n − ηn f (b) (4.3) ∂t n where Db is a diffusion coefficient describing the bacterial movement, Dn is the diffusion coefficient of the nutrient in the agar and a reaction term f (b) which describes the “conversion” of nutrient to bacteria, i.e. cells eating in order to reproduce, with η the conversion rate (3 picogram per bacteria, see Sec. 2.3.2). The point b = 0 should be an unstable fixed R point for the inverted potential F (b) = − f (b0)db0, i.e. f (0) = 0 and f 0 (0) > 0. As was proven in Chapter 3, the DFK model, like the Fisher-Kolmogorov equation, does not exhibit Mullins-Sekerka (diffusive) instability, and a two dimensional growth in a circular geometry can develop into a disk only.

4.1.1.1 A cutoff in the reaction term

As we said in the last chapter, one of reasons of the inadequacy of the DFK model to bacterial colonies is the discreteness of bacteria, for which a continuum description is not always valid. Kessler and Levine [120] argue that when describing a discrete system using continuous models, a cutoff near the ﬁxed point must be imposed, i.e. the reaction term CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 83 must be set to zero when the (bacterial-) density is below some threshold. They have shown that inclusion of such a cutoff leads to a Mullins-Sekerka instability.

To show this, we consider the diffusive Fisher equations with a cutoff:

∂b = D ∇2b + bnΘ(b − β) (4.4) ∂t b ∂n = ∇2n − nbΘ(b − β) (4.5) ∂t where β is the threshold density for growth, and Θ is the Heaviside step function. The food consumption term is of the form f (n,b) = nb, which is the widely used low-nutrient approximation for the Michaelis-Menten law (Sec. 2.3.2). If taken as a model for bacterial colonies, the value of the threshold for bacterial growth, β, should be taken down to one bacterium per 1 − 10µm2, which corresponds to a density of less than uniform layer of bacteria of depth one. Fig. 4.3 depicts the reaction term and the inverted potential for this model. As can be seen in Fig. 4.4, an instability of the front indeed appears, and the compact growth pattern has a surface broken by “fjords”. However, the Mullins-Sekerka instability is not sufﬁcient to produce branches. The emerging dips soon “heal”, so that branches are not formed.

One way to obtain branching growth is to add a “death” term to the model of Kessler and Levine:

∂b = D ∇2b + bnΘ(b − β) − µb (4.6) ∂t b ∂n = ∇2n − nbΘ(b − β) (4.7) ∂t ∂s = µb (4.8) ∂t where µ > 0 is the rate of bacterial differentiation into non-moving state, and s(~x,t) is the density of “frozen” bacteria. This modiﬁed model exhibits distinct branching patterns, as 84

Figure 4.3: Growth term f (b) (above) and inverted potential F(b) (below) for a cutoff (Kessler-Levine) correction. Cutoff value β = 1, nutrient level n = 1.

Figure 4.4: 2D growth pattern of the bacterial ﬁeld b of the model of Kessler and Levine. Parameters are: Db = 0.01, β = 0.25, n0 = 1 CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 85 seen on Fig. 4.5. One can intuitively understand that with a death term present, bacteria left behind the propagating front become non-motile (“dead”). They are unable to move and close the “fjords” (the decrease in density due to death can overweight the inﬂux due to diffusion), thus allowing real branches to form. The death term actually makes the reaction term for the bacteria mate-stable, as the rate of reaction for b < β is −µ < 0.

Consider three possible reaction terms f1 (b,n) = bnΘ(b − β) (from Eq. (4.4) ), f2 (b,n) = bnΘ(b − β) − µb (from Eq. (4.6) ) and f3 (b,n) = (bn − µb)Θ(b − β). Out of the three functions, f2 is the only one that can produce real branching patterns (from the reasons stated above). Yet modeling-wise, f2 is the least sensible of the three; the Θ function arises from the discreetness of the bacteria and should appear in all reaction terms which involve b.

Figure 4.5: 2D growth pattern of the bacterial ﬁelds (b + s) of the model of Kessler and Levine with added death term. Parameters are: Db = 0.01,β = 0.25,n0 = 1,µ = 0.01.

4.1.1.2 A meta-stable reaction term

Mimura et al. [152, 168] proposed a model for branching colonies of Bacillus subtilis, a model which is both biologically reasonable and produces the observed patterns. As was said in Sec. 3.1.4, a growth term with a meta-stable point at b = 0 can lead to branching 86 patterns. Mimura et al. [152, 168] have studied the following model: ∂b µb = ∇D ∇b + εbn − (4.9) ∂t b (b + 1)(n + 1) ∂n = ∇2n − nb (4.10) ∂t ∂s µb = (4.11) ∂t (b + 1)(n + 1) They studied two versions of the model, which they correlated with two regions of experimental conditions. The two cases differ in the functional form of Db. In the ﬁrst case Db is a parameter which does not depend on bacterial density, and it’s value is said to vary with agar concentration. This case models experimental conditions where the agar is soft enough for the bacteria to swim in it. In the second case Db is proportional to the bacterial density b - non linear diffusion. This case models experimental conditions where the agar is hard, the bacteria cannot move, and the colony expands on the face of the agar by accumulation of cells. We will discuss the second case in the last paragraph of this section.

The key feature of this model is that the total bacteria growth term (growth plus death), depicted in Fig. 4.6, gives a reaction term for which b = 0 can be either unstable or meta- p stable ﬁxed point. For large amounts of nutrient, n > nc = −1/2 + 1/4 + 4µ/ε, the reaction term has unstable ﬁxed point at b = 0, and for n < nc this point is meta-stable, µ i.e. in order to initiate bacterial growth, a threshold value of b∗ = − 1 must εn(n + 1) be reached. Indeed the simulations show (Fig . 4.8) that if the initial concentration of nutrient is much larger than nc, then a compact, circular colony grows (as is the case for unstable growth term). On the other hand, if the initial concentration of nutrient is close to nc or smaller (so that the concentration at the edge of the colony is smaller than nc) then a branched colony grows (as is the case for meta-stable growth term).

As we argue before, the existence of a threshold density below which there is no growth CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 87

Figure 4.6: Growth term f (b) (above) and inverted potential F(b) (below) for the model of Mimura et al. . Parameters are: ε = 20, µ = 2400, n = 5. Note the stable point at b = 0.

Figure 4.7: 1D front obtained for the model of Mimura et al. . Parameters are: Db = 0.1, n0 = 10, ε = 20, µ = 2400. 88

Figure 4.8: Various 2D patterns of the bacterial ﬁelds (b + s) obtained for the model of Mimura et al. : a. DLA-like (Db = 0.05, n0 = 10). b. dense branches (Db = 0.09, n0 = 10). c. Concentric rings (Db = 0.05, n0 = 12). d. Compact growth (Db = 0.1, n0 = 14). In all cases, µ = 2400, ε = 20.

is not supported by the observations of T , where even a small number of bacteria, inoculated on a substrate, can multiply and then begin to move. Mimura et al. argue that the model captures the experimental morphology diagram observed for Bacillus subtilis. This is a very crucial point. If indeed the above claim is correct, it implies that the observed patterns can be reproduced with no need for additional biological features. However, using the discrete communicating walkers model [35], Ben-Jacob et al. have concluded hat the additional features of chemotactic response have to be included. So, in order to check this point we have performed more detailed comparison between the model of Mimura et al. and the experimental observations of growth of B. subtilis.

First, we consider the DLA-like growth. In this case, the bacteria do not move on the agar surface, and the growth is indeed very similar to the DLA algorithm, as was proposed CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 89 by Mimura et al. [231, 151]. It is now understood that in a mean-ﬁeld DLA model the particle density (density of bacteria in the present case) can not be described by a diffusion term. Instead, it has to be described by a diffusion multiplied by the nutrients ﬁeld [218], which differs from the linear diffusion in the model of Mimura et al. . Indeed, close inspection of the fractal pattern created by the model reveals that it differs from the observed DLA-like patterns.

Another test of the model is the predicted pattern of concentric rings. It has already been pointed out by Rafols [183] that the model’s pattern differs from the observed one. In the experiment, branching growth slows down. The branches become wider and growth stops. Then, after bacterial differentiation, a new cycle of branches growth starts with thin branches emitted from the stationary wide branches [183]. This description differs from the model patterning shown in Fig. 4.8.

In Fig. 4.9 we exhibit results of numerical simulations for various levels of peptone and for agar concentration for which concentric rings are observed. The sequence of patterns from DLA-like at low peptone to concentric rings at high levels of peptone differ from the similar sequence of observed patterns presented in Ref. [183].

We have also tested the change in patterns as we vary the agar concentration (see Fig. 4.10). When we plot the growth velocity as a function of the agar concentration, it does not show a jump in the velocity or its slope. In other words, the model seems to exhibit a simple crossover between the patterns rather then a morphology transition as the observations seem to indicate.

The above results lead us to conclude that the model of Mimura et al. does capture some of the observed branching patterns, yet the complete description of the observations 90

Figure 4.9: The model of Mimura et al. : The effect of changing the initial nutrient level n0. For all pictures Db = 0.05, µ = 2400, ε = 20. CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 91

Figure 4.10: The model of Mimura et al. : The effect of changing Db. For all pictures n0 = 10, µ = 2400, ε = 20. 92

Figure 4.11: Colonial growth velocity vs. Db for the model of Mimura et al. . Parameter values as in Fig. 4.10. requires additional features to be included. Speciﬁcally, we have proposed to include nonlinear diffusion [59, 93] (Sec. 4.1.2), and we also believe that chemotactic response does play an important role for poor growth conditions [35, 60, 61] (Sec. 4.2). Mimura et al. [152, 168] have proposed to include nonlinear diffusion in their model, which prevents the formation of many of the complex morphologies demonstrated by their model. In fact a nonlinear diffusion with the complex reaction term they propose does not present any im- provement other nonlinear diffusion with linear reaction term (Sec. 4.1.2.2), thus we will not discuss this combination in the following sections of nonlinear diffusion.

4.1.2 Nonlinear diffusion

4.1.2.1 Reaction diffusion with lubrication

The Lubricating Bacteria model (LB model) is a reaction-diffusion model for the bacterial colonies [93, 128]. This model includes four coupled fields. One field describes the bacterial density b(~x,t), the second describe the height of lubrication layer in which the bacteria CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 93 swim l (~x,t), third field describes the nutrient n(~x,t) and the fourth field is the stationary bacteria that “freeze” and begin to sporulate s(~x,t) (see Sec. 4.2).

The dynamics of the bacterial field b has a diffusion term which is coupled to the lubrication field and represents movement, and a reaction part which is coupled to the nutrient field and contains terms for reproduction and sporulation. Following the arguments presented in Sec. 2.3.2 (and assuming that the nutrient is always the factor limiting the bacterial growth) we get the growth term bn − µb (µ constant).

We now turn to the bacterial movement. In a uniform layer of liquid, bacterial swimming can be approximated by linear diffusion. The layer of lubrication in a colony is not uniform, and its height affects the bacterial movement. An increase in the amount of lubrication decreases the effective friction between the bacteria and the agar surface. The term

‘friction’ is used here in a very loose manner to represent the total effect of any force or process that slows down the bacteria. As the bacterial motion is over-damped, the local speed of the bacteria is proportional to the self-generated propulsion force divided by the friction. It is easy to show that variation of the speed leads to variation of the diffusion coefficient, with the diffusion coefficient proportional to the speed to the power of two. We assume that the friction is inversely related to the local lubrication height through some power law: friction∼ lγ and γ < 0. We get an expression for the bacterial flux:

−2γ ~Jb = −Dbl ∇b (4.12) where Db is a positive constant.

The lubrication ﬁeld l is the local height of the lubrication ﬂuid on the agar surface.

The ﬂuid production is assumed to depend on the bacterial density. As the production of lubrication probably demands substantial energy, it also depends on the nutrient’s level. 94

We assume the simplest case where the production depends linearly on both the bacterial density and the concentration of nutrient. The fluid’s dynamics is given by: ∂l = −∇ ·~J + Γbn(l − l) − λl (4.13) ∂t l max where ~Jl is the fluid flux (to be discussed), Γ is the rate of the production of the fluid and λ is the absorption rate of the fluid by the agar. λ is inversely related to the agar dryness.

The lubrication fluid flows by diffusion and by convection caused by bacterial motion. A simple description of the convection is that as each bacterium moves, it drags along with it the fluid surrounding it. The diffusion term of the fluid is assumed to depend on the height of the fluid to the power η > 0 (the nonlinearity in the diffusion of the lubrication, a very complex fluid, is motivated by hydrodynamics of simple fluids).

η ~Jl = −Dll ∇l + j~Jb (4.14) where Dl is a diffusion constant, ~Jb is the bacterial flux and j is the amount of fluid dragged by each bacterium. The nonlinearity in the diffusion term causes the fluid field to have a sharp boundary (singular line) at the front of the colony, as is observed in the bacterial colonies (Fig. 2.6).

The complete model for the bacterial colony is:

∂b ¡ γ ¢ = D ∇ · l−2 ∇b + bn − µb (4.15) ∂t b ∂n = D ∇2n − bn ∂t n ∂l ¡ η γ ¢ = ∇ · D l ∇l + jD l−2 ∇b + Γbn(l − l) − λl ∂t l b max ∂s = µb ∂t The second term in the equation for b represents the reproduction of the bacteria. The reproduction depends on the local amount of nutrient and it reduces this amount. The third CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 95 term in the equation for b represents the process of bacterial entering into pre-spore state.

For the initial condition, we set n to have uniform distribution of level n0, b to have compact support at the center, and the other ﬁelds to be zero everywhere.

Results show that the model can reproduce branching patterns, similar to the bacterial colonies (Fig. 4.12). At low values of absorption rate, the model exhibits dense ﬁngers.

At higher absorption rates the model exhibits ﬁner branches with lower fractal dimension (Fig. 4.13). We also obtain ﬁner branches with lower fractal dimension if we change other parameters that effectively decrease the amount of lubrication. We can relate these conditions to high agar concentration. Note, however, that at the extreme levels of food concentration the patterns of the bacterial colonies are dominated by additional effects (see

Sec. 2.3.1.3) and the model must be modiﬁed to accommodate these effects.

4.1.2.2 Density-dependent diffusion

It is possible to introduce a simplified model, where the fluid field is not included explicitly.

k Instead we introduce a density-dependent diffusion coefﬁcient for the bacteria Db ∼ b [59, 25]. Such a replacement can be justiﬁed assuming few features of the dynamics at small bacterial densities and small lubricant heights:

• The production of lubricant is proportional to the bacterial density to the power α > 0

(α = 1 in the previous model).

• There is a sink in the equation for the time evolution of the lubricant ﬁeld, e.g. ab-

sorption of the lubricant into the agar. This sink is proportional to the height of the lubricant to the power β > 0 (β = 1 in the previous model). 96

Figure 4.12: Growth patterns of the bacterial ﬁelds (b + s) of the LB model, for different values of initial nutrient level n0, and different values of the parameters Db and λ. The different sets of values for these later parameters are related to different values of agar concentration, with lowest concentration at the bottom row. The apparent (though weak) 6-fold anisotropy is due to the underlying tridiagonal lattice. The different colors represent different densities of bacteria, both active and stationary, i.e. different values of b + s. CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 97

Figure 4.13: Fractal dimension and growth velocity as a function of initial food concentrations. The data are for typical runs of the LB model with parameters as in the middle row of ﬁgure 4.12. The growth velocity is presented in arbitrary units.

• The two processes above are much faster than the diffusion process over the bacterial length scale, so the level of lubricant is proportional to the bacterial density to the

power of β/α.

• The friction is proportional to the level of lubrication to the power γ < 0.

Given the above assumptions, the lubrication field can be removed from the dynamics and can be replaced by a density dependent diffusion coefficient. This coefficient is proportional to the bacterial density to the power k ≡ −2γβ/α > 0

A model of this type, the Non-Linear Diffusion (NLD) model was offered by Cohen

[59] and by Kitsunezaki [124]:

∂b ³ ´ = ∇ · D bk∇b + nb − µb (4.16) ∂t 0 ∂n = ∇2n − bn (4.17) ∂t 98

∂s = µb (4.18) ∂t

For k > 0 the 1D model with initial conditions of compact support for b, a sharp front with compact support develops (i.e. b = 0 outside a finite domain for all times t > t0, and on the edge of this domain there is a continuous but non-smooth shock wave). For k > 1 the front has an infinite slope, i.e. the first derivative of b diverges. In 2D, the model exhibits branching patterns for suitable parameter values and initial conditions, as depicted in Fig. 4.14. Increasing the initial nutrient level makes the colonies more dense, similarly to what happens in the other models.

n0=1.0 n0=1.5 n0=2.0

Figure 4.14: Growth patterns of the NLD model, for different values of initial nutrient level n0. Parameters are: D0 = 0.1, k = 1, µ = 0.15. The apparent 6-fold symmetry is due to the underlying tridiagonal lattice. The different colors represent different values of the combined ﬁeld (b + s).

Changes in other parameters of the model can result in similar changes in the pattern.

In Fig. 4.15 we present the effect of µ, the rate in which the bacteria turn stationary. Two strains of the bacteria which differ only in their corresponding values of µ can have different colonial patterns at the same growth conditions, but similar patterns in different growth conditions. The relations between variations in the model’s parameters, bacterial strains and developing patterns are discussed in Chapter 9. CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 99

Figure 4.15: Growth patterns of the bacterial ﬁelds (b + s) of the NLD model, with varying values of µ. All other parameters are as in Fig. 4.14, left pattern. The apparent 6-fold symmetry is due to the underlying tridiagonal lattice. 100

4.1.2.3 Food dependent diffusion

Another state-dependent diffusion coefﬁcient was proposed and studied by Kawasaki et al. [115], which took Db ∼ nb. They justify this form by the observation that bacteria are active mostly at the edge of the colony – the only area where there is high amount of bacteria and food. After appropriate rescaling, the equations of the model are

∂b = ∇ · (D nb∇b) + nb (4.19) ∂t 0 ∂n = ∇2n − bn (4.20) ∂t

This model, too, exhibits branching shapes (Fig. 4.16). This is due to the b dependence of the diffusion coefﬁcient, which leads to front instability, just as in the NLD model. The fact that Db also depends on n prevents bacteria inside the colony from moving – and closing the dips created by the instability. In this way, branches are created without a need for a death term.

Figure 4.16: 2D growth pattern (this case showing b, not b + s) of the model of Kawasaki et al. . Parameters are: D0 = 1.0, n0 = 0.71.

Mathematically, this model is more appealing than the NLD model: there is a conservation of (b + n) over the space, which simpliﬁes the analysis of the model. However, as we will see in the next section, this model as disadvantages as a model for bacterial colonies. CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 101

4.2 Chemotaxis and colonial self-organization

4.2.1 Food chemotaxis and chemotactic signaling

So far we have seen several models for the branching colonies, each with its own mechanism of diffusive instability, which produces patterns resembling the some of the observed ones. In an attempt to differentiate between the models, we rely on results from an atomistic model – the Communicating Walkers model [35, 31, 60] (and see also Sec. 7.1) – for an indication of what are the biological features relevant to the different morphologies. Ben-Jacob et al. [21, 60, 20, 18, 24] showed that many aspects of the patterns are related to chemotaxis (see Sec. 2.3.1.3). Chemotaxis means changes in the movement of the bacteria in response to a gradient of a certain chemical field [2, 42, 133, 40, 77, 209]. The movement is biased along the gradient either in the gradient direction or in the opposite direction. Usu- ally chemotactic response means a response to an externally produced field, like as in the case of chemotaxis towards food. However, the chemotactic response can also be to a field produced directly or indirectly by the bacterial cells. Ben-Jacob et al. claimed that the effect of chemotaxis is most apparent in harsh conditions such as very low levels of nutrients (where the colonies assumed orderly structure) and hard agar (where the colonies assume global chirality), but comparisons of growth velocity can reveal the effects of chemotaxis in all growth conditions. We will now extend the Non-Linear Diffusion model to test for its ability to describe aspects of the bacterial colonies involving chemotaxis and chemotactic signaling.

Before we go into the details, we would like to discuss the food-dependent-diffusion model of Kawasaki et al. [115] (Sec. sec:contT:kawasaki). Repulsive chemotactic signaling, which we will discuss below, requires distinction between active bacteria and inactive 102 bacteria (where the inactive bacteria emit signal and the active bacteria do not). In the NLD model, an integral part of the model is the deﬁnition of the inactive bacteria s. In the model of Kawasaki et al. , some bacteria are more active than others (with different motility), but non of the bacteria is absolutely and irreversibly inactive. We will, therefore, not test the model of Kawasaki et al. [115] with chemotaxis.

4.2.1.1 Chemotaxis in the Non-Linear Diffusion model

Ben-Jacob et al. [21, 60, 20, 18, 24] argued that for the colonial adaptive self-organization T bacteria employs several kinds of chemotactic responses, each dominant in different regime of the morphology diagram. One response is the food chemotaxis we mentioned in

Sec. 2.3.1.3. According to the ”receptor law”, it is expected to be dominant for some range of nutrient levels (the corresponding levels of peptone are determined by the constant K).

The two other kinds of chemotactic responses are signaling chemotaxis. One is long-range repulsive chemotaxis where the chemical is secreted by starved bacteria at the inner parts of the colony. The second signal is a short-range attractive chemotaxis where the chemical is secreted by bacteria at the colony’s front, bacteria which are immersed in toxic waste products. The length scale of each signal is determined by the diffusion constant of the chemical agent and the rate of its spontaneous decomposition.

As we mentioned in Sec. 2.3.1.3, in a continuous model we incorporate the effect of chemotaxis to a chemical ﬁeld R by introducing a chemotactic ﬂux J~chem:

J~chem ≡ ζ(b)χ(R)∇R (4.21)

χ(R)∇R is the gradient sensed by the bacteria (with χ(R) having the units of 1 over chemical concentration). χ(R) is usually taken to be either constant or the “receptor law”. ζ(b) CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 103 is the bacterial response to the sensed gradient (having the same units as a diffusion coefﬁ- cient times the units of the bacterial density b). Chemotaxis is an active response affecting movement, and if there are speed variations in the model (as there are in models with nonlinear diffusion) the same variations in speed effect ζ(b). ζ(b) is proportional to Db times b: ζ(b) = ζ0bDb, where ζ0 is a constant, positive for attractive chemotaxis and negative for repulsive chemotaxis. Thus, we obtain the reaction diffusion chemotaxis equation:

∂b = −∇ · [−D (b)∇b + ζ(b)χ(R)∇R] + f (b,n) ∂t b

= ∇ · {Db (b)[∇b − bζ0χ(R)∇R]} + f (b,n)

If the chemotactic material (R) is not already in the model (in the case of food chemotaxis, the material R is actually the nutrient n), one has to specify an equation describing the dynamics of the material. In the case of chemotactic repulsion from pre-spores [35, 31, 60], the equation describing the dynamics of the chemorepellent is:

∂R = D ∇2R + Γ s − Ω bR − Λ R (4.22) ∂t R R R R

The equation contains (in this order) terms for diffusion, production by pre-spores, decomposition by active bacteria and spontaneous decomposition. DR is the diffusion coefﬁcient for the chemorepellent, ΓR is the emission rate of repellent by pre-spores, ΩR is the decomposition rate of the repellent by active bacteria, and ΛR is the rate of self decomposition of the repellent.

Fig. 4.17 depicts a pattern produced by the NLD model when food chemotaxis is included. All of the parameters values, excluding those of food chemotaxis, are the same as in Fig. 4.14 (no chemotaxis). Although the patterns with and without chemotaxis are very similar, the growth velocity with food chemotaxis is about twice the growth velocity 104 in the absence of chemotactic response. In other words, the velocity is doubled with no signiﬁcant change in the fractal dimension.

2.5

1.5

0.5

Figure 4.17: 2D growth pattern of the bacterial ﬁelds (b + s) of the NLD model with food chemotaxis included. The growth velocity is almost doubled, without signiﬁcant affect of the pattern. χ0n = 3. Values of other parameters are as in Fig. 4.14.

The effect of repulsive chemotactic signaling is demonstrated in Fig. 4.18 – again with parameters values the same as in Fig. 4.14, excluding parameters of repulsive chemotactic signaling. The repulsive chemotactic signaling turns the fractal-like shape into a radial branching pattern with a circular envelope.

1.5

0.5

Figure 4.18: 2D growth pattern of the bacterial ﬁelds (b + s) of the NLD model with repulsive chemotactic signaling included. The pattern has turned into a radial branching pattern with a circular envelope. Parameters are: n0 = 1.0, χ0R = 1, DR = 1, ΓR = 0.25, ΩR = 0, ΛR = 0.001. Values of other parameters are as in Fig. 4.14.

The effect of chemotactic response in the model of Mimura et al. is presented in CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 105

Figs. 4.19 and 4.20. The chemotactic response was added to a previously DLA-like colony

(Fig. 4.8a). The addition of food chemotaxis turns the colony into a densely branched one, with branches much thicker than before. The repulsive chemotaxis makes the branches radially oriented, but they become thicker than before.

The effect of the two types of chemotactic response – food chemotaxis and repulsive chemotactic signaling - in the model of Mimura et al. differs from the effect of chemotaxis on both the Communicating Walkers model [35, 31, 60] and the NLD model. We believe this to stem from the fact that, unlike the former model, the later two models capture the important feature of a lubrication ﬂuid, and in particular the sharpness (localized edge) of the front.

Figure 4.19: 2D growth pattern of the bacterial ﬁelds (b + s) of the Mimura et al. model with food chemotaxis included. χ0n = 0.06. Values of other parameters are as in Fig. 4.8a.

4.2.2 Weak chirality

As was said in Sec. 2.2.1, At high agar concentrations the branches also exhibit a global twist with the same handedness, as shown in Fig. 2.3. Similar observations during growth of other bacterial strains have been reported by Matsuyama et al. [157, 155]. Such growth patterns are referred to as having weak chirality. 106

Figure 4.20: 2D growth pattern of the bacterial ﬁelds (b + s) of the Mimura et al. model with repulsive chemotactic signaling included. Parameters are: χ0R = 0.1, DR = 1, ΓR = 0.2, ΩR = 0, ΛR = 0.01. Values of other parameters are as in Fig. 4.8a.

In a reaction-diffusion model, weak chirality can be obtained by modifying the chemotactic mechanism and causing it to twist: we alter the expression for the chemotactic ﬂux

J~chem (Eq. 4.21) so that it is no longer oriented with the chemical gradient (∇R). Instead it is oriented with a rotated vector Kˆ (θ)∇R, where Kˆ (θ) is the two-dimensional rotation operator and θ is the rotation angle µ ¶ cos(θ) −sin(θ) Kˆ (θ) = sin(θ) cos(θ)

The chemotactic ﬂux is thus written:

¡ ¢ J~chem = ζ(b)χ(R) Kˆ (θ)∇R (4.23)

The effect of rotating the repulsive chemotaxis, as depicted in Fig. 4.21, is to make the pattern chiral, with the degree of chirality determined by the rotation angle θ.

Inspite the apparent resemblance between the experimental patterns and the simulated patterns, we believe that the rotated chemotaxis presented above is at most an approximation to the biological processes causing weak chirality. Rotated chemotaxis is incompatible with the known biological facts: the bacteria cannot modulate their runs as a function of CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 107

Figure 4.21: Growth patterns of the NLD model with a “squinting” repulsive chemotactic signaling, leading to weak chirality. Parameters are as in the previous picture, θ = 43◦.

the difference between their direction and that of the gradient; they do not know what is the direction of the gradient, only the directional derivative along their path. As is demonstrated in Sec. 6.2 and in Sec. 7.2.3, one of the key features for the weak chirality in the spinors model is the correlation in orientation between neighboring bacteria. The twist of the branches is related only indirectly, through the bacterial mean orientation, to the chemorepellent’s gradient. A continuous model of such processes should include information about the orientation of the bacterial cells. Such a model is presented in Chapter 6.

The colonial patterns with weak chirality are indication to the long range interactions between bacteria thorough chemotactic signaling and not merely food chemotaxis (see also next section for support of the claim that weak chirality is not due to short-range interactions): the ﬁeld of chemorepellent chemical is only weakly effected by the local densities of bacteria, and therefore it is not affected by the change of pattern caused by rotated chemotaxis. If we use only food chemotaxis and add to it similar rotation of chemotaxis, the result is not be a chiral pattern: the local gradients of food are strongly affected by the local 108 colonial pattern (due to food consumption), and thus there is no long-range correlation in the deviation of the branches from the radial direction and no weak chirality.

We take the experimental existence of weak chirality to be strong evidence that chemotactic signaling should be included in the model to explain the radial branching patterns with circular envelopes, and not merely a change of parameters in some model without chemotaxis, as some researcher claim [115, 152, 168]

4.2.2.1 Chirality measure

Till now we referred to chirality as a two-valued property – either the pattern is chiral or it is not. There are various attempts in the literature to quantify chirality with a continuous measure. See, for example, the method of Avnir et al. [235, 114], who applied their method also to large disordered objects. While this method is general and can quantify with a single number the measure of asymmetry of any given object, it is not quite satisfactory for our purpose. We would like to know the time evolution of the chirality of a colony, and not just “mean” chirality given by a single number. We sacriﬁce the generality of the measure to that end.

Since the growth velocity of the colonies (both experimental and simulated) is constant, we measure the chirality as a function of radius instead of a function of time. Thus we can work on the chirality of an image, not of a process. The image can be a scanned picture of the real colony or the result of a computer simulation. We look for a mapping of the image to a new one, which in some sense does not distinguish left from right (the ambiguity stems from the fact that a large random object will not have, in general, reflection symmetry, thus CHAPTER 4 – REACTION-DIFFUSION MODELSFOR BRANCHING PATTERNS 109 there is no trivial definition for chirality of such objects). The mapping is defined by:

(r,θ) → (r,θ + ∆θ(r)) (4.24) where each point in the image is described by the polar coordinates (r,θ), measured from the center of the colony. Thus, each point is rotated by an increment ∆θ which depends on the radius r (i.e. the distance from the center).

Working on many experimental patterns, as well as simulated patterns, we have learned that in most cases a linear dependence of ∆θ on r is sufﬁcient to give quite satisfactory results, that is, to transform a chiral pattern into a “normal” branching pattern. The rotating angle is thus written: µ ¶ r ∆θ(r) = θmax (4.25) rmax where rmax is the radius of the colony, and θmax is the rotation angle at that radius.

The fact that this linear angular mapping sufﬁces to “de-chiral” the simulated patterns may not be of much importance (in the case of the continuous model, at least, this is almost a direct result of the way in which we introduce the weak chirality). The same transformation works for images of real colonies of T morphotype, but does not work for chiral colonies of other bacteria (see below). This strengthens our belief in the models.

P. dendritiformis is not the only bacteria whose colonies exhibit chirality. Ben- Jacob et al. discussed in [22] the formation of colonies of Paenibacillus vortex, where each branch is produced by a leading droplet and emits side branches, each with its own leading droplet. Each leading droplet consists of hundreds to millions of bacterial cells that circle a common center (a vortex) at a cellular speed of about 10µm/s (P. vortex is not a close relative of P. dendritiformis and its movement on the agar is swarming, not swimming). In Fig. 4.22 we show a colonial pattern of these bacteria. The chirality we termed ‘weak 110 chirality’ is evident in this ﬁgure. We believe that in this case the chirality is not related to the handedness of the ﬂagella, but to the rotation of the vortices. When “pulled” by attractive food chemotaxis, Magnus force1 acts on the vortices and drive them side-ways from the direction (opposite to the direction) of the food gradient. The chirality of colonies of P. vortex results from mechanism local in nature, and not global as in T morphotype. This difference in mechanisms is expressed in the global pattern: the colonial patterns of

P. vortex cannot be “de-chiraled” by the transformation (linear angular mapping) that “de- chiral” the T morphotype. The fact that the models for weak chirality match in this respect the weak chirality of T morphotype and not the ‘weak chirality’ of P. vortex is another support for their success in describing the bacterial colonies.

Figure 4.22: A colony of Paenibacillus vortex on 10g/l peptone level and 2% agar concentration. The dots at the tips of the branches are bacterial vortices – each is composed of up to millions of bacterial cells rotating around a common center. The twist of the branches results from a Magnus-like force induced by repulsive chemotactic signaling – forward force applied on a rotating body results in a side-way motion

1Magnus force – a force acting on a rotating cylinder whose axis is perpendicular to the flow of fluid in which it is immersed; the force is perpendicular to both the flow direction and the cylinder axis. Chapter 5

Analysis of Models with Non-Linear Diffusion

In this chapter we concentrate on analytical study of the equations we presented in chapters

3 and 4. In the ﬁrst section we show that reaction-diffusion systems with degenerate diffusion term have a unique traveling wave solution with sharp front (”half-ﬁnite” support).

We limit our study to 1-dimensional and 2-dimensional space (RN, N = 1,2), for two reasons, one physical and one mathematical. The physical reason is that bacterial colonies cannot grow unsupported in fully 3-dimensional patterns (fully 3-dimensional patterns means here ”that cannot be approximated by 2-dimensional patterns”). The mathematical reason for the dimensional limitation arises from work of Kamin, Rosenau and Kersner

[113, 188, 112] on diffusion systems with degenerate diffusion term. Kamin, Rosenau and Kersner showed that in RN with N ≥ 3, such diffusion systems with initial condition of

finite support, can give rise, in finite time 0 < T < ∞, to solutions with infinite support. For N = 1,2 , such diffusion systems and initial condition of finite support give rise to solutions of finite support for all t > 0. While this results do not extend readily to reaction- diffusion systems, they of enough concern to influence our choice of spatial dimensions,

111 112 adding weight to the physical considerations.

5.1 Existence and uniqueness of sharp-front traveling wave solutions

In this section we use a dynamical systems approach to prove the existence of traveling wave solutions to the degenerate density-dependent diffusion equation

2 ∂tw = ∂x [D(w)∂xw] + A(w)(∂xw) + G(w) (5.1) where D(0) = 0 and G is a Fisher-like growth term. We prove that there is a critical speed c∗ such that 1) there is no traveling wave solution with speed smaller then c∗ , 2) at speed c∗ there is a traveling wave solution with semi-compact support (the solution is exactly zero beyond a free boundary moving at a speed c∗ ), which we will refer to as sharp solution ,

3) there are traveling wave solutions with exponential decay for all speeds larger than c∗. For the purposes of Chapters 4 and 6, the most relevant functional forms of D, A and G are D(w) ∝ wn , A(w) ∝ wn−1 and G(w) ∝ w(1 − wn).

We also show that the smooth solutions can approximate the sharp solution, so that numerical schemes that can only ﬁnd smooth solutions can approximate the sharp solution.

The proof we present here follow the methodology that Sanchez-Gardu´ no˜ and Maini ¯ d ¯ [190, 191] presented for the special case dw D(w) w=0+ > 0 and A(w) ≡ 0, but our proof ¯ d ¯ differ in many of the details. In Ref. [191] they tried to analyze also the case D(w) + = ¯ dw w=0 d2 ¯ 0, 2 D(w)¯ 6= 0 and A(w) ≡ 0, but fail to recognize the existence of the sharp trav- dw w=0+ eling wave solution. Our generalization allow us to do so easily.

We will prove all of the above for Eq. (5.1) where D , A and G satisfying the following CHAPTER 5 – ANALYSIS OF MODELSWITH NON-LINEAR DIFFUSION 113 conditions:

Deﬁnition 5.1 The functions D , A and G are said to be satisfying conditions set (1) if they satisfy the following conditions:

1. G(0) = G(1) = 0 and G(w) > 0∀w ∈ (0,1) ,

2 0 0 2. G ∈ C[0,1] with G (0) > 0 and G (1) < 0 ,

n m 3. D(w) = w Dn (w) and A(w) = w Am (w) where n > 0 , m ≥ n − 1 and Dn (w) > 0∀w ∈ [0,1] ,

2 2 1−n 0 4. Dn ∈ C[0,1] and Am ∈ C[0,1] with f1 (w) = w A(w) + Dn (w) + wDn (w) > 0∀w ∈ 1−n d (0,1] and f2 (w) = w dw f1 (w) 6= 0∀w ∈ (0,1] ,

5. the limits limw→0+ f1 (w) and limw→0+ f2 (w) exist, with limw→0+ f1 (w) > 0 and

limw→0+ f2 (w) 6= 0 .

In order to proceed we use the following transformations:

√ u(y,s) = w(x,t)n s = t/n y = x/ n (5.2) ∂ ∂ ∂ √ ∂ ⇒ w = u1/n = n = n ∂t ∂s ∂x ∂y and transform Eq. (5.1) into · ¸ ∂ ∂ ³ ´ ∂ ³ ´ u = uD u1/n u + u(n−1)/nG u1/n ∂s ∂y n ∂y µ ¶ 1 h ³ ´ ³ ´i ∂ 2 + (1 − n)D u1/n + u(1−n)/nA u1/n u (5.3) n n ∂y 114

³ ´ ³ ´ ³ ´ 1/n 1−n 1/n 1 (1−n)/n 1/n We deﬁne d (u) = uDn u , a(u) = Dn u + u A u and g(u) = ³ ´ n n u(n−1)/nG u1/n to get the equation

2 ∂su = ∂y (d (u)∂yu) + a(u)(∂yu) + g(u) (5.4)

It is apparent from the transformation that if D , A and G satisfy conditions set (1) then d , a and g satisfy the following conditions:

Deﬁnition 5.2 The functions d , a and g are said to be satisfying conditions set (2) if they satisfy the following conditions:

1. g(0) = g(1) = 0 and g(u) > 0∀u ∈ (0,1) ,

2 0 0 2. g ∈ C[0,1] with g (0) > 0 and g (1) < 0 ,

3. d (0) = 0 and d (u) > 0∀u ∈ (0,1] ,

2 2 0 00 0 4. d ∈ C[0,1] and a ∈ C[0,1] with d (u)+a(u) > 0∀u ∈ [0,1] and d (u)+a (u) 6= 0∀u ∈ [0,1] .

Note that conditions set (2) is a special case of conditions set (1) for n = 1 .

Deﬁnition 5.3 Suppose that there exist a speed c > 0 such that u(x,t) = φ(x − ct) = φ(ξ) satisﬁes

1. d (φ)φ00 + cφ0 + [d0 (φ) + a(φ)](φ0)2 + g(φ) = 0 ∀ξ ∈ (−∞,+∞) ,

2. φ(−∞) = 1 and φ(+∞) = 0 , CHAPTER 5 – ANALYSIS OF MODELSWITH NON-LINEAR DIFFUSION 115 then the function u(x,t) = φ(x − ct) is called a traveling wave solution (t.w.s.) of smooth type for Eq. (5.4)

Deﬁnition 5.4 Suppose that there exist a speed c > 0 and a value ξ0 ∈ (−∞,+∞)such that u(x,t) = φ(x − ct) = φ(ξ) satisﬁes

00 0 0 0 2 1. d (φ)φ + cφ + [d (φ) + a(φ)](φ ) + g(φ) = 0 ∀ξ ∈ (−∞,ξ0) , ¡ ¢ ¡ ¢ φ ∞ φ ξ− φ ξ+ φ ξ ξ ξ ∞ 2. (− ) = 1 , 0 = 0 = 0 and ( ) = 0 ∀ ∈ ( 0,+ ) , ¡ ¢ ¡ ¢ φ0 ξ− φ0 ξ+ 3. 0 < 0 and 0 = 0 then the function u(x,t) = φ(x − ct) is called a traveling wave solution (t.w.s.) of sharp type for Eq. (5.4).

We will prove the following theorem:

Theorem 5.1 Suppose that the functions d, a and g in Eq. (5.4) satisfy conditions set (2), then there exist a unique value of c, c∗ > 0, such that Eq. (5.4) has:

1. No t.w.s. traveling in a speed c, ∀c ∈ (0,c∗) .

2. A t.w.s. of sharp type, traveling in unique speed c∗.

3. A monotonically decreasing t.w.s. of smooth type moving in speed c, ∀c ∈ (c∗,∞).

By substituting u(x,t) = φ(x − ct) ≡ φ(ξ) and φ0 (ξ) ≡ v(ξ) into Eq. (5.4) we get a singular (at φ = 0) ODE system:   φ0 = v −[d0 (φ) + a(φ)]v2 − g(φ) − cv (5.5)  v0 = d (φ) 116

dτ 1 The singularity can be removed by introducing a new variable τ, deﬁned by = , to dξ d (φ) obtain ½ φ˙ = d (φ)v (5.6) v˙ = −cv − [d0 (φ) + a(φ)]v2 − g(φ) where a dot denotes a differentiation with respect to τ. For the validity of this transformation see Refs. [9, 173, 190, 191]. Once solution φ(τ) of the system (5.6) is found, the same transformation can be used to obtain ξ(τ), and the inverse function τ(ξ) can give φ(τ(ξ)) = φ(ξ).

5.1.1 Local analysis

For all positive c there are 3 ﬁxed points to the system (5.6) in the phase plane (φ,v): µ ¶ −c T P = (0,0)T , P = (1,0)T and P = 0, . We start with a local analysis 0 1 c d0 (0) + a(0) around the ﬁxed points.

The Jacobian matrix for all three points is µ ¶ d0 (0)ν d (φ) J φ ν = (5.7) ( , ) −d00 (0)ν2 − a0 (0)ν2 − g0 (φ) −c − 2ν[d0 (0) + a(0)] µ ¶ 0 d (1) The Jacobian matrix for P is J = . Its eigenvalues are λ± = 1 P1 −g0 (1) −c 1 p 2 0 0 λ+ λ− −c/2 ± c /4 − d (1)g (1). Since d (1)g (1) < 0, 1 is positive and 1 is negative. λ+ + P1 is a hyperbolic saddle-point. The eigenvector of the positive eigenvalue 1 is v1 = ³ p ´ 2d (1),−c + c2 − 4d (1)g0 (1) . µ ¶ 0 d (0)vc 0 The Jacobian matrix for Pc is JPc = 0 00 0 2 where vc = −g (0) − [d (0) + a (0)]vc c −c < 0. The eigenvalues of J are λ− = v d0 (0) < 0 and λ+ = c . P is a a(0) + d0 (0) Pc c c c c λ+ T hyperbolic saddle-point. The eigenvector corresponding to c is (0,1) , i.e. along the vertical axis. CHAPTER 5 – ANALYSIS OF MODELSWITH NON-LINEAR DIFFUSION 117

µ ¶ 0 0 The Jacobian matrix for P is J = . Its eigenvalues are λ+ = 0 and 0 P0 −g0 (0) −c 0 λ− λ+ λ− + 0 T 0 = −c. The eigenvectors corresponding to 0 and 0 are v0 = (c/g (0),−1) and − T v0 = (0,−1) respectively. All trajectories in the neighborhood of P0 converge towards λ+ − trajectories near the manifold of 0 (excluding the trajectory along v0 , the vertical axis), and the dynamics near P0 is determined by the dynamics along this manifold. Checking + ε + ε along v0 we ﬁnd that for v = v0 (where | | ¿ 1),  − 0 ( ) ¡ ¢  φ˙ cd 0 ε2 ε3  = 0 + o g" (0) # 2 00 ( ) ¡ ¢  0 c g 0 ε2 ε3  v˙ = − d (0) + a(0) + 2 + o 2g0 (0) " # ∂ ¡ ¢ c2g00 (0) ¡ ¢ εv+ = −ε2d0 (0)v+ + ε2 2d0 (0) + a(0) + v− + o ε3 (5.8) ∂τ 0 0 2g0 (0)2 0 0 − Since d (0) > 0 and perturbations in v0 decay, we ﬁnd that trajectories that pass through ε + ε v0 are attracted by P0 for positive (on the lower right quarter of the plane) and are repelled by P0 for negative ε (on the upper left quarter). P0 is a non-hyperbolic saddle- − point. Since v0 is along the vertical axis we conclude that trajectories near P0 with positive φ λ+ will converge towards the manifold of 0 from the right and move towards P0, while φ λ+ trajectories near P0 with negative will converge towards the manifold of 0 from the left and move away from P0 (see Fig. 5.1). A trajectory with zero φ moves towards P0 (as λ− 0 < 0).

Figure 5.1: Local behavior of the ﬂow near the ﬁxed points P0 , P1 and Pc.

5.1.2 Existence of a continuum of smooth front solutions

In this section we prove that there is a range of velocities for which there is no t.w.s. for Eq. (5.4) and a range of velocities for which there is a t.w.s. of smooth type for Eq. (5.4). 118

We introduce the following notation: for a given speed c we denote the left-unstable manifold of P1 and the right-stable manifold of Pc by Wc (P1) and Wc (Pc) respectively.

Lemma 5.2 For any trajectory U (τ) s.t. U (τ) ∩ (0,1] × (−∞,0) 6= 0/ (a ”relevant” trajectory), the limit of the trajectory is limτ→∞ U (τ) = P where P is either P0 , Pc or (0,−∞).

Proof. Let R0 be the region R0 = (0,1] × (−∞,0) . Let τ0 ∈ (−∞,∞) be a value of τ s.t.

U (τ0) ∈ R0 and let τ1 ∈ (τ0,∞] be the next exit point of U (τ) , i.e. U (τ) ∈ R0 ∀τ ∈ (τ0,τ1) τ τ ∞ φ τ and U ( 1) ∈/ R0 if 1 6= + . Since v˙|v=0,φ∈(0,1) = −g( ) < 0 , U ( 1) cannot be on the ¯ ¯ ¯ ¯ horizontal axis. Since ¯φ˙¯ < 0, U (τ1) must be on the vertical axis. Since ¯φ˙¯ = 0 (φ,v)∈R0 φ=0

, U (τ) cannot reach the vertical axis in ﬁnite time and τ1 must be ∞. So any trajectory in

R0 moves towards the vertical axis but does not reach it in ﬁnite time. U (+∞) can be either a ﬁxed point on the vertical axis (P0 or Pc) or limτ→∞ v = −∞ for the second component of

U (τ), i.e. limτ→∞ U (τ) = (0,−∞)

Proposition 5.3 There exist ε > 0 such that for all 0 < c < ε there is no t.w.s. for Eq. (5.4) and Wc (P1) tends to (0,−∞) as τ → +∞

Proof. The straight line connecting P1 with Pc is the line v = l1 (φ) = −(1 − φ)c/ f (0) where f (φ) = d0 (φ) + a(φ) ≥ 0. We deﬁne the region

R1 = {(φ,v)|0 ≤ φ ≤ 1 , − ∞ < v < l1 (φ)}, and show that for sufﬁciently small values of c , Wc (P1) \ P1 ⊂ R1. Since R1 does not include the ﬁxed points, Wc (P1) cannot be a heteroclinical trajectory and there is no t.w.s. for Eq. (5.4). p ¯ + f (0) 0 d ¯ The slope of Wc (P1) near P1 is the slope of v1 . For c ¿ d(1) −d (1)g (1) , dφWc (P1)¯ = √ √ φ=1 0 −c+ c2−4d(1)g0(1) −d(1)g (1) c 0 2d(1) = d(1) + o(c) > f (0) = l1 (1) and near P1, Wc (P1) enters R1 (see CHAPTER 5 – ANALYSIS OF MODELSWITH NON-LINEAR DIFFUSION 119

Fig. 5.2). Trajectories crossing l1 have the slope ¯ ¯ ¯ dv ¯ v˙¯ −cv − f (φ)v2 − g(φ)¯ ¯ = ¯ = ¯ dφ¯ φ˙ ¯ d (φ)v ¯ v=l1(φ) v=l1(φ) v=−(1−φ)c/ f (0) (1 − φ) f (φ) − f (0) 1 f (0)g(φ) = c + ≡ t (φ) (5.9) f (0)d (φ) c (1 − φ)d (φ) 1

For c ¿ f (0) f (φ)g(φ) t (φ) ' > 0 ∀φ ∈ (0,1) (5.10) 1 (1 − φ)d (φ)c 0 φ ε φ 0 φ φ ε Since l1 ( ) is O(c), there exist 1 > 0 s.t. t1 ( ) > l1 ( ) ∀ ∈ (0,1) ∀c ∈ (0, 1). Trajecto- ries inside R1 (and speciﬁcally Wc (P1) ) does not leave R1 on l1 (φ) φ ∈ (0,1] (see Fig. 5.2). − The only trajectory entering Pc is Wc (Pc) which does so in the slope of vc . For c ¿ f (0) ¯ d ¯ 1 f (0)g0(0) 0 φ φWc (Pc)¯ = 0 +o(c) > l1 ( ) , that is, Wc (Pc) does not enter (in inverse time) d φ=0 c f (0)+d (0) R1 (at least not near Pc) (see Fig. 5.2).

We found that for small positive c ,Wc (P1) enters R1, does not leave it on l1 and does not reach Pc. Therefore according to lemma 5.2 limτ→∞ Wc (P1) must be (0,−∞). There is no heteroclinical trajectory connecting P1 to another ﬁxed point and therefore there is no t.w.s. for Eq. (5.4)

Figure 5.2: Global behavior of the ﬂow for very small values of c.

Proposition 5.4 There exist C > 0 such that for all c > C there is a t.w.s. of smooth type for Eq. (5.4) represented by a trajectory from P1 to P0 .

Proof. We use the line l2 connecting (0,−1) and P1; v = l2 (φ) = −(1 − φ) to deﬁne the region R2 = {(φ,v)|0 < φ ≤ 1 , l2 (φ) ≤ v ≤ 0}. We show that for sufﬁciently large values of c , Wc (P1) ∈ R2 ∪ {P0} and therefore Wc (P1) must connect P1 with P0. 120

+ The slope of Wc (P1) near P1 is the slope of v . For large c , ¯ 1 0 ¡ ¢ 0 d ¯ −g (1) −3 0 −g (1) φWc (P1)¯ = + o c < 1 = l2 (1). Since > 0, Wc (P1) does not leave R2 d φ=1 c c near P1 (see Fig. 5.3).

Trajectories crossing l2 have the slope ¯ ¯ ¯ dv ¯ v˙¯ −cv − f (φ)v2 − g(φ)¯ ¯ = ¯ = ¯ dφ¯ φ˙ ¯ d (φ)v ¯ v=l2(φ) v=l2(φ) v=−(1−φ) c (1 − φ)2 f (φ) + g(φ) = − + ≡ t (φ) (5.11) d (φ) (1 − φ)d (φ) 2

(1 − φ)2 f (φ) + g(φ) g(φ) The function is bounded in φ ∈ [0,1] (since limφ − exists and 1 − φ →1 1 − φ (1 − φ)2 f (φ) + g(φ) it is finite), thus c = maxφ is finite. t (φ) ≤ 0 for c > c , and m ∈[0,1] 1 − φ 2 m 0 φ since l2 ( ) = 1 > 0, trajectories inside R2 (and specifically Wc (P1) ) does not leave R2 on l1 (φ) φ ∈ (0,1] (see Fig. 5.3).

Since Pc and (0,∞) are not in R2 while P0 is, any trajectory in R2 must end in P0 according to lemma 5.2. Wc (P1) is a heteroclinical trajectory connecting P1 with P0 and it represents a t.w.s. of smooth type for Eq. (5.4)

Figure 5.3: Global behavior of the ﬂow for very large values of c.

To summarize, we proved the following theorem:

Theorem 5.5 Suppose that the functions d, a and g in Eq. (5.4) satisfy conditions set (2) above, then Eq. (5.4) has no t.w.s. for sufﬁciently small values of c, and for sufﬁciently large values of c there is a monotonically decreasing t.w.s. of front type u(x,t) = φ(x − ct) satisfying the boundary conditions φ(−∞) = 1 , φ(+∞) = 0 . CHAPTER 5 – ANALYSIS OF MODELSWITH NON-LINEAR DIFFUSION 121

5.1.3 Uniqueness of sharp type solution

In this section we show that if there is a speed c∗ for which there is a heteroclinical trajectory connecting P1 and Pc (representing a t.w.s. of sharp type for Eq. (5.4) ), then this speed is unique.

The ODE for the trajectories of the system (5.6) is

dv −cv − [d0 (φ) + a(φ)]v2 − g(φ) = (5.12) dφ d (φ)v

Lemma 5.6 Let v1 (φ) and v2 (φ) be two solutions of Eq. (5.12) corresponding to two values of c, c1 and c2 respectively, 0 < c1 < c2. If there is φa ∈ (0,1) s.t. v1 (φa) < v2 (φa) then v1 (φ) ≤ v2 (φ) ∀φ ∈ (0,φ0) .

Proof. Let us assume by negation that there exist φc ∈ (0,φa) s.t. v1 (φc) > v2 (φc) . By the mid-value theorem, there exist φb ∈ (φc,φa) s.t. v1 (φb) = v2 (φb) and v1 (φb − ε) > v2 (φb − ε) for any small enough ε > 0 . We will mark (φ0,v0) ≡ (φb,v1 (φb)). At the point

(φ0,v0)

0 φ φ 2 φ 0 −c1 −[d ( 0) + a( 0)]v0 − g( 0) v1 = + d (φ0) d (φ0)v0 0 φ φ 2 φ −c2 −[d ( 0) + a( 0)]v0 − g( 0) 0 > + = v2 (5.13) d (φ0) d (φ0)v0

Therefore for small ε > 0 , v1 (φ0 − ε) < v2 (φ0 − ε). We reached a contradiction, thus our assumption is wrong

Proposition 5.7 If Wc (P1) connects P1 with (0,∞) then Wc (P1) connects P1 with (0,−∞)

∀c ∈ (0,c0) . 122

Proof. Let v(φ) be a solution of Eq. (5.12) with initial condition v(1) = 0 . For small ε > 0 p ¡ ¢ −c + c2 − 4d (1)g0 (1) v(1 − ε) = v(1) − εv0 (1) + o ε2 ' −ε (5.14) 2d (1) √ −c+ c2−4d(1)g0(1) (it can be seen directly from the eigenvector of JP1 ). Note that 2d(1) is a positive, strictly decreasing function of c .

Let v0 (φ) be the solution of Eq. (5.12) with parameter value c = c0 and initial condition v0 (1) = 0 , and let v1 (φ) be the solution of Eq. (5.12) with parameter value c = c1 ∈ (0,c0) and initial condition v1 (1) = 0. For small enough ε > 0 , q 2 0 −c1 + c1 − 4d (1)g (1) v (1 − ε) ' −ε < 1 2d (1) q 2 0 −c0 + c0 − 4d (1)g (1) < −ε ' v (1 − ε) (5.15) 2d (1) 0 therefore, according to Lemma 5.6, v1 (φ) ≤ v0 (φ) ∀φ ∈ (0,1 − ε). We get limφ→0+ v1 (φ) ≤ limφ→0+ v0 (φ) = −∞ . v0 and v1 are the second component of Wc (P1) and Wc (P1) respectively. Thus Wc (P1) and Wc (P1) both tend to (0,−∞) as τ → +∞

Proposition 5.8 If Wc (P1) connects P1 with P0 then Wc (P1) connects P1 with P0 ∀c ∈

(c0,∞)

Proof. Let v0 (φ) be the solution of Eq. (5.12) with parameter value c = c0 and initial condition v0 (1) = 0 , and let v1 (φ) be the solution of Eq. (5.12) with parameter value c = c1 > c0 and initial condition v1 (1) = 0. For small enough ε > 0 , q 2 0 −c1 + c1 − 4d (1)g (1) v (1 − ε) ' −ε > 1 2d (1) q 2 0 −c0 + c0 − 4d (1)g (1) > −ε ' v (1 − ε) (5.16) 2d (1) 0 CHAPTER 5 – ANALYSIS OF MODELSWITH NON-LINEAR DIFFUSION 123

therefore, according to Lemma 5.6, v1 (φ) ≥ v0 (φ) ∀φ ∈ (0,1 − ε). We get 0≥ limφ→0+ v1 (φ)

≥ limφ→0+ v0 (φ) = 0 . v0 and v1 are the second component of Wc (P1) and Wc (P1) respectively. The ﬁrst component of Wc (P1) tend to 0 as τ → +∞ for all c > 0, thus Wc (P1) and

Wc (P1) both tend to (0,0) = P0 as τ → +∞

Theorem 5.9 There exist c∗ > 0 s.t.

∗ 1. Wc (P1) connects P1 with (0,−∞) ∀c ∈ (0,c ) .

∗ 2. Wc (P1) connects P1 with P0 ∀c ∈ (c ,∞) .

∗ 3. If there is c0 for which Wc (P1) connects P1 with Pc ( Wc (P1) = Wc (Pc) ) then c0 = c .

∞ Proof. We define C = {c > 0|Wc (P1) connects P1 with (0,−∞)} and 0 ∞ 0 C = {c > 0|Wc (P1) connects P1 with P0}. We also define c = supC∞ and c = infC0. From proposition 5.5 we can conclude that c∞ is well defined (finite) and that c0 > 0. From ∞ proposition 5.7 we can conclude that irrespective of the nature of Wc (P1), for all c ∈ (0,c ),

Wc (P1) connects P1 with (0,−∞). From proposition 5.8 we can conclude that irrespective ¡ ¢ 0 of the nature of Wc (P1), for all c ∈ c ,∞ , Wc (P1) connects P1 with P0. ¡ ¢ ∞ 0 ∞ 0 ∞ 0 We assume by negation that c 6= c ( c < c ). Let c1,c2 ∈ c ,c , c1 < c2 . Since ∞ 0 c2 > c1 > c , both Wc (Pc) and Wc (P1) do not connect P1 with (0,−∞). Since c1 < c2 < c , both Wc (Pc) and Wc (P1) do not connect P1 with P0. Thus both Wc (Pc) and Wc (P1) connect

P1 with Pc. Let v1 (φ) and v2 (φ) be the solutions of Eq. (5.12) with initial condition v1 (1) = v2 (1) = 0 and parameter value c = c1 and c = c2 respectively. We concluded that 124

−c −c v (0) = 1 and v (0) = 2 . For small ε > 0, 1 d0 (0) + a(0) 2 d0 (0) + a(0) q 2 0 −c1 + c1 − 4d (1)g (1) v (1 − ε) ' −ε < 1 2d (1) q 2 0 −c2 + c0 − 4d (1)g (1) < −ε ' v (1 − ε) (5.17) 2d (1) 2 but v1 (0) > v2 (0) , in contradiction with Lemma 5.6. Thus our assumption is wrong and c∞ = c0 = c∗

5.1.4 Existence of sharp type solution

In order to prove the existence of t.w.s. of sharp type, will look at the system (5.6) in

∗ inverse time we will show that at the speed c , the trajectory starting at Pc∗ (the equivalent of Wc (Pc∗ ) ) must end at P1 .

The system (5.6) in inverse time is simply ½ φ˙ = −d (φ)v (5.18) v˙ = cv + [d0 (φ) + a(φ)]v2 + g(φ)

By deﬁnition, (φ(τ),v(τ)) is a solution of the system (5.18) with initial condition (φ(τ0),v(τ0)) =

(φ0,v0) and parameter value c = c0, iff (φ(−τ),v(−τ)) is a solution of the system (5.6) with

ﬁnal condition (φ(τ0),v(τ0)) = (φ0,v0) and parameter value c = c0.

¡ ¢ f Proposition 5.10 Wc (P1) connects P1 with (0,−∞) iff Wc (Pc) connects Pc with φ (c),0 f , where φ (c) ∈ (0,1) . Furthermore, if 0 < c1 < c2 and Wc (P1) and Wc (P1) both connect f f P1 with (0,−∞) then 0 < φ (c1) < φ (c2) < 1 .

Proof. Since trajectories do not cross if Wc (P1) connects P1 with (0,−∞) then Wc (Pc) must by in the region Rc = {(φ,v)|0 ≤ φ ≤ 1, 0 ≥ v > Wc (P1)} and vise versa. The trajectory CHAPTER 5 – ANALYSIS OF MODELSWITH NON-LINEAR DIFFUSION 125

starting from Pc must exit Rc at some point on the segment (φ,0) where φ ∈ (0,1), which proves the ﬁrst part of the proposition. ¡ ¢ ¡ ¢ f f Let c1 and c2 be two such values of c, 0 < c1 < c2 and let φ (c1),0 and φ (c2),0 their corresponding exit points from Rc. The second component of Pc is monotonically ¡ ¢ φ f decreasing with c . Thus by Lemma 5.6, Wc2 (Pc2 ) passes through the point (c1),v2 , where v2 < 0. Thus the second part of the proposition is proved

¡ ¢ f Proposition 5.11 Wc (P1) connects P1 with P0 iff Wc (Pc) connects Pc with 1,v (c) , f where v < 0 . Furthermore, if c1 < c2 and Wc (P1) and Wc (P1) both connect P1 with f f P0 then 0 > v (c1) > v (c2) .

Proof. Since trajectories do not cross if Wc (P1) connects P1 with P0 then Wc (Pc) must by in the region Rc = {(φ,v)|0 ≤ φ ≤ 1, Wc (P1) > v} and vise versa. The trajectory starting from Pc must exit Rc at some point on the segment (1,v) where v < 0, which proves the ﬁrst part of the proposition. ¡ ¢ ¡ ¢ f f Let c1 and c2 be two such values of c, c1 < c2 and let 1,v (c1) and 1,v (c2) their corresponding exit points. The second component of Pc is monotonically decreasing with f f c . Thus by Lemma 5.6, 0 > v (c1) > v (c2)

Proposition 5.12 The solutions (φ(τ),v(τ)) of the system (5.18) with initial conditions

(φ(τ0),v(τ0)) = (φ0,v0) depend continuously on φ0 , v0 , and c.

Proof. The vector field defined by the system (5.18) depends differentiably on φ, v and c .The corresponding general theorem on continuity of solutions with respect to parameters and initial conditions can be used for the specific case of the system (5.18). Full details can be found in [8] 126

Theorem 5.13 For c = c∗, there is a t.w.s. of sharp type for Eq. (5.4) (there is a heteroclinical trajectory connecting Pc with P1 , Wc (P1) = Wc (Pc) ),

Proof. For c = c∗, there is a t.w.s. of sharp type for Eq. (5.4) iff there is a trajectory from

P1 to Pc∗ in the system (5.6) which is a trajectory from Pc∗ to P1 in the system (5.18). If

Wc∗ (P1) does not connect P1 with Pc∗ , there are two possibilities:

1. Wc∗ (P1) connects P1 with P0

2. Wc∗ (P1) connects P1 with (0,−∞)

We deﬁne the region R1 = [0,1] × [0,−∞). Let us assume by negation that (1) is true. According to proposition 5.11, the solutions (φ(τ),v(τ)) of the system (5.18) with ini- µ ¶ ∗ ¡ ¢ −c f ∗ tial conditions (φ(−∞),v(−∞)) = 0, = P ∗ exit R at a point 1,v (c ) 0 ( ) + ( ) c 1 ¡ ¢ d 0 a 0 where v f (c∗) < 0 . Since 1,v f (c∗) has a smooth neighborhood, according to proposition 5.12, there exist ε > 0 s.t. the solution to the system (5.18) with parameter value ¡ ¢ ∗ ε φ ∞ ∞ f c1 = c − and initial conditions ( (− ),v(− )) = Pc1 exit the region R1 at 1,v (c1) f f ∗ where 0 > v (c1) > v (c ). According to proposition 5.11, Wc1 (P1) connects P1 with P0. ∗ This contradicts theorem 5.9 which proves that for all c < c , Wc (P1) connects P1 with (0,−∞). We reached a contradiction, thus possibility (1) is false.

Let us assume by negation that (2) is true. According to proposition 5.10, the solutions µ ¶ −c∗ (φ(τ),v(τ)) of the system (5.18) with initial conditions (φ(−∞),v(−∞)) = 0, = 0 ( ) + ( ) ¡ ¢ ¡ ¢ d 0 a 0 f ∗ f ∗ f ∗ Pc∗ exit R1 at a point φ (c ),0 where φ (c ) ∈ (0,1) . Since φ (c ),0 has a smooth neighborhood, according to proposition 5.12, there exist ε > 0 s.t. the solution to the sys- ∗ ε φ ∞ ∞ tem (5.18) with parameter value c1 = c + and initial conditions ( (− ),v(− )) = Pc1 ¡ ¢ f f f ∗ exit the region R1 at φ (c1),1 where 1 < φ (c1) < φ (c ). According to proposition CHAPTER 5 – ANALYSIS OF MODELSWITH NON-LINEAR DIFFUSION 127

∞ 5.10, Wc1 (P1) connects P1 with (0,− ). This contradicts theorem 5.9 which proves that ∗ for all c > c , Wc (P1) connects P1 with P0. We reached a contradiction, thus possibility (2) is false.

Both possibilities (1) and (2) proved to be false, thus the solution to the system (5.18)

∗ with parameter value c = c and initial conditions (φ(−∞),v(−∞)) = Pc∗ reaches P1. ∗ Wc∗ (Pc∗ ) connects P1 with Pc∗ ( Wc∗ (P1) = Wc∗ (Pc∗ ) ) thus for c = c , there is a t.w.s. of sharp type for Eq. (5.4).

Theorems 5.9 and 5.13 together prove theorem 5.1.

Using theorem 5.1 and noting the transformation we used to get from Eq. (5.1) to Eq.

(5.4) we also proved the following theorem:

Theorem 5.14 If the functions D, A and G satisfy conditions set (1), then there exist a unique value of c, c∗ > 0, such that Eq. (5.1) has:

1. No t.w.s. ∀c ∈ (0,c∗) .

2. A traveling wave solution w(x,t) = φ∗ (x − c∗t) of sharp type satisfying D(φ∗)φ∗00 +

∗ ∗0 0 ∗ ∗ ∗0 2 ∗ c φ + [D (φ ) + A(φ )](φ ) + G(φ ) = 0∀ξ < ξ0 ( ξ0 ∈ (−∞,+∞) arbitrary), ¡ ¢ ∗ ∗ ∗n−1 ∗0 φ (−∞) = 1 , φ (ξ) = 0∀ξ ≥ ξ0 , limξ ξ− φ φ = → 0 ∗ ¡ ¢ −c ∗n−1 ∗0 limφ∗→0+ and limξ ξ+ φ φ = 0 . φ∗1−n φ∗ φ∗ φ∗ 0 φ∗ → 0 A( ) + Dn ( ) + Dn ( ) 3. ∀c ∈ (c∗,∞) a unique monotonically decreasing traveling wave solution w(x,t) = φ φ φ00 φ0 0 φ φ φ0 2 c (x − ct) of smooth type, the t.w.s. satisfying D( c) c +c c +[D ( c) + A( c)]( c) + φ ξ ∞ ∞ φ ∞ φ0 ∞ φ0 ∞ φ ∞ G( c) = 0∀ ∈ (− ,+ ) , c (− ) = 1 and c (− ) = c (+ ) = c (+ ) = 0 .

We can also see from Proposition 5.12, that as c approaches c∗ from above, for all 128

positive φ the limit of Wc (P1) is Wc∗ (P1). Since the transformations from the manifolds

Wc (P1) to the traveling wave solutions of Eq. (5.1) are all continuous, we can conclude ∗ ∗ that as c approaches c from above, the limit of φc is φ for all ξ < ξ0 (in this context, the ∗ difference between φc and φ should be minimized by “phase” matching. This operation is required as Eq. (5.1) is invariant under spatial transformations and φc as an extra degree of ∗ freedom). Note that we have said nothing on the rate of convergence of φc towards φ . Chapter 6

Mean Orientation Field Model for Chiral Patterns

In this chapter we deal with continuum, reaction-diffusion models for bacteria how grow colonies with special patters: chiral dendritic branching patterns.

Colonial patterns of Paenibacillus dendritiformis var. chiralis bacteria (C morphotype) on semi-solid agar are characterized by dendritic branching patterns, where the branches are very narrow and all the branches in all the colonies are twisted with the same handedness (left handedness or counter-clockwise growth when viewed from above, see Fig. 6.1). As long as the bacteria grow on top of the agar, the handedness is unaffected by external conditions All colonies of this bacteria grown in similar conditions show the same handedness. Side branches are usually emitted to the convex side of the arced branches (see Refs. [30, 37, 27, 24] for morphologies and studies of C morphotype bacteria).

Optical microscope observations indicate that the width of C bacteria is 1µm and their length is 5 − 50µm (on average 4 times longer than those of T morphotype ). Many times they are longer than the branches’ width. During colonial growth the cells move within a well deﬁned envelope of the branches. Inside the envelope the bacteria swim, i.e. perform

129 130

a b

Figure 6.1: Chiral colonial growth of Paenibacillus dendritiformis, strain C. a) Global view of a colony shows thin branches, all twisted with the same handedness. Colony is grown at 2g/l peptone level and 1.25% agar concentration. b) Optical microscope observations of branches of C colony. ×20 magnification of a colony at 1.6g/l peptone level and 0.75% agar concentration, the anti-clockwise twist of the thin branches is apparent. The curvature of the branches is the same throughout the growth. c) ×500 magnification of a colony at 1.6g/l peptone level and 0.75% agar concentration. Each line is a bacterium. the bacteria are long (5-50µm) and mostly ordered. CHAPTER 6 – MEAN ORIENTATION FIELD MODEL 131 straight runs separated by short tumbles. The envelope in which the bacteria swim is a layer of lubricating fluid which is extracted from the agar by the bacteria themselves. In narrow branches, The orientation of neighboring bacteria is usually very correlated (Fig.6.1), and their movement is quasi 1D random walk along their long axis. Each branch tip maintains its shape, and at the same time the tips propagate with a left twist. Electron microscope observations do not reveal any chiral structure on the cellular membrane [34].

Other bacteria display various chiral properties. Mendelson et al. [161, 163, 165, 162] showed that long cells of Bacillus subtilis can grow in helices, in which the cells form long strings that twist around each other. They have also shown that the chiral characteristics affect the structure of the colony. As we discussed is Sec. 2.2.1 and 4.2.2, Ben Jacob et al. [30, 37], found that colonies of Paenibacillus dendritiformis var. dendron can exhibit global twist with the same handedness, as shown in Fig. 2.3. Similar observations during growth of other bacterial strains have been reported by Matsuyama et al. [157, 155].

2D chiral branching patterns are also observed in non-living systems. Shapes resembling Sea Horse, or S, are formed during deposition of thin films of fullerene-tetracyanoquinodimethane (C60-TCNQ) or pure TCNQ [90, 194, 195]. Mod- eling of the system indicates that the apparent curvature of the branches is actually a strong bias in selection of splitting branches, and the branches themselves are not curved. The patterns of bacterial colonies share more resemblance with patterns formed during com- pression of monolayers of various chiral molecules at the air-water interface [227, 100, 7, 99, 79, 225, 111]. Various modeling attempts indicates that the processes forming these patterns are related to processes of solidification (see Refs. [18, 25, 93] for the differences between the processes of solidification vs. colonial growth, and see also Sec. 3.1.4 vs. Sec. 4.1.2). A different model is required for chiral branching growth of bacterial colonies. 132

The Communicating Spinors model (Sec. 7.2 and also [27, 24]) is an atomistic (discrete entities) model which describes chiral branching growth of bacterial colonies. The model presented here is not a mean ﬁeld model of the spinors model. The two models complement each other and highlight different biological features. Each spinor in the spinors model represents a large groups of about 102-104 bacteria. This coarse graining makes simulations computationally feasible, but prevents modeling of processes which cannot be averaged over large groups (speciﬁcally, single-cell events). Continuous model is more appropriate for this purpose (when interpreting densities as probability densities).

6.1 Modeling chiral growth

6.1.1 Orientation Dimension

The bacterial movement in the colonies is orientation-dependent, therefore a good description of the colony must include the orientation of the bacteria (or the direction of their movement). One way to include such details in a continuous description is to use the orientation as an additional dimension, so as to have information about the bacterial orientation.

We deﬁne the bacterial density per angle b(~x,θ,t), where~x ∈ R2 is a position and θ ∈ [0,π] is an angle of orientation. where θ ∈ [0,π] is an angle. Since the bacterial movement is orientation-dependent and not direction-dependent b is periodic in θ with period π. The bacterial density in any given position B(~x,t) is deﬁned as the mean of b over θ. Z 1 π B = b(~x,θ,t) dθ (6.1) π 0

We write the equation for the bacterial dynamics as

∂tb = −∇ · [Jb] − ∂θJθ + G(b,n) (6.2) CHAPTER 6 – MEAN ORIENTATION FIELD MODEL 133

Jb = −Db (B)D0 (θ)∇b (6.3) where ∇ operates in the spatial dimensions, Jb is the bacterial ﬂux in the spatial dimensions,

Jθ is the flux in the angle dimension which represents changes in bacterial orientation, G is a reaction term representing growth and death, and n is the nutrient concentration. In the formulation of the equation there are several implicit assumptions. In the writing of G there are two implicit assumptions about the bacterial reproduction and death – that after division the two daughter cells has the same orientation as the original cell, and that the reproduction and death of bacteria does not depend directly on the local density of bacteria. Another implicit assumption hides in the separation of the flux in the orientation dimension from the flux in the spatial dimensions. This separation is justified because the two types of fluxes represent different bacterial activities: bacterial tumbling causes change in orientation while bacterial runs cause changes in position.

Db is a non-constant diffusion coefficient (see below) D0 is a diffusion matrix which gives different diffusion coefficients for diffusion in the bacterial direction θ and diffusion in the orthogonal orientation. The form of the diffusion matrix is µ ¶ µ ¶ T Dd 0 cosθ −sinθ D0 (θ) = R(θ) R(θ), where R(θ) = is a rotation ma- 0 Dl sinθ cosθ trix. Dd ≥ 0 is a constant coefficient for diffusion in the bacterial direction θ (due to self-propulsion forward and backward) and Dl ≥ 0 is a constant coefficient for diffusion in the lateral direction, due to fluctuations (in all realistic cases Dd ≥ Dl > 0 ). Sources of the lateral diffusion (Dl) could be fluctuations in direction of motion (either thermal fluctuations or fluctuations in the direction of the flagella’s force), or it could be reduced correlations between the direction of straight runs and the orientation of the bacterial body.

Differences between the various processes become important for modeling of chemotaxis (see Sec. 6.2). R(θ), in it’s role in Eq. (6.2), deﬁnes a speciﬁc relation between the ~x 134 axis and θ axis: Eq. (6.2) is invariant under a rotation in any angle φ and translation in the orientation dimension by the same angle.

As we discussed in previous chapters, bacterial movement in a self-produced layer of fluid can be approximated by a non-linear diffusion, where the diffusion coefficient is proportional to the bacterial density to a power greater than one (see Sec. 4.1.2.2). The proportion constant is related to the agar dryness through the rate of absorption of the fluid into the agar. We assume that bacterial drag with the agar is proportional to the agar dryness, and as the bacterial velocity is inversely proportional to the drag, Dd and Dl are k proportional to the agar dryness to the power −2. Hence we take Db (B) = B (k ≥ 2), where the proportion constant is included in Dd and Dl. Later we will relate other parameters to the dryness of the agar and hence to k.

Following [124, 93] (see Sec. 2.3.2) we take a simple form for the growth function, G(b,n) ∝ nb − µb, where µ ≥ 0 is a rate of conversion into immobile sporulating cells. A more accurate form should have saturation for high values of n/B, but the linear form is a reasonable approximation for low levels of nutrients, as is the case in the bacterial colonies.

Bacteria change their orientation (Jθ) in response to their neighbors orientation. It is convenient to use an auxiliary complex orientation ﬁeld p, which is related to the mean local orientation of the bacteria: Z π 1 θ p(~x,t) = b(~x,θ,t)e−i2 dθ (6.4) π 0 ¡ √ √ ¢ The mean orientation of the bacteria is along the vector ±Re p,∓Im p .

The re-orientation of bacteria (whose ﬂux is Jθ) can be spilt into terms which change the mean orientation of the bacteria (terms in the ﬂux which contain even derivatives of b) and terms which does not change the mean orientation. Two processes which does not CHAPTER 6 – MEAN ORIENTATION FIELD MODEL 135

change the mean orientation are fluctuations in bacterial orientation (JD) and co-alignment according to the orientation of neighboring bacteria (JA). Fluctuations in orientation are simply dispersal over the orientation dimension JD = −Dθ∂θb where Dθ is a non-negative constant. Co-alignment of bacterial orientation is an increase in bacterial density around the mean orientation, thus JA = −DA(b,B)∂θb where the orientation diffusion coefficient can be negative. The functional form that we put for DA should be such that the field will not diverge.

The mean orientation of the bacteria is changed by the only process which brakes left- right symmetry: the bacterial bias in their tumbling when they are placed in a layer of liquid on a surface. Viewed from above, this bias is anti-clockwise. In the model, this process is represented by a term in the ﬂux (JT ) which does not depend on derivatives of b. In a uniform layer with a uniform bacterial density JT is a constant, but in the colony the bacterial rotation is constrained by neighboring bacteria and by the boundary of the branches. At the boundary of a branch the gradient of B is the steepest due to the nonlinear diffusion (see Chapters 3 and 5). In fact, ∇B diverges at the boundary for k > 1. The term Bk−1∇B, however, is always ﬁnite and can be used “safely” in the model.

We incorporate the equation of the dynamics of b as detailed thus far in a full model of the colonial growth, the Orientation Dimension (OD) model, which include equation for the dynamics of the oriented bacterial density b, the nutrient concentration n and the density of immobile (sporulating) bacteria s:

h i h ³ ´i k k−1 ∂tb = ∇ · B D0(θ)∇b − ∂θ −Dθ∂θb − DA(b,B)∂θb + JT B ∇B, p,b (6.5)

+nb − µb (6.6)

2 ∂tn = Dn∇ n − nB (6.7) 136

∂ts = µB (6.8)

where n is scaled to units of bacterial mass and Dn is its diffusion coefficient (in this and following equations, all constants and functions are real valued, unless otherwise stated). The OD model is a system of integro-differential equations in four dimensions (two spatial dimensions, temporal dimension, and angular dimension). We found that the model can be greatly simplified by tracking only the dynamics of the two auxiliary fields, B and p.

6.1.2 Mean Orientation Field

Using Fourier expansion of Eq. (6.6) in θ and taking the ﬁrst two resulting equations we derive the Mean Orientation Field (MOF) model, which includes equations for B, p, the nutrient concentration n and the density of immobile (sporulating) bacteria s (at this stage s is used only to record the history of the colonial development): n o k ∂tB = ∇ · B [D1∇B + 2Re(D2∇p)] + (n − µ)B (6.9) n o ∂ ∇ k ∗∇ ∇ t p = · B [D2 B + D1 p] + (n − µ) p ³ ´ +a(B,|p|) p + γ Bk−2∇B, p ip (6.10)

2 ∂tn = Dn∇ n − nB (6.11)

∂ts = µB (6.12)

where n is scaled to units of bacterial mass and Dn is its diffusion coefﬁcient. a and γ are real valued functions, a decomposition into orthogonal elements of the derivation of ∂θJθ

(we specify here the functions a and γ, not the original full functional form of the original µ ¶ D +D D −D 1 i Jθ). Here D = d l and D = d l M , where M = . Eqs. (6.9-6.12) are 1 2 2 4 i −1 invariant under a rotation by any angle φ and a multiplication of p by e−i2φ. CHAPTER 6 – MEAN ORIENTATION FIELD MODEL 137

In our modeling efforts thus far we created one model, the OD model, and immedi- ately converted it to a different model, the MOF model. Modeling wise, the two models have complementary advantages: the MOF model is mathematically (and computationally) more simple than the OD model, but its relation to the biology is more obscure. The spatial diffusion terms demonstrate it most clearly: while the spatial diffusion term in the OD model includes integrals (B as an integral of b), its derivation as a description of the bacterial movement is clear. On the other hand, the spatial diffusion terms in the MOF model include no integrals, but without the model’s relation to the OD model, no one could have guessed the terms’ relation to the bacterial movement. In Sec. 6.2 we modify the models to accommodate changes in the bacterial movement, and we do so by introducing the changes into the OD model and tracking the changes into the MOF model.

For the co-alignment function a in the MOF model we take a phenomenological form which includes JD and JA

B − |p| a(B,|p|) = −4Dθ + ν (6.13) B + |p|

The ﬁrst term in the RHS results from linear diffusion of b in the θ dimension (JD) and the second term is an alignment of non-aligned bacteria with the mean orientation (JA), with ν being the rate of this co-alignment. The exact form of a is not important, as long as a(|p|) has at most one positive root with negative derivative in the range [0,B] and a(|p|) is non-positive outside the range (0,B).

The rotation function γ results from the tumbling ﬂux JT which include the process that brakes left-right symmetry, the bacterial bias in their tumbling when they are placed in a thin layer of liquid. γ can be written as a function of ∆, the angle between the branch’s 138 boundary and the bacterial mean orientation: · ¸ ~αT M~α γ(~α,β) = Re γ0 |β~α| − γ0β |~α| £ ¡ ¢ ¡ ¢ ¤ = |β~α| 2Re γ0 sin2 (∆) + Im γ0 sin(2∆) (6.14)

¡ ¢ where γ0 is a complex constant. Re γ0 is a measure of the anti-clockwise bias of the ¡ ¢ bacteria at the tip of the branch and Im γ0 is a measure of the torque aligning the bacteria in parallel to the boundary. The rotation at the tip of the branch is also restrained by drag with the agar, and from the same reasoning that related Dd and Dl to the agar dry- ¡ ¢ ness, we deduce that Re γ0 is inversely proportional to the agar dryness. We will assume ¡ ¢ that the constrained by neighboring bacteria makes Re γ0 proportional to bacterial length to the power −2 (following [27, 24] we take the standard deviation of the rotation to be proportional to its mean, thus Dθ is proportional to the length to the power −4).

6.1.3 Chiral branching patterns

We solved the equations of the model in a circular 2D geometry. As initial conditions, we set n to have uniform distribution of level n0, B to have compact support at the center where it is positive, and the other ﬁelds to be zero everywhere. We solve the model numerically using a 2nd order explicit scheme. In order to reduce the implicit anisotropy of the scheme, we use tridiagonal lattice and multiply the bacterial diffusion coefﬁcients by a quenched (unchanged in time) noise with mean 1. The quenched noise represent inhomogeneities of the agar surface. We show in Fig. 6.2 that the model can indeed reproduce the microscopic bacterial dynamics and the chiral branching patterns with the local twist.

The Non-Linear Diffusion model (NLD) for colonial growth can reproduce tip-splitting branching patterns [93, 124, 59, 128, 61] of the related morphotype, T morphotype, with CHAPTER 6 – MEAN ORIENTATION FIELD MODEL 139

a b

Figure 6.2: Results of numerical simulation of the MOF model. a) Global view of a simulated colony with a local twist of branches. Densities of B + s are indicated by gray levels. Parameters values are k = 3, Dd = 0.0625, Dl = 0, µ = 0.1, Dθ = 0.001, ν = 0.5, 0 γ = 0.0075 − i0.006, Dn = 1 and n0 = 1.5. b) A close look at the pattern of (a), showing the details of the curved branches. shorter bacterial cells (see Sec. 4.1.2.2). The MOF model reduces to the NLD model if the self-propulsion is not primarily along the bacterial long axis (Dd ' Dl, Fig. 6.4b). In some other cases the model itself does not reduce to the NLD model, yet the dynamics sets |Re(D2∇p)| ¿ |D1∇B| everywhere and the MOF model produce tip-splitting branching patterns. This is the case if the bacteria do not tend to co-align (ν ≤ 4Dθ or

(ν − 4Dθ) ¿ (ν + 4Dθ) ), or if bacteria at the tip of the branch tend to rotate too freely ¯ ¯ ¯ 0¯ ( γ (ν − 4Dθ) > |D2|). This region in parameters space represent weak co-alignment restraints on the bacteria, so they change their orientation signiﬁcantly during tumbling.

Given two strains of bacteria differing only in length, the best representation of their dif- ¡ ¢ ¡ ¢2 ference is in the values of Dθ and Re γ0 , with Dθ scales like Re γ0 . Smaller values of ¯ ¡ ¢¯ Dθ and ¯Re γ0 ¯ correspond to longer bacteria which are more restrained in rotation.

The response of the simulated growth to initial food concentration and agar dryness is shown in Fig. 6.3. In agreement with experimental observations [27, 24], the main effect of these parameters is on the global density of branches, while the curvature of the 140 branches is only weakly affected. Also shown is the growth velocity of the colonies as function of initial food concentration for both long bacteria (C bacteria) and short bacteria (T bacteria). The chiral growth is always faster then the tip-splitting one, also in agreement with experiments.

a b

c d

Figure 6.3: The response of simulated colonies to experimental control parameters. Figures (a) and (b) present change in initial food concentration with n0 = 1 and n0 = 2 in (a) and (b) respectively. All other parameters are the same as in Fig. 6.2a. The expansion time of the colonies, normalized by the expansion time in Fig. 6.2a, is 2.9 and 0.4 in (a) and (b) respectively. Figures (c)¡ and¢ (d) present change in agar dryness A. A is related to the 0 2 model’s parameters with Re γ = 0.009/A and Dd = 0.09/A . In (c) and (d) A = 1 and A = 1.5 respectively, with all other parameters are the same as in Fig. 6.2a, where A = 1.2. The normalized expansion time of the colonies is 0.8 and 1.2 in (c) and (d) respectively. CHAPTER 6 – MEAN ORIENTATION FIELD MODEL 141

a b

Figure 6.4: Simulation of colonies of long and short bacteria. a) Colonies of long bacteria (C morphotype). All parameters are as in 6.3c b) Colonies of short bacteria (T morphotype).¡ ¢ To simulate short bacteria, unrestrained in rotation, we take Dθ = 0.256 and Re γ0 = 0.128. The dendritic, chiral patterns turns into branching, tip-splitting patterns. The weak six-fold asymmetry is due to underlying lattice of the simulations

6.2 Chemotaxis and weak chirality

We showed in Sec. 4.2 that many features of the colonial patterns can be explained by inclusion of food chemotaxis and chemotactic signaling (a chemotactic response to a material emitted by the bacteria themselves) in the model. At the inner parts of the colonies nutrient are deﬁcient. Bacteria lacking nutrient may enter a pre-spore state in which they transform into durable spores. The immobile sporulating cells emit various waste materials, some of which unique [221, 204, 69]. The emitted chemicals may be used by other bacterial cells as signals. Ben-Jacob et al. [35, 27, 60, 24] suggested that such materials are repelling the bacteria – repulsive chemotactic signaling. Bacteria perform chemotaxis by modulating the length of runs according to the gradient of a chemical [2, 42, 133, 40, 77]. They estimate the gradient by measuring temporal derivatives, i.e. by comparing successive measurements of concentration over a time intervals of about 3 seconds. 142

In order to include chemotaxis in our model we bias the spatial ﬂux of b in Eq. (6.2) by adding a chemotaxis term:

Jb = −Db (B)D0 (θ)∇b + Db (B)bζn (n)Z(θ)∇n (6.15) where the first term in the RHS of the equation is a diffusion flux and the second term a food chemotaxis flux (we take food chemotaxis as an example). Z(θ)∇n is the spatial derivative of the food concentration along the direction of movement of the bacteria. ζn (n)Z(θ)∇n is the derivative as sensed by the bacteria, where ζn (n) can be, for example, the “receptor law” [171] or a constant (see Sec. 2.3.2). The chemotaxis is attractive for positive values of

ζn and repulsive otherwise. The diffusion coefﬁcient Db (B) proceeds the chemotaxis term as well as the diffusion term because both the diffusion and chemotaxis result mainly from bacterial movement, and any factor that affects the speed of bacteria affects the two terms in the same manner.

The bacterial chemotactic mechanism is effective only if the bacterial runs are approximately straight [42, 40], thus Z depends on the type of fluctuations which cause the lateral diffusion (Dl). The fluctuations can be thermal fluctuations, fluctuations in the direction of the propulsion force, propulsion which is not aligned with the bacterial long axis, etc. The key difference between the different types of fluctuations is the time correlations between the fluctuations over time scale of few seconds. The bacterial mechanism of chemotaxis is based on the two assumptions that a) there is a strong relation between the temporal derivatives a bacterium measure and the spatial derivative between the bacterium current position and a previous position, and that b) continuing a run without tumbling is taking the bacterium up (or down) the same gradient they measure. If there are large, uncorrelated fluctuations in the bacterial direction of motion, then it undermine both of the above CHAPTER 6 – MEAN ORIENTATION FIELD MODEL 143 assumptions, and it impair both the reliability of the information that the bacteria gather and their ability to act based on that information. If, on the other hand, the bacterial motion is not necessarily along its orientation, but the runs fairly straight (fluctuation which are correlated in time), then the information gathered during runs is reliable and delaying a tumbling event by a bacterium does indeed takes it in a favorable direction.

We generalize both types of fluctuations into the OD model by taking Z(θ) = D (θ) − µ ¶ 0 T T Dd − ξDl 0 ξDlR(θ) R(θ) = R(θ) R(θ) where ξ ∈ [0,1] is a measure of 0 Dl − ξDl the correlations of the fluctuations over time scale of 1 second. Smaller values of ξ represent larger correlation. We could not find any biological estimates to the value of ξ. The NLD model has demonstrated success in modeling chemotaxis and chemotactic signaling [93,

128, 61, 24], thus for consistency we take ξ = 0 (if Dd = Dl then the MOF model with chemotaxis is reduced to the NLD model with chemotaxis).

The bacterial ﬂux due to food chemotaxis translates in the MOF model into additional ﬂux terms in Eqs. (6.9-6.10):

n o k k ∂tB = ∇ · B [D1∇B + 2Re(D2∇p)] + B ζn (n)[BD1I + 2Re(pD2)]∇n

+(n − µ)B (6.16) n o ∂ ∇ k ∗∇ ∇ kζ ∗ ∇ t p = · B [D2 B + D1 p] + B n (n)[BD2 + pD1I] n ³ ´ +a(B,|p|) p + γ Bk−2∇B, p ip + (n − µ) p (6.17) where I is the unit matrix

If a repulsive signaling material R is emitted by sporulating cells [35, 60, 24] then Eqs.

(6.16-6.17) are affected in a similar manner, with ζR (R) replacing ζn (n) and ∇R replacing

∇n. ζR (R) is negative for a repulsive signal. The equation for the dynamics of the signaling 144 material is (similar to Eq. (4.22) )

∂R = D ∇2R + Γ s − Ω BR − Λ R (6.18) ∂t R R R R where DR, ΓR, ΩR and ΛR are non-negative constants. DR is the diffusion coefﬁcient of the repulsive chemical, ΓR is the rate of the chemical production by sporulating bacteria,

ΩR is the rate of chemical digestion by bacteria and ΛR is the rate by which the chemical decompose.

In Fig. 6.5 we show the effect of the two types of chemotaxis on the simulated colonies. Both food chemotaxis and repulsive chemotactic signaling increase the expansion rate of the colonies (up to a factor of 2). Food chemotaxis does not affect the colonial patterns signiﬁcantly, while chemotactic signaling does (Fig. 6.5). For short bacteria, the colonial pattern becomes less ramiﬁed, with radial branches and circular global envelope. For long bacteria, the global envelope becomes circular and the branches acquire an outward bias, changing their appearance from an arc-like to a hook-like appearance.

In Fig. 6.6 we show the phenomena of global twist (weak chirality) [27, 24] (see Sec.

4.2.2). For parameters representing bacteria with intermediate length, repulsive chemotactic signaling can impose global twist on an otherwise tip-splitting pattern. The twist of the branches is relative to the center of the colony, not to the local orientation of the branch.

The global nature of the twist is evident by using the “de-chiraling” method presented by Ben-Jacob et al. in Ref. [24] and in Sec. 4.2.2.1 (see Fig. 4.21). In Sec. 4.2.2 we were able to obtain patterns with a global twist, using the NLD model and applying a rotation operator on the chemotaxis term. As we argue there, such operator is inconsistent with the known biological facts and a more detailed model – like the current MOF model – is required in order to model any type of chirality in the bacterial colonies. CHAPTER 6 – MEAN ORIENTATION FIELD MODEL 145

a b

Figure 6.5: a) The effect of repulsive chemotactic signaling on colonies of short bacteria (T morphotype). For chemotaxis we take ζn(n) ≡ 0, ζR(R) ≡ −1, DR = 1.0, ΓR = 0.25, ΩR = 0 and ΛR = 0.01. All other parameters are as in Fig. 6.4b. Due to chemotaxis the branches are radially oriented with circular global envelope. b) The effect of repulsive chemotactic signaling on colonies of long bacteria (C morphotype). Most chemotaxis parameters are as in (b), ΓR = 0.065 and other parameters are as in Fig. 6.3c (the difference in production rate is related to the fact that long bacteria have more biomass than short ones, hence the same biomass of long bacteria contribute less sporulating bacteria). The branches acquire an outward bias and the global envelope becomes circular.

a b

Figure 6.6: a) Colony showing global twist in response to repulsive chemotactic¡ ¢ signaling. 0 Parameters values are as in Fig. 6.5c, with Dθ = 0.064, Dd = 0.0625 and Re γ = 0.096, representing bacteria of intermediate length on dryer surface. b) “De-chiraling” the pattern in (a) by applying the mapping (r,φ) → (r,φ + r/R), where R is a constant, on the polar coordinates (r,φ), measured from the center of the colony.

Chapter 7

Atomistic Models for Branching Patterns

In this chapter we introduce discrete models for growth of bacterial colonies. We start by presenting the previously published Communicating Walker (CW) model [35, 60, 18, 59]. This presentation serves mainly as a platform for presenting extensions and modiﬁcations of the model. In this chapter we present the Communication Spinors model for modeling chiral branching growth. In chapter 9 we present an extension of the CW model with variable group size. This extension allow for detailed description of cellular-level events, such as mutations, without loosing the computational efﬁciency of the CW model, which treats large groups of bacteria as a single walker.

We can gain many insights into instability mechanisms and nonlinear states from the continuum models of the biological processes, as we have seen thus far. The insights are derived from analysis of the continuum models as well as and from simulation of the time development of the models. Discrete “automata” models are, in some sense, designed to be simulated and often it is more convenient to compute with discrete models than with continuum models. In fact, we will see that perhaps the most convenient approach for studying

147 148 microbial systems seems to be a hybridization of continuum and atomistic methods. In this chapter, we survey discrete analogs of the models discussed so far.

Initially, the simplest discrete analog of systems exhibiting diffusive instabilities was afforded by diffusion-limited-aggregation (DLA) [231, 232]. In DLA model discrete walkers move diffusively in space and attach to a growing cluster. In the limit of taking one walker at a time (i.e. of extremely slow growth) and purely irreversible attachment at any nearest-neighbor site, one obtains the classic DLA fractal [231, 232]. Subsequent workers extended these ideas, producing a variety of systems which could semi-realistically mimic the physics of crystal growth and related processes.

There is also a literature on using discrete models for other, more complex reaction- diffusion processes; see for example the work by Kapral et al. [148] on simulations of 3D knotted labyrinths. In real reaction-diffusion processes, it is almost always the case that a continuum model is a much better approximation of the physical system than a discrete model; with 1023 particles in the system one cannot hope to match the actual number of molecules by discrete simulational entities. The discrete model is used as a simpler stand- in for the ”true” continuum dynamics (recall, though, the cutoff effect for type I systems, see Sec. 3.1.3). One of the problems of a discrete model for such systems is an extra noise CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 149 introduced into the model by having a small number of particles in the simulation. This is the price one pays for a more flexible and more efficiently-coded numerical scheme. Once the basics are understood, one can modify the simulation to encompass more details of the actual system and thereby obtain more reliable results. Similar remarks hold for lattice-gas automata [45], in which discrete objects are used to model systems with fluid flow.

There are good reasons why modeling of biological systems can make good use of discrete entities, aside from computational convenience. First, the numbers match more closely. The number of bacteria in a typical experiment is 109; one can almost approach these numbers computationally and therefore one is not plagued by the extra noise issue. Perhaps more importantly, cells contain large numbers of internal degrees of freedom which modulate their response to external signals form other cells. Hence, describing a population of cells with something as non-informative as a density field is usually insufficient. At the very least, one would have to introduce either new variables (which advect with the cell velocity as these are tied to the cells, see Sec. 8.2 and the Mean Orientation Field model in Sec. 6.1) or even new coordinates (see a model in Ref. [78], where the cell’s age is taken as a relevant coordinate for the density field, or the Orientation Dimension model in Sec. 6.1 where the cells’ orientation is taken as a relevant coordinate for the density field); this makes for ”ugly” reaction-diffusion systems. Tracking cells as individual objects makes it easy to add internal degrees of freedom; we just attach extra labels to the cell and postulate transition rules as to how these labels change in time. This flexibility is, we feel, quite useful and hence many of the models to be discussed keep cells discrete. At the same time, though, continuum analysis is used to shed light on the simulations and forms an indispensable part of an integrated effort to understand microbiological pattern formation. 150

7.1 The Communicating Walkers model

7.1.1 The walkers, the boundary and local interaction

The Communicating Walkers (CW) model describes the growth of colonies of T morphotype. The model was inspired by the diffusion-transition scheme of Shochet et al.

[206, 207, 205], used for the study of solidification from supersaturated solutions. The diffusion-transition scheme is a hybridization of the “continuous” and “atomistic” approaches used in the study of non-living systems. In the CW model, the bacterial cells are represented by discrete walkers that obey dynamic rules. The CW model also consists of at least one chemical field, namely nutrient concentration field, and additional element such as a free boundary of the colony. The model hence falls into the hybrid class of discrete-continuum models alluded to in the last section.

A walker in the CW model does not represent a single bacterium. In a typical experiment, at the end of the growth in a petri-dish there are 109 − 1010 cells – a number impractical to incorporate into a model simulation. Instead, each of the walkers will usually be taken to represent about 104 − 105 cells, so that we work with 104 − 106 walkers in one numerical “experiment”. For the discussion of the dependence of the results on the number of bacteria per walker see Sec. 9.2.2.

Each of the walkers has a position~ri and a metabolic state (‘internal energy’) Hi. The lubrication fluid (Sec. 2.2.1.2) is not incorporated as such into the model, only its effects on the bacterial movement. The area occupied by the colony (wetted by the lubrication fluid) is defined by an on-lattice boundary representing the boundaries of the layer of lubrication fluid. To incorporate the swimming of the bacteria into the model, the walkers perform an CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 151 off-lattice random walk within the area already occupied by the colony.

At each time-step each of the active walkers (motile and metabolizing, see Sec. 7.1.2) attempts to move from its location~ri a step of size d at a random angle Θ (Θ chosen from

0 [0,2π] with uniform distribution), to a new location ~r i given by:

0 ~r i =~ri + d (cosΘ;sinΘ) (7.1)

£ √ ¤ Although d is used in this equation as if it has length units, it’s units are actually length/ time , square root of the units of a diffusion coefﬁcient. These units compensate for fact that the number of steps of a walker per time unit is sensitive to the time-step of the model’s simulation (see page 167 for a model where the number of steps per time unit does not depend of the simulations’ time step). If the units of d would have been length units, then the effective diffusion coefﬁcient of the walkers in the bulk of the colony would have had been sensitive to the time-step of the model’s simulation.

0 If the new location~r i is outside the boundary, the walker does not perform that step, and a counter on the segment of the boundary which would have been crossed by the movement

0 ~ri → ~r i is increased by one. When the segment counter reaches a pre-specified number of hits Nc, the boundary propagates one lattice step and an additional lattice cell is added to the area occupied by the colony. Nc is measured in units of length to the power of −2 (or −D, where D is the spatial dimension of the simulation) - number of hits per boundary length unit which advance the boundary one length unit. The requirement of Nc hits represents the colony propagation through collective production of lubrication fluid and wetting of unoccupied areas. Nc is directly related to the agar concentration, as more lubrication fluid has to be produced (more “collisions” are needed) to push the boundary on a harder (drier) substrate. 152

7.1.2 Food consumption, internal energy, reproduction and sporulation

We represent the metabolic state of the i-th walker by an “internal energy” Hi. The dynamics of this energy is given by

dHi Em = κnconsumed − (7.2) dt τR

∼ 3 where κ is a conversion factor from nutrient to internal energy (κ = 20 · 10 J/g) and Em (about 60nJ) represents the total energy loss for all processes (excluding reproduction) over the minimal time of reproduction τR (about 25 minutes). The nutrient consumption rate nconsumed is ¡ ¢ Ω Ω0 nconsumed ≡ min n, n . (7.3)

Ω Ω0 n is the maximal rate of nutrient consumption of a walker, and n is the rate of nutrient consumption as limited by the local availability of nutrient. The maximal rate of nutrient consumption of a walker equals the consumption rate per cell times the number of cells represented by a single walker. The maximal consumption rate of a single cell is estimated

−15 to be of the order of 1 − 10 f g/sec (1 f g = 10 gram) (see Sec. 2.3.2) hence Ωn is about 10−11 − 10−10g/s.

When sufﬁcient nutrient is available, Hi increases until it reaches a threshold energy Ed and the walker divides into two. When the walker is “starved” for a long interval of time,

Hi drops to zero and the walker “freezes”. This “freezing” represents the transition into pre-spore state. For simplicity (and based on the experimental observations), we assume that in our experiments the cellular density is suitable for sporulation, so that the limiting factor is the supply of nutrients. CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 153

The nutrients is represented by a ﬁeld denoted n(~r,t), and its dynamics is given by the diffusion equation: ∂n = D ∇2n − σ n (7.4) ∂t n a consumed where Dn is a diffusion coefﬁcient and the last term on the RHS includes the consumption of nutrient by the active walkers whose density is denoted by σa,

σa ≡ ∑ δ(~x −~ri) (7.5) i∈{active walkers} where δ(x) is the delta distribution function whose integral is the step function. The diffusion equation has zero-ﬂux boundary conditions and uniform distribution of concentration n0 as initial conditions.

7.1.3 Results of numerical simulations

The diffusion equation is solved on a triangular lattice with a lattice constant ∆x, the same lattice on which the boundary is outlined. For numerical stability the walkers’ step length, √ d ∆t (where ∆t is the simulation’s time-step), must be smaller than the lattice constant. Details of the relationship between the conditions of the bacterial colony and the parameters values of the numerical simulation can be found in Sec. A.3. All the simulations are stopped when the colony reaches a given radius. Results of numerical simulations of the model are shown in Figs. 7.1 (microscopic view) and Fig. 7.2 (Colonial patterns).

As in real bacterial colonies, the simulated patterns are compact at high nutrient concentration levels and become fractal with decreasing nutrient level. For a given nutrient level, the patterns are more ramiﬁed as the agar concentration increases. Fig. 7.3 shows the qualitative dependencies of the fractal dimension and the growth velocity of the nutri- 154

active walker stationary walker

Figure 7.1: A tip of a branch in a simulation of the CW model. The hexagons are those lattice cells that were occupied by walkers and became part of the colony. The reaction- diffusion equations are solved on the whole lattice, weather part of the colony or not. The active and stationary walkers are shown.

Figure 7.2: Colonial patterns of the CW model. Here Nc = 20 and C0 is 6, 8, 10 and 30 from left to right respectively. CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 155 ent concentration levels. As in the real bacterial colonies, in all the simulated colonies the growth velocity doesn’t change signiﬁcantly throughout the growth.

Figure 7.3: Fractal dimension and growth velocity as a function of initial food concentrations. The data are for typical runs of the CW model. The growth velocity is presented in arbitrary units.

Clearly, the results shown in the figures are very encouraging and do capture some features of the experimentally observed patterns. The branching patterns and the constant growth velocity are a manifestation of the diffusion field instability that we have discussed in detail in chapter 3. From this perspective, it is quite reasonable that the effect of the instability is enhanced as the agar concentration is raised and the motion of the bacteria is suppressed. This is analogous to lowering the diffusion coefficient of a bacterial density field (in a continuum description), which leads further into the diffusively unstable region of the parameter space. Comparing the fractal dimension and growth velocity of the simulations of LB model to those of CW model (Fig. 7.3 compared to Fig. 4.13), it seems as if the simulations of the CW model are done for a narrower range of initial nutrient 156 concentration. Note, however, that at the lowest concentrations the patterns of the bacterial colonies are dominated by additional effects such as chemotaxis (see Sec. 2.3.1.3) and both models must be modified to accommodate these effects. Some critical features, such as the ability of the bacteria to develop organized patterns at very low peptone levels (instead of even more ramified structures), the changes in the functional form of the growth velocity as a function of nutrient concentration and the 3-dimensional structures, are not accounted for by the model as is. We therefore propose that chemotactic response should be included in a more complete model in order to explain these additional features [35, 60, 18].

7.1.4 Chemotaxis-based adaptive self-organization

We assume that for the colonial adaptive self-organization the T morphotype employs three kinds of chemotactic responses. One is the food chemotaxis described in Sec. 2.3.1.3. Ac- cording to the ’receptor law’ described in Sec. 2.3.1.3, the gradient of a chemical field C, K ∂C as sensed by bacteria, is , where K is a typical concentration constant. Ac- (K +C)2 ∂x cording to this law, food chemotaxis is expected to be dominant for a range of nutrient levels (the corresponding levels of nutrient are determined by the constant K). The two other kinds are self-induced chemotaxis or signaling chemotaxis, i.e. chemotaxis towards or away from chemical produced by the bacterial cells themselves. As we will show, for efficient self-organization it is useful to have two chemotactic responses operating on different length scales, one regulating the dynamics within the branches (short length scale) and the other regulating the organization of the branches (long length scale). The length scale is determined by the diffusion constant of the chemical agent and the rate of its spontaneous decomposition (the decay time of a chemical is an important factor in its usefulness as information conveyer, thus in its usefulness as a self-organization aid for the bacterial CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 157 colony). If there is also decomposition of the chemical by the cells, it gives them additional control of the length scale. We have reasons to expect that both attractive and repulsive chemotactic responses are employed by the bacteria, so we proceed to find the correspondence between short-range vs. long-range with attractive vs. repulsive. To do so we return to the experimental observations.

7.1.4.1 Repulsive chemotactic signaling

As we have done in previous chapters, we focus on the formation of the fine radial branching patterns at low peptone levels. From the study of non-living systems it is known that, in the same manner that an external diffusion field leads to the diffusion instability, an internal diffusion field can stabilize the growth. We assume that such a field is a chemical agent with a chemotactic effect on the bacteria. In order to be able to regulate the organization of the branches, it must be long-range (with a length scale of the order of the colony’s size). We mentioned in Sec. 2.3.1.3 that sporulation is a drastic and cooperative phenomenon reserved for extreme stress conditions. When the stress is nutrient depletion, it is biologically reasonable that before going through sporulation cells emit (either purposely or as a by-product) a material that causes other cells to move away. Such response is advantageous for the emitting bacteria, which have fewer competitors for the remaining nutrient, and the response is also advantageous for the repelled bacteria, which have better chances of finding nutrient-rich places. To test this idea, a repulsive chemotactic signaling response was included in the model [35].

The equation describing the time evolution of the concentration ﬁeld R(~r,t) of the chemorepellent is: ∂R = D ∇2R + σ Γ − σ Ω R − λ R (7.6) ∂t R s R a R R 158

where σs is the density of the stationary cells, ΓR is the emission rate of the chemorepellent

R, and ΩR is the decomposition rate of R by the active walkers. The last term represents spontaneous decomposition of R at a rate λR.

In the presence of the chemical R the movement of the active walkers changes from pure random walk (equal probability to move in any direction) to a random walk with a bias along the gradient of the communication ﬁeld. The bacteria bias there movement by modulating the length of the steps. In the current CW model it is not trivial to change the step length without changing the walkers’ speed of movement. It is much easier to modulate the direction of the steps. The Communicating Spinors model presented in Sec. 7.2 can easily accommodate both schemes of chemotactic response, and we used it to compare the two schemes. The results indicate that when the branches’ width is much larger then steps’ length (as is the case in simulations of the CW model), the two schemes are equivalent. We therefore follow Refs. [35, 34] and introduce the chemotactic response in the CW model as a higher probability of the walkers to move away from higher concentrations of R (see

0 Refs. [61, 92] for details). Eq. (7.1) is changed so that the i-th walker new position ~r i is now determined by

Θ Θ χ ∇ 0 (cos ;sin ) − R R(~ri) ~r i =~ri + d (7.7) |(cosΘ;sinΘ) − χR∇R(~ri)|

where Θ is random angle in [0,2π] (chosen with a uniform distribution), ∇R(~ri) is the gradient of the attractant R at the walker’s position ~ri and χR is a chemotaxis coefﬁcient which represents the magnitude of the chemotactic response of the bacteria (it is positive for attractive chemotaxis and negative for repulsive chemotaxis). The response towards other chemotactic agents and combinations of chemotactic responses can be incorporated using similar expressions CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 159

In Fig. 7.4a we demonstrate the dramatic effect of the repulsive chemotactic signaling.

The pattern becomes much denser with a smooth circular envelope, while the branches are thinner and radially oriented. This structure enables the colony to spread over the same distance with fewer walkers. Thus the emission of chemorepellent by the stressed walkers also serves the interest of the colony as it can cope better with the growth conditions.

Figure 7.4: The effect of chemotaxis on the colonial growth in the CW model. Growth conditions and model’s parameters are the same in all patterns, excluding parameters related to chemotaxis. In the center: a typical colony without chemotaxis. On the right: chemotaxis towards food is included in the model. The growth velocity is almost doubled, but the pattern is essentially unchanged by food chemotaxis. On the left: repulsive chemotactic signaling is included in the model. The pattern is of ﬁne radial branches with circular envelope, like the bacterial colony presented in Fig. 2.2 d.

The patterns produced by the simulations seem to be in agreement with the experimental observations. Yet the numerical simulation poses a difficulty: as can be expected, the radially outwards “push” of the walkers by the repulsive signaling leads to a faster spread of the colony. This result does not agree with the experimental observations: the velocity of the fine radial branching patterns is lower then that of the fractal growth observed at higher peptone levels. Moreover, in Fig. 2.9 we show a colony with sectors of fractal growth embedded in a fine radial branching structure. Clearly, the fractal growth is the faster one. This dilemma is resolved when food chemotaxis is included as well. 160

7.1.4.2 Ampliﬁcation of diffusive instability due to nutrients chemotaxis

In non-living systems (e.g. many walkers DLA (diffusion limited aggregation), electro- chemical deposition, Hele-Shaw cell), typically, the more ramified the pattern (with a lower fractal dimension) the lower the growth velocity. Now we are looking for a mechanism which is capable of doing just the opposite – a mechanism that can both increase the growth velocity and maintain, or even decrease, the fractal dimension. We expected food chemotaxis to be the required mechanism. It provides an outward drift to the cellular movements; thus, it should increase the rate of envelope propagation. At the same time, being a response to an external field it should also amplify the basic diffusion instability of the nutrients field. Hence, it can support faster growth velocity together with a ramified pattern of low fractal dimension.

Again, we used the communicating walkers model as a research tool to test the above hypothesis. As expected, the inclusion of food chemotaxis led to a considerable increase of the growth velocity without signiﬁcant change in the fractal dimension of the pattern (Fig. 7.4).

7.1.4.3 Attractive chemotactic signaling

Fig. 7.5 shows 3-dimensional structures inside branches of bacterial colony. The length scale of the 3-dimensional structures indicates the existence of a dynamical process with a characteristic length shorter than the branch width. The accumulation of cells in the aggregates brings to mind the existence of a short range self-attracting mechanism. In addition, observations of attractive chemotactic signaling in E. coli [53, 26, 54, 44, 216] indicate that such signaling operates during growth at high levels of nutrients. Under such CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 161 conditions the bacterial cells produce hazardous metabolic waste products. The purpose of the attractive signaling might be to gather cells to help in the decomposition of the waste products. This scenario is consistent with the observation of 3D structures during growth at high peptone levels. Motivated by the above, we assume that the colony employs an attractive self-generated short range chemotaxis during growth at high levels of nutrients. To test this hypothesis we add the new feature to the communicating walkers model and compare the resulting patterns with the observed ones.

Following Ben-Jacob et al. [26, 216], we include the equation describing the time evolution of the concentration ﬁeld A(~r,t) of the attractant:

∂A = D ∇2A + σ Γ − σ Ω A − λ A (7.8) ∂t A T A a A A where σa is the density of the active walkers and σT is the density of the portion of the active walkers triggered to emit the attractant. ΓA is the emission rate of A by triggered walkers, ΩA is the decomposition rate of A by all the active bacteria and λA is the rate of spontaneous decomposition of A. The walkers are triggered either when the level of A is above a threshold (auto-catalytic response) or when the level of a triggering field is above a threshold (see Refs. [26, 216] for details). The triggering field W, which represents the waste products, satisfies the following equation

∂W = D ∇2W + Γ σ n − σ Ω W − λ W (7.9) ∂t W W a consumed a W W

ΓW , ΩW and λW have the meanings corresponding to ΓA, ΩA and λA.

In the presence of A, the movement of the active cells changes from pure random walk to a random walk with a bias to move towards high concentration of A.

In Fig. 7.6 we show an example of the formation of 3D structures when the attractive 162

Figure 7.5: An observed structure of ordered aggregates within branches. The picture shows variation in the height of the branches. The more bacteria are in a unit area, the more layers the bacteria are in, and the higher the area seems. CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 163 chemotactic signaling is included in the model, similar to patterns in the bacterial colonies

(Fig. 7.5)

Figure 7.6: A structure of aggregates within a branch of a simulated colony with attractive signaling chemotaxis. The contours are equi-density contours of the walkers’ “density ﬁeld”.

We note in brief that describing such attractive chemotactic signaling in the context of a continuum model requires inclusion of 3 additional fields in the model: the attractant field A, the triggering waste product W, and a third field for the density of the triggered bacteria

σT . This is the main reason such chemotactic response was not described in previous parts of this work, which presented continuum models.

7.1.4.4 Chemotactic interplay, wetting ﬂuid and morphology transitions

As mentioned earlier, we expect food chemotaxis to dominate the growth for intermediate peptone levels. According to the “receptor law”, its effect should decrease at higher levels of peptone. In this limit we expect the self-generated attractive chemotaxis to become the 164 dominant mechanism. The morphology transition between the fractal branched growth and the radial ﬁngers growth presumably result from switching the leading role between the two kinds of chemotaxis response. The fact that we observe a real transition and not a cross-over means that there is another mechanism at work, a mechanism that regulates the interplay between the two responses. The observations of increased branch width upon transition indicate that regulation of the emission of the wetting ﬂuid might also be part of the transition.

The morphology transition from the ﬁne radial branching growth to the fractal branching growth is probably related to a switching from dominance of repulsive chemotactic signaling to that of food chemotaxis. Again, the sharpness of the transition indicates that an additional regulating mechanism might be involved. Manually switching between the two types of chemotaxis results in a sector formation in the simulation (Fig. 7.7), not unlike the bacterial colonies (Fig. 2.9).

Figure 7.7: Two types of morphologies in the same colony of the CW model. The sectors of faster, less ordered branches are by cells more responsive to food chemotaxis then the others. Compare to Fig. 2.9. CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 165

7.2 Orientation ﬁeld and the Communicating Spinors Model

The Communicating Spinors model was developed in order to explain the chirality of the C morphotype colonies. Our purpose is to show that the flagella handedness, amplified by several mechanisms, leads to the observed chirality. It does so in a manner similar to the manner in which crystalline anisotropy leads to the observed symmetry of snowflakes [17].

We also show that the dominant bacteria-bacteria interactions are conﬁnement of rotation and co-alignment interactions. This last point was not shown by previous versions of the model [27, 24] the only developer of this version).

It is known [76, 211, 202] that flagella have specific handedness. Ben-Jacob et al. [27] proposed that the latter is the origin of the observed chirality. In the bulk of a fluid (which is the state in many experimental setups), as the flagella unfold, the cell tumbles and ends up at a new random angle relative to the original one. The situation changes for quasi 2D motion – motion in a “lubrication” layer thinner than the cellular length. We assume that in this case, of rotation in a plane, the tumbling has a well-defined handedness of rotation. Such handedness requires, along with the chirality of the flagella, the cells’ ability to distinguish up from down. Growth in an upside- down petri- dish shows the same chirality. Therefore, we think that the determination of up versus down is done either via the vertical gradient of the nutrient concentration, the vertical gradient of signaling materials inside the substrate, or the friction of the cells with the surface of the agar. The latter is the most probable alternative; soft enough agar enables the bacteria to swim below the surface of the agar which leads to many changes in the patterns, including reversing the bias of the branches (see Fig. 7.12).

To cause the chirality observed on semi-solid agar, the rotation of tumbling must be, 166 on average, less than 90◦ and relative to a speciﬁc direction. We assume that co-alignment

(orientational interaction) limits the rotation. We further assume that the rotation is relative to the local mean orientation of the surrounding cells.

7.2.1 Oriented particles and orientation ﬁeld

To test the above idea, we included the additional assumed features in the Communicating Walkers model [35], changing it to a ‘Communicating Spinors’ (CS) model (as the particles in the new model have an orientation and move in a quasi-1D random walk). Ben-Jacob et al. have presented in the past two simpliﬁed versions of the CS model for chiral growth [27, 24]. In their models there is an orientation ﬁeld which affects the particles movement.

It represents marks left on the agar by bacteria at the leading edge of the colony. It could be viewed as “zeroth order” approximation to the experiments, as it does not change in time (its local value is set by the particles’ dynamics, but once set it does not change). The model we present here includes both time-independent orientation field which represents marks left on the agar and a dynamic orientation field which is the mean local orientation of the particles. Both fields affect the particles movement (see below). This model could be viewed as “first order” approximation to the experiments as the fields are affected by the bacterial dynamics, but their update is done under the (somewhat unrealistic) assumption that the bacteria tumble one at a time and never simultaneously.

The representation of bacteria as spinors allows for a close relation to the bacterial properties and a more detailed description. At the end of growth in a typical experiment, there are 108–109 bacterial cells in the petri-dish. Thus it is impractical to incorporate into the model each and every cell. Instead, each of the spinors represents about 10–1000 cells, so that we work with 105–107 spinors in one numerical “experiment”. CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 167

Each spinor has a position ~ri, direction θi (a complex number of length 1) and a metabolic state (‘internal energy’) Hi. The spinors perform an off-lattice constrained random walk on a plane within an envelope representing the boundary of the wetting ﬂuid.

This envelope is deﬁned on the same tridiagonal lattice where the diffusion equations are solved. To incorporate the swimming of the bacteria into the model, at each time step each of the active spinors (motile and metabolizing, as described below) recalculate its direction θ0 i and moves a step of length l in this direction.

The direction in which each spinor moves is determined in two steps; ﬁrst the spinor ”decides” whether it should continue the current run, that is to continue in the same direc- θ0 θ tion i = i. In the basic version of the model (see Sec. 7.2.2 for extension of the model) the decision is random with a speciﬁc probability per time unit p to continue the run. The resulting runs have a geometric distribution of lengths, with mean run length of l/p. Once θ0 a spinor “decides” to change direction, the new direction i is calculated using an ODE for the spinor’s angle during tumbling. The ODE is not solved directly, but the long-time solution of the ODE is sought (long-time means much longer than the relaxation time of the ODE, see below). The spinor’s previous direction can be used for the initial conditions of the ODE, but we do not use it as the long-time solution (and the ability of the spinor to move backwards) makes the solution independent of initial conditions. The ODE for the spinor’s angle is

∂τφ = (α + η) − Fa (φ,Φ(~ri,t)) − Fc (φ,Φ(~ri,t)) (7.10)

The terms of the equation are continuous, and they will be explained below. Since φ is an angle, all terms in the equation are either independent of φ or have period 2π in φ. There are two types of solutions to this ODE; if ∂τφ has different signs for different values of φ, then the spinor angle φ will approach some stable ﬁxed-point of the ODE. If, on the other 168 hand, ∂τφ has the same sign for all values of φ, then the spinor keeps rotating for all times.

After some speciﬁc long-time the value of φ can be taken to be a random value taken from a probability distribution whose density is inversely proportional to ∂τφ (it is more likely to

ﬁnd the spinor at angles where it spends longer times, that is, at angles where its angular velocity is smaller in absolute value).

(α + η) in Eq. (7.10) are positive constants representing the bacterial propulsion torque during tumbling. α depends on the spinor’s properties only, but η is different from realization to realization of Eq. (7.10), representing fluctuations in the spinor’s activity. Φ(~ri,t), the local mean orientation, is a complex field which will be defined below.

Fa is a co-aligning function of the spinor with the local mean orientation Φ(~ri,t):

Fa (φ,Φ) = γa |Φ|sin(2[(argΦ) − φ]) = γa |Φ|sin(2∆φ)

∆φ Φ φ θ d 2 θ 2 θ where = [(arg ) − ]. Note that sin2 = dθ sin . sin is a common energy function measuring the co-aligning between two rods [?].

Fc is a conﬁnement function representing the torque applied by the neighborhood that is pushed to make way for the rotating bacterium:

2 Fc (φ,Φ) = γc |Φ|sin (∆φ)

Fa and Fc look very similar, but their effect on the local and global dynamics is different, for their maximum are at different values of ∆φ.

The effect of this difference on the dynamics is as follow. In the case γc = 0 , γa > 0 there are three cases. If 0 ≤ (α + η) ¿ γa |Φ| (case a1) then Eq. (7.10) has stable ﬁx points and hence its long-time solutions are near ∆φ = jπ ( j ∈ N). If (α + η) ' γa |Φ| (case a2) then Eq. (7.10) has stable ﬁx points and its long-time solutions are near ∆φ = jπ + π/4 CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 169

( j ∈ N). If 0 ≤ γa |Φ| ¿ (α + η) (case a3) then Eq. (7.10) has no ﬁxed points and the long-time solutions of the equation are almost uniformly distributed in [0,2π]. If, on the other hand, the case is γa = 0 , γc > 0, there are different solutions. The solutions of the cases 0 ≤ (α + η) ¿ γc |Φ| (case b1) and 0 ≤ γc |Φ| ¿ (α + η) (case b3) are similar to the solutions of cases a1 and a3 respectively. If (α + η) ' γc |Φ| (case b3) then the long-time solutions of Eq. (7.10) are near ∆φ = jπ+π/2 ( j ∈ N). It appears that the solutions for the cases (α + η) ' γc |Φ| and (α + η) ' γa |Φ| are closely related to the angles of emission of side branches (see Fig. 7.8).

Figure 7.8: Zoom-in on side-branching pf real and simulated colonies. a) Branches of a C morphotype colony. Note the pattern of branch splitting. b) Simulation of the CS model with γc = 0 , γa = 1. The angles between the branches and side branches tend to be close to o o 45 or 135 , unlike the colony in (a) c). Simulation of the CS model with γc = 0 , γa = 1. All other parameters are as in (b) The angles between the branches and side branches tend to be close to 90o, like the colony in (a). a b c

θ0 jφ(T) The new orientation of the spinor is deﬁned as i = e where T is some long time, 1 T À α+η , and j is the imaginary root of −1. Once oriented, the spinor attempts to advance θ0 ~0 a step of length l in the direction i , and the new location r i is given by: ¡ ¢ ~0 θ0 θ0 r i =~ri + l Re i,Im i (7.11)

The movement is conﬁned within an envelope which is deﬁned on the tridiagonal lattice.

0 The step is not performed if ~r i is outside the envelope. Whenever this is the case, a counter on the appropriate segment of the envelope is increased by one. When a segment counter reaches Nc, the envelope advances one lattice step and a new lattice cell is occupied (Nc is measured in units of length to the power of −2 – number of hits per boundary length unit which advance the boundary one length unit). The spinor’s direction is not reset upon 170 hitting the envelope, thus it might “bang its head” against the envelope time and time again.

The requirement of Nc hits represent the colony propagation through wetting of unoccupied areas by the bacteria. Nc is related to the agar dryness, as more wetting ﬂuid must be produced (more “collisions” are needed) to push the envelope on a harder substrate.

Next we specify the mean orientation field Φ. To do so, we assume that each lattice cell (hexagonal unit area) is assigned one value of Φ0(~x) (a complex field), representing marks left on the agar by bacteria that first wetted this spot. The value of Φ0 is set when a new lattice cell is first occupied by the advancement of the envelope, and then remains constant. We set the value of Φ0(~r) to be equal to the average over the orientations of the

Nc attempted steps that led to the occupation of the new lattice cell. v u u Nc t Φ0 (~x) = ∑ φk (7.12) k=1 where φk is the orientation of the k-th hit in the boundary. The local orientation field Φ at the center of a lattice cell~x is defined as r Φ Φ 2 θ2 (~x,t) = m0 0 (~x) + m1 ∑w(~ri −~x) i (7.13) i where m0 and m1 are the relative weight of the of the static and dynamic parts of Φ, and w is a spinors weight function and the sum runs over all the spinors (the exact form of w is irrelevant as long as it has a finite support around 0, with the radius of the support smaller than a branch’s width. For computational convenience we used a step function such that w is one for all spinors inside the local lattice cell and zero otherwise). The value of Φ in any given point inside the colony is found by linear interpolation between the three neighboring centers of cells.

The continuum model of the growth of C morphotype , Mean Orientation Field (Chap- CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 171 ter 6) includes terms for changes in mean orientation of the bacterial ﬁeld. Those terms are much more complex and less related to the biology than the terms presented in this section for the spinors’ orientation; as we stated time and time again, representation of the bacteria in an atomistic model enables descriptions which is more easily related to the biology than a representation of the bacteria in a continuum model.

The spinors in the CS model grow, multiply and “freeze” in the same manner as walkers do in the CW model (see Sec. 7.1.2). The i-th spinor has an “internal energy” Hi. The rate of change of the internal energy is given by :

dHi Em = κnconsume/d − , (7.14) dt τR

When sufﬁcient nutrient is available, Hi increases until it reaches a threshold energy. Upon reaching this threshold, the spinor divides into two. When the spinor is “starved” for a long interval of time, Hi drops to zero and the spinor freezes.

As in the CW model, we represent the diffusion of nutrients by solving the diffusion equation for a single agent whose concentration is denoted by n(~x,t): ∂n = D ∇2n − σ n , (7.15) ∂t n a consumed where the last term includes the consumption of nutrient by the active spinors (σa is their density). The equation is solved on the same tridiagonal lattice on which the envelope is deﬁned. The length constant of the lattice ∆x must be larger than the size of the spinors’ step l. The simulations are started with an inoculum of spinors at the center and a uniform distribution of the nutrient. Both Φ and the spinors at the inoculum are given uniformly distributed random directions.

Results of the numerical simulations of the model are shown in Fig. 7.9. These results do capture some important features of the observed patterns: the microscopic torque α leads 172 to a chiral morphology on the macroscopic level. The growth is via stable tips, all of which twist with the same handedness and emit side-branches. The dynamics of the side-branch emission in the time evolution of the model is similar to the observed dynamics.

Figure 7.9: A morphology diagram of the Communicating Spinors model for various values ◦ ◦ of Nc and initial n concentration n0. α = 6 , η = 3 , l = 0.2, p = 0.5.

For large noise strength η the chiral nature of the pattern gives way to a branching pattern (Fig. 7.10). This provides a plausible explanation for the branching patterns produced by C morphotype grown on high peptone levels, as the cells are shorter when grown on a rich substrate. The orientation interaction is weaker for shorter cells, hence the relative noise is stronger. CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 173

a b

Figure 7.10: When the noise η is increased to η = 180◦ the tumbling of the spinors becomes unrestricted. Their movement becomes like that of the T bacteria and accordingly the simulated colonial pattern is like that of T morphotype. On the left η = 3◦, on the right η = 180◦.

7.2.2 Chemotaxis in the Communicating Spinors model

The colonial patterns of C morphotype (e.g. Fig. 7.9) are rarely as ordered as the simulated patterns of the CS model. For example, the branches of the observed colonies usually have varying curvature. In the simulations of C morphotype shown in Fig. 7.9 all the branches have a uniform curvature. We believe that one of the main reasons for this discrepancy is that in experiments there are bacteria with variable length, where the length of the bacteria is related to their nutritional history (the equivalent of Hi). the various lengths give the bacteria various parameters of rotation and different distributions of lengths in different branches give them different curvature and width. Some of the observed variations of curvature do not depend on variable bacterial length and can be explained in the context of the CS model. In some of the observed patterns (Fig. 2.13b), the curvature of the branches has a distinct relation to the branch’s radial orientation (the orientation relative to the radial direction): the curvature is smaller when the branch is in the radial orientation 174 and larger when the branch is orthogonal to that orientation. This brings to mind the radial organization of branches in the T morphotype, and indeed we where able to explain the chiral pattern with the aid of the same concept – repulsive chemotaxis.

Chemotaxis was introduced in previous versions of the Communicating Walkers model by varying, according to the chemical gradient, the probability of moving in different directions [35, 18]. Modulating the directional probability is not the way bacteria implement chemotaxis – they modulate the length of runs. However, the growth of T morphotype is insensitive to the details of the movement. Modulating the directional probability is as good an implementation of chemotaxis as many other implementations (it was chosen for computational convenience). The pattern of the C morphotype is based on ampliﬁcation of microscopic effects (singular perturbation) such as the left bias in the bacterial tumbling. Small differences in the microscopic dynamics of chemotaxis might affect the global pattern. Indeed we found that modulating the directional probability yield unrealistic results in the simulations of C morphotype. We had to resort to the bacterial implementation of chemotaxis – modulating the length of runs according to the chemical gradients.

When modulating the length of runs of walkers or spinors one must be careful not to change the particles’ speed. Such a change is not observed in experiments [42, 196] and it has far reaching effects on the dynamics. Changing the particles’ speed is like changing the diffusion coefﬁcient of the bacterial density ﬁeld, a change that can have undesirable effects on the pattern.

Modulating the length of spinors’ runs without changing their speed can be done by modulating the number of steps that compose a single run (that was our motivation for dividing the runs into steps). Since the mean number of steps in a run is determined by the CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 175 reorientation probability p, chemotaxis should modulate this probability. For chemotaxis, the probability of changing direction by the i-spinor in one time step is (for a repellent R):

∗ χ ∂ p = p + (R) θi R (7.16) where R is measured at the spinor position~ri, χ(R) is the same as in the continuum model ∂ (either constant or the “receptor law”) and θi is the directional derivative in the spinor’s ∗ direction θi. p is truncated to within the range [0,1] as it is a probability. The length of the resulting runs depend on the runs’ direction, where a spinor moving up the gradient of the repellent have shorter mean run length than the same spinor moving down the gradient.

The production and dynamics of the repulsive chemotactic signaling in the CS model is the same as in the Communicating Walkers model, (see Eq. (7.6) ). The patterns resulting from including repulsive chemotaxis in the model have indeed branches with variable curvature, as can be seen in Fig. 7.11. The curvature is smaller for branches in the radial direction. Food chemotaxis also varies the branches’ curvature, but in a less ordered manner, not similar to the observed bacterial patterns.

Under different growth conditions the C morphotype can produce very different patterns. As mention above, if the agar is soft enough the bacteria can move inside it . In such cases, the bias in the bacterial movement might change or even reverse, and it is manifested in the curvature of the branches. Widely changing curvature of the branches can be seen in Fig. 2.13a. The agar hardness was tuned such that in the beginning of the growth the bacteria could swim inside the agar, but they are forced to swim on the agar by the end of growth due to the marginal water evaporation during growth. In Fig. 7.12 we demonstrate the models’ ability to explain such patterns by changing the spinors’ bias α during the simulation. α is set to be a continuous random function of the colonial size, which is 176

a b

Figure 7.11: The effect of repulsive chemotactic signaling on the Communicating Spinors model. a) Without chemotaxis. b) With repulsive chemotaxis. The Spinors are repelled from the inner parts of the colony. The resulting curvature of the branches is reduced when they are in the radial direction. The pattern resembles Fig. 6.1(a), excluding the end of the growth. constrained only at the beginning and end of growth to have certain values. The function for α is the same in all the images of Fig. 7.12, but various types of chemotaxis are used.

As can be seen, repulsive chemotactic signaling is needed to explain the observed bacterial patterns.

7.2.3 Weak chirality in the Communicating Spinors model

As was demonstrated in Sec. 7.2, the CS model is robust enough to reproduce patterns of T morphotype, as well as patterns of C morphotype. Here we use it to model patterns of weak chirality. Two simulated T -like colonies are shown in Fig. 7.13. Fig. 7.13a shows a colony with radial branches while Fig. 7.13b shows a colony with weak chirality and thinner branches. In the two simulations the spinors have exactly the same response to chemotaxis and the same bias α = 9◦. The two runs differ in the freedom of rotation η; in Fig. 7.13a the spinors can rotate freely (η = 180◦) while in Fig. 7.13b spinor rotation is CHAPTER 7 – ATOMISTIC MODELSFOR BRANCHING PATTERNS 177

a b c

Figure 7.12: The snake-like branches observed in Fig. 2.13a can be reproduced by the Communicating Spinors model. α is set to be a continuous function of the colony’s radius (the same function in a, b, and c). Maximal value of α is 8◦, minimal value is −2◦. (a) With repulsive chemotactic signaling. (b) Without chemotaxis. (c) With food chemotaxis. The best resemblance to the observed colony is obtained with repulsive chemotactic signaling.

somewhat limited (η = 35◦, compared to a colony with strong chirality, where η is smaller

than 5◦).

So far we have presented three different models that can exhibit patterns with weak chirality: Non-Linear Diffusion model (Sec. 4.2.2), the Mean Orientation Field model (Sec. 6.2) and the CS model. As was explained in the discussion at the end of Sec. 6.2,

the terms in the NLD model which lead to weak chirality are an unrealistic description of the bacterial movement; they imply bacterial knowledge about the direction of gradient of a chemical ﬁeld, without movement in that direction. The MOF also include explicit

term of the chemical field’s gradient, but as it interacts with the mean orientation field, one can argue that the MOF model includes a reasonable mean field description of the spatial derivatives actually sensed by the bacteria. The CS model is the only one of the

three models whose chemotaxis terms positively include only knowledge accessible to the bacteria, namely the spatial derivatives of the chemical ﬁelds along their path. Only the 178

a b

Figure 7.13: Weak chirality of the T morphotype is modeled by the Communicating Spinors model. Both simulations are with repulsive chemotactic signaling and with bias in the walkers’ rotation. a) With free rotation (η = 180◦) the pattern is branched, without apparent chirality expressed. b) With constrained rotation (η = 35◦) weak chirality is expressed. atomistic description of the bacteria allows for detailed enough description of the behavior of the bacteria. Chapter 8

Vortex Formation

In the previous chapters we focused on bacterial colonies of Paenibacillus dendritiformis (T morphotype and C morphotype ). Our intent was to understand the bacterial cooperation and communication in the context of the colonies, guided by the colonial patterns as a whole, as well as the bacterial dynamics. In this chapter we focus on fascinating bacterial behavior which is revealed in colonies, but it is also an interesting mode of cooperation by itself - ﬂocking and vortex formation.

[83, 208, 233]. The observed phenomena include “turbulent like” collective ﬂow, complicated eddy (vortex) dynamics, merging and splitting of vortices, rotating “bagels” and more. Similar phenomena were observed inside the branches of colonies of Paenibacillus vortex (V morphotype ). Fig. 2.15 shows representative patterns of branching colonies of V morphotype . Each branch is produced by a leading droplet and emits side branches,

179 180 each with its own leading droplet.

Microscopic observations reveals that each leading droplet consists of hundreds to millions of cells that circle a common center (hence the term vortex) at a cellular speed of up to 10µm/sec (Fig. 8.1). Both the size of a vortex and the speed of the cells can vary according to the growth conditions and the vortex location in the colony. Within a given colony, both clockwise and anti-clockwise rotating vortices are observed. The vortices in a colony may consist of either a single or multiple layers of cells. We occasionally observed vortices with an empty core, which we refer to as “bagel” shaped. After formation, the number of cells in the vortex increases, the vortex expands and it translocates as a unit. The movement speed of the vortices is slower than the speed of the individual cells circulating around the vortex center. Bacterial cells are also contained in the trails left behind the leading vortices. Some of them are immobile while others move, swirling with complex dynamics; some form new vortices while other form well-deﬁned groups (containing a few to thousands of cells) moving in various directions and speeds. When two such groups pass by each other they can unite into a single group, or they might remain separate despite the close contact.

The migrating groups of cells are very reminiscent of the “worm” motion of Dictyostelium (cellular slime mold) or schools of multicellular organisms. While the whole intricate dynamics is conﬁned to the trail of the leading vortex, the vortices and the moving groups inside the branches have no ”physical” boundary, and it seems that only their dynamics deﬁne their edge. Only vortices that are formed in the trail can break out of the trail and create a new branch.

a) b) c)

Figure 8.1: Microscope observations of vortices. Magniﬁcation is ×16 in (a), ×50 in (b), and ×500 in (c). Growth conditions are 4g/l peptone level and 2.5% agar concentration. boundary for the branch. We do not observe tumbling motion nor movement forward and backward. Rather, each cell’s motion is forward along the cell’s long axis, and the cell tend to move in the same direction and speed as its surrounding cells, in a synchronized group. Electron microscope observations show that the bacteria have ﬂagella which indicates the movement to be swarming (Sec. 2.3.1.1).

The problem of ﬂocking, in which a large number of moving particles (e.g. ﬁsh, birds or unicellular organisms) remain coherent over long times and distances, has attracted considerable attention. A simulation of a simple numerical model by Vicsek et al. [223] in which each particle has a constant identical speed and a direction of motion that is determined by the average direction of its neighbors, revealed that an ordered phase exists, even in the present of noise and disorder. A subsequent analysis of a continuum model by Toner and

Tu [214, 215] investigated this ordered phase further and derived conditions for its stability.

These models have in common that the ﬂocks have inﬁnite extent and, in simulations,

fill the entire computational box. In reality, however, flocks have a finite size, with its density dropping sharply at the edge of the flock [169]. We present a discrete model con- 182 sisting of self-propelled interacting particles that obey simple rules. We show that self- organization in our model leads to coherent localized states in one dimension (1D) and in two dimensions (2D) that are stable in the presence of noise and disorder. In 1D the coherent state is a localized flock moving in constant velocity. In 2D there is a coherent state of a localized flock moving in constant velocity, and there is also a state of a vortex, where all the particles circle a common center. Furthermore, we present a continuum version of our discrete model which is obtained by coarse-grain averaging the discrete equations. The continuum flock solutions in 1D agree very well with the discrete solutions and are characterized by having a finite extent with densities that drop off sharply at the edges. In 2D, we focus on a vortex state in which the particles are rotating around a common center and show that the discrete model and the continuum model agree well.

8.1 Atomistic model

Our discrete particle model consists of N particles with mass mi, position ~xi and velocity

~vi. Each particle experiences a self-propelling force ~fi with ﬁxed magnitude α. To prevent the particles from reaching large speeds, a friction force with coefﬁcient β is introduced. In addition, each particle is subject to an attractive force which is characterized by an interaction range la. This force is responsible for the aggregation of the particles and corresponds in animal aggregates to an awareness of the position of surrounding animals. In bacterial colonies this force might correspond to the force acting on the bacteria due to the surface tension of the liquid in which the bacteria are immersed. To prevent a collapse of the aggregate, a shorter-range repulsive force is introduced with interaction range lr. Thus, the CHAPTER 8 – VORTEX FORMATION 183 governing equations for each particle is

~ mi∂t~vi = α fˆi − β~vi − ∇U (~xi) (8.1)

∂t~xi = ~vi (8.2)

We have checked that our qualitative results are independent of the explicit form of the interaction potential, and we have chosen here an exponentially decaying interaction: Ã ¯ ¯! Ã ¯ ¯! −¯~xi −~x j¯ −¯~xi −~x j¯ U (~xi) = ∑Ca exp − ∑Cr exp (8.3) j la j lr where Ca, Cr determine the strength of the attractive and repulsive force respectively. The direction of the self-propelling force can be chosen along the instantaneous velocity vector or, similar to the numerical model of Ref. [223], can be determined by aligning it with the average velocity direction of the neighboring particles:

fˆ = uˆ(~v ) without averaging i Ãi Ã ¯ ¯!! −¯~xi −~x j¯ (8.4) fˆi = uˆ ∑~v j exp with averaging j lc where lc is a correlation length and uˆ(~v) is a unity vector in the direction of~v, uˆ(~v) =~v/|v| .

Simulation of the discrete model were done by solving Eqs. (8.1-8.2) using a simple Euler integration routine with time step ∆t = 0.2. The simplest coherent localized solution in our model is a 1D flock in which all particles move with constant velocity v = α/β. An example of this solution is presented in Fig. 8.2 where we have plotted, as solid circles, the 2 density defined as ρi = as a function of the position of the particle (where the xi+1 − xi−1 particles are indexes in the order of their position). The density can be seen to drop abruptly to zero at the edge of the flock. We have checked that this solution is stable in the presence of moderate amounts of noise (added to Eq. (8.2) ) and of disorder in the parameters. 184

Figure 8.2: A coherent moving ﬂock in the one-dimensional version of the model with parameters (all with arbitrary units) m = 1, α = 0.5, β = 1, Ca = 0.45, la = 60, Cr = 2 and lr = 20. The solid circles correspond to the solution of the discrete model for N = 200 and every 10th particle displayed. The solid line shows the solution of the continuum model.

Further simulations revealed that increasing the number of particles does not change the shape of the density function and that the total size of the flock reaches a constant value. This is illustrated in Fig. 8.3 where the dashed line represents the size of the flock as a function of the number of particles. This behavior of the model is due to the soft-core repulsive force which allows the particles to be packed closely. In bacterial colony such condensation can be interpreted as multi-layered flock, as can be seen in Fig. 8.1. This behavior is ob- viously unrealistic for flocks of animals. Our model can easily be extended to incorporate hard-core repulsive forces. In fact, a flock with a size that scales approximately linearly with the total number of particles can be obtained by introducing a hard-core repulsive force in Eq. 8.1. It is also possible to eliminate the soft-core repulsive potential altogether. The soft-core repulsive potential is introduced here represents an approximation to the up- CHAPTER 8 – VORTEX FORMATION 185 wards pressed of the bacteria (dynamics in the third dimension, which we neglected from our model). Mathematically, it facilitates a simple derivation of a continuum model. The specific form of the hard-core potential is not important and we have chosen  ¡¯ ¯ ¢ ¯ ¯  ¯ ¯ 5 ¯ ¯ ∑ Chc xi − x j − lhc if xi − x j ≤ lhc Uhc = j6=i ¯ ¯ (8.5)  ¯ ¯ 0 if xi − x j > lhc

In Fig. 8.3 we show, as a solid line, the size of the flock in the presence of this additional repulsive force as a function of N. In the inset we show the corresponding density of the flock for different N. Naturally, the force has only an effect when the inter-particle separations are smaller than lhc. Hence, for small N the flock solution is unaffected by the additional force. As N is increased and the inter-particle spacing becomes smaller than lhc, the particles in the center of the flock are kept apart by the hard core repulsive force. For large N, the flock reaches a constant density in its center and its size scales linearly with N.

Let us now turn to 2D where we have obtained several ﬂocking states. One, not shown here, is the equivalent of our 1D ﬂock: all particles are moving in the same direction with

|~vi| = α/β. The particles arrange themselves on a disk and this aggregate is stable under small amounts of noise and disorder. The solution which we will focus on here consists of a vortex state where the particles rotate around a common center and which is common in ﬁsh schools [179], bacterial colonies [22] and amoeba aggregates [186, 175, 184]. This solution has been observed previously in models of self-propelling particles but only in the presence of a conﬁning boundary [75, 103] or when a rotational chemotaxis term is invoked [22].

In our model, the vortex solution can be obtained from a wide variety of initial conditions including one in which all particles are randomly placed on a disk with speed α/β 186

Figure 8.3: The size L of a ﬂock as a function of N with (solid line) and without (dashed line) an additional short-range hard-core potential. The inset shows the density of ﬂocks in the presence of the short-range hard-core potential for different N. Parameter values: m = 1, α = 0.5, β = 1, Ca = 0.6, la = 40, Cr = 2, lr = 20, Chc = 1 and lhc = 10.

and random initial velocities. A typical evolution of the particles starting from this initial condition is shown in Fig. 8.4. This ﬁgure was obtained in the absence of the velocity averaging term and illustrates that in this case some particles move clockwise while the others rotate counter clockwise. Some bacterial vortices are composed of ”rings” of bacteria, where some rings rotate clockwise and the others rotate counter clockwise. When the velocity averaging term is included the ﬁnal vortex state consists of all particles turning the same way. An example of this case is shown in the inset of Fig. 8.5. In both cases, the distribution of speeds of each and every particle was found to be sharply peaked around

|~v| = α/β.

The average size of the vortex remains constant in time as shown in Fig. 8.5 where we have plotted the average density (obtained by averaging over 106 iterations) of the vortex CHAPTER 8 – VORTEX FORMATION 187

Figure 8.4: Snapshots of simulations with N = 200 particles for the parameter values α = 10, β = 1, lc = 0, Ca = 0.4, la = 40, Cr = 1 and lr = 20. As initial condition, the particles are placed at random on a disk with velocities that are constant in magnitude (α/β) but random in direction (a). After an initial transient (b, 20 iterations and c, 50 iterations), a stable rotating vortex state is formed (d, 300 iterations). The bar indicates the attraction length la. The position of each particle is denoted by a solid circle and the velocity as a line starting at the particle and pointing in the direction of the velocity. 188 structure. Fig. 8.5 displays several remarkable features. There is a well-deﬁned core which remains void even for extended simulation runs. Using different parameter values one can produce a vortex without an empty core (in which case the particles near the center have speeds much smaller than α/β). As in the one dimensional ﬂock, the density does not decay smoothly to zero at the edges. Instead, it increases at both the inner and outer edge of the aggregate and then drops abruptly. Qualitatively similar vortex solutions were found when an additional hard-core repulsion – like the one discussed above – is added.

Figure 8.5: Average density of a rotating vortex state in the discrete model (solid symbols) and the continuum model for the parameter values N = 400, m = 1, α = 10, β = 1, lc = 0, Ca = 0.5, la = 30, Cr = 1 and lr = 20. The inset shows a snapshot of the discrete model simulation with the bar corresponding to la. As initial condition we used a vortex obtained with lc = 4 which ensured that the angular velocity of all particles has the same sign. CHAPTER 8 – VORTEX FORMATION 189

8.2 continuum model

Unlike atomistic models of entire bacterial colonies, the atomistic flock model is simple enough for a continuum version of the model to be developed. As in the case of traffic models [102], a continuum version is useful for better understanding the model and for better exploration of the parameters space. To the end of developing a continuum version of the model, we simply coarse-grain average Eqs. (8.1-8.2) (taking all the particles to have the same mass m), which results in ³ ´ ~ mρ∂t~v + ρ ~v · ∇ ~v = αρ fˆ− βρ~v + ρG~ (8.6) where ρ is a particles’ density field, fˆ is a self-propulsive force and G~ is an interaction force. Diving the equation by the common factor ρ and adding a conservation equation for the density field ρ result in the continuum model ³ ´ ~ m∂t~v + ~v · ∇ ~v = α fˆ− β~v + G~ (8.7) ~ ∂tρ + ∇ · (~vρ) = 0 (8.8)

The interaction force is given by

Z ¡ ¢ ¡ ¢ G~ (~x) = ~∇ ρ ~x0 U ~x,~x0 d~x0 (8.9) µ 0 ¶ µ 0 ¶ ¡ 0¢ |~x −~x | |~x −~x | U ~x,~x = Ca exp − −Cr exp − (8.10) la lr and the self-propulsive force direction is given by (with or without velocity averaging over

”neighbors’ velocities”):

fˆ = uˆ(~v) without averaging µZ µ 0 ¶ ¶ ¡ 0¢ ¡ 0¢ |~x −~x | 0 (8.11) fˆi = uˆ ρ ~x ~v ~x exp − d~x with averaging lc 190

A comparison between the discrete model and the continuum model can be carried out for the solutions presented here. For a 1D flock, we have simply fˆ = 1, and the solution in the continuum model is given by G = 0 or Z ¡ ¢ ¡ ¢ ρ x0 U x,x0 dx0 = D (8.12) where D is a constant determining the total number of particles. Since the sought after solution has a finite extent with a discontinuity at the edge where the density drops to zero we discretize the integral using M points and discretization step ∆x. The last point corresponds to the edge of the flock. The resulting linear set of M equations for ρ is easily solved using standard linear algebra packages. The size of the flock ( (M − 1)∆x ) was varied until the slope at the center of the flock vanished. The result, with D chosen R such that ρ(x)dx = N, is shown in Fig. 8.2 as a solid line. The density profile in the continuum model is discontinuous at the edge and agrees well with the profile obtained in the discrete model. Note that since the equations are linear in ρ it is not surprising to find that the density profile of the corresponding atomistic model did not change as the number of particles was increased. The simple coarse-grained averaging procedure is not adequate for the hard-core potential case, where higher order terms in the density are important.

The continuum model can also be used to ﬁnd the empty-core vortex solution. To this end, we use the fact that all particles undergo approximately a circular motion with constant speed (α/β). Thus, a continuum vortex solution can be found by changing to polar coordinates and requiring that the force G~ is centripetal:

Z 2π Z ∞ dφ drρ(r)U (r,φ) = D − (α/β)2 ln(r) (8.13) 0 0 A vortex solution with a full core cannot be found in this method because it deviates too much from the constant speed assumption. After integrating Eq. (8.13) over φ, the remain- CHAPTER 8 – VORTEX FORMATION 191 ing integral was discretized as in the 1D case. The resulting matrix was solved for ρ(r) and used in a Newton solver that searched for the size of the hole and the overall size of the vortex (i.e. discretization ∆x) with a condition for smoothness of the solution at both discontinuous edges. This condition simply consisted of requiring that the first and last point can be obtained by linear interpolation from its two neighboring points. We have checked that the solutions we obtained are converged by increasing the number of discretization points. In Fig. 8.5 we compare the discrete solution to the one obtained by Eq. (8.13) R where the integration constant D was varied until ρ(~x)d~x = N. Again, the continuum profile is discontinuous at the edges and the agreement between the continuum profile and the discrete profile is very good.

The continuum equation can be used to explore the (large) parameter space more efﬁ- ciently. An example of such an exploration is shown in Fig. 8.6 where we plot two different solutions found by our Newton solver for the same model parameters and total number of particles (N = 200) but with different integration constant D.

Figure 8.6: Two different solutions of the continuum model for the parameters α/β = 10, Ca = 0.7, la = 40, Cr = 2, lr = 20 and N = 200.

Chapter 9

Mutants Spread in Colonies

9.1 Transitions and sectors formation in bacterial colonies

There is a well-known observed (but rarely studied) phenomenon of bursts of new sectors of

”mutants” during the growth of bacterial colonies (see for example Fig. 9.1 and Refs. [201, 199]). The phenomenon of bursts of new sectors transcends beyond bacterial colonies. Fig. 9.2 (taken from [46]) shows an emerging sector in a yeast colony. Sectors are a form of expression of different growth mode in an expanding colony. The different growth mode is an inheritable change in phenotype. The change might be due to genetic change (as is the usual meaning of ‘mutation’) or due to epi-genetic changes, such as inheritable changes in pattern of gene expression. We will refer to any inheritable change as a ‘mutation’, regardless of change in genetic sequence.

A mutation is not necessarily expressed in a sector burst. If the mutants have the same growth dynamics as the ”normal”, wild-type, bacteria then they will usually go unnoticed.

They might have some property such as coloring that distinguish them (different coloring may result from different enzymatic activity – natural coloring – or from a different re-

193 194 sponse to a staining process – artiﬁcial coloring). In all these cases the mutation is not neutral in the strictest sense, but we will refer to it as neutral because it is neutral as far as the dynamics is concerned. If, however, the mutants have different growth dynamics, a distinguished sector with a different growth pattern might indicate their presence.

Figure 9.1: Emerging sector in a Escherichia coli colony. Picture by James A. Shapiro, from “Bacteria as Multicellular Organisms” edited by J. Shapiro and M. Dworkin [200]. (C) Copyright 1997 by Oxford University Press, Inc. Used by permission of Oxford Uni- versity Press.

In a branching colony, the geometrical structure may aid the bursting of a sector of ”neutral” mutants; once a branch (or a cluster of branches) is effectively detached from his neighboring branches (detached in the sense that bacteria cannot cross from one branch to the other), the effective population is smaller than the colony’s population. In such a ”reduced” population, genetic drift1 is more probable and a neutral mutant may take over the population in some branches. Sectors of ”neutral” mutations usually go undetected – by

1Genetic drift is an evolutionary process which changes the fraction of the population which carry a certain neutral mutation. The population fraction is changing due to the stochastic nature of the reproduction process, and not due to some evolutionary pressure. In general, the bigger the population is, the slower the genetic drift is. CHAPTER 9 – MUTANTS SPREADIN COLONIES 195

Figure 9.2: Emerging sector in a colony of Yarrowia lipolytica. Taken from [46], used with permission.

our deﬁnition their growth dynamics is identical to that of the wild-type (original) bacteria and no geometrical feature highlights the sectors.

Sectors of advantageous mutation are much easier to detect, as they usually grow in a somewhat different pattern. An advantage in this context might be faster multiplication, higher motility or elevated sensitivity to chemotactic materials. In all those cases the mutants have an advantage in accessing or utilizing food resources. In a pre-set geometry (or without spatial structure) advantageous mutants might drive the wild-type bacteria to extinction by depriving them of food. But in a spreading colony each part of the colony is heading in a different direction, reaching for food in different locations, thus the different populations can co-exist. The dynamic process of spreading aids the segregation of the populations.

The first analytical study of spatial spread of mutations was done by Fisher [82]. He studied the spread of advantageous mutation in the limit of large, spatially uniform popu- 196 lation, using the Fisher-Kolmogorov equation. This equation describes the time evolution of a field representing the fraction of the mutants in the local population. The same equation can be taken to describe the spreading of a population into an uninhabited space, in which case the field represents the density of the bacteria (Sec. 3.1.2). To study mutants by this description one must extend the model to include two fields standing for two different types of bacteria. Since these equations are expressed in the continuous limit, it excludes a-priori the effect of genetic drift. As we discuss in Sec. 3.1.2, the Fisher equation has other shortcomings that make it unsuitable for modeling our bacterial colonies.

9.1.1 Observations of colonial development

In this section, we focus on the phenomena observed during the growth of P. dendritiformis var. dendron , or T morphotype (see Sec. 2.2).

9.1.1.1 Growth features

As is shown is Sec. 2, the typical growth pattern on semi-solid agar is a branching pattern, (e.g. Fig. 2.2). The structure of the branching pattern varies for different growth conditions. The various colonial patterns can be grouped into several “essential patterns” or morphologies [35, 37]. In order to explain the various growth morphologies, Ben-Jacob et al. have suggested that bacteria use chemotactic signaling when confronted with adverse growth conditions [35, 60, 31]. Chemotaxis means changes in the movement of the cell in response to a gradient of certain chemical fields [2, 42, 133, 40]. The movement is biased along the gradient either in the gradient direction (attractive chemotaxis, for example towards food) or in the opposite direction (repulsive chemotaxis, for example away from oxidative toxins). Usually chemotactic response means a response to an externally pro- CHAPTER 9 – MUTANTS SPREADIN COLONIES 197 duced field as in the case of chemotaxis towards food. However, the chemotactic response can be also to a field produced directly or indirectly by the bacterial cells. We refer to this case as chemotactic signaling.

At very low agar concentrations, 0.5% and below, the colonies exhibit compact patterns, as shown in Fig. 9.3. Microscopic observations reveal that in this case, the bacteria swim within the agar. Thus, there is no “envelope” to the colony, and hence no branching pattern emerges (see Sec. 3.1 and also [25]).

Figure 9.3: A compact growth pattern of P. dendritiformis var. dendron , obtained when the agar surface is very soft (0.4% agar concentration and 0.1g/l peptone).

9.1.1.2 Observations of sectors

The bursting of sectors can be observed both during compact and branching growth. Exam- ples of the ﬁrst kind are shown in Fig. 9.4, and examples of the latter are shown in Figs. 9.5 and 9.6. As we can see from the pictures, sectors emerging during branching growth have a greater variety of structure and shapes than those emerging from compact colonies. This is demonstrated by Fig. 9.5 depicting colonies at intermediate levels of nutrients and agar, 198 and Fig. 9.6, showing colonies grown at high nutrient level and in the presence of antibiotics. Note that the sector on the left side of Fig. 9.6 is much more expanded than that on the right, probably because it has irrupted at an earlier stage of the colonial development.

Such an early-irrupted sector might indicate a mixed population in the initial inoculum, and not a new mutant.

Figure 9.4: Emerging sectors in compact colonies of P. dendritiformis. (left) var. dendron, 10g/l peptone, 0.5% agar. (middle) var. chiralis, 1.5g/l peptone, 0.4% agar. (right) var. chiralis, 10g/l peptone, 0.4% agar.

a b

Figure 9.5: Emerging sectors in branching colonies of P. dendritiformis var. dendron , obtained at 1g/l peptone and 1.75% agar (a), 0.8g/l peptone and 2% agar (b). CHAPTER 9 – MUTANTS SPREADIN COLONIES 199

a b

Figure 9.6: Emerging sectors in branching colonies of P. dendritiformis var. dendron , obtained at 5g/l peptone and 1.75% agar, in the presence of the antibiotics Stromocine. The difference between the two bursting sectors is presumably because in (a) the mutation occurred at an earlier stage of the colonial development

Throughout this chapter, we use the term “mutant” to describe the population in the emerging sectors. We have not veriﬁed the existence of a genetic difference between the bacteria in the sector and those in the rest of the colony. We have, however, veriﬁed that the phenotypic difference between the two populations is inheritable, using inoculation.

Below we shall see that it is sometimes possible to relate the shape of the sector, and the way it “bursts” out of the colony, with the speciﬁc kind of advantage that the mutants possess over the original bacteria.

9.2 Modeling mutants and mutations

In this study we use many of the models presented in previous chapters, altered to properly accommodate two bacterial types. In some cases (but not all) the two bacterial types have 200 different growth dynamics. We begin each run with a uniform population. The event of mutation is included with some ﬁnite probability of the wild-type strain changing into a mutant during the process of multiplication.

Representing mutations in the models gives rise to possible problems both in discrete and in continuous models. In a continuous model there is a problem representing a single mutation because the equations deal with bacterial area density, not with individual bacterium. In Ref. [61] we have studied the inclusion of ﬁnite size effects in the continuous model via a cutoff in the dynamics. For the study of mutations, we use as our basic “mutation unit” the cutoff density (see below). The value of this density is in the order of a single bacterium in an area determined by the relevant diffusion length (the idea of using a cutoff density to represent discrete entities was ﬁrst raised by Kessler and Levine [120]). In our discrete models there is seemingly a natural unit for mutation – e.g. a single walker in the Communication Walkers model (Chapter 7). However, each “walker” represents not one bacterium, but many bacteria (103 − 106). Thus, a mutation of one walker means the collective mutation of all the bacteria it represents. We therefore must extend the model to enable the representation of a single bacterium to be mutated.

It is not necessary that only the descendants of a single cell (the mutant) will possess the mutation; bacteria have the means [89, 187, 167] to perform ”horizontal inheritance” i.e. to transfer genetic information from cell to cell. If such autocatalytic or cooperative mutation occurs in the experiments, then a mutating walker in the Communicating Walkers model might be an accurate modeling after all. We will not explore this possibility further. CHAPTER 9 – MUTANTS SPREADIN COLONIES 201

9.2.1 Single cell densities in the Non-Linear Diffusion model

Following Sec. 4.1.2.2, we adapt here the Non-Linear Diffusion (NLD) model. The model includes a linear growth term and a non-linear diffusion of the bacteria (representing the effect of a lubricating ﬂuid [59, 128]). In the case of a single strain, the time evolution of the 2D bacterial density b(~x,t) is given by:

∂b ³ ´ = ∇ · D bk∇b + εnb − µb (9.1) ∂t b

The ﬁrst term on the RHS describes density-dependent bacterial movement (Db and k are both positive constants), where the speed of bacterial movement is a function of the local density. The second term on the RHS is a population growth term, which is proportional to food consumption. n(~x,t) is the 2D nutrient density and ε is the nutrient to bacteria conversion factor. The third term on the RHS is a “death” (or sporulation) term, with µ > 0 being the sporulation rate.

We can now check the effect of bacterial discreteness on the observed colonial patterns. Following Kessler and Levine [120] (see Sec. 4.1.1.1), we introduce the discreteness of the system into the continuous model by repressing the growth term at low bacterial densities

(“half a bacterium cannot reproduce”). The growth term is multiplied by a Heaviside step function Θ(b − β), where β is the threshold density for growth. In this version of the NLD model, the time development equations for the bacterial density b(~x,t), nutrient concentration n(~x,t) (after rescaling to have the same units as the bacterial ﬁeld) and the density of stationary bacteria s(~x,t) are given by:

∂b ³ ´ = D ∇ · bk∇b + εnbΘ(b − β) − µb (9.2) ∂t b ∂n = D ∇2n − εbnΘ(b − β) (9.3) ∂t n 202

∂s = µb (9.4) ∂t

In Fig. 9.7 we show the effect of various values of β on the pattern. High cutoff values make the model more sensitive to the implicit anisotropy of the underlying tridiagonal lattice used in the simulation. The result is dendritic growth with marked 6-fold symmetry of the pattern. Increased values of cutoff also decrease the maximal values of b reached in the simulations, and the total area occupied by the colony.

Figure 9.7: The effect of a cutoff correction on growth patterns of the NLD model. On the left, a run without cutoff (β = 0). On the right, a run with a cutoff (β = 0.1). All other parameters are identical in both cases: Db = 0.1, k = 1, ε = 1, µ = 0.15, and initial nutrient concentration n0 = 2.0 . The apparent 6-fold symmetry is due to the underlying tridiagonal lattice. The different gray levels represent different densities of bacteria, both active and stationary, i.e. different values of the combined ﬁeld b + s.

The reason for the pattern turning dendritic is as follows [61]: the difference between tip-splitting growth and dendritic growth is the relative strength of the effect of anisotropy and an effective surface tension [17]. In the NLD model there is no explicit anisotropy and no explicit surface tension (Sec. ??). The implicit anisotropy is related to the underlying CHAPTER 9 – MUTANTS SPREADIN COLONIES 203 lattice, and the effective surface tension is related to the width of the front. The cutoff prevents the growth at the outer parts of the front, thus making it thinner, reduces the effective surface tension and enables the implicit anisotropy to express itself.

We stress that it is possible to ﬁnd a range of parameters in which the growth patterns resemble the bacterial patterns, in spite a high value of cutoff. The effect of the step function is negligible for small β [61], and our main motivation for inclusion of the cutoff is its use in the modeling of mutations.

As in Sec. 4.2, we include the effect of chemotaxis in the model using a chemotactic

ﬂux J~chem, which is written (for the case of a chemorepellent) as:

J~chem = ζ(b)χ(r)∇r (9.5) where r (~x,t) is the concentration of the chemorepellent agent, ζ(b) = b·D(b) ∼ bk+1 is the bacterial response to the chemotactic agent [93], and χr (r) is the chemotactic sensitivity to the repellant, which is negative for a chemorepellent. In the case of a chemoattractant, e.g. a nutrient, χn() has an opposite sign. The sensitivity χr (r) may be a constant, or it may take other functional forms. In the case of the “receptor law”, χ(r) takes the form [171]: χ K χ(r) = 0 (9.6) (K + r)2 with K and χ0 constants.

The equation for r (~x,t) is: ∂r = D ∇2r + Γ s − Ω br − λ r (9.7) ∂t r r r r where Dr is the diffusion coefﬁcient of the chemorepellent agent, Γr is the emission rate of repellant by sporulating cells, Ωr is the decomposition rate of the repellant by active bacteria, and λr is the rate of spontaneous decomposition of the repellant. 204

9.2.1.1 Mutations in the Non-Linear Diffusion model

In order to generalize the model for the of study mutants, we must introduce two ﬁelds, for the densities of the wild-type bacteria (“type 1”) and the mutants (“type 2”), and allow some kind of transition from wild-type to mutants. With chemotaxis included (see Sec. 4.2), the equations for the bacterial density of the two strains are written (with subscript denoting bacterial type): ∂b 1 = ∇ · {D (b)[∇b + b χ (r)∇r]} + ε nb Θ(b − β) − µ b − F (9.8) ∂t 1 1 1 1 1 1 1 1 1 12 ∂b 2 = ∇ · {D (b)[∇b + b χ (r)∇r]} + ε nb Θ(b − β) − µ b + F (9.9) ∂t 2 2 2 2 2 2 2 2 2 12 k where b is deﬁned as b = b1 + b2 and D1,2 (b) = Db1,2 b . We allow the strains to differ in their motility (Dbi), their rate of reproduction (εi), their rate of sporulation (µi) and their response to chemotaxis χi (r).

Note that the mutant strain b2 includes a “source” term F12, which is the rate of transition b1 → b2, and is given by the growth rate of b1 multiplied by a constant mutation rate

(For simplicity, we do not include the process of reverse mutations F21). Like the growth term, the mutation term is also multiplied by the cutoff term Θ(b1 − β), making sure that a mutation can occur in a speciﬁc location only if there are bacteria there.

The full NLD model with food chemotaxis and repulsive chemotactic signaling, adapted to two different types of bacteria, is ∂b n o 1 = ∇ · D bk [∇b − b χ (n)∇n + b χ (r)∇r] (9.10) ∂t b1 1 1 n,1 1 r,1

+ε1nb1Θ(b1 − β) − µ1b1 − F12Θ(b1 − β) (9.11) ∂b n o 2 = ∇ · D bk [∇b − b χ (n)∇n + b χ (r)∇r] (9.12) ∂t b2 2 2 n,2 2 r,2

+ε2nb2Θ(b2 − β) − µ2b2 + F12Θ(b1 − β) (9.13) CHAPTER 9 – MUTANTS SPREADIN COLONIES 205

∂n = D ∇2n − n[ε b Θ(b − β) + ε b Θ(b − β)] (9.14) ∂t n 1 1 1 2 2 2 ∂s = µ b + µ b (9.15) ∂t 1 1 2 2 ∂r = D ∇2r + Γ s − Ω br − λ r (9.16) ∂t r r r r

We study this model in later sections.

9.2.2 Variable group size in the Communication Walkers model

To complement the continuous NLD model we use an atomistic model. We chose to use the Communication Walkers (CW) model (Sec. 7.1) and not Communication Spinors (CS) model (Sec. 7.2) as the additional complexity of the CS might obscure the results of this study. A mutation is usually associated with the change of a single cell. In order to study single-cell events in the model, one must consider groups (walkers/spinors) of size one.

As we said (Sec. 7), it is not practical to model each and every cell in a colony as an individual unit (walker). Therefore we must model the majority of cells in groups (a walker represents a group of bacteria). In order to include single-cell events in a colony, i.e. mutation in a single bacterium, we must generalize the CW model such that we can include in a single run walkers that represents groups of bacteria of different sizes. Most of the changes include linear dependency on the group size, but there are some subtleties to consider.

Each of the walkers has a position ~ri , a metabolic state (‘internal energy’) Hi and a mass Mi which is proportional to the number of bacteria in the group it represents. The rate of change of the internal energy (see Sec. 7.1) scales as the mass of the walker: µ ¶ dHi Em = κnconsumed − Mi (9.17) dt τR 206

where κ is a conversion factor from food to internal energy and Em represents the total energy loss for all processes (excluding reproduction) over the minimal time of reproduction

τR. The food consumption rate nconsumed is ¡ ¢ Ω Ω0 nconsumed ≡ min n, n (9.18)

Ω Ω0 where n is the maximum rate of food consumption, and n is the rate of food consumption as limited by the locally available food. When a walker is “starved” for a long interval of time and Hi drops to zero, the walker freezes.

As in the original CW model, Hi has a threshold for reproduction. The threshold should also scales like Mi and it is EdMi – when sufﬁcient food is available, bacterial division time is independent their number. Upon reaching the threshold, the walker can either multiply into two walkers or double its mass Mi → 2Mi. The choice between those two options is based on the following conditions:

• The number of walkers in our simulations is subject to a stochastic process, where walkers walk randomly in and out areas where they can starve or multiple. The

probability of extinction due to stochastic ﬂuctuations (and not due to too harsh conditions) is related to the number of walkers (or individuals in general). Thus few

walkers, which represent large number of bacteria, have much higher probability of stochastic extinction than the equivalent population of bacteria. To reduce the chances of such stochastic extinction we try to avoid situations of a small number of

walkers. A walker divides into two (and not double in size) if it reaches the reproduction threshold and there are less then 50 walkers of its type in the simulation.

• When there are only few walkers at a tip of a branch, a movement of the biggest of them represents coherent movement of large fraction of the bacteria at a branch CHAPTER 9 – MUTANTS SPREADIN COLONIES 207

tip. Since such coherent movement is unrealistic, we limit the mass of the walkers.

A walker multiple into two (and not double in size) if it reaches the reproduction

threshold and its mass Mi is more than half of a maximal allowed mass Mmax

• If a walker reaches its reproduction threshold and neither of the above two conditions is met, the walker has 1/3 probability to multiply and 2/3 probability to double in

size.

As in the CW model, we represent the diffusion of nutrients by solving the diffusion equation for a single agent whose concentration is denoted by n(~x,t):

∂n = D ∇2n − σ n , (9.19) ∂t n a consumed where the last term includes the consumption of food by the active walkers whose density is denoted σa. The contribution of the i-th walker to σa is proportional to Mi: σa ≡ ∑ δ(~x −~r ) Mi . The diffusion equation is solved on the triangular lattice i∈{active walkers} i Mmax using zero-ﬂux boundary conditions.

The walkers perform an off-lattice random walk within an envelope representing the boundaries of the lubrication ﬂuid. This envelope is deﬁned on the same triangular lattice where the diffusion equations are solved. To incorporate the swimming of the cells into the model, at each time step each of the active walkers attempts to move from its location~ri a

0 step d at a random angle Θ, to a new location ~r i given by:

0 ~r i =~ri + d(cosΘ;sinΘ) . (9.20)

As in the original CW model (Sec. 7.1.1), the units of d are not length units; it’s units are £ √ ¤ length/ time , square root of the units of a diffusion coefﬁcient. These units compensate 208 for fact that the number of steps of a walker per time unit is sensitive to the time-step of the model’s simulation. The ensure that the effective diffusion coefﬁcient of the walkers in the bulk of the colony is not sensitive to the time-step of the model’s simulation.

0 If the new location ~r i is outside the boundary, the walker does not perform the step and a counter on the appropriate segment of the envelope is increased. The increment is proportional to a presumed force exercised by the walker:

F (Mi) = f1v(Mi) (9.21)

f1 is the “force” exercised by a single bacterium (force per bacterial mass), and v(Mi) scales the force to the walker’s mass. We found that it is not satisfactory to use a constant force per bacterium (i.e. v(Mi) = Mi); in such model many small walkers do not affect the envelope as few big walkers do. Big walkers represent coherent movement of groups of bacteria and reduce the noise (variance) of the bacterial effect on the envelope. Instead of taking v proportional to Mi, we take it to be a random variable whose value is chosen from ³ q ´ a Gaussian distribution N Mi ,e Mi with mean Mi and variance e2 Mi . m is the mass m1 0 m1 m1 0 m1 1 of a single bacterium (so that MI is the number of bacteria represented by the walker), and m1 e0 is a measure of the inherit ﬂuctuations in the movement of a single bacterium. Unless otherwise stated, we take f = 1 , m = M vs. e = 0.1. 1 Mmax 1 min 0

When a segment counter on the envelope reaches a threshold Nc, the envelope advances one lattice step and a new lattice cell is occupied. Nc is measured in units of length to the power of −2 (or −D, where D is the spatial dimension of the simulation) - the number of hits per boundary length unit which advance the boundary one length unit. Nc is related to the agar dryness, as more wetting ﬂuid must be produced (more “collisions” are needed) to push the envelope on a harder substrate. In the special case f = 1 and e = 0, the 1 Mmax 0 CHAPTER 9 – MUTANTS SPREADIN COLONIES 209 summation method reduces to the one used by the original CW model. Figure ?? shows the effect of varying the group mass Mi with e0 = 0 and e0 = 0.1. The ﬁgure demonstrates the inadequacy of the summation method of the original CW model for the case of variable group mass.

Chemotaxis is introduced into the model by biasing the angle of movement (see Sec.

7.1.4). In the presence of a chemotactic agent (e.g. a repulsive agent R), we do not use Eq. (9.20). Instead, we calculate the ith walker’s new position as

Θ Θ χ ∇ 0 (cos ;sin ) − R R(~ri) ~r i =~ri + d (9.22) |(cosΘ;sinΘ) − χR∇R(~ri)| where Θ is a random angle (selected uniformly from the range [0,2π)), and χR is a chemotaxis coefﬁcient which represents the magnitude of the response of the bacteria to chemotaxis. The chemotaxis coefﬁcient is positive for attractive chemotaxis (such as food chemotaxis) and negative for repulsive chemotaxis. The response towards other chemotactic agents, and combinations of chemotactic responses, can be incorporated using similar expressions

9.2.2.1 Mutations in the Communication Walkers model

In order to generalize the model for the of study mutants, we must introduce two types of walkers, representing wild-type bacteria (“type 1”) and mutants (“type 2”). The two types

j j can differ in their motility d ( j = 1,2), their rate of food consumption Ωn , the amount of j χ j energy loss Em and their response to chemotaxis R.

In the bacterial colonies, any specific mutation is very rare (occurs once in many colonies, each of which includes 1010 reproduction events). It is not practical to simulate the actual rate of mutations. Instead, we chose to impose a mutation event at a pre-specified 210 stage in the colony’s development: when the simulated colony reaches some specific radius, one (and only one) of the reproduction events is randomly chosen to be a mutation event. The mutation event occurs when a walker of type 1 divides into three walkers. First, a walker of mass Mmin is split from the “mother” walker. This new small walker converted to type 2. It represents a small group of bacteria (possibly a group of size 1) that mutates from type 1 to type 2. The remaining of the energy of the “mother” walker is used for its division, as in a normal reproduction event.

9.3 Modeling formation of sectors

In order to study the formation of sectors, we run simulations of the modiﬁed Non-Linear Diffusion (NLD) model and modiﬁed Communicating Walkers (CW) model. Mutations are included in the simulations as described in Secs. 9.2.1.1 and 9.2.2.1, and we study the development of the colonies.

9.3.1 Mutation during compact growth

We start with simulations of sector formation of neutral mutations. One could expect that such mutations will not form a segregated sector. The simulations show, however, that many times the mutants can increase their relative part of the total population in a sector of the colony. A change in fraction of the mutants without any practical (“evolutionary”) difference can result only from a genetic drift, i.e. from random fluctuations in number. Due to the expansion of the colony, parts of the colony are effectively separated from each other: for any two points on the colony’s outer boundary, the distance between them grow until a walker cannot visit the two points within is lifetime. Thus, a mutant is exposed to CHAPTER 9 – MUTANTS SPREADIN COLONIES 211 an “effective population” which is much smaller than the entire colonial population. The genetic drift in a smaller population is stronger and there is a higher chance that a neutral mutant will “take over” a part of an expanding colony, compared to a static colony. In other words, due to the expansion of the colony, an initially tiny number of mutants can gradually become a significant part of the total population in a specific area. Experimentally, this means that if the mutant has some distinguishable feature – e.g. color – a sector will be observed.

Next we study the more interesting case of superior mutants. In this case one observes a sector that grows faster than the rest of the colony. In the simulations of the modiﬁed NLD model, a sharp segregation is obtained when the mutant is endowed with a higher growth rate ε or a higher motility D0. Figure 9.8 demonstrate this result, and also depicts the results obtained by simulations of the modiﬁed CW model. As can be seen, both models exhibit a fan-like sector of mutants, very similar to the one observed in the experiments.

The “mixing area”, where both strains are present, is narrow, and its width is related to the width of the propagating front of the colony (the region where the bacteria are active).

9.3.2 Mutation during Branching growth

We now turn to the case of sectoring during branching growth. In this case the models can show a slow process of segregation even for a neutral mutation. This results from the fact that a particular branch may stem from a small number of bacteria, thus allowing an initially insigniﬁcant number of mutants quickly to become the majority in some branches, and therefore in some area of the colony (genetic drift in small populations).

Mutants superior in motility or growth rate (Fig. 9.9) form segregated fan-like sectors, 212

Figure 9.8: Mutants in a compact colony: Results of numerical simulation. (a) Mutant with a higher growth rate in NLD model. The mutant (dark) erupts in a fan-like sector from ε the wild-type (light) colony. Model parameters for the wild-type: D01 = 0.2,k = 0, 1 = 1.0,β = 0.0001,µ1 = 0.1, mutant has ε2 = 5.0. Initial nutrient level n0 = 1.0 (b) Mutant with a higher motility in NLD model. wild-type as in (a), mutant has D02 = 0.3. (c) Mutant with a higher motility (larger step size in the Communicating Walker model). which burst out of relatively slow advancing colony.

9.3.3 The effect of chemotaxis

As we have said, an important feature of the bacterial movement is chemotaxis. We start by studying neutral mutations in the case of a colony that employs repulsive chemotactic signaling (Chemotactic signaling is a chemotactic response to an agent emitted by the bacteria. See Refs. [21, 60, 20, 18, 24] and Sec. 7.1.4.1). The chemotactic response en- hances the segregation of sectors neutral mutants. This results from branches being thinner in the presence of repulsive chemotaxis, and the reduced mixing of bacteria because of the directed motion.

In the case of mutants with superior motility, a segregated sector is formed which is not fan-like (as opposed to the case without chemotaxis), probably because of the biased, radially-oriented motion of the bacteria, coming from the long range repulsive chemotaxis. CHAPTER 9 – MUTANTS SPREADIN COLONIES 213

Figure 9.9: Mutant with a higher growth rate in a branching colony: Numerical simulation of NLD model. The mutant (dark) erupts in a fan-like sector from the wild-type (light) ε β colony. Model parameters for the wild-type: D01 = 0.1,k = 1, 1 = 1.0, = 0.0001,µ1 = 0.15, mutant has ε2 = 5.0. Initial nutrient level n0 = 1.0

A fan-like sector does appear when the mutant has a higher sensitivity to the chemotac- χ2 χ1 tic signals ( r > r , both of which are constant here) see Fig. 9.10. In this case, however, the sector is composed of a mixture of the “wild-type” and the mutants. Figure 9.11 displays the result of Communicating Walker simulation for this case. Note the similarly of both models’ results with experimental observations (Figs. 9.6b and 9.5b respectively).

Figure 9.10: Mutant with a higher sensitivity to repulsive chemotactic signaling, in a branching colony: Numerical simulation of the modiﬁed NLD model. The mutant erupts in a fan-like sector from the colony of wild-type bacteria. However, the sector is not a segregated area, and contains wild-type bacteria as well. Model parameters for the wild-type: ε β χ χ D01 = 0.1,k = 1, 1 = 1.0, = 0.0001,µ1 = 0.2, r1 = 0.5, mutant has r2 = 1.0. Initial nutrient level n0 = 1.0 214

Figure 9.11: Mutant with a higher sensitivity to repulsive chemotactic signaling, in a branching colony: Results of the modiﬁed CW model. As in the NLD model, the mutant erupts in a fan-like sector from the colony of wild-type bacteria.

The inﬂuence of chemotaxis towards food on the sectors is similar to that of repulsive chemotaxis (not shown).

9.4 Mutants Spread in Colonies – Conclusions

In this chapter we presented our study of the appearance of segregated sectors of mutants in expanding bacterial colonies. After reviewing the experimental observations of this phenomenon, we showed the results of simulations performed using two different models, a modiﬁed version of the discrete “Communicating Walker” model, and a modiﬁed version of the continuous Non-Linear Diffusion model. Using these models as an aid to analytical reasoning, we are able to understand what factors – geometrical, regulatory and others – favor the segregation of the mutant population. These factors include:

1. Expansion of the colony – in the form of a ﬁnite front propagating away from areas of

depleted nutrient, so that a mutant is exposed to a population which is much smaller than the colony’s population. CHAPTER 9 – MUTANTS SPREADIN COLONIES 215

2. Branching patterns, where the population in each branch is much smaller than the

colony’s population, making a genetic drift more probable, so that a mutant can take over the whole population in a sector of the colony.

3. Chemotaxis: Food chemotaxis and repulsive chemotactic signaling cause the bacterial motion to become less random and more directed (outward and towards nutrients), thus lowering mixing of populations.

4. An advantageous mutant, having e.g. a higher motility or a faster reproduction rate, will probably conquer a sector of its own and quickly become segregated. This sector

will usually be fan-like, bursting out of the colony, owing to the faster expansion of the mutants, as compared to that of the wild-type population.

The observed segregation of mutant population raises some interesting evolutionary questions. Faster movement (for example) is an advantage for the bacteria, as the burst of sectors show. Why then does this mutation not take over the general population and becomes the wild-type? In other words, why were there any of the current wild-type bacteria for us to isolate in the ﬁrst place? One possible answer is that this advantage might turn out to be a disadvantage at different environmental conditions (e.g. inability to remain con-

ﬁned to some small toxin-free oasis). Another possibility is that this that advantages of this mutation are in a trade-off with some disadvantages, thus hampering its long-term survival

(e.g. wasting too much energy on movement when it is not advantageous).

Beyond the study of sectoring in bacterial colonies, intuition about the basic mechanisms of spatial segregation of populations might be useful for other problems. Such problems may include the important issues of growth of tumors and the diversiﬁcation of 216 populations on a macroscopic scale. Both may employ similar geometrical features and communication capabilities, leading to segregation.

Notwithstanding the above-mentioned reservations, we believe this study demonstrates once more the capability of generic models to serve as a theoretical research tool, not only to study the basic patterns created by bacterial colonies, but also to gain deeper understanding of more general phenomena, such as the segregation of mutants in an expanding colony. Chapter 10

Discussion

I have demonstrated in this work how mathematical models and their construction can be used as a theoretical tool for understanding a biological system – bacterial colonies. Patterns formation in bacterial colonies is shown to be a good study subject. It is simple enough to allow progress, yet it is also well motivated by the signiﬁcance of the results.

In the research presented here, I studied three types of colonial patterns. Each type of pattern is formed by a different type of bacteria, and each one of them received a mathematical treatment suited to its unique characteristics.

The bacteria P. dendritiformis var. dendron (T morphotype) creates colonies with tip- splitting branching patterns. The colonial patterns are branched, and new branches are formed during development as wide, old branches split into two. The bacteria in the colony perform two-dimensional random walk within the boundaries of the colony. I used the Lu- bricating Bacteria model (LB model) and its simplified version, the Non-Linear Diffusion (NLD) model to serve as continuum models for the bacterial colonies (chapter 4). Using these models, I showed that a continuum model of the bacterial density field is suited for describing the colonial growth if it has a diffusion term with a diffusion coefficient that

217 218 depends on the bacterial density field. The diffusion coefficient describes a bacterial motion which is limited by a lubricating material that is created by the bacteria themselves. In such a description, the diffusion coefficient should decay to zero where the bacterial density decays to zero. The boundary of a colony is a singularity line of a weak solution to the colonies’ system of equations.

The continuum models for colonies of T bacteria is matched by the Communicating Spinors model (chapter 7), which is an extension of the Communicating Walkers model

I presented in my M.Sc. thesis. In this model, the bacteria are represented by discrete entities, the spinors, which move within a boundary they push. Using this model I explain additional aspects of the bacterial colonies, aspects that affect the colonial growth in very limiting conditions. I show that in conditions of low nutrients, the bacteria use repulsive chemotactic signaling in order to arrange the colonies. The starved bacteria emit chemicals that repel other bacteria, thus driving them away from the nutrient-depleted areas to look for food sources.

The same model, the Communicating Spinors model with different parameters values, is used for explaining the colonial growth of different bacteria, the P. dendritiformis var. chiralis bacteria (C morphotype) (section 7.2). These bacteria also create colonies with branching patterns, with thin chiral branches. The branches are narrow, elongated, and twist sideways as they grow (always in the same direction, in all parts of the colony). New branches break from sides of branches of old ones. The bacteria in the colony perform quasi one-dimensional random walk in the branches, as the branches are too narrow for the bacteria to rotate in.

The C bacteria are closely related to the T bacteria (they are classified as variants of the CHAPTER 10 – DISCUSSION 219 same species), but they differ in their length. C bacteria are much longer than T bacteria. I used the Communicating Spinors model, and two continuum models, the Orientation Dimension model and its simplified version, the Mean Orientation Field model (chapter 6) to study C bacteria. I show that this single difference in length is sufficient to explain the big difference in the colonial patterns. Long bacteria can match large contact between them if they are aligned side by side. When they are aligned, correlation in orientation tends to be maintained, as they limit each other rotation. The models show that sustention of orientation correlation is the leading factor in the creation of long, narrow branches with new branches coming out as side branches. The twisting of the branches is added as an extra element to the models, and it is thought to be related to the propeller-like action of the bacterial engine, and its interaction with the surface of the substrate.

The models for the C bacteria also give an additional support to the claim that the T bacteria respond the signaling chemotaxis. In some conditions where nutrient is scarce, the T bacteria become somewhat more elongate than they usually are. The branches of the colony become narrower and somewhat twisted to the side. This phenomenon is referred to as weak chirality. The models for the C bacteria, adapted for elongated T bacteria, show that the chirality is due to bacterial response to signaling chemotaxis.

A third type of bacteria that was studied in this work is P. vortex (V morphotype). These bacteria do not perform random walk, they always move forward. They arrange themselves into rotating vortices. The difference in motion dictates different models. In a discrete model, the difference in motion is introduces in a straight forward manner. The difference in motion dictates different continuum models. The equation for the bacterial density ﬁeld does not include diffusion term (which is used to describe the random walk of other bacteria). It does include a self-propulsion term to describe the bacterial motion, and 220 a coupling term between velocities of neighboring bacteria. The coupling term represents a contact interaction between neighboring bacteria. The models show that the exact form of the coupling is not important.

Communication and cooperation in bacterial colonies

The models and their analysis reveal several modes of communication and cooperation in the bacterial colonies. All the bacteria studied here produce lubricating material in order to facilitate their motion on the hard surface. A bacterium cannot produce enough material for its own movement, and cooperation is needed. This is cooperation in distance of the order of 10−4m (width of the edge of a branch, where lubrication production is expressed in large slopes of the lubrication layer). T and C bacteria have means for long range communication and cooperation – chemotactic signaling. This mean of chemical communication operates in a range of 10−2m meters – communication between seemingly unrelated parts of a colony. C bacteria have another mean of cooperation – contact interactions, which align them and facilitate faster growth of branches (all the bacteria “push” only a small tip of a branch). This cooperation operates in a distance range of 10−5m - the width of a bacterium, but its effects are visible in the range of 10−3m - the structure of the branches. ’Communication’ of seemingly the same type - contact interactions - governs the dynamics of colonies of V bacteria. These interactions operates in the order of 10−5m, but its effects are most conspicuous in the order of 10−3m - size of a vortex, and also in the order of

10−1m - the entire colony.

We can see here different types of cooperation operating over 5 orders of spatial mag- nitud , where at least three types of cooperation can operate in the same colony. Chapter 9 shows yet another feedback loop in bacterial colonies. Different patterns of different CHAPTER 10 – DISCUSSION 221 colonies impose different modes of selection of mutants in the colonies. Branched colonies require a mutation to be established in a much smaller population than compact colonies require. The same mutation can be unnoticeable in some colonies, and can slowly or quickly form a sector in other colonies – all depending on its effect on the colonial growth in spe- ciﬁc conditions.

The research presented here can be viewed as a part of an on-going large scale effort, aimed at exposing interaction mechanisms in biological systems. How many interaction levels and feedback loops can there be in a complex biological system such as bacterial colony? It seems almost unlimited:

• Chemical commutation mechanisms (e.g. chemotactic signaling) can be switched on and off in different conditions (in [26, 22], we have shown a previously unknown

chemotactic signaling mechanism, used by E. coli bacteria to change a very hostile environment of oxidative stress).

• Effectiveness of different sensors (which are different proteins) is affected by external conditions such as acidity. The external conditions can be changed by the colony, as is evident in the case of E. coli in oxidative stress.

• Chemical commutation mechanisms can initiate drastic changes in the bacteria (e.g. the effect of quorum sensing on sporulation of bacteria, see [87, 140]

• Bacteria can exchange genetic material [89, 187, 167], thus completely altering their nature and behavior. They can use extensive chemical and genetic communication to

redesign genes, pack them in an “off the shelf” resistance cassettes ( ’plasmids’) and distribute them among peers [141, 167]. Bacteria can pass resistance traits to others 222

by giving them a useful plasmid. Viruses occasionally transfer resistance genes by

extracting a gene from one bacterial cell and inject it into a different one. In addition, after a bacterium dies and releases its contents into the environment, another

bacterium will occasionally take up a free-ﬂoating gene for itself. All these methods of transferring genetic information enable bacteria to communicate with bacteria of the same species, as well as with relative and foreign species [167, 198, 144].

Future work

One can not hope to participate in all parts of the endeavor sketched above, especially when taking into account all the feedback loops that where not mentioned here, and the many more whose very existence is unknown. In the following paragraphs, I will focus on future work that is related to the current research.

The research presented here can be extended in several directions. One type of possible extension to the study presented here is to detach the mathematics from the biology. The motivation for this research was to study the biological systems, and many mathematical models were proposed for that end. There is much mathematical research that can be done on these models, research that is not motivated by the study of the biological systems, and presumably will not contribute directly for our understanding of the biological systems. This avenue of research diverges from the theme of this work, and as it is not my intent in the current research, I did not pursue this path of research.

Another type of extension to the study presented here is to apply the approach of ’generic modeling’ to other types of biological systems. The study of the V bacteria, presented in chapter 8, was also motivated by vortices generated by the slime mold Dic- CHAPTER 10 – DISCUSSION 223 tyostelium [185] (note that Dictyostelium is an eukaryotic organism and it is no more related to bacteria than we are).

Yet another type of possible extension is the use of the same models (or closely related) and the same bacteria in order to study various biological phenomena that are expressed in colonial pattern. Indeed the study of sector formation (chapter 9) is an example of study of an interesting phenomenon using models developed in earlier chapters. I was part of another such research [23], where the models were used to study the effects of antibiotics on bacterial colonies. In this research we showed that some types of antibiotics affect the growth of the individual bacterium, yet other types also interfere with the communication between the bacteria. We speculated that materials that are explicitly designed to target the communication can increase the affect of antibiotics on bacteria that grow in colonies.

Appendix A

Biological Units and Dimensionless Equations

Our goal in this appendix is to relate the dimensionless equations appearing throughout this work with biophysical values of the parameters. The procedure is to set the dimensional units to be the natural scales.

A standard procedure is to rescale length and time such to obtain dimensionless equations. Such procedure is necessary (i) to reduce the number of independent parameters, thus simplifying the space of solutions; (ii) to identify small parameters needed for asymptotic treatments.

A.1 The Fisher-Kolmogorov equations

A.1.1 Deriving the equations

The diffusive Fisher-Kolmogorov model (see Sec. 3.8) in dimensional form is:

∂bˆ = D ∇2bˆ + E nˆbˆ (A.1) ∂tˆ b b

225 226

∂nˆ = D ∇2nˆ − E nˆbˆ ∂tˆ n n bˆ is the density field of bacteria and nˆ is the concentration field of nutrition. The field bˆ is measured in units of number of bacteria per cm2 of area. The field nˆ is measured in units of grams per cm2 of area. Experimentally nutrients are usually measured in gram/litre of agar. Taking into account the agar (constant) thickness, 1g/l corresponds to 0.3mg/cm2.

2 Db and Dn are the corresponding diffusion coefﬁcients in units of cm /sec. Eb is the bac- −1 terial reproduction rate in units of sec per nutrition concentration. En is the nutrition consumption rate in units of sec−1 per bacterial density.

We change to new variables:

tˆ = tT xˆ = xX (A.2) bˆ = bB nˆ = nN the new variables are dimensionless and the capitals are their corresponding convention coefﬁcients, measured in the same units as their corresponding ”hatted” variables. We let the temporal and spatial units be the natural scales. The (microscopic) time scale of the model is the bacterial reproduction time τR, so T = τR. The length scale is the diffusion length of the nutrition during reproduction time. The nutrition available for bacteria during reproduction time is proportional to the square of the diffusion length. The length unit is:

√ X = DnτR (A.3)

After replacing the dimensionless variables into Eq. (A.1) we obtain:

∂ b Db 2 = ∇ b + (TN)Ebbn (A.4) ∂t Dn ∂n = ∇2n − (TB)E bn ∂t n APPENDIX A – BIOLOGICAL UNITSAND DIMENSIONLESS EQUATIONS 227

We deﬁne a relative diffusion coefﬁcient D ≡ Db/Dn and impose the relations:

1 TBEn = 1 ⇒ B = (A.5) TEn 1 TNEb = 1 ⇒ N = (A.6) TEb

The dimensionless equations obtained are:

∂b = D∇2b + bn (A.7) ∂t ∂n = ∇2n − bn ∂t

A.1.2 Evaluation of the parameters

We will estimate the values of the parameters Eb, En and Dn, and derive from them the dimensional units. Following Sec. 2.3.2, we make the following points:

• The bacterial reproduction time, when bacteria grow under optimal conditions, is

about τR = 25min. Colonies which exhibit branching patterns grow under limited nutrition supply. Therefore the reproduction time will be longer, but in the same

order of magnitude. We set the time unit to be:

T = τR = 25min (A.8)

• A typical value for the diffusion coefﬁcient of chemicals in agar is 10−7cm2/sec. So

we assume that Dn, the diffusion coefﬁcient of the nutrition in the agar, is similar. We can ﬁnd the length unit using Eq. (A.3):

X = 0.01cm = 100µm (A.9) 228

• The nutrition concentration in the experiments conducted by Ben-Jacob et al. [18],

Rafols [183] and others was 0.1-5 mg/cm2. We set N to have a similar value:

N = 1mg/cm2 = 10 × 10−12g/µ2 = 10−7g/X2 (A.10)

• The reproduction rate per bacterium (Eq. (A.1) ) is EbNn, where n is the dimensionless concentration. The rate is the inverse of the reproduction time, which depends

on the nutrition concentration. We assume that N is the concentration for which the

reproduction time is τR. Therefore:

EbN = 1/τR (A.11)

and Eq. (A.6) is satisﬁed.

• Similarly, EnN is the nutrition consumption rate per bacterium. We suggest that during reproduction time, a single bacterium consumes an amount of nutrition three times its mass, which is about 3 × 10−12g. Therefore, the rate of nutrition consumption is: 3 × 10−12g 1 E N ∼ (A.12) n bacteria 25min

From Eqs. (A.5),(A.10) we obtain the bacterial concentration:

2 4 2 B = N/TEnN = 3bacteria/µ = 3 × 10 bacteria/X (A.13)

The discrete time step of the numerical integration is measured in units of T. Sometimes numeric stability demands that the time step will be less than one. Then, as an example, a time step of 0.001 will correspond to 0.001T ∼ 1sec APPENDIX A – BIOLOGICAL UNITSAND DIMENSIONLESS EQUATIONS 229

A.2 Reaction-diffusion models for branching patterns

A.2.1 A cutoff in the reaction term

The scaling performed in the previous section is also suitable for the cutoff equations (see

Sec. 4.1.1.1). We are interested in the meaning of the cutoff β. Since β is measured in units of B, we use Eq. (A.13) to translate numerical values to bacteria concentrations. For example, β = 3 × 10−5 corresponds to 1bacteria/X2 = 100bacteria/mm2, while β = 1 corresponds approximately to bacteria covering the agar surface in a uniform monolayer.

A.2.2 The model of Mimura at al.

Mimura et al. model’s equations in dimensional form (see Sec. 4.1.1.2) are:

∂ b 2 a0b = Db∇ b + Ebbn − (A.14) ∂t (an + n)(ab + b) ∂n = D ∇2n − E bn ∂t n n ∂s a b = 0 ∂t (an + n)(ab + b) where an and ab are constants which have the same units as n and b respectively. We introduce dimensionless variables as previously (Eq. (A.2) ), and impose relation (A.5) to obtain:

∂b D E N a T b = b ∇2b + b bn − 0 (A.15) ∂t Dn EnB BN (an/N + n)(ab/B + b) ∂n = ∇2n − bn ∂t ∂s a T b = 0 ∂t BN (an/N + n)(ab/B + b) 230

We deﬁne new parameters:

D E N a T D ≡ b ε ≡ b µ ≡ 0 (A.16) Dn EnB BN

We set an/N = 1 and ab/B = 1 in the last term of the bacterial equation. (The assignment is acceptable since Mimura et al. does not justify the exact form of that term, rather they state that it is only its general properties that matters.) Changing to the new parameters gives the dimensionless equations.

Scaling this model requires adjusting the values of the units. Relation (A.6) is replaced by relation (A.16), which is equivalent to:

TNEb = ε (A.17)

Since Eq. (A.11) is still valid, we leave N unchanged. The other units re-scale according to:

T → εT (A.18) √ X → εX

B → B/ε compared to the values of the units evaluated in Sec. (A.1.2)

A.3 Atomistic models for branching patterns

The basic atomistic model described here is the Communicating Walker (CW) model (see Sec. 7.1). It as much more parameters then the basic NLD model, so it might look as if it less associated with the bacterial colonies. However, the atomistic nature of the CW model makes it easy to relate its details to the actual biological details. In the following discussion APPENDIX A – BIOLOGICAL UNITSAND DIMENSIONLESS EQUATIONS 231 we do not repeat the details of the model, as they were repeated time and time again in other parts of this work.

In the CW model the walker assume new direction each time step (Eq. (7.1) ). Hence one time step in the simulations is the bacterial tumbling time τT (Sec. 2.3.1) which corresponds to about 1sec. A typical numerical run is up to about 104 − 106 time steps which translates to about two days of bacterial growth. Initial concentration of the nutrient of n0 = 10 corresponds approximately to 1g/l; in the petri- dish [37] we pour about 22ml of the agar mixture. Hence 1g/l can support growth of about 109 bacterial cells. In a typical

−7 run we use a lattice of size 500 × 500. Thus n0 = 10 corresponds to about 10 g per lattice cell or about one walker per lattice cell. During the numerical growth there are about ten walkers per lattice cell, which means that indeed for 1g/l peptone level the growth is diffusion limited. The diffusion coefﬁcient Dn (Eq. (7.4) ) is typically (depending on agar dryness) 10−4 − 10−6cm2/sec, which is comparable with the bacterial effective diffusion constant (Sec. 2.3.1). To complete the correspondence between the model’s parameters and real growth we should estimate Nc, the number of walker collisions needed to push the colony’s boundary. The lattice constant a0 is about ten times (about 100µm) the walkers’ tumbling step. Hence for ten walkers in one lattice cell we have about 0.25−2 collisions on each segment of the boundary. From time- lapse measurements of the growth, the envelope of the bacterial colony propagates at a rate of 2−20µm per minute for 1.5% agar. Thus the corresponding Nc is about 20.

Bibliography

[1] J. Adler. Chemotaxis in bacteria. Science, 153:708–716, 1966.

[2] J. Adler. Chemoreceptors in bacteria. Science, 166:1588–1597, 1969.

[3] G. Albrecht-Buehler. In defense of “non-molecular” cell biology. Int. Rev. Cytol., 120:191–241, 1990.

[4] D. G. Allison, P. Gilbert, H. M. Lappin-Scott, and M. Wilson, editors. Community Structure and Co-operation in Bioﬁlms. Cambridge University Press, 2001.

[5] J. W. Anderson. Bioenergetics of Autotrophs and Heterotrophs. Studies in Biology. E. Arnold, London, 1980.

[6] K. Arima, A. Kakinuma, and G. Tamura. Surfactin, a crystalline peptidelipid surfactant produced by Bacillus subtilis: isolation, characterization and its inhibition of ﬁbrin clot formation. Biochem. Biophys. Res. Commun., 31:488–494, 1968.

[7] E. M. Arnett, N. G. Harvey, and P. L. Rose. Acc. Chem. Res., 22:131, 1989.

[8] V. I. Arnol’d. Ordinary Differential Equations. Springler-Verlag, Heidelberg, 1992.

[9] D. G. Aronson. Density-dependent interaction-diffusion systems. In W. E. Stewart, W. H. Ray, and C. C. Conley, editors, Dynamics and Modeling of Reactive Systems, pages 161–176. Academic Press, New York, 1980.

[10] D. G. Aronson and H. F. Weinberger. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In A. Dold and B. Eckmann, editors, Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, pages 5–49. Springer-Verlag, Berlin, 1975.

[11] V. A. Avetisov, V. I. Goldanskii, and V. V. Kuzmin. Handedness, origin of life and evolution. Phys. Today, 44:33–41, 1991.

233 234

[12] M. N. Barber, A. Barbieri, and J. S. Langer. Dynamics of dendritic sidebranching in the two-dimensional symmetric model of solidiﬁcation. Phys. Rev. A, 36:3340– 3349, 1987.

[13] A. Barbieri, D. C. Hong, and J. S. Langer. Velocity selection in the symmetric model of dendritic crystal growth. Phys. Rev. A, 35:1802–1808, 1987.

[14] J. Bechhoefer and J. L. Hutter. Morphology transitions in a liquid crystal. Physica A, 49:82–87, 1998.

[15] M. Ben-Amar and E. Brener. Theory of pattern selection in three-dimensional nonaxisymmetric dendritic growth. Phys. Rev. Lett., 71:589–592, 1993.

[16] M. Ben-Amar and Y. Pomeau. Theory of dendritic growth in a weakly undercooled melt. Europhys. Lett., 2:307–314, 1986.

[17] E. Ben-Jacob. From snowﬂake formation to the growth of bacterial colonies. part I: Diffusive patterning in non-living systems. Contemp. Phys., 34:247–273, 1993.

[18] E. Ben-Jacob. From snowﬂake formation to the growth of bacterial colonies. part II: Cooperative formation of complex colonial patterns. Contemp. Phys., 38:205–241, 1997.

[19] E. Ben-Jacob, H. Brand, G. Dee, L. Kramer, and J. S. Langer. Pattern propagation in nonlinear dissipative systems. Physica D, 14:348–364, 1985.

[20] E. Ben-Jacob and I. Cohen. Cooperative formation of bacterial patterns. In J. A. Shapiro and M. Dworkin, editors, Bacteria as Multicellular Organisms. Oxford Uni- versity Press, New-York, 1997.

[21] E. Ben-Jacob, I. Cohen, and A. Czirok.´ Smart bacterial colonies. In H. Flyvbjerg, J. Hertz, M. H. Jensen, O. G. Mouritsen, and K. Sneppen, editors, Physics of Biolog- ical Systems: From Molecules to Species, Lecture Notes in Physics, pages 307–324. Springer-Verlag, Berlin, 1997. BIBLIOGRAPHY 235

[22] E. Ben-Jacob, I. Cohen, A. Czirok,´ T. Vicsek, and D. L. Gutnick. Chemomodulation of cellular movement and collective formation of vortices by swarming bacteria and colonial development. Physica A, 238:181–197, 1997.

[23] E. Ben-Jacob, I. Cohen, I. Golding, D. L. Gutnick, M. Tcherpakov, D. Helbing, and I. G. Ron. Bacterial cooperative organization under antibiotic stress. Physica A, 282:247–282, 2000.

[24] E. Ben-Jacob, I. Cohen, I. Golding, and Y. Kozlovsky. Modeling branching and chiral colonial patterning of lubricating bacteria. In P. Maini and H. Othmer, editors, Mathematical Models for Biological Pattern Formation, Volume 121 in IMA series “Frontiers in Applications of Mathematics”, pages 211–253. Springer, New-York, 2001.

[25] E. Ben-Jacob, I. Cohen, and H. Levine. Cooperative self-organization of microorganisms. Adv. Phys., 49:395–575, 2000.

[26] E. Ben-Jacob, I. Cohen, O. Shochet, I. Aronson, H. Levine, and L. Tsimering. Com- plex bacterial patterns. Nature, 373:566–567, 1995.

[27] E. Ben-Jacob, I. Cohen, O. Shochet, A. Czirok,´ and T. Vicsek. Cooperative formation of chiral patterns during growth of bacterial colonies. Phys. Rev. Lett., 75:2899– 2902, 1995.

[28] E. Ben-Jacob and P. Garik. The formation of patterns in non-equilibrium growth. Nature, 343:523–530, 1990.

[29] E. Ben-Jacob, P. Garik, T. Muller, and D. Grier. Characterization of morphology transitions in diffusion-controlled systems. Phys. Rev. A, 38:1370, 1988.

[30] E. Ben-Jacob, H. Shmueli, O. Shochet, and A. Tenenbaum. Adaptive self- organization during growth of bacterial colonies. Physica A, 187:378–424, 1992.

[31] E. Ben-Jacob, O. Shochet, I. Cohen, A. Tenenbaum, A. Czirok,´ and T. Vicsek. Co- operative strategies in formation of complex bacterial patterns. Fractals, 3:849–868, 1995. 236

[32] E. Ben-Jacob, O. Shochet, and A. Tenenbaum. Bakterien schließen sich zu bizarren formationen zusammen. In A. Deutsch, editor, Muster des Ledendigen: Faszination inher Entstehung und Simulation. Verlag Vieweg, 1994.

[33] E. Ben-Jacob, O. Shochet, A. Tenenbaum, and O. Avidan. Evolution of complexity during growth of bacterial colonies. In P. E. Cladis and P. Palffy-Muhoray, editors, Spatio-Temporal Patterns in Nonequilibrium Complex Systems, Santa-Fe Institute studies in the sciences of complexity, pages 619–634. Addison-Weseley Publishing Company, 1995.

[34] E. Ben-Jacob, O. Shochet, A. Tenenbaum, I. Cohen, A. Czirok,´ and T. Vicsek. Com- munication, regulation and control during complex patterning of bacterial colonies. Fractals, 2:15–44, 1994.

[35] E. Ben-Jacob, O. Shochet, A. Tenenbaum, I. Cohen, A. Czirok,´ and T. Vicsek. Generic modeling of cooperative growth patterns in bacterial colonies. Nature, 368:46–49, 1994.

[36] E. Ben-Jacob, O. Shochet, A. Tenenbaum, I. Cohen, A. Czirok,´ and T. Vicsek. Re- sponse of bacterial colonies to imposed anisotropy. Phys. Rev. E., 53:1835–1845, 1996.

[37] E. Ben-Jacob, A. Tenenbaum, O. Shochet, and O. Avidan. Holotransformations of bacterial colonies and genome cybernetics. Physica A, 202:1–47, 1994.

[38] D. Bensimon, L. P. Kadanoff, S. Liang, and B. I. Shraiman. Viscous ﬂows in two dimensions. Reviews of Modern Physics, 58:977–999, 1986.

[39] W. A. Bentley and W. J. Humphreys. Snow crystals. Dover Publications, 1962.

[40] H. C. Berg. Random Walks in Biology. Princeton University Press, Princeton, N.J., 1993.

[41] H. C. Berg and D. A. Brown. Chemotaxis in Escherichia coli analyzed by three- dimensional tracking. Nature, 239:500–504, 1972. BIBLIOGRAPHY 237

[42] H. C. Berg and E. M. Purcell. Physics of chemoreception. Biophysical Journal, 20:193–219, 1977.

[43] H. C. Berg and P. M. Tedesco. Transient response to chemotactic stimuli in Es- cherichia coli. Proc. Natl. Acad. Sci. USA, 72:3235–3239, 1975.

[44] Y. Blat and M. Eisenbach. Tar-dependent and -independent pattern formation by Salmonella typhimurium. J. Bact., 177:1683–1691, 1995.

[45] J. P. Boon, D. Dab, R. Kapral, and A. Lawniczak. Lattice gas automata for reactive systems. Phys. Reports, 273:55–147, 1996.

[46] E. Boschke and Th. Bley. Growth patterns of yeast colonies depending on nutrient supply. Acta Biotechnol., 18:17–27, 1998.

[47] D. Bray, M. D. Levin, and C. J. Morton-Firth. Receptor clustering as a cellular mechanism to control sensitivity. Nature, 393:85–88, 1998.

[48] E. Brener, H. Levine, and Y. Tu. Mean-ﬁeld theory for diffusion-limited aggregation in low dimensions. Phys. Rev. Lett., 66:1978–1981, 1990.

[49] E. Brener and D. Temkin. Noise-induced sidebranching in the three-dimensional nonaxisymmetric dendritic growth. Phys. Rev. E, 51:351–359, 1995.

[50] E. A. Brener and V. I. Mel’nikov. Pattern selection in two-dimensional dendritic growth. Adv. Phys., 40:53–97, 1991.

[51] E. Brunet and B. Derrida. Shift in the velocity of a front due to a cutoff. Phys. Rev. E, 56:2597–604, 1997.

[52] J. Buckmaster. Viscous sheets advancing over dry beds. J. Fluid Mech., 81:735–756, 1977.

[53] E. O. Budrene and H. C. Berg. Complex patterns formed by motile cells of Esh- erichia coli. Nature, 349:630–633, 1991.

[54] E. O. Budrene and H. C. Berg. Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature, 376:49–53, 1995. 238

[55] G. Caginalp. An analysis of a phase ﬁeld model of a free boundary. Arch. Rat. Mech. Anal., 92:205, 1986.

[56] G. Caginalp and P. C. Fife. Higher order phase-ﬁeld models and detailed anisotropy. Phys. Rev. B, 34:4940, 1986.

[57] G. Caginalp and P. C. Fife. Phase-ﬁeld methods for interfacial boundaries. Phys. Rev. B, 33:7792–7794, 1986.

[58] Cell 73(5), Special issue, Development in Bacteria, June 4 , 1993.

[59] I. Cohen. Mathematical modeling and analysis of pattern formation and colonial organization in bacterial colonies, 1997. M.Sc. thesis, Tel-Aviv University, ISRAEL.

[60] I. Cohen, A. Czirok,´ and E. Ben-Jacob. Chemotactic-based adaptive self organization during colonial development. Physica A, 233:678–698, 1996.

[61] I. Cohen, I. Golding, Y. Kozlovsky, E. Ben-Jacob, and I. G. Ron. Continuous and discrete models of cooperation in complex bacterial colonies. Fractals, 7:235–247, 1999.

[62] I. Cohen, I. Golding, I. G. Ron, and E. BenJacob. Bioﬂuiddynamics of lubricating bacteria. Math. Meth. Appl. Sci., 24:1429–1468, 2001.

[63] I. Cohen, I. G. Ron, and E. BenJacob. From branching to nebula patterning during colonial development of the Paenibacillus alvei bacteria. Physica A, 286:321–336, 2000.

[64] J. B. Collins and H. Levine. Diffuse interface model of diffusion limited crystal growth. Phys. Rev. B, 31:6119, 1985.

[65] D. G. Cooper and J. E. Zajic. Surface-active compounds from microorganisms. Adv. Appl. Microbio., 26:229–251, 1980.

[66] J. W. Costerton and P. S. Stewart. Batteling bioﬁlms. Sci. Am., 285:60–67, 2001.

[67] M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium. Rev. Mod. Phys., 65:851, 1993. BIBLIOGRAPHY 239

[68] A. Czirok,´ E. Ben-Jacob, I. Cohen, and T. Vicsek. Formation of complex bacterial colonies via self-generated vortices. Phys. Rev. E, 54:1791–1801, 1996.

[69] E. Deak, I. Szabo, A. Kalmaczhelyi, Z. Gal, G. Barabas, and A. Penyige. Membrane- bound and extracellular beta-lactamase production with developmental regulation in Streptomyces griseus NRRL B-2682. Microbiol., 144:2169–2177, 1998.

[70] J. D. Desai and I. M. Banat. Microbial production of surfactants and their commer- cial potential. Microbiol. Mol. Biol. Rev., 61:47–64, 1997.

[71] A. Deutsch, editor. Muster des Ledendigen: Faszination inher Entstehung und Sim- ulation. Verlag Vieweg, 1994.

[72] P. Devreotes. Dictyostelium discoideum: a model system for cell-cell interactions in development. Science, 245:1054–1058, 1989.

[73] C. D’Souza, M. M. Nakano, and P. Zuber. Identiﬁcation of comS, a gene of the srfA operon that regulates the establishment of genetic competence in Bacillus subtilis. Proc. Natl. Acad. Sci. USA, 91:9397–9401, 1994.

[74] A. Dukler. Isolation and characterization of bacteria growing in patterns on the surface of solid agar. Senior project under the supervision of D. Gutnick., Tel-Aviv University, 1993.

[75] Y. L. Duparcmeur, H. Herrmann, and J. P. Troadec. Spontaneous formation of vortex in a system of self motorized particles. J. de Physique I, 5:1119–1128, 1995.

[76] M. Eisenbach. Functions of the ﬂagellar modes of rotation in bacterial motility and chemotaxis. Molec. Microbiol., 4:161–167, 1990.

[77] M. Eisenbach. Control of bacterial chemotaxis. Mole. Microbiol., 20:903–910, 1996.

[78] S. E. Esipov and J. A. shapiro. Kinetic model of Proteus Mirabilis swarm colony development. J. Math. Biol., 36:249–268, 1998.

[79] X.M. Yang et al. Phys. Lett. A, 193:195, 1994. 240

[80] J. Feder. Fractals. Plenum, New York, 1988.

[81] P. C. Fife, editor. Dynamics of Internal Layers and Diffusive Interfaces. Society for Industrial and Applied Mathematics, Philadelphia, 1988.

[82] R. A. Fisher. The wave of advance of advantageous genes. Annual Eugenics, 7:255– 369, 1937.

[83] W. W. Ford. Studies on aerobic spore-bearing non-pathogenic bacteria. part II: Mis- cellaneous cultures. J. Bacteriol., 1:518–526, 1916.

[84] H. Frahm, S. Ullah, and A. T. Dorsey. Flux dynamics and the growth of the superconducting phase. Phys. Rev. Lett., 66:3067–3070, 1991.

[85] H. Fujikawa and M. Matsushita. Fractal growth of Bacillus subtilis on agar plates. J. Phys. Soc. Jap., 58:3875–3878, 1989.

[86] H. Fujikawa and M. Matsushita. Bacterial fractal growth in the concentration ﬁeld of nutrient. J. Phys. Soc. Jap., 60:88–94, 1991.

[87] W. C. Fuqua, S. C. Winans, and E. P. Greenberg. Quorum sensing in bacteria: the LuxR-LuxI family of cell density-responsive transcriptional regulators. J. Bact., 176:269–275, 1994.

[88] Y. L. Fuxman. Reproduction rate, feeding process, and liebig limitations in cell populations–II. Bul. Math. Biol., 57:749–782, 1995.

[89] T. Galitski and J. R. Roth. Evidence that F plasmid transfer replication underlies apparent adaptive mutation. Science, 268:421–423, 1995.

[90] H. J. Gao, G. S. Canright, S. J. Pang, I. M. Sandler, Z. Q. Xue, and Z. Y. Zhang. Unique dendritic patterns in organic thin ﬁlms: Experiments and modeling. Fractals, 6:337–350, 1998.

[91] P. J. Gardina and M. D. Manson. Attractant signaling by an aspartate chemoreceptor dimer with a single cytoplasmic domain. Science, 274:425–426, 1996. BIBLIOGRAPHY 241

[92] I. Golding, I. Cohen, I. G. Ron, and E. Ben-Jacob. Adaptive branching during colonial development of lubricating bacteria. In V. Fleury, J. F. Gouyet, and M. Leonetti, editors, Branching in Nature: Dynamics and Morphogenesis of Branching Struc- tures, from Cell to River Networks. Springer-Verlag, 2001.

[93] I. Golding, Y. Kozlovsky, I. Cohen, and E. Ben-Jacob. Studies of bacterial branching growth using reaction-diffusion models of colonial development. Physica A, 260:510–554, 1998.

[94] A. D. Grossman and R. Losick. Extracellular control of spore formation in Bacillus subtilis. Proc. Natl. Acad. Sci. USA, 85:4369–4373, 1988.

[95] M. E. Gurtin and R. C. MacCamy. On the diffusion of biological populations. Math. Biosci., 33:35–49, 1977.

[96] H. Haken. Information and self-organization. Springer-Verlag, Berlin, 1988.

[97] L. W. Hamoen, H. Eshuis, J. Jongbloed, G. Venema, and D. van Sinderen. A small gene, designated comS, located within the coding region of the fourth amino acid- activation domain of srfA, is required for competence development in Bacillus subtilis. Mol. Microbiol., 15:55–63, 1995.

[98] R. M. Harshey. Bees aren’t the only ones: swarming in gram-negative bacteria. Molecular Microbiology, 13:389–394, 1994.

[99] J. G. Heath and E. M. Arnett. J. Am. Chem. Soc., 114:4500, 1992.

[100] W. M. Heckl, M. Losche, D. A. Cadenhead, and H. Mohwald. Europ. Biophys. J., 14:11, 1986.

[101] R. A. Hegstrom and D. K. Kondepudi. The handedness of the universe. Sci. Am., 262:108–115, 1990.

[102] D. Helbing. Gas-kinetic derivation of Navier-Stokes-like trafﬁc equations. Phys. Rev. E, 53:2366–2381, 1996.

[103] J. Hemmingsson. Modellization of self-propelling particles with a coupled map lattice model. J. Phys. A, 28:4245–4250, 1995. 242

[104] J. Henrichsen. Bacterial surface translocation: a survey and a classiﬁcation. Bact. Rev., 36:478–503, 1972.

[105] T. H. Henrici. The Biology of Bacteria: The Bacillaceae. D. C. Heath & company, 3rd edition, 1948.

[106] G. M. Homsy. Viscous ﬁngering in porous media. Annual Review of Fluid Mechan- ics, 19:271–311, 1987.

[107] D. Horvath and A. Toth. Diffusion-driven front instabilities in the chlorite- tetrathionate reaction. J. Chem. Phys., 108:1447–1451, 1998.

[108] G. P. Ivantsov. Dokl. Akad. Nauk SSSR, 58:567, 1947.

[109] D. Kaiser. Control of multicellular development: Dictyostelium and Myxococcus. Ann. Rev. Genetics, 20:539–566, 1986.

[110] D. Kaiser and R. Losick. How and why bacteria talk to each other. Cell, 73:873–887, 1993.

[111] R. Kam and H. Levine. Phase-ﬁeld model of spiral dendritic growth. Phys. Rev. E, 54:2797–2801, 1996.

[112] S. Kamin and R. Kersner. Disappearance of interfaces in ﬁnite time. Meccanica, 28:117–120, 1993.

[113] S. Kamin and P. Rosenau. Propagation of thermal waves in an inhomogeneous medium. Comm. Pure Appl. Math., 34:831–852, 1981.

[114] O. Katzenelso, H. Z. Hel-Or, and D. Avnir. Chirality of large random supramolecular structures. Chem.-Euro. J., 2:174–181, 1996.

[115] K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda, and N. Shigesada. Modeling spatio-temporal patterns created by Bacillus subtilis. J. Theor. Biol., 188:177–185, 1997.

[116] D. A. Kessler, J. Koplik, and H. Levine. Steady-state dendritic crystal growth. Phys. Rev. A, 33:3352–3357, 1986. BIBLIOGRAPHY 243

[117] D. A. Kessler, J. Koplik, and H. Levine. Pattern selection in ﬁngered growth phenomena. Adv. Phys., 37:255, 1988.

[118] D. A. Kessler and H. Levine. Pattern selection in three dimensional dendritic growth. Acta Metallurgica, 36:2693–2706, 1988.

[119] D. A. Kessler and H. Levine. Pattern formation in Dictyostelium via the dynamics of cooperative biological entities. Phys. Rev. E, 48:4801–4804, 1993.

[120] D. A. Kessler and H. Levine. Fluctuation-induced diffusive instabilities. Nature, 394:556–558, 1998.

[121] D. A. Kessler, Z. Ner, and L. M. Sander. Front propagation: Precursors, cutoffs, and structural stability. Physical Review E, 58:107–14, 1998.

[122] J. O. Kessler. Co-operative and concentrative phenomena of swimming microorganisms. Cont. Phys., 26:147–166, 1985.

[123] S. K. Kim and D. Kaiser. Cell alignment required in differentiation of Myxococcus xanthus. Science, 249:926–928, 1990.

[124] S. Kitsunezaki. Interface dynamics for bacterial colony formation. J. Phys. Soc. Japan, 66:1544–1550, 1997.

[125] R. Kobayashi. Modeling and numerical simulations of dendritic crystal growth. Physica D, 63:410–423, 1993.

[126] A. I. Kolmogorov, I. Petrovsky, and N. Piscounov. etude´ de l’equation´ de la diffusion avec croissance de la quantite´ de matiere` et son application a` un probleme` biologique. Moscow Univ. Bull. Math., 1:1–25, 1937.

[127] M. Kowall, J. Vater, B. Kluge, T. Stein, P. Franke, and D. Ziessow. Separation and characterization of surfactin isoforms produced by Bacillus subtilis OKB 105. J. Collid. Interf. Sic., 204:1–8, 1998.

[128] Y. Kozlovsky, I. Cohen, I. Golding, and E. Ben-Jacob. Lubricating bacteria model for branching growth of bacterial colonies. Phys. Rev. E, 59:7025–7035, 1999. 244

[129] R. Kupferman. Morphology, coexistence and selection in interfacial pattern formation. PhD thesis, Tel-Aviv University, 1995.

[130] R. Kupferman, O. Shochet, and E. Ben-Jacob. Numerical study of morphology diagram in the large undercooling limit using a phase-ﬁeld model. Phys. Rev. E, 50:1005, 1994.

[131] R. Kupferman, O. Shochet, E. Ben-Jacob, and Z. Schuss. Studies of a phase-ﬁeld model: Boundary layer, selected velocity and stability spectrum. Phys. Rev. B, 49:16045, 1992.

[132] A. Kuspa, L. Plamann, and D. Kaiser. A-signaling and the cell density requirement for Myxococcus xanthus development. J. Bact., 174:7360–7369, 1992.

[133] J. M. Lackie, editor. Biology of the chemotactic response. Cambridge Univ. Press, 1986.

[134] L. D. Landau, A. I. Akhiezer, and E. M. Lifshitz. General Physics : Mechanics and Molecular Physics. Pergamon Press, Oxford, 1967.

[135] J. S. Langer. Instabilities and pattern formation in crystal growth. Rev. Mod. Phys., 52:1–28, 1980.

[136] J. S. Langer. Models of pattern formation in ﬁrst-order phase transitions. In G. Grin- stein and G. Mazenko, editors, Directions in condensed matter physics, page 167. World Scientiﬁc, 1986.

[137] J. S. Langer. Chance and Matter, Les Houches lectures XLVI. Elsevier, 1987.

[138] J. S. Langer. Dendrites, viscous ﬁngering, and the theory of pattern formation. Sci- ence, 243:1150–1154, 1989.

[139] E. W. Larsen and G. C. Pomraning. Asymptotic analysis of nonlinear Marshak waves. SIAM J. Appl. Math., 39:201–212, 1980.

[140] A. Latiﬁ, M. K. Winson, M. Foglino, B. W. Bycroft, G. S. Stewart, A. Lazdunski, and P. Williams. Multiple homologues of LuxR and LuxI control expression of virulence BIBLIOGRAPHY 245

determinants and secondary metabolites through quorum sensing in Pseudomonas aeruginosa PAO1. Molecular Microbiology, 17:333–343, 1995.

[141] S. B. Levy. The challenge of antibiotic resistance. Sci. Am., March, 1998.

[142] F. Liu, M. Mondello, and N. Goldenfeld. Kinetics of the superconducting transition. Phys. Rev. Lett., 66:3071, 1991.

[143] W. F. Loomis and P. W. Sternberg. Genetic networks. Science, 269:649, 1995.

[144] R. Losick and D. Kaiser. Why and how bacteria communicate. Sci. Am., February, 1997.

[145] R. M. Macnab and M. K. Ornston. Normal-to-curly ﬂagellar transitions and their role in bacterial tumbling. stabilization of an alternative quartenary structure by mechanical force. J. Molec. Biol., 112:1–30, 1977.

[146] J. Mai, I. M. Sokolov, and A. Blumen. Front propagation and local ordering in one- dimensional irreversible autocatalytic reactions. Phys. Rev. Lett., 77:4462, 1996.

[147] J. Mai, I. M. Sokolov, V. N. Kuzovkov, and A. Blumen. Front form and velocity in a one-dimensional autocatalytic A+B to 2A reaction. Phys. Rev. E, 56:4130, 1997.

[148] A. Malevanets and R. Kapral. Links, knots, and knotted labyrinths in bistable systems. Phys. Rev. Lett., 77:767–770, 1996.

[149] B. B. Mandelbrot. The Fractal Geometry of Nature. Freeman, San Francisco, 1977.

[150] B. B. Mandelbrot. Fractals: Form, Chance and Dimension. Freeman, San Francisco, 1977.

[151] M. Matsushita and H. Fujikawa. Diffusion-limited growth in bacterial colony formation. Physica A, 168:498–506, 1990.

[152] M. Matsushita, J.-I. Wakita, H. Itoh, I. Rafols, T. Matsuyama, H. Sakaguchi, and M. Mimura. Interface growth and pattern formation in bacterial colonies. Physica A, 249:517–524, 1998. 246

[153] M. Matsushita, J.-I. Wakita, and T. Matsuyama. Growth and morphological changes of bacteria colonies. In P. E. Cladis and P. Palffy-Muhoray, editors, Spatio-Temporal Patterns in Nonequilibrium Complex Systems, Santa-Fe Institute studies in the sciences of complexity, pages 609–618. Addison-Weseley Publishing Company, 1995.

[154] T. Matsuyama, A. Bhasin, and R. M. Harshey. Mutational analysis of ﬂagellum- independent surface spreading of Serratia marcescens 274 on a low-agar medium. J. Bacteriol., 177:987–991, 1995.

[155] T. Matsuyama, R. M. Harshey, and M. Matsushita. Self-similar colony morphogenesis by bacteria as the experimental model of fractal growth by a cell population. Fractals, 1:302–311, 1993.

[156] T. Matsuyama, K. Kaneda, Y. Nakagawa, K. Isa, H. Hara-Hotta, and I. Yano. A novel extracellular cyclic lipopeptide which promotes ﬂagellum-dependent and - independent spreading growth of Serratia marcescens. J. Bact., 174:1769–1776, 1992.

[157] T. Matsuyama and M. Matsushita. Fractal morphogenesis by a bacterial cell population. Crit. Rev. Microbiol., 19:117–135, 1993.

[158] T. Matsuyama and M. Matsushita. Morphogenesis by bacterial cells. In P. M. Ian- naccone and M. K. Khokha, editors, Farctal Geometry in Biological Systems, an Analytical Approach, pages 127–171. CRC Press, New-York, 1995.

[159] T. Matsuyama, M. Sogawa, and Y. Nakagawa. Fractal spreading growth of Serratia marcescens which produces surface active exolipids. FEMS Microbiol. Lett., 52:43– 246, 1989.

[160] D. Meiron. Selection of steady states in the two-dimensional symmetric model of dendritic growth. Phys. Rev. A, 33:2704–2715, 1986.

[161] N. H. Mendelson. Helical Bacillus subtilis macroﬁbers: morphogenesis of a bacterial multicellular macroorganism. Proc. Natl. Acad. Sci. USA, 75:2478–2482, 1978.

[162] N. H. Mendelson. Bacterial macroﬁbres: the morphogenesis of complex multicellular bacterial forms. Sci. Progress, 74:425–441, 1990. BIBLIOGRAPHY 247

[163] N. H. Mendelson and S. L. Keener. Clockwise and counterclockwise pinwheel colony morphologies of Bacillus subtilis are correlated with the helix hand of the strain. J. Bacteriol., 151:455–457, 1982.

[164] N. H. Mendelson and B. Salhi. Patterns of reporter gene expression in the phase diagram of Bacillus subtilis colony forms. J. Bacteriol., 178:1980–1989, 1996.

[165] N. H. Mendelson and J. J. Thwaites. Cell wall mechanical properties as measured with bacterial thread made from Bacillus subtilis. J. Bacteriol., 171:1055–1062, 1989.

[166] R. Mesibov, G. W. Ordal, and J. Adler. The range of attractant concentrations for bacterial chemotaxis and the threshold and size of response over this range. Weber law and related phenomena. J. Gen. Physiol., 62:203–223, 1973.

[167] R. V. Miller. Bacterial gene swapping in nature. Sci. Am., 278, 1998.

[168] M. Mimura, H. Sakaguchi, and M. Matsushita. Reaction-diffusion modelling of bacterial colony patterns. Physica A, 282:283–303, 2000.

[169] A. Mogilner and L. Edelstein-Keshet. A non-local model for a swarm. J. Math. Biol., 38:534–570, 1999.

[170] W. W. Mullins and R. F. Sekerka. Stability of a planar interface during solidiﬁcation of a dilute binary alloy. J. Appl. Phys., 35:444, 1964.

[171] J. D. Murray. Mathematical Biology. Springer-Verlag, Berlin, 1989.

[172] M. Muskat. The Flow of Homogeneous Fluids Through Porous Media. McGraw- Hill, New-York, 1937.

[173] W. I. Newman. Some exact solutions to a non-linear diffusion problem in population genetics and combustion. J. Theor. Biol., 85:325–334, 1980.

[174] W. I. Newman. The long-time behavior of the solution to a non-linear diffusion problem in population genetics and combustion. J. Theor. Biol., 104:473–484, 1983. 248

[175] A. Nicol, W.-J. Rappel, H. Levine, and W. F. Loomis. Cell-sorting in aggregates of Dictyostelium discoideum. J. Cell Sci., 112:3923–3929, 1999.

[176] G. Nicolis and I. Prigogine. Exploring Complexity. W. H. Freeman and company, New-York, 1989.

[177] O. A. Oleinik, A. S. Kalashnikov, and C. Yui-Lin. The cauchy problem and boundary problems for equations of the type of unsteady ﬁltration. Izv. Akad. Nauk. SSR ser. Mat., 22:667–704, 1958.

[178] G. C. Paquette, L.-Y. Chen, N. Goldenfeld, and Y. Oono. Structural stability and renormalization group for propagating fronts. Phys. Rev. Lett., 72:76–79, 1994.

[179] J. K. Parrish and L. Edelstein-Keshet. Complexity, pattern, and evolutionary trade- offs in animal aggregation. Science, 284:99–101, 1999.

[180] L. Pechenik and H. Levine. Interfacial velocity corrections due to multiplicative noise. Phys. Rev. E, 1998. (in press).

[181] P. Pelce. Dynamics of curved fronts. Academic Press, 1988.

[182] P. Pelce and A. Pocheau. geometrical approach to the morphogenesis of unicellular algae. J. Theor. Biol., 156:197–214, 1992.

[183] I. Rafols. Formation of Concentric Rings in Bacterial Colonies. M.Sc. thesis, Chuo University, Japan, 1998.

[184] W.-J. Rappel, H. Levine, A. Nicol, and W. F. Loomis. Modeling self-propelled deformable cell motion in the Dictyostelium mound; a status report. In P. Maini and H. Othmer, editors, Mathematical Models for Biological Pattern Formation, Volume 121 in IMA series “Frontiers in Applications of Mathematics”, pages 255– 267. Springer, New-York, 2001.

[185] W.-J. Rappel, A. Nicol, A. Sarkissian, and H. Levine. Self-organized vortex state in two-dimensional Dictyostelium dynamics. Physical Review Letters, 83:1247–1250, 1999. BIBLIOGRAPHY 249

[186] W.-J. Rappel, A. Nicol, A. Sarkissian, H. Levine, and W. F. Loomis. Self-organized vortex state in two-dimensional Dictyostelium dynamics. Phys. Rev. Lett., 83:1247– 1250, 1999.

[187] J. P. Rasicella, P. U. Park, and M. S. Fox. Adaptive mutation in Escherichia coli : a role for conjugation. Science, 268:418–420, 1995.

[188] P. Rosenau and S. Kamin. Nonlinear diffusion in a ﬁnite mass medium. Comm. Pure Appl. Math., 35:113–127, 1982.

[189] E. Rosenberg, editor. Myxobacteria; Development and Cell Interactions. Springer series in molecular biology. Springer-Verlag, 1984.

[190] F. Sanchez-Gardu´ no˜ and P. K. Maini. Existence and uniqueness of sharp traveling waves in degenerate non-linear diffusion Fisher-KPP equations. J. Math. Biol., 33:163–192, 1994.

[191] F. Sanchez-Gardu´ no˜ and P. K. Maini. Travelling wave phenomena in some degenerate reaction-diffusion equations. J. Diff. Equ., 117:281–319, 1995.

[192] F. Sanchez-Gardu´ no,˜ P. K. Maini, and E. Kappos. A review of traveling wave solutions of one-dimensional reaction-diffusion equations with non-linear diffusion term. Forma, 11:45–59, 1996.

[193] L. M. Sander. Fractal growth processes. Nature, 322:789–793, 1986.

[194] I. M. Sandler, G. S. Canright, H. J. Gao, S. J. Pang, Z. Q. Xue, and Z. Y. Zhang. Chiral patterns arising from electrostatic growth models. Phys. Rev. E, 58:6015– 6026, 1998.

[195] I. M. Sandler, G. S. Canright, Z. Y. Zhang, H. J. Gao, Z. Q. Xue, and S. J. Pang. Spontaneous chiral symmetry breaking in two-dimensional aggregation. Phys. Lett. A, 245:233–238, 1998.

[196] J. E. Segall, S. M. Block, and H. C. Berg. Temporal comparisons in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA, 83:8987–8991, 1986. 250

[197] J. A. Shapiro. Bacteria as multicellular organisms. Sci. Am., 258:62–69, 1988.

[198] J. A. Shapiro. Natural genetic engineering in evolution. Genetica, 86:99–111, 1992.

[199] J. A. Shapiro. The signiﬁcances of bacterial colony patterns. BioEssays, 17:597– 607, 1995.

[200] J. A. Shapiro and M. Dworkin, editors. Bacteria as Multicellular Organisms. Oxford University Press, New-York, 1997.

[201] J. A. Shapiro and D. Trubatch. Sequential events in bacterial colony morphogenesis. Physica D, 49:214–223, 1991.

[202] C. H. Shaw. Swimming against the tide: chemotaxis in Agrobacterium. BioEssays, 13:25–29, 1991.

[203] J. D. Sheppard, C. Jumarie, D. G. Cooper, and R. Laprade. Ionic channels induced by surfactin in planar lipid bilayer membranes. Biochim. Biophys. Acta, 1064:13–23, 1991.

[204] N. J. Shih and R. G. Labbe. Characterization and distribution of amylases during vegetative cell growth and sporulation of Clostridium perfringens. Can. J. Micro- biol., 42:628–33, 1996.

[205] O. Shochet. Study of late-stage growth and morphology selection during diffusive patterning. PhD thesis, Tel-Aviv University, 1995.

[206] O. Shochet, K. Kassner, E. Ben-Jacob, S. G. Lipson, and H. Muller-Krumbhaar.¨ Morphology transition during non-equilibrium growth: I. study of equilibrium shapes and properties. Physica A, 181:136–155, 1992.

[207] O. Shochet, K. Kassner, E. Ben-Jacob, S. G. Lipson, and H. Muller-Krumbhaar.¨ Morphology transition during non-equilibrium growth: II. morphology diagram and characterization of the transition. Physica A, 187:87–111, 1992.

[208] R. N. Smith and F. E. Clacrk. Motile colonies of Bacillus alvei and other bacteria. J. Bacteriol., 35:59–60, 1938. BIBLIOGRAPHY 251

[209] P. A. Spiro, J. S. Parkinson, and H. G. Othmer. A model of excitation and adaptation in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA, 94:7263–7268, 1997.

[210] R. Y. Stainer, M. Doudoroff, and E. A. Adelberg. The Microbial World. Prentice- Hall and Inc., N.J., 1957.

[211] J. B. Stock, A. M. Stock, and M. Mottonen. Signal transduction in bacteria. Nature, 344:395–400, 1990.

[212] M. Tcherpakov, E. Ben-Jacob, I. Cohen, and D. L. Gutnick. Paenibacillus vortex sp. nov., proposal for a new pattern-forming species and its localization within a phylogenetic cluster. submitted to Int. J. Syst. Bacteriol.

[213] M. Tcherpakov, E. Ben-Jacob, and D. L. Gutnick. Paenibacillus dendritiformis sp. nov., proposal for a new pattern-forming species and its localization within a phylogenetic cluster. Int. J. Syst. Bacteriol., 49:239–246, 1999.

[214] J. Toner and Y. Tu. Long-range order in a two-dimensional dynamical XY model: How birds ﬂy together. Phys. Rev. Lett., 75:4326–4329, 1995.

[215] J. Toner and Y. Tu. Flocks, herds, and schools: A quantitative theory of ﬂocking. Phys. Rev. E, 58:4828–4858, 1998.

[216] L. Tsimring, H. Levine, I. Aranson, E. Ben-Jacob, I. Cohen, O. Shochet, and W. N. Reynolds. Aggregation patterns in stressed bacteria. Phys. Rev. Lett., 75:1859–1862, 1995.

[217] N. Tsukagoshi, G. Tamura, and K. Arima. A novel protoplast-bursting factor (surfactin) obtained from Bacillus subtilis IAM 1213. II. the interaction of surfactin with bacterial membranes and lipids. Biochim. Biophys. Acta, 196:211–214, 1970.

[218] Y. Tu, H. Levine, and D. Ridgway. Morphology transition in a mean-ﬁeld model of diffusion-limited growth. Phys. Rev. Lett., 71:3838–3841, 1993.

[219] W. van Saarloos. Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection. Phys. Rev. A, 37:211–229, 1988. 252

[220] W. van Saarloos. Front propagation into unstable states: II. linear versus non-linear marginal stability and rate of convergence. Phys. Rev. A, 39:6367–6389, 1989.

[221] D. van Sinderen, R. Kiewiet, and G. Venema. Differential expression of two closely related deoxyribonuclease genes, nucA and nucB, in Bacillus subtilis. Mol. Micro- biol., 15:213–223, 1995.

[222] T. Vicsek. Fractal Growth Phenomena. World Scientiﬁc, New York, 1989.

[223] T. Vicsek, A. Czirok,´ E. Ben-Jacob, I. Cohen, O. Shochet, and A. Tenenbaum. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett., 75:1226–1229, 1995.

[224] D. Vollenbroich, G. Pauli, M. Ozel, and J. Vater. Antimycoplasma properties and application in cell culture of surfactin, a lipopeptide antibiotic from Bacillus subtilis. Appl. Environ. Microbio., 63:44–49, 1997.

[225] D. Vollhardt, G. Emrich, T. Gutberlet, and J.-H. Fuhrhop. Chiral discrimination and pattern formation in N-Dodecylmannonamide monolayers at the air-water interface. Langmuir, 12:5659–5663, 1996.

[226] J. Y. Wakano, A. Komoto, and Y. Yamaguchi. Phase transition of traveling waves in bacterial colony pattern. Physical Review E, 69:051904, 2004.

[227] R. Weis and H. McConnell. Nature, 310:47, 1984.

[228] A. A. Wheeler, W. J. Boettinger, and G. B. McFadden. Phase-ﬁeld model for isother- mal phase transitions in binary alloys. Phys. Rev. A, 45:7424, 1992.

[229] A. A. Wheeler, B. T. Murray, and R. J. Schaefer. Computation of dendrites using a phase-ﬁeld model. Physica D, 66:243–62, 1992.

[230] N. Wiener. Cybernetics: or Control an Communication in The Animal and Machine. Wiley, New-York, 1948.

[231] T. A. Witten and L. M. Sander. Diffusion–limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett., 47:1400–1403, 1981. BIBLIOGRAPHY 253

[232] T. A. Witten and L. M. Sander. Diffusion-limited aggregation. Phys. Rev. B, 27:5686–5697, 1983.

[233] G. Wolf, editor. Encyclopaedia Cinematographica. Institut fur¨ Wissenschaftlichen Film, Gottingen,¨ 1968.

[234] A. J. Wolfe and H. C. Berg. Migration of bacteria in semisolid agar. Proc. Natl. Acad. Sci. USA, 86:6973–6977, 1989.

[235] H. Zabrodsky and D. Avnir. Continuous symmetry measures .4. chirality. J. Am. Chem. Soc., 117:462–473, 1995.

תקציר iv iv

בפרק האחרו של עבודה זו אני מראה שימוש במודלי למחקר נוס% . אני מטפל בתופעה של פריצת סקטורי מתו! מושבות גדלות . אני מראה כיצד התופעה ניתנת להסבר כשינוי קט בחיידקי שבסקטור , שינוי המתואר על ידי שינוי של פרמטר יח יד בתיאור החיידקי . תיאורי שוני של החיידקי , על ידי מודלי שוני , , משמשי לחקירת סוגי שוני של סקטורי . . לסיכו , ומגו התבניות שמושבות חיידקי מציגות נראה כמעט אינסופי , בגלל האופני שחיידקי שוני מגיבי לגורמי סביבתיי שוני . מחקר מתמטי , כגו המחקר המ וצג בעבודה זו , יכול לבודד את התכונות הביולוגיות הקריטיות להתנהגות ה מושבות . בנייה של מודלי גנריי מתאימי היא צעד חשוב לקראת הבנת החיידקי . ביכולתנו להיות יותר בטוחי בטענות הביולוגיות שנובעות מהמודלי א נבצע בדיקה חוזרת בעזרת מודלי מסוגי שוני , כמו מוד לי רציפי ובדידי . .

תקציר iii

אלא שהגבול קשור לחומר סיכה שהחיידקי יוצרי לש תנועה על משטח קשה . ההשפעה של חומר הסיכה על תנועת החיידקי מתוארת על ידי משוואת ריאקצי ה ה דיפוזיה לשדה צפיפות החיידקי , כשהדיפוזיה היא לא לינארית ומושפעת מצפיפות החיידקי . במקרה כזה גבול המוש בה הוא קו ייחודיות של פתרו חלש של מערכת המשוואות . תכונות נוספות של תבניות המושבות נגרמות על ידי כמוטקסיס של החיידקי ( כמוטקסיס הוא תגובה של החיידקי לשינויי מרחביי בריכוז כימיקל כלשהו . החיידקי משני את התנועה שלה כ! שתהיה בה הטיה לתנועה בכיוו ריכוז ג בוה או ריכוז נמו! ). בעזרת מודל בדיד אני מראה שצורת הכמוטקסיס החשובה היא לא תגובה למזו , שהיא הצורה הנפוצה ביותר , אלא כמוטקסיס לאותות ( איתות בעזרת כמוטקסיס ) . החיידקי מגיבי לחומרי שנפלטי על ידי חיידקי אחרי . תקשורת כימית בי החיידקי מובילה להתנהגות שית ופית שמשפיעה על מבנה המושבה . . חיידקי T שיוצרי את תבניות הענפי המתפצלי בקצוות קרובי מאוד לחיידקי C שיוצרי את תבניות הענפי המסולסלי . ג מודל רצי% וג מודל בדיד מראי שההבדל הקריטי ביניה הוא אור! החיידקי . חיידקי C ה הרבה יותר ארוכי מחיידקי T . אורכ מגביל את הסיבוב שלה במקומות צרי . כשענפי קטני נוצרי , אפשרות הסיבוב של החיידקי מוגבל .ת הענפי נהיי ארוכי יותר וצרי יותר , והחיידקי מוגבלי לתנועת קדימה אחורה בלבד . הסלסול של הענפי קשור לתנועה הסיבובית של " מנוע " החיידקי , והאינטראקציה שלו ע המשטח . כמו ע קרוביה , חיידקי T , תוצאות המודלי מראות שהתפתחות המושבות מושפעת על ידי כמוטקסיס . המודלי נותני ג תמיכה נוספת לטענה שחיידקי T מוש פעי על ידי איתות בעזרת כמוטקסיס . קיימי תנאי של מחסור במזו שבה חיידקי T נהיי ארוכי יותר משה בדר! כלל . הענפי של המושבה נהיי צרי יותר , ומעט נוטי בסיבוב . תופעה זו נקראת סלסול עדי . המודלי לחיידקי C , כשה מותאמי לחיידקי T ארוכי , מראי שהסלסול נובע מתגובה כמוטקטית לאותות . . לחיידקי V שיוצרי את המערבולות יש צורת תנועה שונה מאשר T וC . ה אינ נעי בתנועת הילו! מי קרי , אלא בת נועה קדימה . מודלי מראי שמערבולות נוצרות כתוצאה מאינטראקציות של מגע בי החיידקי . האינטראקציות גורמות לתאו בתנועת חיידקי שכני . בחלקי גדולי של מרחב הפרמטרי , יש שני פתרונות יציבי למשוואות התנועה של החיידקי . פתרו אחד הוא תנועה אחידה בכיו ו אחד . הפתרו השני הוא תנועה סיבובית מתואמת . .

תקציר ii ii

שלושה סוגי של תבניות מושבתיות נחקרי בעבודה זו . כל אחד מה נוצר על ידי סוג שונה של חיידקי , וכל אחד מה מקבל טיפול מ תמטי מיוחד המסביר את התפתחות מושבות החיידקי . הסוג הראשו של תבניות הוא תבניות ענפי המתפצלי בקצוות ( ראה תמונה (a 1.1 .) ) תבניות אלה נוצרות על ידי חיידקי מסוג Paenibacillus dendritiformis var. dendron , אליה נתייחס כסוג T . המושבה בנויה בתבנית ענפי , וענפי חדשי נוצרי בזמ הגידול כאשר ענפי ישני ורחבי מתפצלי לשניי . החיידקי במושבה מבצעי הילו! מקרי דו מימדי המוגבל על ידי גבולות המושבה . . הסוג השני של תבניות הוא תבניות ענפי מסולסלי ( ראה תמונה (b 1.1 ) ). תבניות אלה נוצרות על ידי חיידקי מסוג .Paenibacillus dendritiformis var chiralis , אליה נתייחס כסוג C . הענפי דקי , ארוכי , ומסתלסלי הצידה בזמ גדילת ( תמיד לאותו צד , בכל חלקי המושבה ). ענפי חדשי גדלי מתו! ציד של ענפי קיימי . החיידקי בתו! הענפי מבצעי הילו! מקרי כמעט חד מימדי , מכיוו שהענפי צרי מ ידי מכדי שהחיידקי יסתובבו בתוכ . . הסוג השלישי של תבניות הוא תבניות ענפי המובלי על ידי מערבולות ( ראה תמונה (c 1.1 ) ). תבניות אלו נוצרות על ידי חיידקי מסוג Paenibacillus vortex , אליה נתייחס כסוג V . תבניות אלו מתייחדות בתנועת החיידקי : בקצה כל ענ% גדל ( ולע יתי ג בתוכו ) יש קבוצה של חיידקי המסתחררת יחד , כמו מערבולת . . אני חוקר כל סוג של חיידקי בעזרת שני מודלי לפחות . זוהי בדיקה כפולה המספקת יותר ביטחו לפירוש הביולוגי של הטענות המתמטיות . המודלי משתייכי לשתי קטגור יות שונות : קטגוריה :1 מודלי בדידי , שבה החיידקי מיוצגי על ידי ישויות בדידות הממ משות תהלי! סטוכסטי . ישויות כאל ה צורכות מזו ומייצרות כימיקלי , מתרבות , זזות , ומגיבות לחומרי שוני . קטגוריה 2 : מודלי רציפי של ריאקצי ה דיפוזיה , ע כמה שדות דיפיוסיביי מצומדי . . במודלי אלה הח יידקי מיוצגי על ידי צפיפות דו מימדית , וההשתנות בזמ של שדה הצפיפות מתוארת על ידי משוואת דיפרנציאליות חלקיות מסוג ריאקצי ה ה דיפוזיה . משווא ת הצפיפות של החיידקי מצומדת למשוואות ריאקצי ה דיפוזיה של ריכוזי כימיקלי שוני ( בפרט מזו ). בחלק מהמודלי תנועת החיידקי מתוארת ג על ידי שדות אחרי , כגו שדה מהירות או שדה אוריינטציה . . עבור חיידקי T , אני מוצא שקיו גבול כלשהו למושבה הוא חיוני להווצרות תבניות הענפי המתפצלות בקצוות . בעזרת מודל רצי% אני מראה שהגבול אינו מתואר על ידי שינוי פאזה כלשהו באיבר הריאקציה ( צריכת מז ו , התרבות ומוות ) , )

תקציר מוקד עבודה זו הוא חקירה של מושבות חיידקי . בעבר , גידול מושבות חיידקי במעבדה נעשה בדר! כלל על מצעי מוצקי למחצה ( וריכוזי אגא ר בינוניי ) ע רמות גבוהות של חומרי מ זיני ב. תנאי ' ידידותיי ' כאלה מושבות החיידקי מתפתחות בצ וור ת פשוט ות וקומפקטי .תו צורות אלה תואמות את הדעה המקובלת שמושבות חיידקי ה אוס% של יצורי חד תאיי בלתי תלויי '( חלקיקי בלתי תלויי ` , בניסוח המקובל אצל פיסיקאי ). בטבע, לעומת זאת, חיידקי מתמודדי לעיתי קרובות ע סביבות קשות . לש התמודדות זו יפ, תחו החיידקי יכולת לפעילות שיתופית מתוחכמת ותקשורת מורכבת , כ! ש בתגובה לתנאי סביבה קשי , המושבות גדלות בתבניות מורכבות במרחב ובזמ . כשיוצרי במעבדה תנאי קשי בצלחות פטרי , כגו ריכוזי מזו נמוכי או מצע י קש י ( או שניה ,) נוצרות מושבות הגדלות בתבניות מורכבות כמו ב תמונה 1.1 בפרק Introduction . . בעבודה זו אני מפתח תיאור מתמטי של מושבות חיידקי מורכבות ונעזר בו ודילק ההב נה של ה מערכות ה ביולוגיות ה אלה , ואת העקרונות השולטי בהתפתחות . אני מאמ' את הגישה שהכלי התיאור טי היעיל ביותר שברשותנו הוא הניסוח הפורמאלי של מודלי מתאימי '( מודלי גנריי ' ) . לפי גישה זו , המודלי צריכי להתאי לעובדות ביולוגיות ידועות ולתפוס את תמצית המערכת הביולוגית . המודלי צריכי להתאי ל עובדות ביולוגיות ידועות , א! לא לכלול עובדות ביולוגיות שא ינ רלוונטיות לתופע ות הביולוגיו ת הנחקר .תו .תו משנוסח מודל שכזה , יוו דא נו שאכ הוא מתאר את התופע ות ה ביולוגי תו , הוא יכול לספק ל נו ידע על המערכת הביולוגית בשתי דרכי שונות . ראשית , עצ היכולת להבדיל בי עובדות ביולוגית רלוונטיות ולא רלוונטיות היא בעלת חשיבות רבה, מ שו ש הבנה של המשמעות הביולוגית של הביטויי המתמטיי מראה לנו את התכונות החיוניות לש הווצרות התופעות הביולוגיות . שנית , ניתוח של המודל מאפשר לנו לנסח תחזיות על התנהגות המערכת הביולוגית , כגו גילוי תכונות ביולוגיות לא ידועות הכלולות ב מודל , התנהגות המערכת בתנ אי חדשי , או ההשפעה של הכללת אלמנטי נוספי במערכת . . מושבות חיידקי מציגות עושר רב של תבניות יפות בזמ התפתחות המושבות בתנאי גידול שוני . בעזרת רעיונות שנלקחו מ תורת ה יה ווצרות של תבניות במ ערכות לא חיות , ובעזרת מודלי ' גנריי ', ביכולתנו לחשו% אסטרטגיו ת ת הישרד תו חדשות הקשורות לפרטי התבניות המתפתחות . בעזרת המודלי , אני מראה כיצד תקשורת מובילה לארגו עצמי של המושבות דר! התנהגות שיתופית של התאי . .

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