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 By 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 individ- ual 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 cor- respond 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 calcu- lated 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 move- ment, which is necessary for nutrient regulation, pH balance, temperature regulation, evading threats (e.g. toxins), oxygen regulation, reacting to a light source and pro- ducing 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).