A Two-Component Model for Bacterial Chemotaxis

Home , Chemotaxis

A TWO-COMPONENT MODEL FOR BACTERIAL CHEMOTAXIS

THESIS

Presented in Partial Fulﬁllment of the Requirements for the Degree of Master of

Science in the Graduate School of the Ohio State University

Clinton H. Durney, BS

Graduate Program in Mathematics

The Ohio State University

2013

Dissertation Committee:

Dr. Chuan Xue, Advisor

Dr. Ching-Shan Chou

Dr. Daniel Wozniak c Copyright by

Clinton H. Durney

2013 ABSTRACT

Chemotaxis is the directed cell movement in response to chemical signals. It is crucial in many multicellular processes, such as wound healing, cancer metastasis, embryonic development and bioremediation. Quantitative descriptions of chemotaxis will lead to a better understanding of the mechanisms of chemotaxis and are essential in understanding the aforementioned processes.

In this thesis, we study chemotaxis of the run-and-tumble bacteria E. coli. We develop a mathematical model of E. coli chemotaxis at the single cell level. The model consists of two modules: The ﬁrst describes how the cell transduces the external signal into an internal signal (i.e. the change of the concentration of the intracellular protein

CheYP ). Our description is based on an existing “trimers of dimers” model and we improve the parameter estimation in this work. The second module is the change of cell movement in response to the internal CheYP change. We propose a new method to interpret existing data on ﬂagellar rotation. Finally we couple these two modules and use the full model to simulate a population of cells.

ii ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Chuan Xue for her patience, expertise, guidance and encouragement during this process. I would also like to acknowledge my committe members Dr. Daniel Wozniak and Dr. Ching-Shan Chou for their contributions to this project.

iii VITAE

1989 ...... Born, Cumberland, MD

2011 ...... B.S. Physics, B.S. Mathematics, Virginia Tech, Blacksburg, VA

2011-Present ...... Graduate Teaching Associate, The Ohio State University

PUBLICATIONS

C.H. Durney, S.O. Case, M. Pleimling, and R.K.P. Zia; Stochastic Evolution of Four Species in Cyclic Competition, J. Stat. Mech., 06, P06014, (2012)

C.H. Durney, S.O. Case, M. Pleimling, and R.K.P. Zia; Saddles, Arrows, and Spirals: Deterministic Trajectories in Cyclic Competition of Four Species, Phys. Rev. E. 83, 051108, pp.1-13 (2011)

S.O. Case, C.H. Durney, M. Pleimling, and R.K.P. Zia; Cyclic Competition of four species: mean ﬁeld theory and stochastic evolution, EPL 92, 58003, pp.1-6 (2010)

iv FIELDS OF STUDY

Major Field: Mathematics

Specialization: Mathematical Biology

v TABLE OF CONTENTS

Abstract ...... ii

Acknowledgments ...... iii

Vitae ...... iv

List of Figures ...... viii

CHAPTER PAGE

1 Introduction ...... 1

1.1 Motivations ...... 1 1.2 Biological Background ...... 2 1.3 Deterministic Models of Reaction Networks ...... 6 1.4 Stochastic Models for Biological Processes ...... 8

2 An ODE Model of E. coli Signal Transduction ...... 11

2.1 The Trimer’s of Dimers Model by Xin & Othmer ...... 11 2.1.1 Reduction of the Model ...... 15 2.2 An Alternative Parameter Estimation ...... 18

3 A New Stochastic Model of E. coli Chemotaxis ...... 22

3.1 Experimental Results of Cluzel et. al...... 22 3.2 A Two-Rate Model of Run and Tumble ...... 23

4 An Integrated Model of E. coli Movement ...... 29

4.1 A Simple Case with a Single Rate ...... 29 4.2 The Full Model ...... 32

5 Conclusions and Future Work ...... 35

Bibliography ...... 38

vi Appendix: Glossary ...... 40

vii LIST OF FIGURES

FIGURE PAGE

1.1 A schematic diagram of the signal transduction network of an E. coli cell. The ﬁgure outlines the interactions of the various components necessary for bacterial chemotaxis. Figure reproduced with permission from [22]...... 5

2.1 Top: The transition network of the signaling complex T from [1]. Top left describes the notation of the signaling complex, the top shows the transitions with vertical being ligand binding, front-to-back is activity regulation and phosphotransfer and horizontal being methylation and demethylation. Bottom: Shows a typical transition chain of the above matrix. Reproduced with permission from [1]...... 13

2.2 Comparison of [1]’s full 158 ODE model (Blue Line) versus the reduced 4 ODE model (Green Line). The system is simulated for 1000 seconds. The top(bottom) panel corresponds to the addition of a 1.0(10) µM α-methyl-aspartate signal at t=200 and removed at t=600...... 17

2.3 Example of the alternative method of parameter ﬁtting. [1] uses a linear relationship between the two extremum values (blue line). Our method uses a quadratic approximation (green line) better capturing the convex shape of the parameters. (a) Probability unphosphorylated m complex is active, pm, 1 (b) Rate constant in activity regulation, k−1. 19 2.4 Comparison of [1]’s Full 158 ODE Model (Blue Line), the reduced 4 ODE model (Green Line) and the reduced model with alternative parameter ﬁtting (red line). The system is simulated for 1000 seconds. The top(bottom) panel corresponds to the addition of a 1.0(10) µM α-methyl-aspartate signal at t=200 and removed at t=600...... 20

viii 3.1 Experimental results of Cluzel et. al. showing the response of individual motors as a function of CheYP concentration. Red dots correspond to experimental data points and green lines correspond to calculated functions (a) Clockwise bias of the motor versus CheYP . The function is a Hill function with coeﬃcient NH = 10.3 ± 1.1 and KM = 3.1µM. −1 (b) Switching Frequency versus CheYP (s ). The function is the derivative of the Hill function of (a). (c) k1 calculated from the data points and the functions of (a) and (b). (d) k2 calculated from the data points and the functions of (a) and (b)...... 24

3.2 Red dots correspond to experimental data points and green lines correspond to calculated functions (a) Clockwise bias for ﬂagella rotation as a function of CheYP concentration (µM). Function corresponds to calculating BCW using k1 and k2 of (c) and (d) below. (b) Switching frequency of given cells based on CheYP concentration (µM), SW (y). Function corresponds to calculating SW using k1 and k2 of (c) and (d) below. (c) k1 as a function of CheYP concentration between 0 and 6.0 µM. Data points calculated directly using (3.3.8) and the data points −(b−y)4 of [6]. The function is the exponential curve k1(y) = ae c with a = 12.0809, b = −5.83762 and c = 2892.12. (d) k2 as a function of CheYP concentration between 0 and 4.5 µM. Red data points calculated directly using (3.3.9) and the data points of [6]. The function is by the exponential curve k2(y) = ae with a = 0.0174001 and b = 1.32887. 28 4.1 Simulation of a population of 500 cells. The two-components of the model were comprised of the reduced 4 ODE transduction model and a simple linear turning rate for ﬂagella rotation. Inset: The exponential signal the cells were subjected to...... 30

4.2 Intracellular concentration (µM) of CheYP of a a typical cell in Figure 4.1...... 32

4.3 Scatter plot revealing the intracellular concentration (µM) of CheYP and X position of all cells in Figure 4.1 taken at t = 1 (red), t = 50 (blue), t = 300 (magenta), t = 600 (green), t = 850 (cyan) and t = 1000 (black) ...... 33

4.4 Simulation of a population of 5 cells. The two-components of the model were comprised of the reduced 4 ODE transduction model and the results of Section 3.3 for ﬂagella rotation subjected to the same exponential signal 4.1.1...... 34

ix CHAPTER 1

INTRODUCTION

1.1 Motivations

One of the fundamental mechanisms cells utilize to sustain life is controlled movement, which is necessary for nutrient regulation, pH balance, temperature regulation, evading threats (e.g. toxins), oxygen regulation, reacting to a light source and producing colonies that provide advantageous results through cooperation. This directed cellular movement is known as taxis. Whenever, this cellular movement is in response to a chemical signal, the result is chemotaxis [1, 2, 8, 11].

Chemotaxis is ubiquitous in biological systems and is necessary to create and sustain life. For example, a sperm cell follows a chemoattractant secreted by the egg to direct its movement for fertilization [17]. Continually, during fetal development chemotaxis is vital for properly outlaying the developing fetus [19]. Another example, is the immune response as neutrophils track a chemical excreted as waste by bacteria.

The neutrophil eventually catches the bacteria, engulfs and neutralizes the invader

[20]. Continually, lymphocytes play an integral part in wound healing and are directed by chemotaxis [18]. Lastly, this process can be utilized for bioremidiation, as bacteria can be used to eliminate waste from ecosystems [21].

There are several methods by which cells move. Eukaryotic cells crawl or swim by

1 changing their shape. Bacteria can move by swimming, swarming, gliding or twitch- ing. In particular, E. coli bacterium swim in a run-and-table strategy by changing the rotation of its ﬂagella. This is the result of a cascade of chemotactic proteins when the cell is presented with an external chemical signal. At the end of the protein network are 6-8 ﬂagella that rotate clockwise or counter-clockwise causing the cell to either continue moving in the same direction or “tumble” into a new direction chosen almost randomly (there is a slight bias to the previous direction) [1, 12]. Due to this behavior, chemotaxis can simply be thought of as an input - output system, as the cell is presented with a signal (input) and then the cell either runs or tumbles

(output).

Bacteria, being one of the simplest living organisms makes them paradigmatic models for unlocking the mysteries of life. E. coli, the simplest bacteria, is the most widely studied due to its accessibility and relative simplicity. For this study we will be studying E. coli cells at the individual cellular level in order to understand the signal transduction network responsible for chemotaxis. Understanding this process can have many applications from responding to fertility issues, developmental errors, immune response, and gaining a better understanding of how cancer spreads.

1.2 Biological Background

The signal transduction network for chemotaxis of a single E. coli cell begins with the signaling complex that transcends the cellular membrane. A single cell has four

Membrane Cofactor Proteins (Tsr, Tar, Tap and Trg) and a ﬁfth that functions by mimicking an MCP protein (Aer). These unites serve as the signaling complex of an individual cell and are able to sense a chemical signal in the cytoplasmic domain.

2 These complexes have methylation sites that are responsible for controlling the activity of the transduction network [1]. The signaling complex is described by three main components, that is its:

• Ligand binding state

• Activity state

• Methylation state

As mentioned, the methylation level controls the chemotactic (Che) proteins involved in this network, CheA, CheW , CheY , CheZ, CheR and CheB. Each Che protein has a unique role in transferring the external stimuli to the biomechanical motor that controls the movement of the cell. Before describing the signaling cascade, we will brieﬂy list the role of each Che protein involved.

• CheA and CheW - form a stable ternary signaling complex in the cytoplasmic

domain of the chemoreceptor

• CheA - autophosphorylates and transfers a phosphoryl group to CheY and

CheB

• CheY - When in the phosphorylated state (CheYP ), it diﬀuses to the ﬂagellar motor, binds to the FliM protein biasing the rotation direction of the biome-

chanical motor

• CheZ - catalyzes the dephosphorylation of CheYP

• CheR - methylates glutamate, unregulated

• CheB - demethylates glutamate and deamidated glutamine to glutamate, highly

regulated by CheAp

3 The signaling cascade begins when the signaling complex is stimulated by a ligand.

This triggers the regulation of the autophosphorylation of CheA. As CheA phosphorylates, it transfers a phosphoryl group to CheY and CheB, that is

CheAP + CheY → CheA + CheYP and

CheAP + CheB → CheA + CheBP

In order for CheY to regulate the flagellar motor, it must be in the phosphorylated state, CheYP . Once phosphorylated, CheYP diffuses to the flagellar motor and binds to the FliM protein. Once successfully bound, the probability of clockwise (CW) rotation increases, therefore increasing the tumbling rate. This mechanism is then reversed by CheZ which is responsible for catalyzing the dephosphorylation of CheA,

CheYP + CheZ → CheY + CheZ.

In the following, we brieﬂy summarize the cellular network thus far. If a cell is presented with a positive signal, the amount of CheYP will decrease, consequently decreasing the probability of a tumble. If a cell is less likely to tumble, its runtime will increase. If a positive signal begins to decrease, then CheYP will begin to increase leading to an increase in tumble and a decrease in run time. The converse holds for a negative signal [1].

The second segment of the chemotactic pathway involves the adaptation inherent to the network. As stated earlier, CheA also phosphorylates CheB which aﬀects the methylation activity of the MCP receptors. The chemoreceptors have four or more sites that are occupied by either a glutamine (Q) or a glutamate (E) in the intracellular domain. The activity of CheA (which leads to phosphorylating CheY ) is regulated by the methylation level of the MCP receptors. The higher the methylation 4 Figure 1.1: A schematic diagram of the signal transduction network of an E. coli cell.

The ﬁgure outlines the interactions of the various components necessary for bacterial chemotaxis. Figure reproduced with permission from [22].

5 level, the higher the activity of CheA. On the receptor, CheR methylates glutamate

(E → EM ) and CheB demethylates this (EM → E). CheB also has another role by deamidating glutamine to glutamate (Q → E). Therefore, this activity is controlled by ligand binding of the extracellular domain and the methylation level in the intracellular domain. Since, CheB is activated by CheAP , this causes a negative feedback loop of the system. Thus, CheAP causes CheYP levels to increase, but also negatively feeds back through CheB and the demethylation of the receptors.

One important feature of the transduction network is the time scale of the diﬀer- ent reactions. The initial response to a signal is characterized by the ligand binding and initial kinase response which happens on the order of milliseconds. This is then followed by the transfer of the phosphoryl groups which happens in the span of about tenths of seconds to seconds. Lastly, the methylation and demethylation is the slowest and occurs on the scale of seconds to minutes and is much slower than the former processes. Continually, it is this interplay of an initial fast response of CheYP and slow adaptation through CheBP that creates the variety in CheYP concentrations necessary for a cell to respond to varying signals. The overarching concept of the chemotactic network is summarized in [1] nicely as “The overall response to an attractant can be summarized as a rapid decrease in kinase activity and CheYP , followed by a slower restoration of the kinase activity due to methylation of occupied receptors.”

1.3 Deterministic Models of Reaction Networks

The ﬁeld of mathematical biology, particularly computational cellular biology, has exploded in the last decade. The increasing power of computers and availability of large amounts of memory, have enabled a “bottom-up” approach to biological systems. Starting from the building blocks of life, we can simulate cellular models from

6 their fundamental features. Conversely, the same power computers provide can also provide a “top-down” approach and draw conclusions from population based models, much like experiments. From a mathematical perspective, this provides many different mathematical challenges to answer the questions that nature poses. Top-down models generally involve partial differential equations (PDEs) which model entire cellular compositions in a continuous medium. Bottom-up models typically involve ordinary (or stochastic) differential equations, concerning themselves with the specific concentrations of proteins and how they interact with each other. Biological experiments have revealed the need for greater quantitative results which are obtainable by creating mathematical models based on the essential features of the system and then carrying out the calculations [13].

Deterministic models are based on mass action kinetics, where a basic chemical reaction follows the form

k+ A + B C (1.3.1) k− where A,B,C could be concentrations of chemicals, proteins or even species populations. Motivating this framework is if given large enough quantities of A and B in a well-mixed system, they will form compound C at a constant rate, k+ while C dissociates into A and B at rate k−. Therefore, we can write the time evolution of the concentrations of A,B and C as such:

d A = −k AB + k C (1.3.2) dt + − d B = −k AB + k C (1.3.3) dt + − d C = k AB − k C (1.3.4) dt + −

This yields 3 Ordinary Diﬀerential Equations that can reveal the derivative (time- evolution) of each species of interest; A, B and C. Oftentimes these cannot be solved 7 analytically to obtain closed form solutions A(t), B(t) and C(t) and we must therefore rely on numerical solving techniques such as Euler’s Method, Runge-Kutta Methods or solving routines available in MATLAB such as ODE15s or ODE45.

In more general cases, not only is the concentration important but also the spatial structure of the component of interest. This results in a Partial Diﬀerential Equa- tion (PDE). This paper is not concerned with spatial structures and instead will assume that the chemotactic proteins are evenly distributed and ubiquitous within the system.

1.4 Stochastic Models for Biological Processes

The most fundamental stochastic model is the one-dimensional random walk. The most colorful scenario is that a drunkard stumbles out of a bar and is trying to make it home, while others include a particle that move randomly. We will consider a particle that has a probability of stepping to the right and to the left and that each step is independent of the previous steps that it has taken. We want to concern ourselves with questions such as on average where does(do) the particle(s) end up?

On average how long does it take it for a particle to make it to a specific boundary or an absorbing state? And on average how long will the particle stay in motion? 1 Therefore, lets consider a particle who steps to the right with probability + q 2 1 1 and to the left with probability − q where |q| ≤ . The case with q = 0 would 2 2 1 1 be considered unbiased, − < q < 0 biased to the left and 0 < q < biased to 2 2 the right. We will also assume the particle to be uncorrelated, meaning each step is of length 1 and independent of the previous steps taken. Therefore, the particle only takes on integer values. That is, if we let XN denote his position after N steps, we have XN = X0 + X1 + ... + XN−1 where each Xj (j = 1, ..., N − 1) is a random 1 1 variable with P (X = 1) = + q and P (X = −1) = − q. Choosing an initial j 2 j 2 8 position, say X0 = 0 for simplicity, where will he end up? We define, W (m, N) to be the probability that a walker will end up m steps to the right of the starting point after N steps. Therefore we have the initial condition,   1 if m = 0 W (m, 0) =  0 if m 6= 0 and now we are looking for arbitrary N > 0. Noticing that the ordering of the steps does not matter, we realize we only need to consider the total steps to the right and to the left. For example, the sequence of steps right, right, left results in the same m = 2 as the sequence left, right, right. The only important values are (2 steps right) N + m - (1 step left) = 1. Thus, we let the total steps to the right be and to the left 2 N − m . Therefore, 2 N + m N − m + = N (1.4.1) 2 2 N + m N − m − = m (1.4.2) 2 2 Where 1.4.1 is the sum of steps to right and left yielding total steps (N) and 1.4.2 is the number of steps to the right minus the steps to the left yielding the final location N + m (m) as desired. The probability of taking steps to the right is 2

1 1 1 1 N+M ( + q)( + q)...( + q) = ( + q) 2 2 2 2 2 N − m and taking steps to the left is 2

1 1 1 1 N−M ( − q)( − q)...( − q) = ( − q) 2 2 2 2 2 N Since these steps can occur in any order we include the binomial coeﬃcient N−m 2 to account for all possible combinations. This yields the probability of landing at location m after N total steps as N 1 N+M 1 N−M 2 2 W (m, N) = N−m ( + q) ( − q) if N − m ≥ 0 (1.4.3) 2 2 2 9 Using a generating function, one can obtain the expectation value and variance of the displacement, m to be

E[m] = 2Nq

1 1 E[(m − E[m])2] = 4N( − q)( + q) 2 2 Now we wish to consider a population of cells and how their density changes through time. Using the construction previously established we write the probability of a single cell to be at location m at the next time step N + 1 using 1.4.3

1 1 W (m, N + 1) = ( + q)W (m − 1,N) + ( − q)W (m + 1,N) (1.4.4) 2 2

We now ﬁnd the diﬀerence between successive discreet time steps,

W ( m , N + 1) − W (m, N) = 1 1 ( + q)W (m − 1,N) + ( − q)W (m + 1,N) − W (m, N) (1.4.5) 2 2

Deﬁning the average probability density as

W (m, N) u(x, t) = (1.4.6) 2∆x allows us to write the diﬀerence equation for the probability density 1.4.6 as follows

u(x, t + ∆t) − u(x, t) = ∆t ( 1 + q)u(x − ∆x, t) + ( 1 − q)u(x + ∆x, t) − u(x, t) ∆x 2 2 ( )2 (1.4.7) ∆t ∆x

Taking the limit of 1.4.7 as ∆x → 0 and ∆t → 0 we obtain the diﬀusion equation,

ut = Duxx − Sux (1.4.8)

(∆x)2 2q∆x where and → D, a diﬀusion term and → S, a drift coeﬃcient and are 2∆t ∆t properties of the biomechanics of the particles in motion as well as the medium they are in. 10 CHAPTER 2

AN ODE MODEL OF E. coli SIGNAL TRANSDUCTION

It is often stated that mathematics is the language of the universe, which has rang true in the past as physics ﬂourished through the use of sophisticated mathematical techniques. Now in increasing numbers, biologists are wishing to make use of similar mathematical methods to further explain the world in which we live. Now, problems from biology are inspiring new mathematics and are helping to solve some complex problems. One of the ﬁrst steps when looking at a biological process is discerning the relevant components needed in developing an accurate mathematical representation of the phenomenon. In chemotaxis of E. coli, much of this work has already been developed with several competing mathematical models. This reaction network is now in a stage that the models need to be improved upon and utilized to draw broad biological conclusions. The following chapter will detail the mathematical model that will be used during this study.

2.1 The Trimer’s of Dimers Model by Xin & Othmer

Now that we have an understanding of the biological components of the chemotactic pathway, we wish to extend our knowledge of the biology to a mathematical framework to gain a deeper understanding of this delicate process. There are many

11 diﬀerent models already existing for this pathway, each based on slightly diﬀerent assumptions. These range from classical Ising models to the Monod-Wyman-Changeux

(MWC) model. Therefore, at this point to recreate a model from scratch based on different assumptions would be superfluous. Consequently, we will base this study off of an already existing model, which is based on a complete dynamic model for a single trimer of dimers. In the following section, I will outline the model used for this study as developed in [1].

The idea behind the model is to think of it as an input-ouput process. With this framework we can use the concentration of ligands as the input and the output being the concentration CheYP which controls the rotation direction of the biomechanical motor. In order to connect the two, they considered a signaling complex consisting of a trimer of identical receptor dimers ([1] discusses pure-type and mixed-type, we will limit ourselves speciﬁcally to pure-type). Each signaling complex is characterized by the three main components of the biological network, its ligand binding state, activity state speciﬁcally of CheA and its methylation state. They assume that each receptor dimer can only accept one ligand at a time. Consequently, the trimer has three possible states – 0, 1, 2 and 3. The next component, the activity of CheA, will be motivated by its primary responsibility of transferring a phosphoryl group to CheY . Henceforth, if CheA is active it will undergo the necessary autophosphorylation and phosphotransfer. This process will be represented by a transition from inactive to active-unphosphorylated to active-phosphorylated. Lastly, each trimer has

6 monomers that can be methylated to a maximum level of 12, yielding 13 possible states, 0, 1, 2, ..., 12. This leads us to describe the signaling complexes, denoted by

T as such:

[s:activitystate] T[m:methylationstate], [n:ligandbindingstate]

12 Figure 2.1: Top: The transition network of the signaling complex T from [1]. Top left describes the notation of the signaling complex, the top shows the transitions with vertical being ligand binding, front-to-back is activity regulation and phosphotransfer and horizontal being methylation and demethylation. Bottom: Shows a typical transition chain of the above matrix. Reproduced with permission from [1].

13 where

s = i(inactive), a(active, unphosphorylated), p(active, phosphorylated)

m = 0, 1, ..., 12

n = 0, 1, 2, 3

Figure 2.2 shows the diﬀerent transitions possible among the signaling complex. Verti- cal transitions describe ligand binding, front-to-rear transitions illustrate the activity state of CheA and horizontal transitions portray the methylation and demethylation of the signaling complex. Therefore, in response to a signal there is a shift down as the ligand binds. This is followed by a shift backwards as autophospohorylation begins to happen. This in turn also increases the levels of CheYP . The last step is a shift from left to right as the methylation level increases as the cell begins the slower adaptation phase.

From Figure 2.2, the methodology of Section 1.2.1 was followed to relate the network to rate equations. The rate equations relate every component necessary in each reaction and these are tabulated in [1]. Mass action kinetics are then utilized forming time-dependent differential equations for each component of the signal transduction network. This leads to 158 differential equations that can be used to simulate the sensory pathway, obtaining the necessary output of a CheYP concentration. The next step in the process was to establish the parameters of this model so that the differential equations could be numerically solved yielding the time evolution of each component.

Their method wasn’t to use data ﬁtting to establish the model parameters, but rather base them oﬀ of the available experimental data and estimates. Several of the parameters such as ligand binding rates are known experimentally. The rate constants for activity regulation have no known values, so they were estimated from observed

14 kinase activity of signaling complexes in the absence of a signal and making a quasi- steady-state approximation. Ligand binding and release was harder to approximate due to the stochastic nature of the process and a simple linear mapping was used.

2.1.1 Reduction of the Model

The full trimer of dimers model consists of 158 diﬀerential equations for a single cell. While this is feasible for simulating one cell, extending this to a population of cells would prove computationally taxing. Furthermore, the inherent nature of the time-dependent quantities begs the question whether or not, the dimension (number of time-dependent quantities) can be reduced to produced admissible output of

CheYP concentration. Therefore, [1] used two subsequent methods to reduce the dimensionality of the transduction network.

As already described, the chemotactic pathway includes reactions on three explicit time scales; fast, intermediate, and slow. The slow methylation and demethylation step occurs on the time of seconds to minutes, intermediate speed which involves the transfer of the phosphoryl groups on tenths of seconds to seconds, and the fastest being the initial response of ligand binding and initial kinase response on the order of milliseconds. Due to the diﬀerent time scales, it suggests a multi-time-scale analysis to reduce the dimensionality of the system.

The idea behind multi-time-scale analysis exploits the fact that the reactions occur on diﬀerent time scales, fast and slow reactions. Suppose that x is a slow variable and y is a fast variable. Following perturbation theory, we obtain the equations:

dx = f(x, y, ) (2.1.1) dt dy = g(x, y, ) (2.1.2) dt where is small and t is the time scale for the slow variable, x. Then following the 15 quasi-steady-state assumption (QSSA) the time derivative of the fast variable y is small and assumed to be 0. Following this method with the given example, we make dy the assumption that = 0. Now instead of having two-differential equations, we dt have one differential equation and one algebraic equation to solve. Carrying out this method on the 158 differential equation system, the dimensionality of the system is reduced to 16 ordinary differential equations and several algebraic equations. The 16 variables include 13 methylation states, CheBP , the active phosphorylated complexes

P (T ), and CheYP [8]. The second phase of dimension reduction of the system uses mean-ﬁeld theory speciﬁcally on the dynamics of the methylation states of the trimer of dimer complex.

In this method, simpliﬁcations are made by using the average behavior of all individ- uals of the system, in our case kinase activity as a response of average methylation levels. This essentially collapses the cube in Figure 2.1 into a square by averaging over all of the methylations states. In using the average instead of incorporating each individual component the dimensionality of the system was further reduced to

4 ordinary diﬀerential equations by combing the 13 ODEs of each methylation state

P into 1 and keeping CheBP , the active phosphorylated complexes (T ), and CheYP as before.

For a full explanation of model development, parameter estimation, the reduction of the model and results validating the methods, please refer to [1].

Looking at the comparison of the full model to the reduced model in Figure 2.2, we see that the reduced model produces a fairly good representation of the time evolution of CheYP . Elucidating the details, we ﬁrst see the initial fast kinase response to a ligand being added (removed) at t = 200(600). We then see the slower adaptation of

CheYP as levels return to the steady state level of CheYP = 3.12 for 200 < t < 600. Another notable feature is that the L = 10µM signal provides a much larger kinase

16 + Attr − Attr 3.4 Full Model 3.35 Reduced Model

3.3

3.25

3.2 M) µ (

P 3.15

CheY 3.1

3.05

2.95

0 200 400 600 800 1000 Time

(a) L = 1 µM

+ Attr − Attr 14 Full Model 12 Reduced Model

M) 8 µ ( P

6 CheY

0 0 200 400 600 800 1000 Time

(b) L = 10 µM

Figure 2.2: Comparison of [1]’s full 158 ODE model (Blue Line) versus the reduced 4 ODE model (Green Line). The system is simulated for 1000 seconds. The top(bottom) panel corresponds to the addition of a 1.0(10) µM α-methyl-aspartate signal at t=200 and removed at t=600.

17 response than the smaller L = 1µM signal. This indicates that the magnitude of the signal is important and can greatly inﬂuence the behavior of the cell. Additionally, if the initial response is small the adaptation phase is much shorter.

2.2 An Alternative Parameter Estimation

The ﬁrst possible augmentation to Xin’s 4 ODE model is to attempt to make the reduced system more accurate. The adaptation component of this model is very important as it is responsible for the cell to properly follow a dynamic signal. With an adaptation too fast, CheYP concentration would quickly return to steady state levels. This would result in the cells performing a random walk with no bias, as there would not be enough variability in CheYP concentration for the cell to act accordingly. Furthermore, if the adaptation is too slow, the cell would fail to adapt and would be set in a run or tumble for much longer than necessary.

In Figure 2.2 we see that the reduced system fails to fully capture the adaptation of CheYP . Henceforth, we look at the parameter estimation technique, namely where a linear approximation for the probability of the unphosphorylated signaling complex with n ligands bound and m methyl groups to be active, pm, n and the rate

m constant in activity regulation k−i. In the original, a linear approximation between the two extremum values was used despite the clear convex shape of these parameters.

Henceforth, we will ﬁt each with a quadratic curve to better capture the nature of these parameters. Figure 2.3 shows the quadratic approximation in comparison for two of the parameters.

In order to ﬁnd the time evolution of CheYP considering these changes, we simulate the system in the same manner as before. We are going to consider an α-methyl- aspartate signal of 1µM and 10µM which will be added at t = 200 and removed at t = 600. Doing so for each of the three systems yields Figures 2.4a and 2.4b.

18 1.4 0.03

1.2 0.025

1 0.02

0.8 0.015 m,0 kd1 p 0.6 0.01

0.4 0.005

0.2 0

0 −0.005 0 2 4 6 8 10 12 0 2 4 6 8 10 12 m−bar m−bar

(a) (b)

Figure 2.3: Example of the alternative method of parameter ﬁtting. [1] uses a linear relationship between the two extremum values (blue line). Our method uses a quadratic approximation (green line) better capturing the convex shape of the parameters. (a) Probability unphosphorylated complex is active, pm, 1 (b) Rate constant in

m activity regulation, k−1.

When comparing the time evolution of CheYP in the three models, we see an improvement for L = 1µM and a slight improvement for L = 10µM . In L = 1µM, we see an entire order of magnitude improvement for the peak values at t = 200 and t = 600. Furthermore, we see an improvement in the adaptation. These same improvements can also be seen for L = 10µM. However, more detailed comparison with smooth signal functions needs to be done in the future.

19 3.4

3.35 Full Model Reduced Model 3.3 Alternate Reduced Model

3.25

3.2 P 3.15 CheY 3.1

3.05

2.95

0 200 400 600 800 1000 Time

(a) L = 1 µM

14 Full Model Reduced Model 12 Alternate Reduced Model

8 P

CheY 6

0 0 200 400 600 800 1000 Time

(b) L = 10 µM

Figure 2.4: Comparison of [1]’s Full 158 ODE Model (Blue Line), the reduced 4 ODE model (Green Line) and the reduced model with alternative parameter ﬁtting (red line). The system is simulated for 1000 seconds. The top(bottom) panel corresponds to the addition of a 1.0(10) µM α-methyl-aspartate signal at t=200 and removed at t=600.

20 Parameter a b c

pm,0 0.00968866 0.0311727 0.00435078

pm,1 0.00484433 0.0155864 0.00217539

pm,2 0.000968866 0.00311727 0.000435078

m k−1 0.0258594 -0.00601228 0.00119842

m k−2 0.129297 -0.0300614 0.00176406

m k−3 1.29297 -0.300614 0.0176406

Table 2.1: Coeﬃcients of quadratic approximations to the probability of the unphos-

m phorylated complex as active pm,n and the rate constant in activity regulation, k−i of the form a + bx + cx2.

21 CHAPTER 3

A NEW STOCHASTIC MODEL OF E. coli CHEMOTAXIS

In order to truly understand the cell, a connection needs to be made between the biochemical concentrations and the behavior of the cell. This is not a trivial link as each biochemical parameter involves its own level of uncertainty. The behavior of a cell is probabilistic at best, but still achieves desired results with remarkable precision.

Thus far, we have concerned ourselves with the transduction network upstream of the ﬂagellar motor. This resulted in developing an accurate representation of the key chemotactic protein CheYP which is responsible for biasing the rotation of the motor. Now we wish to focus on the latter portion of the transduction network, namely the motor response to CheYP .

3.1 Experimental Results of Cluzel et. al.

The goal of this section is to use “An Ultrasensitive Bacterial Motor Revealed by

Monitoring Signaling Proteins in Single Cells” Cluzel et. al. (2000) to couple the physical behavior of the ﬂagellar motor with a CheYP concentration. Doing so with an appropriate model for motor rotations will allow us to simulate the run and tumble of a single cell. The importance of this work is that it may be able to explain the high ampliﬁcation gain exhibited by the chemotactic sensory system through incorporating all aspects of the signal transduction network [14].

22 Cluzel et. al. quantiﬁed the ﬂagellar motor’s output with CheYP concentrations. This was achieved by relating the bias that a motor is in the clockwise state as a function of CheYP and the switching frequency of a motor as a function of CheYP . Conceptually, the former is explained by taking a set amount of time and calculating the fraction of time spent in the clockwise state and calculating its intracellular CheYP concentration during this time frame (which is held constant). Likewise, switching frequency is found by taking a time span and counting how many switches were made.

The obtained data for bias of clockwise rotation was then fitted with a Hill function with parameters Hill coefficient NH = 10.3 ± 1.1 and KM = 3.1µM. The switching frequency was fit with the derivative of the Hill function used for the bias. This data is recreated in Figures 3.1a and 3.1b.

3.2 A Two-Rate Model of Run and Tumble

Simplifying the chemotactic transduction network for the purposes of this section, we concern ourselves only with the intracellular concentration of CheYP and the rotation of the ﬂagellar motor. This is especially novel in conjunction with Chapter

2, because we have a system of four ordinary diﬀerential equations, whose output is CheYP concentration when given an external signal (e.g. nutrition gradient of α-methyl-aspartate) and initial cellular parameters. Therefore, using the 4 ODE system in conjunction with the experimental results of [6] will give us a more complete description of how a single cell goes from translating an external signal to a biomechanical response.

We first characterize a single flagellar motor to be in one of two conformational states: clockwise (CW) and counter-clockwise (CCW) . During a CCW rotation, the multiple flagella of a single cell form a bundle propelling the cell in a smooth swim. When the motors rotate CW, the flagella unwind and the cell tumbles in place

23 1.2 1.6

1.4 1 1.2 0.8 1

0.6 0.8

CW Bias 0.4 0.6

Switching Frequency 0.4 0.2 0.2 0 0

−0.2 −0.2 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 CheY (µM) CheY (µM) P P

(a) (b)

6 4

3.5 5

4 2.5

1 3 2 2 k k

1.5 2

1 0.5

0 0 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 CheY CheY P P

Figure 3.1: Experimental results of Cluzel et. al. showing the response of individual motors as a function of CheYP concentration. Red dots correspond to experimental data points and green lines correspond to calculated functions (a) Clockwise bias of the motor versus CheYP . The function is a Hill function with coeﬃcient NH =

−1 10.3 ± 1.1 and KM = 3.1µM. (b) Switching Frequency versus CheYP (s ). The function is the derivative of the Hill function of (a). (c) k1 calculated from the data points and the functions of (a) and (b). (d) k2 calculated from the data points and the functions of (a) and (b).

24 choosing a new direction quasi-randomly – there is a small bias towards the direction of previous velocity [12]. We now assume that a cell switches between the two states with rates k1 and k2 respectively (also note that the rates are functions of CheYP ), namely

k1 CW CCW (3.2.1) k2

We now use the experimental results of [6] to ﬁnd the bias that a cell is in the CW state (denoted BCW). Continually, its safe to assume that in a given time span, the time spent in the CW state is BCW, which we denote by T1. Thus,

BCW = T1 (3.2.2)

BCW ∗ k1 = (1 − BCW ) ∗ k2 (3.2.3)

Using (3.2.2) in (3.2.3), we can solve for k2 in terms of k1 which yields

k T k = 1 1 (3.2.4) 2 (1 − T 1)

Combining the two rates we are able to deﬁne the switching frequency of ﬂagella from

CW to CCW and vice-versa as

2 SW (y) = 1 + 1 k1 k2 2 = 1 + 1−T1 k1 T1k1

= 2T1k1 (3.2.5)

where y is the CheYP concentration. SW (y) is analogous to Fig. 2b of [6]. Rear- ranging we obtain,

SW (y) k1 = (3.2.6) 2T1

25 and from (3.2.4)

SW (y) k2 = (3.2.7) 2(1 − T1)

Expressions 3.2.6 and 3.2.7 are the goal, as we now have related the rates k1 and k2 to the experimental values of [6] as seen in Figures 3.1a and 3.1b which are the bias for CW rotation and the switching frequency based on CheYP magnitude. Using the Hill function presented in [6] shown in Figures 3.1a and 3.1b we can calculate explicitly k1 and k2 as shown in Figures 3.1c and 3.1d. Also represented in these two ﬁgures are the data points generated directly from the data points of the experiment. However, we see that these do not provide accurate representations of the data. Furthermore, this reveals k1 and k2 not to be monotonic functions, creating a biological impasse for this method. Therefore, this motivates generating our own functions for k1 and k2. Using the data points generated from [6] we calculate k1 and k2 using 3.2.6 and 3.2.7 to create new data points that can be plotted in a CheYP vs. k1,2 scatter plot. We then use Gnuplot’s Non-Linear Least Squares algorithm to ﬁt an exponential function to the data. Exponential curves were chosen due to the apparent nature of k1 and k2. It is also important to note that only the interior points were used in calculating the curves. Namely, those data points with CheYP < 1 or

CheYP > 6 were omitted. When the neglected points were plugged into the formula 3.2.6 and 3.2.7 they are very close to producing a singularity in these equations.

Hence, small errors in these measurements results in large variations in k1 and k2. Therefore we dropped these data points in curve ﬁtting, but made sure the function values of the ﬁtted function at the corresponding CheYp ranges are realistic in biology and consistent with the experimental data. Doing so produced,

−(b−y)4 k1(y) = ae c (3.2.8)

26 with a = 12.0809, b = −5.83762 and 2892.12 and

by k2(y) = ae (3.2.9) with a = 0.0174001 and b = 1.32887.

To verify this procedure, we use these expressions in 3.2.6 and 3.2.7 to calculate the bias for clockwise rotation and the switching frequency. We then compare these results with the experimental data of [6]. Figure 3.2a shows how using k1 and k2 recreates the clockwise bias. We see that it is indeed a valid representation of the data, even more so than the Hill function presented earlier. Figure 3.2b uses k1 and k2 to recreate a fit for the switching frequency. Before analyzing if our method represents the data, lets look at the important features of this data set. We see that for 0 < CheYP < 1 and CheYP > 4.75 switching frequency is close to 0. Furthermore, there is a peak centered around CheYP = 3.2. Other than these features, its hard to say that the data follows any obvious or distinct trend. As for our line, we see we properly capture the peak, the most important feature. However, the tails of our trendline fail to go to zero as needed a tradeoff necessary so that k1 and k2 are monotonic functions. Since the tails are very small and exponentially decaying, we can assume they are a good fit nonetheless. From Figure 3.2a and 3.2b we see that our analysis is consistent with the true biological nature of the switching rates between the two conformational states of flagellar rotation - clockwise and counterclockwise.

27 1.2 1.6

1.4 1 1.2 0.8 1

0.6 0.8

0.6 CW Bias 0.4

Switching Frequency 0.4 0.2 0.2 0 0

−0.2 −0.2 0 2 4 6 8 10 0 2 4 CheY (µM)6 8 10 CheY (µM) P P

(a) (b)

9 7

8 6

5 6

5 4 1 2 k k

4 3

3 2 2

1 1

0 0 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 CheY CheY P P

Figure 3.2: Red dots correspond to experimental data points and green lines correspond to calculated functions (a) Clockwise bias for ﬂagella rotation as a function of CheYP concentration (µM). Function corresponds to calculating BCW using k1 and k2 of (c) and (d) below. (b) Switching frequency of given cells based on CheYP concentration (µM), SW (y). Function corresponds to calculating SW using k1 and k2 of (c) and (d) below. (c) k1 as a function of CheYP concentration between 0 and 6.0 µM. Data points calculated directly using (3.3.8) and the data points of [6]. The

−(b−y)4 function is the exponential curve k1(y) = ae c with a = 12.0809, b = −5.83762 and c = 2892.12. (d) k2 as a function of CheYP concentration between 0 and 4.5 µM. Red data points calculated directly using (3.3.9) and the data points of [6]. The

by function is the exponential curve k2(y) = ae with a = 0.0174001 and b = 1.32887. 28 CHAPTER 4

AN INTEGRATED MODEL OF E. coli MOVEMENT

Now that we have a complete description of the chemotactic network, we are able to simulate cells in an environment in much the same manner as the methodology of the random walk of Section 1.4. Integrating the reduced model for CheYP and the model for ﬂagellar response to form a two-component model at the individual cellular level allows us to go from interpreting the environment (ligand sensing) to biomechanical response (run or tumble) . Furthermore, we can apply this model to each individual of a population allowing us to draw conclusions of population level behavior of E. coli cells. The following chapter will present two separate models for incorporating the two-component model to individual cells.

4.1 A Simple Case with a Single Rate

The ﬁrst two-component model we wish to use combines Section 2.1.1 with a simple linear turning rate. This will serve to elucidate the dynamics of a population of cells in response to a concentration gradiant. A simple linear turning rate is characterized by the probability of turning (P(turn)) as a function of CheYP (y) as,

P (turn) = max(1 + a(y − b), 0)dt

29 where a tunes the sensitivity of turning and b is the steady state CheYP value. There- fore, if the current CheYP value is greater than the steady state value then the probability of turning increases. We will begin by simulating a population of 500 cells for

Figure 4.1: Simulation of a population of 500 cells. The two-components of the model were comprised of the reduced 4 ODE transduction model and a simple linear turning rate for ﬂagella rotation. Inset: The exponential signal the cells were subjected to.

1000 seconds subject to an exponential concentration gradient of α-methyl aspartate following the form,   x e if 0 ≤ x ≤ 3 S(x) = (4.1.1)  3 e if 3 < x ≤ 4 30 Analyzing the behavior of the population of cells in Figure 4.1, we can make conclusions about how the CheYP values work in the stochastic setting. First, we notice all but one of the cells eventually follows the gradient. Next, we notice that cells perform a random walk until their CheYP values get suppressed enough locking them into a run in the direction of the signal. This can be explained by the nature of the turning probability. Once the cell begins to move to the right, its CheYP value decreases, further decreasing the probability of turning. This causes the cell to continue moving in that direction, further suppressing CheYP keeping the probability of turning very close to zero. That is why we see the cells appear to “march” in the direction of the signal. This is further reinforced if we look into the CheYP proﬁle of a typical cell in Figure 4.1. In Figure 4.2 we see that the CheYP concentration shows much variation until time, t ≈ 110 and then begins monotonically decreasing while the cell traverses with the signal gradient. Then at t ≈ 675, we see adaptation occur as the cell makes it past x = 3 and begins experiencing a constant signal. After adaptation, we see that CheYP holds at steady state for t > 750. Continually, once the cells reach the constant signal, we observe that an unbiased random walk occurs once the intracellular proteins fully adapt back to steady state.

We now turn our attention to how the entire population of cells CheYP concentrations change in time. Figure 4.3 plots CheYP versus its position at various snapshots of time. We see that all cells start fully adapted at t = 1. Then as they begin to random walk, we see the cells’ CheYP values begin to space out also causing the spatial positions to vary. We see an overall trend down in CheYP and right in location, both of which are consistent with the exponential signal. As time goes on, we then see very little noise in the system as the cells “march” to the right (2.2 < x < 3.0).

This can be attributed to the deterministic ODEs that are simulating each cell. Next, we see at x = 3, where the cell becomes constant, adaptation back to steady state

31 3.3

3.2

3.1

2.9 P 2.8 CheY 2.7

2.6

2.5

2.4

2.3 0 200 400 600 800 1000 Time

Figure 4.2: Intracellular concentration (µM) of CheYP of a a typical cell in Figure 4.1.

occurs. This elucidates the importance of a signal to have a non-zero derivative which is consistent with [3].

4.2 The Full Model

The conclusion of this work will be uniting the reduced system upstream of the

ﬂagellar motor with the two-rate model for biomechanical rotation forming a two- component model. This is novel as it combines a robust 4 ODE model of chemotactic concentrations with a novel way of relating experimental switching rates with intracellular CheYP concentration. We will proceed in much the same way as the previous

32 3.3

3.2

3.1

2.9 P 2.8 CheY 2.7 t = 1 2.6 t = 50 t = 300 2.5 t = 600 2.4 t = 850 t = 1000 2.3 1.5 2 2.5 3 3.5 X

Figure 4.3: Scatter plot revealing the intracellular concentration (µM) of CheYP and X position of all cells in Figure 4.1 taken at t = 1 (red), t = 50 (blue), t = 300

(magenta), t = 600 (green), t = 850 (cyan) and t = 1000 (black) .

section. Considering the same exponential signal in equation 4.1.1 we choose to simulate ﬁve individual cells for a more detailed look at each individual trajectory.

Analyzing the figure, we find qualitatively different results than that of the two- component model using a simple linear turning rate. Primarily we notice that the cells do follow the gradient however not nearly as quickly or definitively as those involving the simple linear turning rate. It is apparent that these cells have a bias with the signal, however it is not as pronounced and quite noisy. This noise causes the cells to move to in the +x direction with the signal much slower than that of the simple one turning rate probability. This is much more meaningful biologically because nature is inherently noisy.

33 x 104 X Position 5

4.5

3.5

2.5

1.5

0.5

0 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Figure 4.4: Simulation of a population of 5 cells. The two-components of the model were comprised of the reduced 4 ODE transduction model and the results of Section

3.3 for ﬂagella rotation subjected to the same exponential signal 4.1.1.

34 CHAPTER 5

CONCLUSIONS AND FUTURE WORK

Chemotaxis is the directed cell movement in response to chemical signals. This process is fundamental to many multicellular processes, such as wound healing, cancer metastasis, fertilization, embryonic development and bioremdiation. Quantitative descriptions of chemotaxis will lead to a better understanding of the mechanisms of chemotaxis and are essential in understanding the aforementioned processes.

We studied E. coli cells which specifically utilize a strategy of running and tumbling powered by 6-8 flagella. This is accomplished by the cell sensing a signal through cell surface receptors. This triggers an intracellular cascade of chemotactic protein activity that has two competing modes. The first directly affects the bias of the rotation through CheYP . Meanwhile, there is a much slower feedback where CheB demethylates the receptors causing CheYP activity to increase. This interaction allows the cell to experience rapid kinase activity followed by slower adaptation. The biasing of the flagellar motor by CheYP is what causes the cell to run or tumble at any given moment. The mathematical model used to describe the transduction network was an already existing model based on a “trimer of dimers.” This model produces a robust representation of the complex reaction network necessary for chemotaxis.

We improved upon this model by conducting an alternative parameter estimation to several parameters. Next, using existing experimental data, a novel way of describing

ﬂagellar rotation was developed using two rates for the switching of the rotation of

35 the motor. The conclusion of this work was uniting the “trimer of dimers” model with our description of motor rotation to form a complete two-component model that fully describes the chemotactic process of a single E. coli cell. The two-component model allows us to go from signal to biomechanical response. This model can then be used for a single cell or used to investigate population level behavior.

The final model presented here provides the foundations for conducting broader simulations to make general conclusions about the biological mechanisms of cellular chemotaxis. A nuance of the presented model here, is that there is one flagella which is not consistent with the biology of the cell. We would like to broaden the model to consist of 6-8 flagella that work in a voting process or other coordinated manner.

Altering the conditions for how many flagella need to be in agreement might alter the dynamic behavior of the cell. Comparing different scenarios to see which matches up with experimental results of similar parameters, conclusions can be drawn about how the flagella work – either independently or dependently. The next use of the model can be to consider a variety of spatio-temporally varying signals. Applying more complex signals to this simulation can help to elucidate how cells deal with an environment that is inherently noisy. Additionally, solving the macroscopic limit for the population behavior can prove to be useful. It can be used to compare with stochastic simulations and even possibly be used to reduce computing time by having a much easier equation to be used.

Chemotaxis is but one mechanism a cell utilizes to determine its movement. Other forms of taxis include phototaxis, aerotaxis, thermotaxis and hydrotaxis just to name a few. Exactly how a cell weighs each of these external factors and then makes a

“decision” of movement is still far from being known. However, by tackling each of these components individually we get closer to one day possibly having a complete description of all cellular processes – not just movement. This work has chosen

36 to speciﬁcally work with chemotaxis, but there are similarities amongst all of the diﬀerent processes of a cell. Uniting this work with others can possibly be used to obtain a complete description of exactly how E. coli cells function.

37 BIBLIOGRAPHY

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39 APPENDIX: GLOSSARY

• Bias Clockwise (BCW) - The bias for a ﬂagellar motor to be rotating in the

clockwise direction based on CheYP concentration

• Chemotaxis - Directed cell movement in response to a chemical signal.

• Che Proteins - Proteins utilized in the chemotactic transduction network.

• Demethylation - The removal of a methyl group (CH3) from the MCP complex.

• Flagella - The component of a cell that dictates the movement. Each E. coli cell

has 6-8 ﬂagella that rotate in either the counter-clockwise (CCW) or clockwise

(CW) direction.

• Methylation - The addition of a methyl group (CH3) to the MCP complex.

• Ordinary Diﬀerential Equation (ODE) - An equation involving a function and

any number of its derivatives.

• Partial Diﬀerential Equation (PDE) - An equation involving a function and its

partial derivatives.

• Random Walk - A random process involving a sequence of discrete time steps

of ﬁxed length.

• Receptors - A molecule found on the outside of the cell that detects a chemical

40 signal in the extracellular membrane. For chemotaxis, four membrane cofactor

proteins (MCP) comprise the receptors of chemotaxis.

• Run and Tumble - Terms used to describe the possible state of a cell. A run

occurs when the ﬂagella rotate in the counter-clockwise (CCW) direction as

they form a bundle that propels the cell in a run-like fashion. A tumble occurs

when the ﬂagella rotate in the clockwise (CW) direction as the ﬂagella unwind

causing the cell to tumble in place.

• Switching Frequency (SW) - The switching frequency of a ﬂagellar motor to

switch between clockwise rotation and counter-clockwise rotation in a set amount

of time. SW is dependent on CheYP concentration.