Mathematical Studies of Pattern Formation and Cooperative Organization in Bacterial Colonies

Mathematical Studies of Pattern Formation and Cooperative Organization in Bacterial Colonies Thesis Submitted for the Degree ”Doctor of Philosophy” by 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 (intermedi- ate 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 intri- cate 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, be- cause 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.

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