Stochastic Modelling of a Chemotactic Microswimmer

Stochastic Modelling of a Chemotactic Microswimmer

ABSTRACT

Key to Escherichia coli (E-coli) bacteria survival is its ability to direct its movement to greener pasture and ee harmful environment also known as chemotaxis. This thesis focuses on the modelling of E-coli chemotaxis in two- dimensions with emphasis on trying to understand the basic physics of how such a tiny micro-swimmer swim up a concentration gradient despite the enormous thermal fluctuations in its environment. E-coli strategically employs
near straight swimming (also known as run) often interrupted by random re-orientations (also known as tumble). How often this interruptions happens is the swimmer tumbling frequency. This chemotaxis strategy is here modeled as random telegraph process, which is a dichotomous stochastic process. The swimmer tumbling frequency is represented as the transition rate from run phase to tumble phase. The transition rate is a function of swimmer specific trait (known as response kernel) and the environmental condition – concentration parole. Furthermore, the random telegraph process is coupled to the swimmer Langevin equations in which the system was solved analytically making judicious approximations. Important chemotaxis parameter expression was obtained for a swimmer with arbitrary trait and a simple swimmer case scenario analyzed. Even though, this framework describes E-coli chemotaxis excellently, it can as well serve as a base framework for study of other interesting models that exhibit two state swimming strategy.

TABLE OF CONTENTS

Abstract 3
List of Figures 6
1 Introduction 7
2 Background Literature 10
2.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 E-coli Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Scallop Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 2D Micro-swimmer Dynamics 15
3.1 The chemotaxis strategy . . . . . . . . . . . . . . . . . . . . . 17
3.2 Associated Langevin Equations . . . . . . . . . . . . . . . . . 20
3.3 Chemotaxis parameter and new noise . . . . . . . . . . . . 27
4 Discussion 34
Appendix I 37
Appendix II 42

CHAPTER ONE

Introduction

The ability of a living organism to directs its movement up or down a chemical gradient is termed chemotaxis. This thesis focuses on modelling Escherichia coli (E-coli) chemotaxis in two dimensions. E-coli bacteria lives in aqueous environment such as our intestine. It is quite challenging for a tiny organism like bacteria that knows nothing about inertia to and food in such aqueous environments. Motile (agellated) E-coli evolves strategies to maneuver themselves to regions of high concentrations of a chemical attractant (chemoattractant) and moves away from chemical repellent (chemorepellent) [1{3]. The strategy involves runs; which are near-straight swimming along its principal axis, and tumble; which are random re-reorientation (erratic movement) in same neighborhood. This mystic mechanism of bacterial chemotaxis had attracted a lot of attention in the scientific community in last few decades [1{5]. Currently, there is intensive thrust towards design and fabrication of artificial microswimmers (micro- and nano-swimmers) – mimicking nature to accomplish important tasks [6,8]. Man-made microswimmers holds the promise of revolutionising medicine, such as in drug-delivery and early disease detection [6].

Chemical molecules in the hydrodynamic environment binds to chemoreceptors distributed over the E-coli cell membrane.

The cell analyses the signal through its biochemical pathways (feedback loop) and respond via its motility machinery [1{3]. It changes the direction of the agella rotation from clock-wise direction (CW) to counter-clock-wise direction (CCW) periodically depending on the chemical (chemoattractant or chemorepellent) sensed or intercepted over some characteristic time scale. Smooth runs corresponds to CCW- and tumbles corresponds to CW-rotations of its agella [1].

Consequently, the switching of the agella direction of rotation results in longer runs in case its moving towards a region of high concentration and shorter when moving in the opposite. Through tethering assays of E-coli cell, it was found that it is unlikely the cell will make spatial measurements of concentration gradient but rather it will sample the space and average the detections over time, in other words it makes temporal comparisons of the most recent measurements [1, 5]. The response function that its good with the experimental data was determined by Celani and Vergassola [9] using game theoretic approach. Celani and Vergassola showed that the response was selected as a maximin strategy of the E-coli, i.e highest minimum uptake of chemoattractant for any concentration prole. This strategy ensures good response to any concentration parole for a microswimmer in a complex environment (where the microswimmer is subject to conflicting response requirements).

In molecular scales, the microswimmer exhibit stochastic motion due to fluctuations in its environment. In addition, random torques can significantly affect the swimmer motion [1]. Artificial microswimmers recently developed (e.g spherical Janus particles) with additional asymmetry demonstrates significant gain in translational displacement [10]. However, these microswimmers do not mimick the E-coli chemotaxis strategy directly. Hence, we have so far achieved only partial control of these swimmers. The question remains, how a swimmer swim up the chemoattractant concentration gradient despite the thermal fluctuations? – taking note of the fact that engineered devices usually have small number of degrees of freedom. What strategy a swimmer should employ to extract directed motion from the ceaseless noise with minimal internal degrees of freedom? Here, random telegraph process (as decision making process) is incorporated into the microswimmer’s associated Langevin equations (for a microswimmer moving with a net deterministic velocity subject to Brownian randomization) and the resulting dynamics analyzed under different environmental conditions (concentration proles).