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Mathematical Models of Pattern Formation in Cell Biology

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Mathematical Models of Pattern Formation in Cell Biology MATHEMATICAL MODELS OF PATTERN FORMATION IN CELL BIOLOGY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University by Xige Yang, B.S, Mathematics Graduate Program in Mathematics The Ohio State University 2018 Dissertation Committee: Dr. Chuan Xue, Advisor Dr. Avner Friedman Dr. Janet Best ©Copyright by Xige Yang 2018 ABSTRACT This thesis provides a study of mathematical models about pattern formation phenomena in cell biology. Chapter2 introduces necessary background on mathematical methods of develop- ing and analyzing mathematical models at the cellular and molecular scales. I then considered two detailed cases related to pattern formation in cell biology, chemotaxis of bacterial popu- lations which generates wave-like patterns, and axonal cytoskeleton segregation which exhibits peak-valley patterns. In Chapter3, we derived a PDE model for E. coli chemotaxis that incorpo- rates detailed intracellular signaling. Unlike previous PDE models, the new model can approxi- mate the population dynamics accurately even if the external signal changes rapidly. In Chapter 4, we developed and analyzed a nonlocal PDE model for the polymer segregation phenomena. Reaction-diffusion equations have been widely used to describe biological pattern formation. In Chapter5, we developed numerical methods for computing multiple steady-state patterns of reaction-diffusion models. For that we combined homotopy tracking method in numerical alge- braic geometry and multigrid method, and as an example, we computed multiple patterns for the Gray-Scott model. ii VITA 1992 . Born, Chengdu, China 2014 . B.S. in Mathematics, Zhe- jiang University, China 2014-2018 . Graduate Associate, The Ohio State University, USA FIELDS OF STUDY Major Field: Mathematics Specialization: Mathematical Biology iii To my parents, You always support me to do whatever I like. iv ACKNOWLEDGMENTS I would like to express my gratitude to those who fostered by personal and professional devel- opment. First, sincere thanks are due to my advisor, Dr. Chuan Xue for her invaluable assistance, instructions and insights leading to the writing of this dissertation. Second, I would like to thank Dr. Wenrui Hao for his dedicated instructions on Chapter5. For the computational part, I also thank Jonanthan Toy for sharing his codes used in Chapter4. Third, I also want to thank the Mathematical Bioscience Institute (MBI), where I met many incredible mathematicians. Among them, I especially thank Dr. Marty Golubitsky for his invalu- able discussions and guidance, and Dr. Thomas Woolley for sharing his thesis draft and outreach works as well as his passion in mathematical biology. Fourth, I would like to thank my committee members Dr. Avner Friedman and Dr. Janet for agreeing to serve on my committee, and for their various valuable suggestions and discussions on improving the dissertation draft. Thanks are also due to the Department of Mathematics for providing me with various GTA and SGA supports. I also express my special thanks to Dr. Fangyang Zheng for their valuable suggestions and help all throughout my graduate life. Finally, this dissertation was partially supported by NSF DMS 0931642 and NSF CAREER Award 1553637. v TABLE OF CONTENTS Abstract........................................... ii Vita ............................................. iii Dedication.......................................... iv Acknowledgments...................................... v List of Tables ........................................ xii List of Figures........................................ xiii 1 Introduction....................................... 1 1.1 Statement of main results.............................2 2 Modeling Background in Cell Biology......................... 5 2.1 Some physical background............................6 2.1.1 Conservation laws............................6 2.1.2 Chemical reactions............................6 2.1.3 Diffusion.................................8 2.1.4 Advection................................. 10 2.2 Useful tools in mathematical modeling...................... 11 2.2.1 Reaction-diffusion systems........................ 11 2.2.2 Parameter estimation........................... 14 vi 2.2.3 Nondimensionalization.......................... 17 2.3 Modeling using stochastic approach....................... 18 2.3.1 Diffusion and randomness........................ 19 2.3.2 Advection and randomness........................ 20 2.3.3 Reaction and randomness........................ 23 2.4 Summary and discussion............................. 28 3 Chemotaxis patterns driven by large signal gradient.................. 30 3.1 Biological background.............................. 31 3.1.1 Chemotaxis................................ 31 3.1.2 Biochemical details of E. coli chemotaxis................ 32 3.1.3 Pattern formation and bacteria chemotaxis................ 34 3.2 The individual-based model and previous PDE models............. 34 3.2.1 Overview................................. 35 3.2.2 The individual-based model....................... 38 3.2.3 Previous PDE models for the population dynamics........... 41 3.2.4 Numerical comparisons......................... 45 3.3 New moment-flux models for chemotaxis in large signal gradients....... 47 3.3.1 The moment closure method....................... 48 3.3.2 The case with linear turning frequency.................. 49 vii 3.3.3 The case with a nonlinear turning frequency............... 54 3.4 Numerical simulations and conclusions..................... 57 3.4.1 Linear turning frequency......................... 58 3.4.2 Nonlinear turning frequency....................... 59 3.5 Summary and discussion............................. 64 4 Cross-sectional segregation of axonal cytoskeleton in neurodegenerative diseases . 67 4.1 Biological background.............................. 68 4.1.1 Neuron.................................. 68 4.1.2 Cytoskeletons and axonal transport................... 69 4.1.3 Neurodegenerative disease and cross-sectional segregation of axonal cy- toskeleton................................. 72 4.2 Previous deterministic and stochastic models of neurofilament transport and MT- NF segregation.................................. 77 4.2.1 Models of neurofilament transport.................... 77 4.2.2 An agent-based model for MT-NF interaction.............. 79 4.3 A nonlocal PDE model for the cross-section dynamics............. 81 4.3.1 Continuum approach for nonlocal interactions.............. 82 4.3.2 Model mechanisms and derivation.................... 83 4.3.3 The model equations........................... 88 viii 4.4 Mathematical results............................... 88 4.4.1 Parameter estimation and nondimensionalization............ 88 4.4.2 Linear stability analysis......................... 93 4.4.3 A simplified model in 2D......................... 98 4.4.4 An investigation to circular domain model................ 101 4.5 Numerical simulations and biological implications............... 104 4.5.1 Simulations in 1D............................ 104 4.5.2 The impact of domain size........................ 106 4.5.3 MT-NF phase separation is reversible due to IDPN washout...... 108 4.5.4 Nonlinear diffusion is important for homogeneity, but not for MT-NF phase separation............................. 110 4.6 Summary and discussion............................. 112 5 Homotopy tracking method – finding multiple pattens of a reaction-diffusion system . 115 5.1 Motivation – the Gray-Scott model........................ 116 5.1.1 Numerical simulations.......................... 118 5.1.2 Multiple patterns arising with the same parameters........... 120 5.2 Basics on numerical algebraic geometry..................... 121 5.2.1 Solving polynomial systems....................... 121 5.2.2 Homotopy tracking method....................... 123 ix 5.2.3 Bootstrapping method.......................... 126 5.3 Multiple steady-states of Gray-Scott system................... 127 5.3.1 Introduction................................ 127 5.3.2 Discretization in 1D........................... 127 5.3.3 Solving the 1D system via Homotopy tracking............. 129 5.3.4 Multigrid refining............................. 129 5.4 Stability of the steady-states........................... 131 5.4.1 Linear stability.............................. 133 5.4.2 Nonlinear stability............................ 133 5.5 Parameter regime – proposed work........................ 136 5.5.1 Homotopy tracking............................ 136 5.6 Summary and discussion............................. 137 References.......................................... 141 Appendices ......................................... 159 A Numerical methods of Chapter 5............................ 160 A.1 Finite difference scheme............................. 160 A.1.1 Numerical sheme in 1D......................... 160 A.2 Finite volume scheme............................... 162 A.2.1 Radially symmetric case......................... 164 x A.2.2 Square domain and periodic boundary condition case.......... 166 A.2.3 Circular domain with noflux boundary condition case.......... 167 A.2.4 Calculating integral terms........................ 169 xi LIST OF TABLES 4.1 Parameter values of (4.3.7)-(4.3.10)......................... 92 5.1 Computational complexity of the polynomial system (5.3.3) versus subinterval num- ber N.......................................... 140 5.2 Number of solutions versus number of grid in the polynomial system (5.3.3).... 140
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