Mathematical modelling and analysis of aspects of planktonic bacterial motility Gabriel Aaron Rosser St Anne's College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Michaelmas 2012 Contents 1 The biology of bacterial motility and taxis 8 1.1 Bacterial motility and taxis . .8 1.2 Experimental methods used to probe bacterial motility . 14 1.3 Tracking . 20 1.4 Conclusion and outlook . 21 2 Mathematical methods and models of bacterial motility and taxis 23 2.1 Modelling bacterial motility and taxis: a multiscale problem . 24 2.2 The velocity jump process . 34 2.3 Spatial moments of the general velocity jump process . 46 2.4 Circular statistics . 49 2.5 Stochastic simulation algorithm . 52 2.6 Conclusion and outlook . 54 3 Analysis methods for inferring stopping phases in tracking data 55 3.1 Analysis methods . 58 3.2 Simulation study comparison of the analysis methods . 76 3.3 Results . 80 3.4 Discussion and conclusions . 86 4 Analysis of experimental data 92 4.1 Methods . 92 i 4.2 Results . 109 4.3 Discussion and conclusions . 124 5 The effect of sampling frequency 132 5.1 Background and methods . 133 5.2 Stationary distributions . 136 5.3 Simulation study of dynamic distributions . 140 5.4 Analytic study of dynamic distributions . 149 5.5 Discussion and conclusions . 159 6 Modelling the effect of Brownian buffeting on motile bacteria 162 6.1 Background . 163 6.2 Mathematical methods . 166 6.3 A model of rotational diffusion in bacterial motility . 173 6.4 Results . 183 6.5 Discussion and conclusion . 197 7 Discussion and conclusions 202 7.1 Further work . 206 Appendices A Mathematical methods 209 A.1 Algorithms for simulating random walks . 209 A.2 Derivation of the one-dimensional Telegraph equation . 211 A.3 Deriving expressions for the spatial moments of the general velocity jump process . 213 A.4 Kernel density estimation . 218 A.5 Conditional distribution of interarrival times in a Poisson process . 219 ii B Analysis of experimental data 220 B.1 Links to online video clips . 220 B.2 Computational method for finding the minimum bounding circle . 221 B.3 Results from the surface swimming dataset . 221 B.4 Bayes factor for hypothesis testing . 227 C Modelling the effect of Brownian buffeting on motile bacteria 230 C.1 Implementation of the Euler-Maruyama algorithm . 230 C.2 Derivation of second moments for the run-only model . 230 Bibliography 234 iii Acknowledgements The past five years that I have spent at Oxford have seen some of the most rewarding, and most difficult, times of my life. There are so many people whom I would like to thank for their help and encouragement along the way. Many will regrettably remain nameless to prevent this thesis expanding by another chapter, but I would particularly like to acknowledge a few. I am deeply grateful to my supervisors, Professor Philip Maini, Dr Ruth Baker, Dr Alexander Fletcher and Dr Mark Leake. Your enduring patience, support and commit- ment have spurred me on throughout my time here. Thank you for helping to make my project so interesting and enjoyable. I also want to thank the Doctoral Training Centre for a fantastic first year, and the con- tinued support after that. Without the DTC, I would never have found my way to such an excellent project and department. Thank you for doing things differently. I am indebted to the many members of the Centre for Mathematical Biology who made it such a great place to work and socialise. I particularly wish to acknowledge the vast amount of help I have received from Kit, Louise, Aaron S and Aaron L, without which my project would be so much poorer. Thank you for your unerring altruism. My involvement in a successful and fun collaboration played a positive formative part in my time at the CMB. I feel lucky to have been part of the `bacteria boyz' with Kit, Trev and Dave, whom I count as friends in addition to collaborators. Thank you for showing me the best face of interdisciplinary research. To all of my friends, both those who have been with me from the start and those whom I have met along the way, your encouragement and/or derision have bolstered my spirits. Thank you for the laughs and adventures. Finally, to my mum, dad and sister, whose contribution is beyond words. Thank you can never be enough. 2 Abstract The motile behaviour of bacteria underlies many important aspects of their actions, in- cluding pathogenicity, foraging efficiency, and ability to form biofilms. In this thesis, we apply mathematical modelling and analysis to various aspects of the planktonic motility of flagellated bacteria, guided by experimental observations. We use data obtained by tracking free-swimming Rhodobacter sphaeroides under a microscope, taking advantage of the availability of a large dataset acquired using a recently developed, high-throughput protocol. A novel analysis method using a hidden Markov model for the identification of reorientation phases in the tracks is described. This is assessed and compared with an established method using a computational simulation study, which shows that the new method has a reduced error rate and less systematic bias. We proceed to apply the novel analysis method to experimental tracks, demonstrating that we are able to suc- cessfully identify reorientations and record the angle changes of each reorientation phase. The analysis pipeline developed here is an important proof of concept, demonstrating a rapid and cost-effective protocol for the investigation of myriad aspects of the motility of microorganisms. In addition, we use mathematical modelling and computational sim- ulations to investigate the effect that the microscope sampling rate has on the observed tracking data. This is an important, but often overlooked aspect of experimental de- sign, which affects the observed data in a complex manner. Finally, we examine the role of rotational diffusion in bacterial motility, testing various models against the analysed data. This provides strong evidence that R. sphaeroides undergoes some form of active reorientation, in contrast to the mainstream belief that the process is passive. Statement of Originality This thesis is submitted to the University of Oxford in fulfillment of the requirements of the degree of Doctor of Philosophy. This thesis contains no material which has been previously submitted for a degree or diploma at this University or any other institution. All experimental data presented in this thesis were obtained by Dr David A. Wilkinson whilst working in the laboratory of Prof. Judith P. Armitage at the Department of Bio- chemistry, University of Oxford. The specific datasets considered here and all results derived from their analysis have not previously been published at the time of writing. The algorithm used to obtain tracks from the data was written by Dr Trevor M. Wood. Otherwise, this thesis represents my own original work towards this research degree. Gabriel Aaron Rosser University of Oxford October 2012 Foreword In this thesis, we use mathematical modelling and analysis of experimental data to in- vestigate the process of planktonic bacterial motility, also known as bacterial taxis. This widely-studied phenomenon is implicated in bacterial infection and industrial biofouling, which are responsible for severe health risks and vast financial losses. Furthermore, the motile behaviour of bacteria may provide important information on how they are adapted to inhabit a specific environmental niche. All of the present work is motivated by the availability of new, unpublished tracking data, obtained using a novel high-throughput experimental protocol. The subject organism is Rhodobacter sphaeroides, a rod-shaped bacterium that occurs naturally in soil, freshwater and marine environments. The bio- chemical systems underlying the motile behaviour of R. sphaeroides are more complex than those of the more widely studied model bacterium Escherichia coli [158]. The tracking data represent a rich source of information on the motility of R. sphaeroides, but robust analysis techniques are required to extract reliable statistics. We dedicate part of this thesis to the development of novel methodology to achieve this result. The re- maining parts are concerned with modelling aspects of bacterial tracking data, including a study of the effect that sampling frequency has on the information we extract from tracking data, and an investigation of the effect of Brownian rotation on bacterial motil- ity. The work is motivated, guided and verified throughout by comparison with the experimental data. This approach ensures that the problems we investigate in this thesis are biologically relevant and realistic. 5 The field of bacterial motility, though the subject of a great number of studies, contains many key areas in which further work is required. We now briefly identify the areas in which, to our knowledge, this thesis has made novel contributions. First, the state of the art in the analysis of bacterial tracking data remains, with a few exceptions, predomi- nantly restricted to early work carried out by Berg and Brown [24] and contemporaries. The analysis methods developed in Chapter 3 are a novel approach to the quantitative study of tracking data. Our comparative assessment of their performance with simulated data is the first such study for tracking analysis methods. The application of our meth- ods to real experimental data leads to novel insight into the motion of R. sphaeroides, as shown in Chapter 4. Furthermore, the analysis process presented in Chapters 3-4 demon- strates an important proof of principal for the new experimental protocol, constituting a new, low-cost, high-throughput methodology that is applicable to myriad biological in- vestigations in which bacterial motility is a factor. Second, the study by Codling and Hill [50] on the effect of sampling frequency on tracking data suggests several further areas of investigation.