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
xii LIST OF FIGURES
1.1 Spatial biological patterns in nature...... 1
2.1 Fick’s law illustrated in 1d case...... 9
2.2 Simulation of 2000 particles with total time T = 20, ∆x = 0.1 ...... 20
2.3 A comparison between the true solution of (2.1.11) and statistics of Brownian motion
at different time slices...... 21
2.4 Comparison of advection effect based on hyperbolic form (2.3.12), advection-
diffusion form (2.3.14) and stochastic simulation when s = 5, λ+ = 30, λ− = 10
...... 24
2.5 Comparison of advection effect based on hyperbolic form (2.3.12), advection-
diffusion form (2.3.14) and stochastic simulation when s = 1, λ+ = 0.55, λ− = 0.45. 25
2.6 A comparison between the true solution of (2.1.11) and statistics of Brownian motion
at different time slices with a), c): λ = 0.5, b), d): D = 10...... 26
2.7 Comparison of stochastic and deterministic simulation of (2.3.15) under the identical
initial condition...... 26
2.8 Comparison of stochastic and deterministic simulation of (2.3.16) under the same
initial condition...... 27
2.9 A sample path of stochastic simulation of (2.3.15) and its relation to the potential
function of (2.3.16)...... 28
xiii 3.1 The signal transduction pathway for E. coli chemotaxis...... 33
3.2 Patterns generated by chemotactic E. coli bacteria in the presence of succinate, a
chemoattractant. (a) swarm ring in 1 mM succinate; (b) spots on a pseudo-rectangular
lattice in 2 mM succinate; (c) spots on a pseudo-hexagonal lattice in 3 mM succinate;
(d) spots with tails on a pseudo-hexagonal lattice in 3 mM succinate with a mutant
strain.. Picture is adapted from [28]...... 35
3.3 The response to a step change of the signal S. te = 0.01, ta = 5...... 40
3.4 Previous PDE models for bacterial chemotaxis in small and large signal gradients.. 46
3.5 Comparison of the moment-flux models with the individual-based model for a static
signal with large gradient...... 59
3.6 Comparison of the moment-flux model with the individual-based model for the os-
cillating signal (3.4.1) with large gradients...... 60
3.7 Initial conditions for the comparison of the closed moment system with different K
with the cell-based model shown in Figure 3.8...... 62
3.8 Comparison of the closed moment system with different K with the cell-based model. 62
3.9 Comparison of the moment-flux model with the individual-based model for the os-
cillating signal (3.4.1) with large gradient (µ0 = 2, c = s0/4 = 5 µm/s)...... 63
3.10 A numerical comparison between a particle undergoing (Left) regular Brownian mo-
tion and (Right) Levy flight...... 66
4.1 A cartoon description of a nerve cell...... 69
xiv 4.2 Fluorescence image of a cortical axon and its growth cone...... 70
4.3 The kinesin dimer attaches to and moves along microtubules towards its + end... 72
4.4 Schematic description of microtubule-neurofilament (MT-NF) phase separation in
cross-sectional view of the axon, under different neuron conditions...... 76
4.5 MT-NF segregation and remixing after washout...... 82
4.6 Simplifying assumption of the model...... 85
4.7 A plot of the H(z)...... 95
4.8 Decreasing MT-NF interaction destabilize the uniform steady states...... 97
p 4.9 Max eigenvalue of A with different k1 and k2 values in 1D case...... 100
p,q 4.10 Max eigenvalue of A with different k1 and k2 values in 2D case...... 101
4.11 Numerical simulations of microtubule and neurofilament distribution in 24 hours in
1D with periodic condition...... 105
4.12 Numerical simulations of microtubule and neurofilament distribution in 24 hours in
1D with noflux condition...... 106
4.13 Microtubule and neurofilament distribution in 80 hours in 2D with k1 = 0.6 and
different values of k2...... 107
4.14 Comparison of microtubule-neurofilament distribution patterns computed in 50 hours
in 1D with different sizes...... 107
4.15 Surface plot of Fig. 4.14...... 108
xv 4.16 Microtubule-neurofilament distribution patterns computed in 50 hours in 1D, IDPN
washout takes place at t = 30 hours...... 109
4.17 Plot of Dirichlet energy of Fig. 4.16. Dirichlet energy is a measure of inhomogene-
ity of a function, a sharp drop of Dirichlet energy is indicated by the simulation,
regardless of the domain size...... 109
4.18 Linear stability for linear-diffusion-only cases in 1d and 2d, periodic boundary con-
ditions are imposed...... 111
4.19 Microtubule-neurofilament distribution patterns computed in 20 hours in 1D, without
nonlinear diffusion terms...... 111
4.20 Plot of Dirichlet energy of Fig. 4.19...... 112
4.21 An extended bifurcation diagram for Fig. 4.9...... 113
4.22 Simulation result for large MT-NF interaction in T = 40 hours...... 113
5.1 Classification of static patterns developed in (5.1.1)...... 117
5.2 Numerical simulation of 2D Gray-Scott system with different µ and ρ values when
DS = 1,DA = 0.5...... 118
5.3 Concentration of chemical A...... 119
5.4 Various steady-state patterns formed an RDS with identical parameters...... 121
5.5 Polynomials can be regarded as algebraic curves, their roots are the intersections of
these curves...... 122
5.6 A graphical explanation of the homotopy tracking method...... 124
xvi 5.7 A graphical explanation of the prediction-correction method...... 125
5.8 A test of refined results when DA = 0.1,DS = 1, µ = 1, ρ = 0.01...... 130
5.9 Refined solutions when DA = 0.1,DS = 1, µ = 1, ρ = 0.01...... 131
5.10 Refined solutions when DA = 0.1,DS = 1, µ = 0.062, ρ = 0.055...... 132
5.11 Linear stability test of (5.1.1)...... 134
5.12 Linear stability test of (5.1.1)...... 135
A.1 Visual interpretation of notations...... 169
xvii CHAPTER 1
INTRODUCTION
A cell is the smallest unit of life that can replicate independently, and cells are often called the “building blocks of life”. Cell biology is a branch of biology aiming at studying behaviors, functions, structure, and activities of different kind of cells. Cell biology is undergoing rapid development during the last 30 years, and this relatively young subject is attracting more and more scientists and researchers from different areas including mathematics, physics, computer science, statistics and engineering [46, 101]. Albeit these effort, we are far from a full understanding of the intricate cellular world.
Pattern, on the other hand, is a concept widely used and observed in physics, chemistry, biology, geogra- phy and economics. For example, spontaneous symme- try breaking, Belousov-Zhabotinsky reaction, Turing’s pattern, landmark formation and economic oscillations
[48, 51, 98, 23]. Although used in different contexts and sometimes may be misleading, the theory of pattern Figure 1.1: Spatial biological patterns in formation has different roots for each branch. nature.
This thesis is to derive, develop and analyze mathe- matical models on pattern formation in cell biology. Specifically speaking, we study three differ- ent cases in detail:
1 1. Bacterial chemotaxis, in which we derive a macroscopic model from a microscopic;
2. Cytoskeleton segregation in an axon, in which we develop a nonlocal model;
3. Mutilple patterns arising in reaction-diffusion systems, in which we carry out computa-
tional analysis of a model.
1.1 STATEMENT OF MAIN RESULTS
Collective movement of bacteria due to large external signal gradient
Chemotaxis is a fundamental process in the life of many prokaryotic and eukaryotic cells.
Collectively bacteria can generate wave-like patterns and cell aggregates given different exper- imental conditions. To understand the role of chemotactic signaling in the population pattern formation, it is crucial to develop quantitative models to describe chemotaxis of the cell popula- tion. Individual-based stochastic models that take into account the biochemistry of intracellular signaling have been developed previously, and continuum PDE models have been derived from these individual-based models. However, the derivations rely on quasi-steady state approxima- tions of the internal ODE system. While this assumption is valid if cell movement is subject to slowly changing signals, it is often violated if cells are exposed to rapidly changing signals. In the latter case current PDE models break down and do not match the underlying individual-based model quantitatively.
In Chapter3, we derive new PDE models for bacterial chemotaxis in large signal gradients that involve not only the cell density and flux, but also moments of the intracellular signals as a measure of the deviation of cell’s internal state from its steady state. The derivation is based
2 on a new moment closure method without calling the quasi-steady state assumption of intracel- lular signaling. Numerical simulations suggest that the resulting model matches the population dynamics quantitatively for a much larger range of signals.
Cross-sectional microtubule-neurofilament segegation
The shape and function of an axon depend critically on the organization of its cytoskeleton, which is a dynamic system of intracellular polymers including microtubules, neurofilaments and actin. Under normal conditions, microtubules and neurofilaments align longitudinally in axons and are interspersed in axonal cross-sections. However, in many neurodegenerative disorders, they separate radially with microtubules clustered centrally and neurofilaments located near the periphery. This striking polymer segregation proceeds to focal accumulations of neurofilaments and/or organelles that are early hallmarks of nerve degeneration. A recent stochastic model sug- gests that this segregation is a consequence of the disruption of neurofilament transport along mi- crotubules, and in the absence of neurofilament transport, axonal organelles pull microtubules to- gether to the center and displace neurofilaments to the periphery. Despite insightful, the stochastic model is computationally intensive and
In Chapter4 we introduce an alternative approach to model the cytoskeleton segregation phenomena. Specifically, we developed a nonlocal PDE model based on the insights we ob- tained from the stochastic model. We used the new model to systematically analyze how the cross-sectional organization of microtubules and neurofilaments depends on the microtubule- neurofilament ratio, the size distribution of the organelles, and the size and boundary effect of the axon. Our results highlight the importance of incorporating the interactions of fast and slow
3 cargoes in addressing biological questions related to axonal transport.
Multiple solutions in reaction-diffusion systems
Reaction-diffusion systems have been widely used to describe phenomena arising in cell and developmental biology. In Chapter5, I present a method to compute multiple nonuniform equi- libriums for reaction-diffusion models. These solutions correspond to stationary spatial patterns supported by such models, thus characterize the underlying potential of these models in describ- ing relevant biological phenomena. The method is based on homotopy continuation and uses tools from numerical algebraic geometry. I illustrate the method using the Gray-Scott model which was proposed in 1980’s to describe autocatalytic glycolysis reactions. I compute all possi- ble nonuniform steady-state solutions and find their stability for a 1D situation.
4 CHAPTER 2
MODELING BACKGROUND IN CELL BIOLOGY
In the past half century, mathematical models have been widely used in biological and med- ical science. Many of these models help people to further understand the underlying biological processes involved in pattern formation phenomena. For example, as emphasized in Chapter
1, Turing’s reaction-diffusion system [154] was the theoretical foundation of modeling pattern- forming systems in developmental biology. One key reason for the huge success of Turing’s model, though may under much criticisms in modern times, is that the equations satisfies basic physical principles on chemical reaction and diffusion. For this reason, as well as serving as a modeling framework of Chapter3-5, I think it necessary to summarize some commonly used modeling concepts, terminologies and ideas in an independent chapter.
The organization of this chapter is as follows. In Section 2.1, we introduce some basic concepts and principles in mathematical modeling, all the materials are important for deriving models in the consequent chapters. In Section 2.2, we summarize some useful tools in mathematical modeling. In Section 2.3, as an alternative to deterministic models, we briefly introduce how stochastic models are developed. Moreover, the comparison between deterministic models and stochastic models has important implications to the modeling process of Chapter3 and Chapter
4.
5 2.1 SOME PHYSICAL BACKGROUND
2.1.1 Conservation laws
Conservation laws are probably among the most widely-used physical laws. Conservation of
Mass, Conservation of momentum, Conservation of Energy, Conservation of Electric Charge etc.
All these laws can be written into a general form
∂ρ + ∇ · J = 0, (2.1.1) ∂t for some quantity ρ and corresponding flux J of interest.
Take Conservation of Mass (which was first termed by Waage and Guldberg In 1864) as an example. If we denote n(x, t) to be the density of some matter at position x and time t, and v(x, t) to be the velocity of that matter, then conservation of mass yields
d Z n(x, t)dx = 0 (2.1.2) dt Ω or
∂n n(x, t) + ∇ · (vn) = 0 (2.1.3) ∂t
2.1.2 Chemical reactions
Chemical reactions dominates the whole biological world – every breath we take, fireflies shining in the night, and followers blooming during spring, are all involved with complicated chemical reactions.
Let’s start with a simple synthetic reaction of three chemicals A, B and C
k1 A + 2B C (2.1.4) k−1
6 where k1 and k−1 are respectively the reaction rates of positive direction and negative direction respectively. If we denote [X] to be the density of chemical then we can derive a corresponding system of ODEs based on (2.1.4) d[A] = −k [A][B]2 + k [C], dt 1 −1 d[B] = −2k [A][B]2 + 2k [C], (2.1.5) dt 1 −1 d[C] = k [A][B]2 − k [C]. dt 1 −1 In a more general case, a system of K chemical reactions takes the following form
k1 a11X1 + a12X2 + ... + a1M XM b11Y1 + b12Y2 + ... + b1N YN k−1
k2 a21X1 + a22X2 + ... + a2M XM b21Y1 + b22Y2 + ... + b2N YN k−2 (2.1.6) ......
kK aK1X1 + aK2X2 + ... + aKM XM bK1Y1 + bK2Y2 + ... + bKN YN k−K or in short
M N X ki X aijXj bimYm, 1 ≤ i ≤ K. (2.1.7) k j −i m where Xj, 1 ≤ j ≤ M are reactants and Ym, 1 ≤ m ≤ N are products.
Every chemical reaction system (2.1.7) can be written into ODE systems by kinetic equations, in a similar fashion with (2.1.5). For system (2.1.7), we have
And corresponding chemical kinetic equation yields K d[Xm] X Y Y = a (−k [X ]ajk + k [Y ]bjl ), dt jm j k −j l j k≤M l≤N K d[Yn] X Y Y = b (k [X ]ajk − k [Y ]bjl ), (2.1.8) dt jn j k −j l j k≤M l≤N
1 ≤ m ≤ M, 1 ≤ n ≤ N.
7 2.1.3 Diffusion
Diffusion is the process by which matter (here, a matter can either be molecule, heat, charge or fluid flow) is transported from one part of a system to another as a result of random molecular motions [35]. The net motion of the matter is driven by the gradient difference of it, making it moving from higher concentration part to lower difference part.
Suppose we want to model diffusion phenomena of some uniform molecule, the first thing is to define the diffusion flux J which is produced by molecular concentration gradient. If we denote u(t, x) to be the concentration of the molecule at time t location x, then the corresponding diffusion equation of u(t, x) writes
∂u = −∇J (2.1.9) ∂t if there is no external source. Applying Fick’s First Law [35]
∂u J = D (2.1.10) ∂x to (2.1.9), we can get the basic diffusion equation.
∂u = ∆u (2.1.11) ∂t note that up to this point, we don’t focus on technical details like initial or boundary condition.
Remark 2.1.1. In mathematical terminology, (2.1.11) is called a parabolic equation of second order, and it’s also a subclass of evolution equation.
Remark 2.1.2. There are other physical laws that have very similar mathematical formulation as Fick’s Law, but under different context. These laws are
8 Figure 2.1: Fick’s law illustrated in 1d case. The diffusion flux from higher density part (up) to lower density part (down) is proportional to the local density gradient.
κ • Darcy’s law of fluid flow in porous media: Jp = − µ ∇p, where Jp is the flow flux, κ is the
permeability (of the fluid), µ is the drag coefficient (of the fluid), and p is the pressure;
• Fourier’s law of heat transfer: JH = −k∇T , where JH is heat flux, k is the heat conduc-
tivity (of the material) and T is the temperature;
• Ohm’s law of electronic motion: Je = −ρ∇Φ, where Je is electric flux, ρ is electric
conductivity (of the material) and Φ is the electric potential.
Although in most text books in physics, all the parameters above are assumed to be constant, they are not in the most general case.
9 2.1.4 Advection
Unlike diffusion, advection is the transport of a matter by collective motion instead of random motion. Although in pure reaction-diffusion systems or Turing’s systems, advection terms may not appear, these terms are important too because they dictates how much bias a system may have.
Denote u(x, t) to be a scalar function that measures some conserved property of the matter
(e.g. u can be density, charge distribution etc) and v(x, t) to be the velocity field of that matter.
Then by conservation of u:
Du ∂u 0 = := + ∇ · (uv) (2.1.12) Dt ∂t
Advection phenomena are widely found in physics, engineering, and earth sciences, especially in the context of meteorology and oceanography, see [52] for a detailed mathematical introduction to advection. It is worth noticing that, there is a huge difference between a advection equation
(2.1.12) and a diffusion one (2.1.11) from mathematical point of view.
∂ Z Z u2dx = − u2dx (2.1.13) ∂t Ω Ω by simply using integration by part formula, the prove can be found in many standard PDE textbook (e.g. [47, 57, 149]).
10 2.2 USEFUL TOOLS IN MATHEMATICAL MODELING
2.2.1 Reaction-diffusion systems
In Section 2.1, we studied some Now we are ready to introduce the rigorous framework of a reaction-diffusion system.
Denote Ω ⊂ Rn to be the domain, u(x, t) ∈ Rn the density of n types of molecules or chemical species locating at x ∈ Rn and time t ∈ (0, ∞), p ∈ Rk represents k parameters (e.g. reaction rate, interaction rate etc), then a general reaction-diffusion system usually takes the form
∂u = ∇ · (D(u) · ∇u) + F(u, p) (2.2.1) ∂t | {z } | {z } diffusion reaction where D(u) is the diffusion matrix; F(u, p) is the reaction term, denoting reactions or transportation-independent interactions between different species.
More explicitly, in accordance with the chemical kinetic system (2.2.2), we can write (2.2.1) into
K ∂um X Y a Y b = ∇ · (D(u)(u, v)u) + a (−k u jk + k v jl ), ∂t m jm j k −j l j k≤M l≤N K ∂vn X Y a Y b = ∇ · (D(v)(u, v)v) + b (k u jk − k v jl ), (2.2.2) ∂t n jn j k −j l j k≤M l≤N
1 ≤ m ≤ M, 1 ≤ n ≤ N.
(u) (v) where Dm (u, v), 1 ≤ m and Dn (u, v), 1 ≤ n ≤ N are diffusion coefficients.
If, in addition, advection terms are to be considered, then (2.2.1) is modified to be
∂u = ∇ · (D(u) · ∇u) + ∇ · (vu) + F(u, p) (2.2.3) ∂t | {z } | {z } | {z } diffusion advection reaction which is also called an advection-reaction-diffusion system.
11 Typically, advection-diffusion-reaction models deal with the time evolution of chemical or biological species in a flowing medium such as water or air. For example:
• The Fokker-Planck equation of density evolution [59]:
∂ ∂ ∂2 p(x, t) = − [µ(x, t)p(x, t)] + [D(x, t)p(x, t)] . (2.2.4) ∂t ∂x ∂x2
• The Keller-Segel equation of bacteria chemotaxis (with growth terms) [74]:
∂u = Du∆u − ∇(χ(u, c)∇c) + f(u, c), ∂t (2.2.5) ∂c = D ∆c + g(u, c). ∂t c
There are many other pattern-forming PDE systems, especially in the context of physics. we list some of the most widely studied pattern-forming systems below and discuss their basic features. In Chapter5, we introduce computational methods to compute multiple patterns of reaction-diffusion systems, and the methods can be further applied to study the following sys- tems.
1. Fisher-KPP equation. Fisher-KPP (Fisher-Kolmogorov-Petrovsky-Piskunov) equation is
one of the simplest semilinear reaction-diffusion equation, and it is originally used by
Ronald Fisher in modeling collective migration of a single specie [49].
∂u ∂2u − D = ru(1 − u). (2.2.6) ∂t ∂x2
This equation is also famous because it admits traveling-wave solutions which are not
oscillating.
12 2. Fokker-Planck equation. The Fokker-Planck equation is is also known as the Kolmogorov
forward equation, it describes the dynamics of the probability density function (p) of the
velocity of a particle under the influence of drag forces and expressing Brownian motion.
∂ ∂ ∂2 p(x, t) = − [µ(x, t)p(x, t)] + [D(x, t)p(x, t)] . (2.2.7) ∂t ∂x ∂x2
3. Lotka-Volterra equation. Lotka-Volterra equations , also called a Predator-prey system, are
used to describe the dynamics of biological systems in which two species interact, one as
a predator (u) and the other as prey (v).
∂u = ∇ · (Du∇u) + αu − βuv, ∂dt (2.2.8) ∂v = ∇ · (D ∇v) + δuv − γv, ∂dt v
4. Keller-Segel system. A (classical) Keller-Segel system is a two-compartment reaction-
diffusion system describing how cell motion is guided by environmental chemicals. If we
denote u to be the cell density and c the concentration of the chemical signal, then
∂u = Du∆u − ∇(χ(u, c)∇c) + f(u, c), ∂t (2.2.9) ∂c = D ∆c + g(u, c). ∂t c
5. FitzHugh-Nagumo system. A FitzHugh-Nagumo system is a two-compartment PDE alter-
native to the Hodgkin-Huxley model. Denote membrane voltage to be v, and replace all
the detailed probabilities of ion channels by a slow linear recovery variable w, then the
FitzHugh-Nagumo system reads
∂v v3 = Dv∆v + v − − w + Iext ∂t 3 (2.2.10) ∂w τ = D ∆w + v + a − bw. ∂t w
13 6. Activator-inhibitor system. An activator-inhibitor system is also called a Gierer-Meinhardt
system, gives more intuitions into the original Turing’s system by introducing the interac-
tion mechanism between a short-range activator (A) and a long-range inhibitor (H). The
model reads
∂A A2 ∂2A = ρ − µAA + DA 2 + ρA, ∂t H ∂x (2.2.11) ∂H ∂2H = ρA2 − µ H + D + ρ ∂t H H ∂x2 H
7. Activator-substract system.
∂A 2 = DA∆A + SA − (µ + ρ)A, ∂t (2.2.12) ∂S = D ∆S − SA2 + ρ(1 − S). ∂t S
2.2.2 Parameter estimation
Once the model is constructed, the next step would be to estimate the parameter values. In general, this step can be proceeded in two methodologies, i.e.
1. Statistical methods. If experimental data is sufficient, all the parameters can be selected
by fitting the model with data, which essentially transformed the parameter estimation
problem into an optimization problem. Depending on the problem and the model itself,
frequently used methods are linear/nonlinear regression [105], max likelihood methods
and Bayesian methods [14]. For biological systems, penalty terms are often adopted to
both avoid ill-poseness as well as increase optimization efficiency.
2. Physical methods. If only partial experimental data is available, or the data is inaccurate, a
more robust way is to determine these model parameters by making use of physical laws.
14 This method is always better when underline physical laws are accurate, for example, con-
servation laws listed in Section 2.1.1. If the underline physical laws are not first principles,
or were derived due to approximation methods (e.g. Bolzmann’s equation is an approxi-
mation for dilute particle systems [31]), then sensitivity analysis is favored.
In this thesis, we only focus on the physical methods, these methods are used in Chapter3 through Chapter5. To follow up, I will give some examples to demonstrate how to apply physical principles.
Estimation of diffusion coefficients
First of all, we are interested in estimating the diffusion coefficients in the reaction-diffusion system (2.2.1). It is frequently archived by using the fluctuation-dissipation relations, which re- lates thermal fluctuations (uncertainties) in a physical variable predict the response quantified by the admittance or impedance of the same physical variable, when the system is in thermodynamic equilibrium (steady-state).
For example, the Einstein relation takes the following form
D = νkBT (2.2.13)
where D is the diffusion coefficient, ν is particle motility, kB is Boltzmann’s constant, T is abso- lute temperature. The choice of ν depends on the detailed physical environment, two examples are given as following.
• Diffusion of charged particles:
ν k T D = q B e (2.2.14) q
15 where q is the drag coefficient and r is the radius of the particle.
• Diffusion of spherical particles through low Reynold number fluid:
k T D = B (2.2.15) 6πµr
where ν is the drag coefficient and r is the radius of the particle.
The proof of general fluctuation-dissipation relations is proceeded by balancing the drift flux with the diffusion flux under thermodynamic equilibrium, i.e. Jdrift + Jdiffusion = 0, where
Jdrift(x) = ν(x)F (x)ρ(x), (2.2.16)
Jdiffusion(x) = −D(x)∇ρ(x). (2.2.17)
ν(x)F (x) yielding D(x) = ∇ ln (ρ(x)) .
Another frequently used method of estimating diffusion coefficient of some molecule is by relating the coefficient to its accessible surface area A [67, 68]. Since the surface area of a large class of molecules, like globular proteins, are proportional to M 2/3 where M is the weight of the molecule and is measurable [80], the diffusion coefficient can be calculated in this way.
Estimation of reaction coefficients
According to the Arrhenius equation, the interaction rates ki in (2.1.7), are taken to be the following form
∆µi ki = A exp ( ), (2.2.18) kBT where kB is the Boltzmann Constant, T is the temparature (in Kelvin degree), ∆µi is the activa- tion energy for specie i, and A is a rescale constant for the whole reaction system.
16 Since the activation energy are fixed for a specific specie, temperature is the only environmen- tal variable that may infect the interaction rates.
We refer the readers to Section 4.4 for more examples on parameter estimation.
2.2.3 Nondimensionalization
Variables and coefficients of a physical equation often involves various units, and the com- bination of them may become complicated and not of mathematical interest. For example, for physical constants like Bolzmann’s constant the Gravitational constant all have complex dimen- sions, and one need to be very careful if these constants are used at different scales in which different dimensions are used. In reaction-diffusion systems, physical dimensions are often taken to be length scale L0, the time scale T0, the mass (or density) scale N0 and the temperature scale
T0 with dimensions.
After choosing a proper physical scales, one can then use scale transformation to reduce the original variables to nondimensionalized ones. For example, after rescaling, the nondimension- alized form position x, time t and diffusion coefficient D are
x ¯ t ¯ T0 x¯ = , t = , D = D 2 . (2.2.19) L0 T0 L0
Thanks to the Buckingham π theorem, we can always reduce a physically-meaningful equation in terms of a set of p = n−k dimensionless parameters π1, π2, ..., πp., where n is the total number of variables and k is the number of physical dimensions.
17 2.3 MODELING USING STOCHASTIC APPROACH
Both reaction and diffusion are related to molecular dynamics. For example, Albert Einstein
[41] and Marian Smoluchowski (1906) were the first people who found the underlying connection between Brownian motion and diffusion, and presented it as a way to indirectly confirm the existence of atoms and molecules. Their fundamental work had popularized the rigorous studies addressing on Brownian motion [88, 130] and stochastic differential equations [79, 113] in both physical and mathematical context.
While the diffusion equation describes the probability distribution that a particle is at a particu- lar place at some time, it does not describe how the particle actually move. The actual movement of molecules are governed by Brownian motion, making them constantly moving in random di- rections and colliding at random location. The challenge in modeling Brownian motion using differential equation is that, even though sample paths of Brownian motions are continuous, they may not be differentiable [113]. The classical way is to use Ito (or Stratonovich) calculus to express Brownian as a Stochastic differential equation. For example, the Ito calculus takes the form
√ dx = 2DdW (2.3.1) where D is the diffusion coefficient, W is the infinitesimal change of a standard Wiener process.
One may roughly arguing that dW ≈ N(0, dt).
From the SDE description of Brownian motion (2.3.1), one may expect that there is a natural way to be recognize diffusion and reaction as stochastic processes. In this section, we will treat diffusion, reaction and advection affects individually, to see how these different terminologies are
18 raised from randomness.
2.3.1 Diffusion and randomness
Now, I will show how to rigorously derive the diffusion equation (2.1.11) from spatial temporal discretization of the probability density function of a homogeneous particle. For simplicity, we only consider the 1D case, derivation in higher dimensions can be extended in a similar fashion.
Assume a particle moving along a straight line by making steps of size ∆x at fixed discrete time step ∆x, and set xi = i∆x(i = 0, ±1, ±2, ...). We assume that the particle is initially located at x = 0, and we denote by ui(t) the probability that the particle will appear at xi in time
± t . We denote by Pi the probability of jumping from xi to xi ± 1. Then, for any t,
+ + − ui(t + ∆t) − ui(t) = Pi−1(t) + P + −i+1 − (Pi + Pi )ui(t) (2.3.2)
+ − We further assume that the diffusion is homogeneous so that Pi = Pi ≡ α (α ∈ [0, 1] is some constant), divide the discretized equation (2.3.2) by ∆t then we arrive at a the limit equation
u(x, t + ∆t) − u(x, t) α = [u(x − ∆x, t) + u(x + ∆x, t) − 2u(x, t)], ∆t ∆t (2.3.3) (∆x)2 ∂2u(x, t) ∆x ≈ α + O(( )3), ∆t ∂x2 ∆t ∂u(x, t) for ∆x small enough. Moreover, the left hand side of (2.3.3) can be approximated by + ∂t O(dt).
To further simplify (2.3.3), we want to take limit with ∆t → 0, ∆x → 0 in a proper way. If we assume
(∆x)2 lim α = D, D > 0, (2.3.4) ∆t→0 ∆t
19 then (2.3.3) reduces to
∂u ∂2u = D (2.3.5) ∂t ∂x2 and that’s exactly the same as (2.1.11).
Numerical results can help to validate the above analysis. In Figure. 2.2, I simulated a total number of 2000 particles initially located at the origin and undergo Brownian motions. and the
Figure 2.2: Simulation of 2000 particles with total time T = 20, ∆x = 0.1 and a): D = 1, b): D = 10.
comparison between Equation. (2.1.11) and Brownian motion is displayed in Figure. 2.3 as follows.
2.3.2 Advection and randomness
As we had seen in Section. 2.1.4, advection is very different from diffusion in that 1) instead of possibly moving to every direction, advection is a transportation process which admits directional movement; 2) advection conserves energy and therefore is typically reversible. Surprisingly, advection can still be realized as a large population limit of some particle system, just as diffusion
20 Figure 2.3: A comparison between the true solution of (2.1.11) and statistics of Brownian motion at different time slices with a), c): D = 1, b), d): D = 10.
did! Although one may expect that the property of such a system is far from unruled random motion.
Let’s take velocity-jump process as an example. This process can be achieved by the following transport equation [146, 119, 116]:
∂p −z1 − z2 h z2 ˙ i + ∇x · vp + ∇v · Fp + ∇z1 · p + ∇z2 · − − S p (2.3.6) ∂t te ta Z 0 0 0 = −λ(z1)p + λ(z1) T (v , v)p(x, v , y, t)dv . V
In [116], the authors studied the regime in which a diffusion equation can be obtained from
(2.3.6) arises in one spatial dimension. I will consider a slightly more general case here. Assume
21 a particle moves along the x-axis at a speed s (taken to be constant for simplicity), and it reverses direction at random times according to a Poisson process with rates λ±, where + and − respec- tively represents the turning rate of right and left moving particles. Let p±(x, t) be the density of particles that are at (x, t), and are moving to the right (+) and left (?). Then p±(x, t) satisfy the following system
∂p+ ∂p+ + s = −λ+p+ + λ−p−, (2.3.7) ∂t ∂x ∂p− ∂p− − s = λ+p+ − λ−p−. (2.3.8) ∂t ∂x
It’s not difficult to see that the above system is just the conservation of mass. If we plus and minus
(2.3.7) and (2.3.8) together and denote n = p+ + p−, J = s(p+ − p−), we get two telegraph-type equations
∂n ∂J + = 0, (2.3.9) ∂t ∂x ∂J ∂n + s2 = −2λ J − 2sλ n. (2.3.10) ∂t ∂x 0 b
+ − + − where λ0 = (λ + λ )/2, λ0 = (λ − λ )/2.Taking t derivative w.r.t (2.3.9), then minus the x-derivative of (2.3.10), we then get a second order hyperbolic equation of n:
∂2n ∂n ∂2n ∂n + 2λ = s2 + 2sλ (2.3.11) ∂t2 0 ∂t ∂x2 b ∂x
Dividing both sides by s2,(2.3.11) becomes
1 ∂2n 2λ ∂n ∂2n 2λ ∂n + 0 = + b (2.3.12) s2 ∂t2 s2 ∂t ∂x2 s ∂x
We can further analyze the behavior of right moving and left moving particles by properly rescaling equation (2.3.12). For example, if the velocity and turning rates of the particle are large
22 enough, and such that
2λ 2λ 0 → D, b → b, as s, λ± → ∞ (2.3.13) s2 s where D, b are some constants, then the hyperbolic equation (2.3.12) behaves like a diffusion- advection equation (a parabolic one) by simply dropping O(s2) terms
∂n ∂2n ∂n D = + b (2.3.14) ∂t ∂x2 ∂x
To check the validity of equation (2.3.14) as well as to illustrate the difference between hy- perbolic systems and parabolic systems, the author did some numerical experiments whose re- sults are displayed in Fig. 2.8 and Fig. 2.5. In Fig. 2.8, we chose (all variables are unitless) s = 5, λ+ = 30, λ− = 10, both (2.3.14) and (2.3.12) match well with stochastic simulations.
In Fig. 2.5, we change s = 1, λ+ = 0.55, λ− = 0.45, we can see that only (2.3.12) matches with stochastic simulation, indicating a violation of diffusion-advection approximation of the transportation system (2.3.10).
2.3.3 Reaction and randomness
Because of their nonlinearity, chemical reactions are different from diffusion and advection.
For this reason, one needs to be careful when he/she attempts to model a chemical reaction using stochastic approach, because large number theorems may fail to apply in these cases.
As an illustrative example, consider the well-know Michaelis-Menten kinetics
k1 k E + S )−*− ES →3 E + P (2.3.15) k2
Where E,S,ES and P are respectively number of enzyme, substrate, enzyme-substrate com- plex and product, and k1, k2, k3 are reaction rates. For stochastic approach, we can use Gillespie
23 Figure 2.4: Comparison of advection effect based on hyperbolic form (2.3.12), advection-diffusion form
(2.3.14) and stochastic simulation when s = 5, λ+ = 30, λ− = 10 . algorithm to simulate (2.3.15).
A comparison between Glimpse algorithm and ODE-based simulation is shown in Fig. 2.7. In the Michaelis-Menten example, stochastic and deterministic simulations consist with each other for system (2.3.15). In fact, for a large class of chemical reactions, even if the system is stiff, stochastic simulation and deterministic simulation are often consistent.
On the other hand, if a reaction system admits multiple steady-states, there can be a huge bias between stochastic simulations results and their deterministic counterparts. For example, the well-known noise-induced bistability phenomenon is a characteristic of stochasticity only – without noise, the corresponding system has unique steady-state; when noise is present, on the other hand, the system can jump between two different steady-states.
24 Figure 2.5: Comparison of advection effect based on hyperbolic form (2.3.12), advection-diffusion form
(2.3.14) and stochastic simulation when s = 1, λ+ = 0.55, λ− = 0.45.
To see how randomness can induce bistability, consider the reaction equation studied in [132,
45].
dx = −0.00025x3 + 0.18x2 − 37.5x + 2200 (2.3.16) dt which is characterized by the following reaction
k1 k 2A )−*− 3A, 2A ←→3 A (2.3.17) k2 k4 with k1 = 0.18, k2 = 0.00025, k3 = 37.5, k4 = 2200.
The following is a comparison between Gillespie algorithm and ODE-based simulation of chemical reaction.
As shown in Fig. 2.8, the deterministic simulation of the reaction (2.3.17) converges quickly
25 Figure 2.6: A comparison between the true solution of (2.1.11) and statistics of Brownian motion at different time slices with a), c): λ = 0.5, b), d): D = 10.
Figure 2.7: Comparison of stochastic and deterministic simulation of (2.3.15) under the identical initial
condition. Stochastic and deterministic simulations match well with each other.
to one of the steady-states of (2.3.16), the initial condition determines to which steady-states the
solution will converge. In contrast, however, Gillespie’s algorithm based simulation can make
the solution switch between the two different steady-states of (2.3.16), although it behaviors in
26 Algorithm 1: Gillespie algorithm
Input : Initial number of molecules E0,S0,ES0,P0, reaction rates k1, k2, k3, total
simulation time T .
Step 1: Generate two uniformly distributed random numbers r1, r2 to determine the next reaction
type j = 1, 2, 3 as well as waiting time τ. The probability of a given reaction to be chosen
is proportional to the number of substrate molecules, the waiting time τ is exponentially
distributed with mean − log(r2);
Step 2: Increase the time step by τ in Step 1. Update the molecule count of each specie based on
the reaction that occurred;
Step 3: Go back to Step 1, until the number of reactants is zero or T has been exceeded.
Figure 2.8: Comparison of stochastic and deterministic simulation of (2.3.16) under the same initial con- dition. For stochastic version, I chose only one sample path, which seems to match with the deterministic in the short run (left), but it deviates away from the deterministic case in the long run. concordance with ODE simulation in a short run.
If we look at the corresponding “potential” function of (2.3.16), we can see that the sample path simulated by Gillespie’s algorithm wanders around the two local minimums of the potential
27 (2.3.16), switching between these minimums in a random fashion (Fig. 2.9 below).
Figure 2.9: A sample path of stochastic simulation of (2.3.15) and its relation to the potential function of
(2.3.16). The potential function is given by the integral of the right hand side of (2.3.16) after normaliza- tion.
A biochemical implication to Fig. 2.9 is that, if we want to study how a protein or a DNA can undergo conformal changes, we can denote the two local minimums of corresponding potential function (Gibbs free energy) to be two different conformal state. See [129] for detail.
2.4 SUMMARY AND DISCUSSION
Mathematical modeling is an important tool for understanding the biological world. In this chapter, we summarized some useful concepts, ideas and methods of mathematical models, with specific focus on the models derived by using physical methods.
Another main point of this chapter is that, as proposed in Section 2.3, mathematical models can either be derived using deterministic approach or stochastic approach. Both stochastic and deter- ministic modeling approaches have been used and compared in various context of mathematical biology. Roughly speaking, stochastic models are easier to parameterize, but hard to analyze
28 mathematically and they are usually computationally expensive. Deterministic models, on the other hand, are amenable to mathematical analysis, and corresponding computational methods are well-developed. However, since biological processes are usually complicated, one may not
find it an easy task to derive deterministic equations out of these processes. Therefore, it is important to find connections between both approaches and compare those two strategies.
In the three subsections of Section 2.3, we see that for most of the cases, both approaches are consistent with each other. In some special cases, however, it may not be true. This is another reason why both approaches are used interchangeably even in identical situations. It is noticeable that, in Chapter3 and Chapter4, our proposed models are all deterministic, but both of them are inspired by stochastic models (individual or agent-based models) due to the limitation of stochastic models. For this reason, in our future work, we may go back to stochastic models and will further compare both approaches.
29 CHAPTER 3
CHEMOTAXIS PATTERNS DRIVEN BY LARGE SIGNAL GRADIENT
Chemotaxis is a fundamental process in the life of many prokaryotic and eukaryotic cells.
One important feature of chemotaxis is the collective behavior of cells, revealing the pattern- forming nature of a chemotactic system. Chemotaxis of bacterial populations has been modeled by both individual-based stochastic models that take into account the biochemistry of intracellular signaling, and continuum PDE models that track the evolution of the cell density in space and time.
In this chapter, we derive new PDE models for bacterial chemotaxis in large signal gradients that involve not only the cell density and flux, but also moments of the intracellular signals as a measure of the deviation of cell’s internal state from its steady state. The derivation is based on a new moment closure method without calling the quasi-steady state assumption of intracel- lular signaling. Numerical simulations suggest that the resulting model matches the population dynamics quantitatively for a much larger range of signals.
The organization of this chapter is as follows. Section 3.1 introduces biological background of chemotaxis. Section 3.2 summarizes previous works and states our motivation. Specifically, we describe the individual based model and review previous methods to extract PDE models from it in detail. We use numerical simulations to illustrate that while these PDE models provide good approximations to the individual-based model with small signal gradients, they fail to match the individual-based model quantitatively for situations that involve large signal gradients. In Section
3.3, we derive new PDE models that can be used for signals with much larger gradients.We
30 demonstrate the accuracy of these models using numerical simulations. Finally, in Section 3.5, we summarize our results and discuss other problems that should be addressed in future research.
3.1 BIOLOGICAL BACKGROUND
3.1.1 Chemotaxis
The motility of cells is fundamental during many biological functions, including embryoge- nesis and wound healing (see Albert’s book [46] or the review article [90]). Chemotaxis is an example of cell motility which is guided by external signal gradients.
Chemotaxis is the active movement of cells or organisms in response to external chemical sig- nals. It is crucial in many multicellular processes such as biofilm formation, embryonic develop- ment, wound healing, tumor metastasis, and immune responses [136, 93]. For example, chemo- taxis can greatly enhance the ability of microorganisms to clean contaminated environments in a process called bioremediation [140], and is critical for neutrophils to recognize bacteria and fight infection [82].
Bacterial chemotaxis is important for bacteria to find food by swimming toward the highest concentration of food molecules, or to flee from poisons. After a sequence of experiments done by Henry Harris in 1950s, related works on bacterial chemotaxis were developed quickly by both mathematicians and biologists, due to the revolution of modern cell biology, biochemistry and mathematical biology. Up to now, probably the best understood chemotaxis behavior was on Escherichia coli (E. coli for short). Depending on external (or environmental) conditions,
E. coli can be induced to form a variety of spatial patterns. Such reaction can be simulated by
Keller-Segel model, an example of reaction-diffusion system.
31 On the other hand, amoeboid cells such as Dictyostelium discoideum and neutrophils, adopt much more complicated chemotaxis strategy than bacteria such as E. coli. Two of the differences are
1. Unlike E. coli which can only detect temporal changes of extracellular signal, amoeboid
cells are large enough to extract spatial variation of the signal over their body, amplify it,
and become polarized [121, 32, 148].
2. in contrast to E. coli which move by alternating runs and tumbles, amoeboid cells move by
rearranging their cytoskeleton and generating forces on the substrate [103, 50]. When they
change direction of movement, they tend to change more slowly and smoothly rather than
having sharp reorientations as observed in bacterial tumbling.
Despite the different details across cell types, chemotaxis of a single cell involves three major steps: (1) detection of extracellular signal by receptors on the cell membrane, (2) transduction and amplification of the extracellular signal into an intracellular signal, and (3) change of cell move- ment in response to the intracellular signal. A fundamental biological question is to investigate the role of single-cell signal transduction and movement in chemotaxis of cell populations.
3.1.2 Biochemical details of E. coli chemotaxis
Many bacteria such as E. coli swim by rotating their flagella in different directions [13, 171].
When rotated counterclockwise (CCW), the flagella form a bundle and the cell is propelled for- ward with a speed 10−30µm/s; when rotated clockwise (CW), the bundle flies apart and the cell
‘tumbles’ in place. After a tumble, the cell picks a new direction randomly and runs again. The
32 new direction has a slight bias in the previous direction. In the absence of signal gradients, the overall cell movement is an unbiased random walk, with a mean run time 1 s and a mean tumble time 0.1 s. However, when exposed to a signal gradient, depending on whether it is a chemoat- tractant (e.g. sugar) or chemorepellent (e.g. toxin), the cell increases (or decreases) the mean run time when moving upwards (or downwards) the signal direction, and thus the net movement of the cell has an overall drift [97].
Figure 3.1: The signal transduction pathway for E. coli chemotaxis. Chemoreceptors (MCPs) span the cytoplasmic membrane (hatched lines), with a ligand-binding domain on the periplasmic side and a signal- ing domain on the cytoplasmic side. The cytoplasmic signaling proteins are represented by single letters, e.g., A = CheA. Adapted from [143].
The signal transduction pathway that orchestrates bacterial chemotaxis has been studied exten- sively in E. coli [18, 97]. Figure 3.1 depicts the major processes involved. The transmembrane chemoreceptors (MCPs) form stable ternary complexes with the signaling proteins CheA and
CheW. CheA is a kinase for the response regulators CheY and CheB. The phosphorylated form
CheYp binds to the flagella motor, increases the probability of CW rotation, and thus triggers tumbling. CheBp and CheR change the methylation state of the receptor at a slower rate: CheR methylates is and CheBp demethylates it. This methylation-demethylation cycle change the ac- tivity of the associated CheA.
33 Upon ligand binding, the kinase activity of CheA is reduced, thus CheYp decreases rapidly, and the cell runs for longer. This process, called excitation, occurs within fractions of seconds.
In the mean time, CheBp is reduced but CheR is not affected, thus the receptor methylation level increases, until the activity of CheA is restored to its pre-stimulus level. This process, called adaptation, takes from seconds to minutes, depending on the nature of the signal. Adaptation is crucial for cells to respond to further signal changes. E. coli chemotaxis system can detect signals that span over 5 orders in concentration and respond to a change in the occupancy of the receptor as little as 0.1% [12, 33].
3.1.3 Pattern formation and bacteria chemotaxis
The collective response of swimming bacteria to a chemoattractant can lead to intricate spatial patterns. Adler found that E. coli cells move up the gradient of a nutrient and form traveling bands in a capillary assay [1]. Budrene and Berg [28, 29] found that in semi-solid agar, when E. coli cells move up the gradient of a nutrient (succinate), they also secrete another stronger signal
(aspartate), and self-organize into patterns from outward-moving concentric rings to symmetric arrays of spots and stripes. In a thin layer of liquid medium, E. coli cells also form aggregates and hollow ring patterns [104, 21]. Other complex patterns driven by bacterial chemotaxis were also observed in similar bacterial species [162, 11, 166].
3.2 THE INDIVIDUAL-BASED MODEL AND PREVIOUS PDE MODELS
In this section, we first give a general overview to previous models on chemotaxis. We then give detailed description of the individual-based model for bacterial chemotaxis and then review
34 Figure 3.2: Patterns generated by chemotactic E. coli bacteria in the presence of succinate, a chemoat- tractant. (a) swarm ring in 1 mM succinate; (b) spots on a pseudo-rectangular lattice in 2 mM succinate;
(c) spots on a pseudo-hexagonal lattice in 3 mM succinate; (d) spots with tails on a pseudo-hexagonal lattice in 3 mM succinate with a mutant strain.. Picture is adapted from [28]. previous PDE models that were derived from the individual-based model.
3.2.1 Overview
The chemotactic movement of cell populations has been described using both continuum mod- els and cell-based models.
In continuum models, the evolution of cell density is described by partial differential equations
(PDEs), such as the classical Keller-Segel (KS) chemotaxis equation [122, 85, 86, 86], or its vari- ant forms [118, 56, 74]. We had actually introduced the two-compartment Keller-Segel equation
35 in Section 2.2. For the convenience of the reader, we rewrite the equation as follows. ∂u = Du∆u + ∇(χ(u, c)∇c) + f(u, c), ∂t (3.2.1) ∂c = D ∆c + g(u, c). ∂t c Mathematical analysis of these equations and numerical methods to solve them have been studied extensively in the past 40 years [74, 155]. However, these models are often derived from phenomenological descriptions of cell fluxes, and their relation to fundamental cellular mechanisms such as signal transduction and movement is not well-understood. On the other hand, cell-based models that incorporate many details of signal transduction and movement have been used to address population dynamics and pattern formation [36, 20, 87, 53, 166].
However, these models often involve large numbers of cells (106 and above), which make the computation extremely expensive and parameter exploration extremely time-consuming. In order to obtain a complete understanding of the chemotactic behavior of cells, it is crucial to establish connections between these two distinct approaches by deriving continuum models of cell density from cell-based models of signal transduction and movement. In this chapter, we study this question for chemotaxis of swimming bacteria such as E. coli.
Macroscopic equations have been derived for bacterial chemotaxis in small signal gradients.
Early work derived PKS equations from heuristic descriptions of cell movement as biased random walks with signal-dependent parameters [122,2, 114, 73]. Recently, simplified mechanisms of signal transduction have also been incorporated.
The derivation of (3.2.1) in previous studies relied on a “small gradient assumption” [44] or
“small signal variation assumption” [167, 170]. Either of these mean that the Lagrangian deriva- tive of the signal, i.e., the gradient of the signal along the cell trajectory, is sufficiently small that
36 intracellular signaling remains close to its steady state. When true, this justifies the approxima- tion of the intracellular ODE system by its quasi-steady states. However, if the signal changes rapidly such that most cells are far from their adapted states, the solutions of 3.2.1 do not match the corresponding individual-based model predictions quantitatively [165, 170]. This occurs, for example, in self-organized population dynamics such as moving band formation in bacterial pop- ulations due to the consumption of the signal by the cells [54]. This problem has been known as the “large gradient problem” [167] and little progress has been made on it to date. We note that other PDE models have been derived for bacterial chemotaxis in [139, 138], and these models are also based on similar assumptions of intracellular signaling and do not address the large gradient problem.
Two major features of bacterial chemotaxis in large signal gradients pose difficulties in the derivation of macroscopic PDE models. The first is the presence of overlapping time scales for intracellular signaling and external signal variation. Previous methods that assume separation of time scales, namely the time scale for intracellular signaling is much faster than the time scale for external signal variation, no longer apply in this context. The second, which derives from the first, involves the nonlinearities of the internal cell dynamics, either in the signal transduction or in the motor response, when far from equilibrium. A fundamental question is how the nonlinearities at the single cell level are reflected in the population-level descriptions of the dynamics.
In [169], we derive new PDE models for bacterial chemotaxis that can be used to model bacte- rial chemotaxis in large signal gradients. We consider individual-based models with the cartoon internal dynamics as in [44, 139, 167], and derive moment-flux models using a new moment clo- sure method that does not rely on the quasi-steady state assumption of intracellular signaling. The
37 method is based on the observation that higher-order moments of the internal dynamics equili- brate much faster than lower-order moments. The variables of the resulting PDE models not only include the cell density and flux, but also moments of the intracellular signaling variables. Nu- merical simulations show that these models match the population dynamics of individual-based models quantitatively for a much larger range of signals.
3.2.2 The individual-based model
Quantitative modeling has considerably advanced our understanding of E. coli signal trans- duction and chemotactic behavior (see [151, 150] for recent reviews). In particular, ODE models based on detailed biochemistry were used to model the entire network (Figure 3.1)[72, 143, 20].
These models share the general form
dy = f(S(x(t), t), y). (3.2.2) dt
q where y = (y1, y2, ··· , yq) ∈ R are the concentrations of the intracellular signaling proteins, x(t) is the position of the cell, S(x(t), t) is the signal along the cell trajectory, and f depends on the specific model.
The chemotactic movement of swimming bacteria has been described using velocity jump processes (VJPs) [73, 115, 44, 43, 166]. A VJP is a stochastic process in which the velocity of an individual jumps instantaneously according to a turning rate λ and a turning kernel T (v0, v), specifying the probability of transition from v0 to v [114]. When applied to bacterial swimming and if we ignore the relatively short tumbling, λ specifies the frequency of cell reorientations. It depends on the extracellular signal gradient either directly as λ = λ(∇S) [73, 115], or indirectly through the internal states y as λ = λ(y), in particular, the component that represents CheYp
38 [44, 43]. In the latter case, let p(x, v, y, t) be the probability density of a cell with position x ∈ RN , velocity v ∈ V ⊂ RN , and internal states y ∈ Rq at time t, then based on the general theory in [119], p satisfies the following transport equation Z ∂p 0 0 0 + ∇x · (vp) + ∇v · (Fp) + ∇y · (f(S, y)p) = −λ(y)p + λ(y) T (v , v)p(x, v , y, t)dv . ∂t V (3.2.3)
Here F is the force acting on the cell and f(S, y) is given by (3.2.2). The left-hand side of (3.2.3) models convection in the state space during a run and the right-hand side models the reorien- tation of the cell during a tumble. When cells are sufficiently separated, cell-cell mechanical interactions can be neglected and p(x, v, y, t) can be treated as the cell density in the (2N + q) dimensional phase space.
In this chapter, we consider the cartoon cell-based model studied in [44, 43, 167]: dy S(x(t), t) − y − y 1 = 1 2 , dt t e (3.2.4) dy S(x(t), t) − y 2 = 2 . dt ta
Here te and ta with te << ta are the excitation and adaptation time scales, S is measured as the fraction of receptor occupancy, and y1 is the variable that excites and adapts to the signal, which roughly represents the negative of the deviation of CheYp from its steady state. The exact solution of the system (3.2.4) is: Z t − t 1 − t−τ y (t) = e te y (0) + S(x(τ), τ) − y (τ) e te dτ, 1 1 t 2 e 0 (3.2.5) Z t − t 1 − t−τ y2(t) = e ta y2(0) + S(x(τ), τ)e ta dτ. ta 0 This cartoon model captures the main dynamics of excitation and adaptation involved in bac- terial chemotaxis. Given a step change of the signal S, we obtain the evolution of y1 and y2 as in
Figure 3.3.
39 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 1 2 G 0.4 y 0.4 y 0.4 0.2 0.2 0.2 0 0 0 0 20 40 0 20 40 0 20 40 t t t
Figure 3.3: The response to a step change of the signal S. te = 0.01, ta = 5.
After the change of variables z1 = y1, z2 = y2 − S, the system (3.2.4) was reduced to
dz −z − z dz z 1 = 1 2 , 2 = − 2 − S,˙ (3.2.6) dt te dt ta
˙ where S = ∂tS + ∇S · v is the Lagrangian derivative of S.
Lemma 3.2.1. Suppose that the cells move according to the VJP with internal dynamics (3.2.4).
˙ Denote Z0 = min{kSkL∞ , takSkL∞ }. Assume that |z2(0)| ≤ Z0, then we have
|z2| ≤ Z0.
Proof. The proof of this lemma is straightforward and thus omitted here.
Following [167], we assume the turning rate of each cell to be λ = λ(y1). It is a strictly decreasing function thus cells turn less when moving in favorable directions. We assume the turning rate can be expanded as
2 3 λ = λ0 − a1y1 + a2y1 − a3y1 + ···
2 3 = λ0 − a1z1 + a2z1 − a3z1 + ··· (3.2.7)
Let r0 be the radius of convergence for λ. It is necessary to assume
r0 ≥ Z0, (3.2.8)
40 so that the series (3.2.7) converges and remains bounded. This follows directly from Lemma
3.2.1.
The turning kernel was assumed to be symmetric, T (v, v0) = h(θ), where θ is the angle
0 between v and v , and the cell speed was a constant s0. Under the above assumptions, the transport equation (3.2.3) become
∂p −z1 − z2 z2 ˙ + ∇x · vp + ∇v · Fp + ∇z1 · p + ∇z2 · − − S p (3.2.9) ∂t te ta Z 0 0 0 = −λ(z1)p + λ(z1) T (v , v)p(x, v , y, t)dv . V
In [167], the authors applied regular perturbation methods directly to the infinite moment sys- tem (3.3.2) and (3.3.3) by recursively inverting an infinite matrix operator. However, the tech- nique used there is specific to that type of matrix, and the analysis cannot be generalized to the large gradient problem. In order to obtain population-level approximations of the cell-based model, we need to introduce a new moment closure method and study further reductions of the closed moment system in Section 3.3.
3.2.3 Previous PDE models for the population dynamics
In [44], a hierarchy of macroscopic models were derived from the above individual-based model in 1D. The signal was assumed to be a given function that does not vary over time, that is,
S = S(x). To understand the scopes of applicability of these models, we describe the key ideas of their derivation and the main underlying assumptions in the following. The notations that we use here are consistent with the rest of the chapter but slightly different from those in [44].
41 The stochastic movement of the cell is governed by the following master equations
∂p+ ∂p+ ∂ z 1 + s + − − s S (x) p+ = λ(z) −p+ + p− , ∂t 0 ∂x ∂z t 0 x 2 a (3.2.10) − − ∂p ∂p ∂ z − 1 + − − s0 + − + s0Sx(x) p = λ(z) p − p . ∂t ∂x ∂z ta 2
Here p± = p±(x, z, t) are the probability densities for a cell to be at position x with internal state z and velocity ±s0 at time t. The cell is unbiased in choosing a new direction of movement.
Since the individual-based model neglects the mechanical interactions of the cells, p± can also be thought as the microscopic cell densities at position x with state z at time t. In [44], the turning frequency was assumed to be a linear function,
λ(z) = λ0 + a1z, (3.2.11)
By properly rescaling z and S, one can assume that λ0 and a1 have the same order of magnitudes.
Define the moment variables as follows: Z Z + − + − n(x, t) = (p + p )dz, j(x, t) = s0(p − p )dz, R R Z Z k + − k + − nk(x, t) = z (p + p )dz, jk(x, t) = z s0(p − p )dz, k ≥ 1 R R Here n(x, t) and j(x, t) are the macroscopic cell density and flux.
The following lemma holds for the individual-based model.
Lemma 3.2.2. (i) Suppose that a cell moves according to the individual-based model with inter- nal dynamics (3.2.6). Assume St = 0 and denote Z0 ≡ s0 maxx |Sx(x)|ta. If |z(0)| ≤ Z0, then one has
|z(t)| ≤ Z0, ∀t > 0.
42 ± ± (ii) If p (x, z, 0) = 0 for all |z| > Z0, then the solution of (3.2.10) satisfies p (x, z, t) = 0 for all |z| > Z0, and the following estimates holds
k k nk ≤ Z0 n, |jk| ≤ s0Z0 n.
Proof. The proof of this lemma is straightforward and thus omitted here.
Taking the sum and difference of the two components of (3.2.10) and integrating over z, one obtains ∂n ∂j + = 0, ∂t ∂x (3.2.12) ∂j ∂n + s2 = −λ j − a j , ∂t 0 ∂x 0 1 1 We note that after integration the z-derivative terms in (3.2.10) vanishes because Lemma 3.2.2(ii)
± implies that limz→±∞ p (x, z, t) = 0.
Multiplying (3.2.10) by z and then performing the same calculations, one obtains
∂n ∂j 1 1 + 1 = −S (x)j − n , ∂t ∂x x t 1 a (3.2.13) ∂j1 2 ∂n1 2 1 + s0 = −s0Sx(x)n − λ0 + j1 − a1j2, ∂t ∂x ta
Note that the moment-flux system (3.2.12) and (3.2.13) is not closed because the 2nd-order moment j2 in (3.2.13) is unknown. To obtain a closed moment-flux system, j2 was estimated using the quasi-steady state approximation for the intracellular variable y2. Specifically, y2 in
(3.2.4) was estimated to equal S(x), which leads to z = y2 − S(x) ≈ 0 and thus
j2 ≈ 0. (3.2.14)
This estimation can be justified when the external signal changes slowly along the cell trajectory,
s0 max |Sx(x)|taa1/λ0 1, x
43 Since λ0 and a1 have the same order of magnitudes, the above condition is equivalent to
s0 max |Sx(x)|ta 1. (3.2.15) x
The condition (3.2.15) was called the “small gradient assumption”. Mathematically, if (3.2.15) holds, then j2 is much smaller than j1, and this follows immediately from Lemma 3.2.2.
The moment-flux system (3.2.12) and (3.2.13) with the closure (3.2.14) represents a macro- scopic model for the bacterial population dynamics. Under the diffusion space and time scale, it was further reduced to the PKS equation (3.2.1) for the cell density n(x, t) with
2 2 a1s0ta Dn = s0/λ0, χ = . (3.2.16) λ0(1 + λ0ta)
This was done using regular perturbations against the small parameter .
Alternative moment closure methods were also suggested in [44] based on quasi-steady state approximations of z. Replacing one or both z in the definition of j2 by its quasi-steady state
+ − zqss = ∓s0Sx(x)ta, “-” for right-moving cells (p ) and “+” for left-moving cells (p ), one obtains
2 j2 ≈ −s0Sx(x)tan1, (3.2.17) or
2 2 2 j2 ≈ s0Sx(x) taj. (3.2.18)
Under these closure assumptions, the moment-flux system reduces to the same PKS equation after taking the diffusion limit. These analysis were extended to 3D in [43].
A new method was developed to derive the PKS model for the situation with a nonlinear turning frequency in the most general form
∞ X i λ(z) = λ0 + aiz (3.2.19) i=1
44 in [167]. The derivation did not involve any moment closure step, but instead employed regular perturbation to the infinite moment system for n, j, nk and jk with k ≥ 1 on the diffusion space and time scale. Perturbation of an infinite moment system involves the inversion of an infinite matrix operator that is not feasible in general. In this case a special technique was developed based on the structure of the equations as well as the small gradient assumption.
3.2.4 Numerical comparisons
In this section, we numerically compare the moment-flux models (3.2.12) and (3.2.13) under the three moment closure assumptions (3.2.14), (3.2.17) or (3.2.18), the PKS model (3.2.1) with
(3.2.16), and the individual-based model governed by (3.2.10) and (3.2.11). We consider cell movement in the domain 0 ≤ x ≤ L with L = 5 mm. We use periodic boundary conditions to eliminate any potential boundary effects. We consider the piecewise linear signal
S(x) = µL/2 − µ|x − L/2|, (3.2.20) which reaches its maximum at the center of the domain and minimum at the boundary (Figure
3.4A). We assume that cells initially form an aggregate in x ∈ [0, 1] with the distribution (see
Figure 3.4B) 3π 3 | sin(πx)| 0 < x < 1, n(x, 0) = 4 0 x ≥ 1. We further assume that initially the direction of cell movement is random and cell signaling is at its steady state, i.e., z = 0 for each cell. Numerical simulations for bacteria movement in small and large signal gradient cases are shown in Fig. 3.4.
In Fig. 3.4 we investigated previous approximations of the cell-based model given a linear
45 Figure 3.4: Previous PDE models for bacterial chemotaxis in small and large signal gradients. A: the signal function with small (µ = 1) or large (µ = 4.5) gradient; B. the initial cell distribution for both cases; C: chemotaxis in small gradient; D: chemotaxis in large gradient. Gray bars: stochastic simulation of the individual-based model. Blue, red, and green curves: the moment-flux model (3.2.12), (3.2.13) with three different closure assumptions (3.2.14), (3.2.17). Black curves: the PKS model with (3.2.16).
Parameters used: s = 0.02 mm/s, a1 = 2/s, ak = 0 for k ≥ 2, λ0 = 1/s, ta = 5 s.
turning rate λ = λ0 − a1z1 and externally imposed signal. The solid curve shows comparisons of the macroscopic equations derived previously with the stochastic simulations of the cell based model under small signal gradients (taH = 0.3). We see that both methods show a good fit with the stochastic simulation. In addition, as shown in [139], the derived PKS equation tends to overestimate the amplitude of the aggregate.
The dashed curve compares these representations for chemotaxis in large signal gradients
(taH = 0.9). We see that none of these representations fit the dynamics of the cell density in a satisfactory manner. More specifically, as expected, the former case captures the steady state
46 solution better, but have a significant slowing effect for the transient dynamics. The latter case fits the average dispersal speed better but the solution profile is more diffusive tends to underestimate the amplitude of the peak. In these simulations, we used second order upwind scheme in space and Runge-Kutta 4 in time. We used small enough time step to make the scheme stable.
From above comparisons, we can see that under large gradient case, we need new higher-level approximations for the transient dynamics and steady state solutions of the cell-based model for chemotaxis in large signal gradient. The new method is introduced in the following section.
3.3 NEW MOMENT-FLUX MODELS FOR CHEMOTAXIS IN LARGE SIGNAL GRA-
DIENTS
In this section, we derive the macroscopic equation in 1D. Without loss of generality we assume there is no external force acted on bacterial cells. We further assume that fast excitation te ≈ 0 thus z1 = −z2 and neglect tumbling time of cells, and consider time-independent signal with large spatial gradients.
Under these assumptions, the velocity jump process (3.2.10) reduces to
+ + ∞ ∂p ∂p ∂ h z2 i 1 X + s + − − s S0(x) p+ = λ + a zj − p+ + p− , ∂t 0 ∂x ∂z t 0 2 0 j 2 2 a j=1 (3.3.1) − − ∞ ∂p ∂p ∂ h z2 i 1 X − s + − + s S0(x) p− = λ + a zj p+ − p− . ∂t 0 ∂x ∂z t 0 2 0 j 2 2 a j=1
By properly rescaling z2 and S, we can assume that a1/λ0 ∼ O(1).
We define the moments as Z Z + − + − n = (p + p )dz2, j = s0(p − p )dz2, R R Z Z k + − k + − nk = z2 (p + p )dz2, jk = z2 s0(p − p )dz2, k = 1, 2, ··· R R 47 Here n(x, t) and j(x, t) are the macroscopic cell density and flux.
k By multiplying(3.3.1) by 1 and z2 /k, k ≥ 1, integrating over z2, and taking the sum and difference of the two components, we obtain the infinite system of moment equations in the following form: ∂n ∂j + = 0, ∂t ∂x ∞ (3.3.2) ∂j ∂n X + s2 = − λ j + a j , ∂t 0 ∂x 0 i i i=1
1 ∂nk 1 ∂jk h 0 1 i + = − S (x)jk−1 − nk , k ≥ 1 k ∂t k ∂x ta 2 ∞ (3.3.3) 1 ∂jk s ∂nk h 1 i 1 X + 0 = − s2S0(x)n − j − λ j + a j , k ≥ 1. k ∂t k ∂x 0 k−1 t k k 0 k i k+i a i=1 In the above, the scaling factor k is taken so that all coefficients are at most O(1). The following lemma shows that (3.3.2) and (3.3.3) are well-defined and remain to be bounded.
Lemma 3.3.1. Under the conditions of Lemma 3.2.1, the z2-moments nk, jk have the following estimates:
k k nk ≤ Z0 n0, |jk| ≤ s0Z0 n0.
Proof. The proof of this lemma is straightforward and thus omitted here.
The infinite system (3.3.2) and (3.3.3) in general cannot be solved directly. In order to get a closed, finite system of equations, we need to properly approximate higher-order moments in terms of lower-order moments.
3.3.1 The moment closure method
In this section, we introduce a new moment closure method to obtain macroscopic PDE models from the individual-based model in Sec. 3.2.2. We assume that the external signal is a given
48 function of space and time, i.e., S = S(x, t). The derivation is not dependent on the small signal gradient assumption (3.2.15) but instead the signal is allowed to have a much larger gradient,
max |St| + s0|Sx| taa1/λ0 = O(1). (3.3.4) x,t
Here St and Sx are the partial derivatives of S(x, t). If the signal does not depend on time
(St = 0), this condition reduces to
s0 max |Sx|taa1/λ0 = O(1). x
To illustrate the essential ideas of the method, we first consider a simple situation with the linear turning frequency (3.2.11) in Section 3.3.2. We then extend the analysis to allow nonlinear turning frequencies in the form of (3.2.19) in Section 3.3.3.
3.3.2 The case with linear turning frequency
In this section, we consider the case that the external signal is a function of space and time and the turning rate is given by (3.2.11). The microscopic cell density p± satisfies the following system
∂p+ ∂p+ ∂ z + s + − − S − s S p+ ∂t 0 ∂x ∂z t t 0 x a (3.3.5) 1 = (λ + a z) −p+ + p− , 2 0 1 ∂p− ∂p− ∂ z − s + − − S + s S p− ∂t 0 ∂x ∂z t t 0 x a (3.3.6) 1 = (λ + a z) p+ − p− . 2 0 1
49 Multiplying (3.3.5) and (3.3.6) by 1 and zk/k for all k ≥ 1, taking the sum and difference of the two components and integrating over z, we obtain the following moment equations
∂n ∂j + = 0, (3.3.7) ∂t ∂x ∂j ∂n + s2 = −λ j − a j , (3.3.8) ∂t 0 ∂x 0 1 1 1 ∂nk 1 ∂jk 1 + = −Stnk−1 − Sxjk−1 − nk , k ≥ 1 (3.3.9) k ∂t k ∂x ta 1 ∂j s2 ∂n 1 k + 0 k = −s2S n − S j − j k ∂t k ∂x 0 x k−1 t k−1 t k a (3.3.10) 1 − (λ j + a j ) , k ≥ 1, k 0 k 1 k+1 where n0 ≡ n and j0 ≡ j. Here the k-th order moments nk, jk depend on the k + 1-th order moments because (3.3.10) depends on jk+1. We note that under the condition (3.3.4) the moments n, nk and j, jk have the same order of magnitudes and this follows immediately from Lemma
3.2.2. Therefore all the terms in the above equations are at most O(1).
The infinite moment system (3.3.7)-(3.3.10) cannot be solved directly, and in order to get a closed finite system of equations one needs to properly approximate higher-order moments in terms of lower-order moments. For the case with St(x, t) = 0, the approximations (3.2.14),
(3.2.17) and (3.2.18) were used in [44]. However, these approximations are only valid under the small gradient assumption (3.2.15). If the signal gradient becomes large, then j2 will have the same order of magnitude as j1 and thus cannot be neglected as in (3.2.14). Moreover, intracellular signaling can be far away from equilibrium so z cannot be approximated by its steady state as in
(3.2.17) and (3.2.18). For this reason, it is not surprising that previous PDE models do not match the individual-based model as illustrated in Figure 3.4.
We propose a new moment closure method here to close the infinite moment system. We note
50 that the k-th order moment equations (3.3.9) and (3.3.10) have the factor 1/k in all terms except the terms in the square brackets that are integrated from the ∂/∂z terms in (3.3.5) and (3.3.6) using integration by parts. When k becomes large, the factor 1/k becomes very small. Because n, nk, j, jk have the same orders of magnitude, this motivates the estimation that for k large, the terms in the square brackets dominates, i.e., high-order moments nk and jk equilibrate relatively fast. Based on this observation, we pick an integer K large enough such that 1/(K + 1) ≡
1, and approximate all the moments with order higher than K by dropping the terms with the coefficient 1/k in their equations. This leads to
1 − S n − S j − n = 0 ∀k > K, t k−1 x k−1 t k a (3.3.11) 2 1 − s0Sxnk−1 − Stjk−1 − jk = 0 ∀k > K, ta which implies the following moment closure assumption
2 jK+1 = −s0SxtanK − SttajK . (3.3.12)
With this approximation, the infinite system (3.3.7)-(3.3.10) reduces to a closed moment system of (2K+2) moment equations for
T MK = (n, j, n1, j1, ··· , nK , jK ) . (3.3.13)
For any matrices E = (eij)1≤i,j≤q1 and F = (fij)1≤i,j≤q2 , the Kronecker product of these two
51 matrices is defined as the following q1q2 × q1q2 matrix e F e F ··· e F 11 12 1q2 e F e F ··· e F 21 22 2q2 E ⊗ F = (eijF )1≤i,j≤q2 = . . . . . . . . . . . . . eq21F eq22F ··· eq2q2 F
We denote 0 1 S S t x A = ,B = , 2 2 s0 0 s0Sx St 0 0 0 0 C = ,E = . 2 0 1 s0Sxta Stta and the 2 × 2 and (K + 1) × (K + 1) identity matrix by I2 and IK+1 respectively.
We define J to be the following (K + 1) × (K + 1) matrix 0 1 0 ··· 0 0 0 1 ··· 0 J = , . . . . . . . . . . . . . . . 0 0 0 ··· 0
Using these notations, the closed moment system can be rewritten in the following compact form: ∂ ∂ Λ M + A M = Q M + R M , (3.3.14) ∂t K K ∂x K K K K K K
52 where ΛK , AK and QK are (2K + 2) × (2K + 2) matrices with the following form
ΛK = diag{1, 1, 1/2, . . . λ, 1/K} ⊗ I2,
AK = diag{1, 1, 1/2, . . . λ, 1/K} ⊗ A,
1 T h i QK = − diag{0, 1, 1, ··· , 1} ⊗ I2 − J ⊗ B − ΛK (λ0IK+1 + a1J) ⊗ C , ta a R = 1 diag{0, 0, ··· , 0, 1} ⊗ E. K K
We call (3.3.14) the K-th order moment-flux model. We note that for the situation with St = 0, if K = 1 is taken then the moment closure assumption (3.3.12) reduces to the approxima- tion (3.2.17). If the signal gradient is small, the approximation is good because the magni- tudes of nk and jk decrease fast (by Lemma 3.2.2). However, if the signal gradient is large,
= 1/(K + 1) = 0.5 is not small enough to guarantee the accuracy of the approximation
(3.3.11) or (3.3.12). This observation is reflected in the numerical results in Figures 3.4C and D which demonstrate agreement of the derived PDE models and the individual-based model given a small signal gradient (3.4C) but not a large signal gradient (3.4D).
53 3.3.3 The case with a nonlinear turning frequency
In this section, we generalize the above analysis and computation to situations with nonlinear turning frequency given in the general form (3.2.19). In this case, the equations for p± are
+ + ∂p ∂p ∂ z + + s0 + − − St − s0Sx p ∂t ∂x ∂z ta ∞ (3.3.15) 1 X = (λ + a zi) −p+ + p− , 2 0 i i=1 − − ∂p ∂p ∂ z − − s0 + − − St + s0Sx p ∂t ∂x ∂z ta ∞ (3.3.16) 1 X = (λ + a zi) p+ − p− . 2 0 i i=1
Without loss of generality we assume that ai . O(1), which can be achieved by rescaling z and the internal dynamics of the model.
Multiplying (3.3.15) and (3.3.16) by 1 and zk/k for all k ≥ 1, taking the sum and difference of the two components and integrating over z, we obtain the following moment equations
∂n ∂j + = 0, (3.3.17) ∂t ∂x ∞ ∂j ∂n X + s2 = −λ j − a j , (3.3.18) ∂t 0 ∂x 0 i i i=1 1 ∂nk 1 ∂jk 1 + = −Stnk−1 − Sxjk−1 − nk , k ≥ 1 (3.3.19) k ∂t k ∂x ta 2 1 ∂jk s0 ∂nk 2 1 + = −s0Sxnk−1 − Stjk−1 − jk k ∂t k ∂x ta ∞ ! (3.3.20) 1 X − λ j + a j , k ≥ 1. k 0 k i k+i i=1
If we set ai = 0 for i ≥ 2, then the above infinite moment system reduces to (3.3.7)-(3.3.10).
Because Eqns (3.3.18) and (3.3.20) for j and jk depend on the higher order moments jk+i for all i ≥ 1 to obtain a closed moment-flux system of K-th order, we need to estimate all the
54 variables jK+i in terms of n, j, nk, jk with k ≤ K. To do that we use the same method as (3.3.11), which can be rewritten as s n s n 0 k ¯ 0 k−1 = −taB ∀k > K. (3.3.21) jk jk−1
Here B¯ S s S t 0 x B¯ = s0Sx St is a symmetric matrix and has the decomposition B¯ = PDP −1 with St − s0Sx 0 1 1 1 −1 −1 1 D = ,P = ,P = . 2 0 St + s0Sx −1 1 1 1
The approximation (3.3.21) leads to the following moment closure method: take K large enough such that 1/(K + 1) 1 and approximate jK+i by s0nK s0nK i ¯i i i −1 jK+i = (−ta) B = (−ta) PD P , i ≥ 1, (3.3.22) jK jK 2 2 where the subscript “2” means taking the second component of the vector.
Plugging (3.3.22) into (3.3.18) we obtain
∞ K ∞ ∂j ∂n X X X + s2 = −λ j − a j = −λ j − a j − a j ∂t 0 ∂x 0 i i 0 i i K+i K+i i=1 i=1 i=1 K ∞ s0nK X X i i −1 = −λ0j − aiji − aK+i(−ta) PD P i=1 i=1 jK 2 K ∞ ! s0nK X X i i −1 = −λ0j − aiji − P aK+i(−ta) D P i=1 i=1 jK 2
55 We denote ∞ ¯ X i λk(z) = ak+iz , (3.3.23) i=1
± dS = St ± s0Sx, and ¯ + ¯ − ¯e λk(−tadS ) + λk(−tadS ) λk = , 2 (3.3.24) λ¯ (−t dS+) − λ¯ (−t dS−) λ¯o = k a k a . k 2 Using these notations, we have
∞ X i i n¯ − ¯ + o aK+i(−ta) D = diag λK (−tadS ), λK (−tadS ) (3.3.25) i=1 and ∞ ! s0nK X i i −1 ¯o ¯e P aK+i(−ta) D P = λK s0nK + λK jK . (3.3.26) i=1 jK 2
Using these notations (3.3.18) reduces to
K ∂j ∂n X + s2 = −λ j − a j − λ¯o s n − λ¯e j . (3.3.27) ∂t 0 ∂x 0 i i K 0 K K K i=1
Similarly, plugging (3.3.22) into (3.3.20) with k ≤ K we obtain
2 1 ∂jk s0 ∂nk 2 1 + = −s0Sxnk−1 − Stjk−1 − jk k ∂t k ∂x ta K−k ! 1 X − λ j + a j + λ¯o s n + λ¯e j . (3.3.28) k 0 k i k+i K−k 0 K K−k K i=1
The equations (3.3.17), (3.3.19), (3.3.27) and (3.3.28) form the K-th order closed moment-flux system and can be written in the compact form
∂ ∂ Λ M + A M = Q¯ M + R¯ M , (3.3.29) ∂t K K ∂x K K K K K K
56 ¯ ¯ where MK , ΛK and AK are the same as before, and QK , RK are defined as
¯ 1 T QK = − diag{0, 1, 1, ··· , 1} ⊗ I2 − J ⊗ B ta " K ! # X i −ΛK λ0IK+1 + aiJ ⊗ C , i=1 ¯o 0 s0λ ¯ T i RK = −ΛK (EK ,EK−1, ··· ,E1,E0) (0, 0, ··· , 0,I2) with Ei = . ¯e 0 λi
Here the last term on the RHS comes from the approximations using the moment closure method
(3.3.22).
¯ ¯ We note that λk defined by (3.3.23) satisfies λk(0) = 0 and
∞ k ! ¯ X i X i k λk(z) = ak+iz = λ(z) − λ0 − aiz /z . i=1 i=1 ¯ For z 6= 0, λk(z) can be calculated sequentially as
¯ ¯ ¯ λ0(z) = λ(z) − λ0, λk(z) = (λk−1(z) − akz)/z, k ≥ 1.
¯e ¯o This in turn can be used to calculate λk and λk using (3.3.24). In the numerical examples in Sec
3.4.1, we found that the above recursion is stable and accurate.
3.4 NUMERICAL SIMULATIONS AND CONCLUSIONS
In this section, we numerically compare the new moment-flux models introduced in Section
3.3.2 and 3.3.3 under the three moment closure assumptions (3.2.14) and (3.2.17), the PKS model (3.2.1) with (3.2.16), and the individual-based model governed by (3.2.10) and (3.2.11).
We investigated numerically how the K-th order moment-flux model (3.3.14) approximates the individual-based model as K increases.
57 3.4.1 Linear turning frequency
For linear turning frequency equations, we first compared the K-th order moment-flux model
(3.3.14) with K = 1, 2, 3, 4 with the individual-based model by using the static signal (3.2.20) with a large gradient (a = 4.5) as in Figure 3.4D. The initial cell distribution, the turning rate and all the parameters were taken to be the same as in Figure 3.4D. Furthermore we used the initial conditions j = nk = jk = 0 for all x and k ≥ 1, which is consistent with the initialization of the cells using z = 0 in the individual-based model.
We see that as K increases, the accuracy of the moment-flux model increases dramatically in its ability to match the individual-based model in space and time, and for the cases with K = 3 or 4 we obtain very tight fits to the individual-based model. We also see that starting with biologically reasonable initial conditions, the variables n and R p±dz remain positive in these models.
Finally we compared these models by using a spatially and temporally oscillating signal,
8π(x − ct) S(x, t) = µ − µ cos , (3.4.1) L
with µ = 3/π and c = s0/4 = 5µm/s, and the same initial cell density as the previous example.
All the other conditions and parameters are the same as the previous example. In this case, we have
max(|St| + s0|Sx|)ta = 8πµ(s0 + c)ta/L = 1. x,t
In Fig. 3.6, we see that the cell population breaks into several aggregates in response to the wave-like signal. Similar to the previous example, the accuracy of the moment-flux models increases as K increases, K = 3 or 4 give very tight fits to the individual-based model, and the cell density n as well as R p±dz (not shown) remain positive for all K given biologically
58 Figure 3.5: Comparison of the moment-flux models with the individual-based model for a static signal with large gradient. Same signal and parameters are used as in Figure 3.4D. The gray bar plot is one realization of the individual-based model with 104 cells. The blue, red, green and black curves are the numerical solutions of the moment-flux models with order K = 1, 2, 3, 4. Row A: t = 2 min; B: t = 4
min. Left: the macroscopic cell density n; middle and right: the macroscopic density of cells moving to
the right R p+dz and to the left R p−dz calculated from n and j, which are always positive. The model
with K = 1 uses comparable moment closure as in previous work. The cyan curves in the plots of n(x, t)
are solutions of the PKS model.
reasonable initial conditions.
3.4.2 Nonlinear turning frequency
We rewrite the closed moment system as follows
∂ ∂ Λ M + A M = Q M + R M , (3.4.2) ∂t K K ∂x K K K K K K
59 Figure 3.6: Comparison of the moment-flux model with the individual-based model for the oscillating signal (3.4.1) with large gradients. In each plot, the gray bars come from one realization of the individual-
based model with 104 cells; the black curve is the numerical solutions of the moment-flux models with order K = 4; the cyan curve is the solution of the PKS model. Same parameters are used as in Figure
3.4D.
Similar to the linear turning frequency case, for linear turning frequency equations, we used the highly nonlinear turning rate
−1 λ(z) = λ0 + α tan (βz) (3.4.3)
which has Tylor expansion (3.2.7) with coefficients
(−1)iαβ2i+1 a = , a = 0, i ∈ . (3.4.4) 2i+1 2i + 1 2i N
¯ that is, λk are odd functions for even k and even functions for odd k. We chose the parameter
4 α = π and β = 0.9 to ensure (3.4.3) is well-defined, i.e., λ ≥ 0. Besides, we enforced the
60 following condition for the signal
s0max(|St| = s0|Sx|)ta ≤ tan(λ0/α)/β (3.4.5)
−1 so that |z| ≤ tan(λ0/α)/β = λ (0). We first considered the situation with a static signal (14).
In this case we have dS± = s0Sx, and ¯ λ¯ (t s S ) + λ¯ (−t s S ) λk(tas0Sx), k odd λ¯e = k a 0 x k a 0 x = (3.4.6) k 2 0, k even λ¯ (−t s S ) − λ¯ (t s S ) 0, k odd λ¯o = k a 0 x k a 0 x = (3.4.7) k 2 ¯ λk(−tas0Sx), k even
We first considered the situation with a static signal. We compared the moment-flux model
(3.4.2) for K = 1, 2, 3, 4 and the individual-based model by using the following static signal with a large gradient (µ = 5π/2).
S(x) = µ(L/2 − |x − L/2|), (3.4.8) which reaches its maximum at the center of the domain and minimum at the boundary. Besides, we assume that cells initially form an aggregate in x ∈ [0, 1] with the distribution 3π 3 4 | sin (πx)| 0 < x < 1 n(x, 0) = (3.4.9) 0 x ≥ 1
We further assume that initially the direction of cell movement is random and cell signaling is at its steady state, i.e., z = 0 for each cell. The result is shown in Fig. 3.7 and Fig. 3.8 below.
We then considered the situation with the oscillating wave signal (3.4.1). As in the case with a linear turning rate, the moment-flux model with K = 3, 4 give tight fit to the individual-based
61 Figure 3.7: Initial conditions for the comparison of the closed moment system with different K with the cell-based model shown in Figure 3.8.
Figure 3.8: Comparison of the closed moment system with different K with the cell-based model. The blue, orange, yellow, and purple curves are numerical solutions of the system with K = 1, 2, 3, 4. The gray bars are simulated from the cell-based model. The first column is the macroscopic cell density n, the second column is the macroscopic flux j. The third and fourth columns are the macroscopic density of cells moving to the right p+ and to the left p−, which are calculated from n and j. The purpose is to show that p± solved from the closed moment system are positive.
62 model but the PKS model can only roughly capture the number and location of the peaks but deviates from the individual-based model significantly in terms of the spatial-temporal dynamics.
The moment-flux model with K = 1, 2 also do not match with the individual-based model in a satisfactory manner. This suggest that in order to predict the spatial-temporal dynamics of chemotaxis in large signal gradients using a deterministic model, one must take into account information about the higher order moments of the internal dynamics.
Figure 3.9: Comparison of the moment-flux model with the individual-based model for the oscillating signal (3.4.1) with large gradient (µ0 = 2, c = s0/4 = 5 µm/s). (A) - (F): t = 0, 4, 8, 12, 16, 20 min.
In each plot, the gray bars come from one realization of the individual-based model with 104 cells; the blue, red, green and thick black curves are the numerical solutions of the moment-flux models with order
K = 1, 2, 3, 4; the cyan curve is the solution of the PKS model.
63 3.5 SUMMARY AND DISCUSSION
In this Chapter, we derived a class of PDE models that involve a hyperbolic system compris- ing equations for densities, fluxes and moments thereof for bacterial chemotaxis in large signal gradients. This system is based on a new moment closure method that allows large deviations of the internal state from the steady state. The method utilizes the fact that higher-order moments of the internal state variables equilibrate faster than lower-order moments, and approximates higher- order moments with k ≥ K when the turning frequency is either linear or for a general one. The order K is chosen in the way that 1/(K + 1) is significantly smaller than 1. The variables of the model include the density and flux moments of order 0 ≤ k ≤ K. Numerical simulations show that as K increases, the moment-flux models become more accurate in approximating the popula- tion dynamics governed by the individual-based model. Indeed, K = 3 and 4 match moderately to the individual-based model for, either static or time-dependent signals. In contrast, the PKS equation and previous momentflux models deviate from the spatial-temporal dynamics of the individual-based model significantly. Previous moment-flux models discussed in Erban and Oth- mer [44], Simons and Milewski [139], Si et al. [138] are similar to the case with K = 1. While the accuracy of approximation increases with increasing K for K > 4, the number of equations of the model is (2K + 2) and thus the computational complexity also increases. Therefore it is desirable to use K no bigger than four.
When the external signal gradient is large, the intracellular states of the population can be far from equilibrium and broadly distributed instead of locally concentrated. We note that the moment-flux model derived here highlights that in order to capture the population dynamics, it is necessary to track multiple moments of the internal states which inform us of the distribu-
64 tion of the internal variables. We also note that if the signal gradient becomes large, then the moments nk and jk/s0 are essentially of the same order and there is no obvious way to further simplify the moment-flux model. For cases with even larger signal gradients, the external signal would change faster than the intracellular dynamics, in which case simulating the individual- based model is more favorable than using a PDE model. If such large signals only occur on part of the domain, then hybrid methods or domain decomposition methods that bridge continuum
PDEs and individual-based models need to be developed.
Finally, it is noticeable that, all the analyses presented in this chapter are based on the individual velocity-jump process. As we had seen, as we take large population limit, the macroscopic equation can give us various pattern-forming phenomena featuring the run-and-tumble multiscale behavior of bacteria, and can be treated as a powerful mathematical tool in studying pattern formation, especially in the context of cell and molecular biology. Recent studies suggested that the run-and-tumble nature of some bacteria can be modeled by levy flight [117], leading to a macroscopic fractional-differential approximation [40, 133, 172]. Fractional differential equations can also be achieved via taking expectation of proper heavy-tail distributed stochastic processes, and I expect them to display more interesting patterns compared to classical ODE or
PDE systems.
65 (a) T = 1 (simulation at early time)
(b) T = 80 (long time behavior)
Figure 3.10: A numerical comparison between a particle undergoing (Left) regular Brownian motion and
(Right) Levy flight. In the long run, the particle undergoing Levy flight displays a run-and-tumble strategy, which is adopt by bacteria like E.Coli.
66 CHAPTER 4
CROSS-SECTIONAL SEGREGATION OF AXONAL CYTOSKELETON IN
NEURODEGENERATIVE DISEASES
The shape and function of an axon depend critically on the organization of its cytoskeleton, which is a dynamic system of intracellular polymers including microtubules, neurofilaments and actin. Under normal conditions, microtubules and neurofilaments align longitudinally in axons and are interspersed in axonal cross-sections. However, in many neurodegenerative disorders, they separate radially with microtubules clustered centrally and neurofilaments located near the periphery. This striking polymer segregation proceeds to focal accumulations of neurofilaments and/or organelles that are early hallmarks of nerve degeneration. A recent stochastic model sug- gests that this segregation is a consequence of the disruption of neurofilament transport along microtubules, and in the absence of neurofilament transport, axonal organelles pull microtubules to the center and displace neurofilaments to the periphery [168, 172].
Motivated by these results, we develop a nonlocal PDE model to systematically analyze how different balances of organelle transport and neurofilament transport affect the cross-sectional organization of microtubules and neurofilaments. Through modeling we highlight the importance of incorporating cargo interactions in addressing biological questions related to axonal transport.
Numerical results of the PDE model also led to new insights into the biological phenomena. The
first numerical simulations are done in Jonathan Toy’s thesis [152], in this thesis we followed
Jonathan’s work on numerical simulations as well as discussed stability analysis in different situlations.
67 The organization of this chapter is as follows. Section 4.1 will be the biological introduction and Section 4.2 reviewed previous models on axonal cytoskeleton dynamics. In Section 4.3, a novel continuum model will be constructed, giving rise to an integro-differential PDE system which describes the cross-sectional distribution of microtubule and neurofinaments. In section
4.4, we carry out mathematical analysis with respect to the nonlocal model developed in Section
4.3. In section 4.5, we carry out some numerical simulations of that model in 1D and 2D under various boundary conditions. Finally, Section 4.6 is the summary and discussion section.
4.1 BIOLOGICAL BACKGROUND
4.1.1 Neuron
Neurons are able to communicate rapidly with one another in networks that can span vast distances in response to sensory input using a combination of chemical and electrical signals.
While the finer details of the structure can vary to some degree, all neurons possess a similar overall form, as shown in Figure 4.1. The large, roughly circular object forming the center of the neuron is the cell body, and it contains the neuron’s nucleus, a portion of its organelles, along with the many of the other cellular components shared among eukaryotic cell types. The small projections away from the cell body are dendrites; they serve as the cell’s receivers, as they house the chemical receptors that allows the neuron to receive chemical signals sent by nearby neurons using neurotransmitters. While similar in visual form to the dendrite, the long, winding appendage to the cell is called the axon, and its function is to transmit outgoing information away from the cell body in the form of rapid short electrical impulses towards the axon terminal which will then be converted into a chemical messages and sent to nearby neurons.
68 Figure 4.1: A cartoon description of a nerve cell. It is a highly polarized cell that can be briefly divided into three major parts – cell body, axon and axon terminal. Cell body is equipped with numerous spike- like dendrites, which are accounting for receiving signals; axon is the longest part of a nerve, which is responsible for transmitting signals between cell body and axon terminal. The figure is cited from http://hyperphysics.phy-astr.gsu.edu/hbase/Biology/nervecell.html.
4.1.2 Cytoskeletons and axonal transport
Axons are long slender projections of nerve cells that permit fast and specific electrical com- munication with other cells over long distances. The ability of nerve cells to extend and maintain these processes is critically dependent on the cytoskeleton.
Cytoskeletons are dynamic scaffolds of microscopic protein polymers found in the cytoplasm of evert eukaryotic cell that give the cell its shape and help organize the cell’s parts. Cytoskeletons comprise microtubules, intermediate filaments (neurofilaments in neurons) and actin filaments.
Actins are generally tiny compared to the former two types and sometimes are neglectable in certain mathematical models).
Microtubules and neurofilaments are both long polymers that align in parallel along the long axis of the axon, forming a continuous overlapping array that extends from the cell body to
69 the axon tip. In healthy axons, microtubules and neurofilaments align along the axon and are interspersed in axonal cross-sections [153, 127, 78]. However, in many toxic neurodegenerative disorders these two populations of polymers separate from each other, with microtubules and organelles located near the long axis of the axon and neurofilaments displaced to the periphery near the axonal membrane.
Figure 4.2: Fluorescence image of a cortical axon and its growth cone. Actin filaments are shown in red, and microtubules composed of beta tubulin are in green. Picture is from [84], reprinted with permission under CC-BY licence by Frontier.
Microtubules are stiff hollow cylindrical structures about 25 nm in diameter that serve as tracks for the long-range bidirectional movement of intracellular membranous organelles and macro- molecular cargo complexes. In axons, this movement is known as axonal transport. The cargoes of axonal transport are conveyed by microtubule motor proteins: kinesins in the anterograde di- rection (towards the axon tip), and dyneins in the retrograde direction (towards the cell body).
Neurofilaments, which are the intermediate filaments of nerve cells, are flexible rope-like poly- mers that measure about 10 nm in diameter. These polymers function as space-filling structures
70 that expand axonal cross-sectional area, thereby maximizing the rate of propagation of the nerve impulse. In large axons, neurofilaments are the single most abundant structure and occupy most of the axonal volume. Mutant animals that lack neurofilaments develop smaller caliber axons and exhibit delayed conduction velocities.
Neurofilaments are long non-polarized polymers that function as space-filling structures to increase the axonal caliber [125], and occupy most of the axonal volume in large axons [55].
In addition to their structural role, they also undergo infrequent bidirectional transport along microtubules, which consists of short bouts of movements interrupted by prolonged pauses [158,
24]. This is different from the axonal transport of organelles which is usually more frequent and more persistent [25].
A molecular motor is a protein molecule in the cell that generates force and maintains trans- portation of certain cargoes. A loose analogy can be made between conventional molecular mo- tors, such as myosin and kinesin, and trains. Myosin and kinesin travel along actin filaments and tubulin microtubules respectively, similar to the way that trains travel on tracks. The chemical process of ATP hydrolysis acts as the fuel that myosin and kinesin motors burn in order to move along their tracks. An important feature shared by trains and molecular motors is that, both of them only move unidirectionally on average over time, and may stop and waiting for a long time
(trains stop to wait for cargoes).
Another mode for generating motion or transport in the cell is via polymerization (or de- polymerization) of microfilaments or microtubules themselves. Classical examples include the depolymerization of the spindle pulling apart sister chromatids during anaphase, and sickle hemoglobin polymerization creating deformed ?sickle? red blood cells. If we represent the
71 position of the end of the polymer as idealized points, the analysis of the growth or shrinking of a polymer and the motion of a conventional molecular motor become closely related. See [156] for details on how kinesin “walks” along microtubules, and [26] a review on axonal transport process.
Figure 4.3: The kinesin dimer attaches to and moves along microtubules towards its + end. The process is fueled by ATP hydrolysis.
4.1.3 Neurodegenerative disease and cross-sectional segregation of axonal cytoskeleton
The striking polymer separation found in many neurodegenerative diseases has been stud- ied most extensively in axons treated by the toxin 3,3’-iminodiproprionitrile (IDPN) and 2,5- hexanedione. Systematic administration of IDPN to rats causes selective impairment of neuro-
filament transport, massive focal accumulations of neurofilaments, and neurological disorders
72 similar to amyotrophic lateral sclerosis (ALS) in humans [120, 96, 95]. Microtubules and neu- rofilaments have been observed to segregate within a few hours after IDPN injection, preceding local accumulations of neurofilaments and axonal swelling by hours or days. Moreover, the seg- regation has been shown to be reversible, as has the disruption of neurofilament transport [120].
These phenomena have been discovered for over 30 years now, but the underlying mechanisms are still largely unexplored. One important question is the relation between impairment of neu- rofilament transport and their segregation from microtubules. It is clear that if neurofilaments are separated from microtubules, they cannot be transported along microtubule tracks. However, it is not known if extra mechanisms are needed to explain the polymer segregation.
If the functionality of neurons in the human brain were damaged, neurodegenerative disease may take place. Neurodegenerative disease is an umbrella term including well-known examples like Parkinson?s, Alzheimer?s, and Huntington?s diseases.
For example, Alzheimer’s disease (AD) has a history of over 100 years, and is the most com- mon form of dementia in the elderly and chronic neurodegenerative disease [94]. It leads to the progressive loss of mental, behavioral, functional decline and ability to learn [3]. It is estimated that as many as 5 million people in USA, and nearly 44 million people worldwide have AD [7].
Pathologically, AD is characterized by intracellular neurofibrillary tangles (NFTs) and extra- cellular amyloidal protein deposits contributing to senile plaques. The NFTs are intracellular aggregation of hyperphosphorylated tau proteins, while the extracellular plaques are primarily developed from β-peptide (Aβ)[94, 134].
Axonal swelling and spheroids have been observed in many different neurodegenerative dis-
73 eases. Axonal transport can be disrupted by a variety of mechanisms including damage to: ki- nesin and cytoplasmic dynein, microtubules, cargoes, and mitochondria, see [37] for a dedicated review. Therefore, it is important to study axonal transport and corresponding cytoskeleton dy- namics.
An intermediate state between swelled axon, spheroids and their healthy counterpart is the ap- pearance of cross-sectional microtubule clusters. In Chapter4, we will use mathematical models to study why blocked microtubule-neurofilament (MT-NF) interaction may lead to cross-sectional clusters of microtubules, and MT-NF phase separation.
In many neurodegenerative disorders, microtubules and neurofilaments separate from each other, forming inhomogeneous patterns in a cross-sectional view of the axon. These phenom- ena have been discovered for over 30 years now, but the underlying mechanisms are still largely unexplored. One important question is the relation between impairment of neurofilament trans- port and their segregation from microtubules. It is clear that if neurofilaments are separated from microtubules, they cannot be transported along microtubule tracks. However, it is yet an open questions if extra mechanisms are needed to explain the polymer segregation. Besides, driven by these outstanding in vivo and in vitro studies about polymer separation, one may naturally ask the following two questions: How does the aggregation occur? How do we quantify these phenomena using mathematical models?
Recently, significant insights into this question have been obtained using mathematical model- ing. In [168], we developed a stochastic particle-based model for the cross-sectional distribution of microtubules, neurofilaments and organelles in a cross-section of an axon. In this model, the axonal transport of organelles and neurofilaments are incorporated as stochastic arrival and
74 departure of these cargoes. While present in the domain, these cargoes interact with nearby mi- crotubules randomly through elastic spring forces that models the cross-bridges by molecular mo- tors. Finally, the model also includes volume exclusion nature of cytosol and Brownian motion of all particles. Simulations of the model demonstrated that in the absence of neurofilament trans- port, organelles can pull microtubule together and segregate them from neurofilaments within hours, in a similar way as observed in experiments. This suggests that microtubule-neurofilament
(MT-NF for short) segregation can be explained as a consequence of impairment of neurofilament transport and additional mechanisms is not needed. The model also suggest that the extent and rate of MT-NF segregation depends on the flux rate and size of the moving organelles.
To figure out the answer, Xue et. al (2015) constructed a mathematical model that involves a system of stochastic differential equations [168]. Besides, in Silico studies in the paper indicated that a stochastic multiscale mechanism is necessary for MT-NF phase separation [24, 25].
From mathematical point of view, it’s a dynamic system of intracellular polymers including microtubules, neurofilaments and microfilaments actin. In [168], Xue et. al (2015) constructed a mathematical model that involves a system of stochastic differential equations [168]. Besides, in Silico studies in the paper indicated that a stochastic multiscale mechanism is necessary for
MT-NF phase separation [24, 25]. To my knowledge, [168] was the first mathematical model addressing on MT-NF segregation.
However, due to the stochastic nature of the model, mathematical analysis cannot be carried out easily. Moreover, particle binding, unbinding, addition and removal occurs on a time scale of seconds or fractions of a second, while polymer segregation occurs on a time scale of hours.
Therefore, the model has to be simulated on a time scale of milliseconds to ensure accuracy and
75 thus is computationally expensive. In this chapter, we develop a continuum model for the MT-
NF segregation phenomena which is amenable for mathematical analysis and fast computation.
The model consists of a system of two nonlocal PDEs that describe the densities of microtubules and neurofilaments. The pulling effect of organelle transport is included implicitly as a means of indirect microtubule-microtubule (MT-MT for short) interaction. The nonlocal integrals comes from the fact that particle interactions occur within a capturing radius. We use the model to sys- tematically analyze and compute how different balances of organelle transport and neurofilament transport affect the distribution of microtubules and neurofilaments.
Figure 4.4: Schematic description of microtubule-neurofilament (MT-NF) phase separation in cross- sectional view of the axon, under different neuron conditions. (A) In health neurons, microtubules (MT) and neuronfilaments (NF) appear to be uniformly distributed. (B, C) In IDPN-treated neurons, micro- tubule form cluster(s) near the center of the axon, while neurofilament are accumulated towards periphery of the axon. (B) and (C) indicate that different separation patterns can appear in IDPN-treated neurons, probably depending on the positions of the organelles
76 4.2 PREVIOUS DETERMINISTIC AND STOCHASTIC MODELS OF NEUROFILA-
MENT TRANSPORT AND MT-NF SEGREGATION
The first mathematical models addressing axonal transport are [15] for fast transport and [16] for slow transports, These models assumed the existence of a hypothetical engine that moves at a constant velocity. As reviewed in Chapter2, axonal transport along a microtubule (MT) is guided by antegrade (forward-moving) motor proteins kinesin and retrograde (backward-moving) proteins dynein, so a more reasonable mathematical model should reflect the bidirectional nature of axonal transport of neurofilaments (neurofilaments).
4.2.1 Models of neurofilament transport
The first mathematical models addressing axonal transport are [15] for fast transport and [16] for slow transports, These models assumed the existence of a hypothetical engine that moves at a constant velocity. However, Blum-Reed models only considered unidirectional movement of the molecular motors. Axonal transport along a microtubule (MT) is guided by antergrade (forward- moving) motor proteins kinesin and retrograde (backward-moving) proteins dynein [158, 92], so a more reasonable mathematical model should reflect the bidirectional nature of axonal transport of neurofilaments (neurofilaments).
Since the development of the Blum-Reed model [15, 16] , considerable progress has been made in the experimental study of neurofilament transport in axons. For example, the “stop-and-go” hypothesis builds directly on our experimental measurements of the movement of single neurofil- aments [27]. Consequently, a master equation based model of the “stop-and-go” hypothesis has been derived in [34], with biological applications to mouse sciatic nerve [83] and mouse optic
77 nerve [92]. From pure mathematical point of view, the “stop-and-go” models also lead to new re- sults and implications on reaction-diffusion-hyperbolic systems [58]. See [172] for a more recent review on neurofilament transport models.
Here we briefly introduce the dynamical system model developed in [34], using master equa- tion approach. To our knowledge, this is the first deterministic model addressing axonal transport of neurofilament featuring bidirectional movement. In their paper, the authors used the following notations:
u2 : concentration of neurofilaments bound to anterograde motors, moving anterogradely, on
MT;
u1 : concentration of neurofilaments bound to anterograde motors, pausing, on MT;
u0 : concentration of neurofilaments bound to anterograde/retrograde motors, pausing, off
MT;
u−1 : concentration of neurofilaments bound to retrograde motors, pausing, on MT;
u−2 : concentration of neurofilaments bound to retrograde motors, moving retrogradely, ofn
MT; with the following transition diagram (the idea is similar to a reaction network)
k−1,−2 k0,−1 k1,0 k2,1 u−2 )−−−−−−−−* u−1 )−−−−−−* u0 )−−−−* u1 )−−−−* u2 k−2,−1 k−1,0 k0,1 k1,2
78 Therefore, we can derive a corresponding advection-reaction equation as done in [34] ∂u ∂u 2 = −v 2 + k u − k u , A 1,2 1 2,1 2 ∂t ∂x ∂u1 = k u + k u − (k + k )u , ∂t 0,1 0 2,1 2 1,2 1,0 1 ∂u0 (4.2.1) = k1,0u1 + k−1,0u−1 + k2,1u2, ∂t ∂u −1 = k u + k u − (k + k )u , 0,−1 0 −2,−1 −2 −1,−2 −1,0 −1 ∂t ∂u−2 ∂u2 = v + k u − k u , ∂t R ∂x −1,−2 −1 −2,−1 −2 where vA, vR are respectively anterograde and retrograde motor speed. It is known experimentally that vA = 0.56µm/s, vR = 0.62µm/s.
Except for the above deterministic models, stochastic models on axonal transport have also been studied in recent 10 years [109, 110]. To the authors’ knowledge ,however, these models focus more on mRNA and intermediate filament related transport. See [22] for a detailed view on stochastic models of intermediate transport.
4.2.2 An agent-based model for MT-NF interaction
While Section 4.2.1 only considered axonal transport of neurofilaments rather than cross- sectional MT-NF segregation, we should go back to the cross-sectional MT-NF interaction ex- periment studied in Chapter4.
A MT-NF interaction model is proposed in [168], modeling the system using agent-based stochastic equations. In this paper, the authors described microtubules, neurofilaments and or- ganelles as non-deformable particles in Ω, and track their center positions over time. In this way, positions of the particles are governed by the following stochastic differential equations
79 (Langevin equations)
k k k dxi = Fi /µkdt + σkdWi (4.2.2)
Here k is the index for the type of the particle, M for microtubule, N for neurofilament , O for organelle; i is the index for the particle of the same type, F is the total force acted on that particle,
µ is the drag coefficient, and W is the standard 2D brownian motion. The movement is slow and viscous dominated, therefore we simplified out the velocity equation using quasi-steady-state analysis (QSSA).
To explicitly determine the force terms F , three key mechanisms are incorporated in (4.2.2), i.e.
1. Excluded volume effect, which is dictated by passive interactions of these particles;
2. MT-NF interactions are exerted through molecular motors (kinesin and dynein as intro-
duced in Chapter2), neurofilament exhibit slow axonal transport ;
3. MT-organelle interactions are also exerted through molecular motors, but in contrast to
MT-NF binding, they exhibit fast axonal transport.
Since we are interested in the radial distribution of neurofilaments, they considered the follow- ing rules according to the above listed mechanisms. First, A neurofilament can bind to a nearby microtubule and move along the microtubule. This binding is through molecular motors, and thus the binding radius is given by the length of the molecular motor. A neurofilament bound
N N N to a microtubule can either unbind with rate koff or arrive/leave the domain with rate kin/kout,
N and a free neurofilament can bind with a microtubule with rate kon. When a microtubule and
80 neurofilament are tethered through motors, we let them interact with each other through elastic spring forces. Consequently, the authors in [168] derived the following form of (4.2.2) k 1 P k k k k dxi = j(Rij + sijSij)dt + σi dWi, µ Lr Rij = − − 1 sij, dij + (4.2.3) N O Sij = −κdijsij, κ = κ or κ , k sij jumps between 0 and 1.
Since MT-NF segregation occurs on a time scale of hours without visible swelling, we can as- sume the total number of neurofilament is constant, thereby neglecting the axonal arriving/leaving process of neurofilament . The numerical simulation is shown in Fig. 4.5[168].
Aa a comparison between the above figure with Fig. 4.16 in which the washout effect is introduced t = 30 hours, we can see that the agent-based simulation approach allows 7 hours response before fully remixing and hence fits more with experimental data.
4.3 A NONLOCAL PDE MODEL FOR THE CROSS-SECTION DYNAMICS
In this section, we describe a new continuum model for the distribution of microtubules and neurofilaments in a cross-section of an axon, denoted as Ω. A similar nonlocal continuum model was first theoretically derived in [5] in 2006 for modeling nonlocal interaction feature of cell sorting experiments. To my own knowledge, however, this work is the first nonlocal continuum model addressing on axonal transportation and cross-sectional cytoskeleton dynamics.
81 Figure 4.5: MT-NF segregation and remixing after washout. Neurofilament transport is blocked starting at t = 1 h and restored at t = 13 h. Above are snapshots of the positions of microtubules, neurofilaments and organelles at different time. Simulation is taken for a single realization.
4.3.1 Continuum approach for nonlocal interactions
Agent-based models (or individual-based models) offer a relatively straightforward path to model the impact of direct contacts on movement and are widely applied in modeling nonlocal interactions. This includes discrete-lattice models based on the cellular Potts models [63, 62], cellular automaton models [66] and many others. However, as I will comment in Section ??, two of the greatest drawbacks of agent-based models lie in 1. their difficulty to analytical rigor – it can be difficult to draw out general properties and conclusions 2. extensive and time-consuming numerical simulations are involved.
An alternative way of modeling nonlocal behavior of cellular and molecular is by continuum
82 approach, which was initiated in [5], and extended by both theoretical studies [30] and applica- tions to chemotaxis [75], morphogenesis [6, 64] and cancer invading [60, 137]. The basic model is an integro-advection-diffusion equation in which the integral represents the effect of adhesion between cells on their movement:
∂u(x, t) Z = D∆u(x, t) − η∇ · u(x, t) A(u(y, t))G(y − x)dy + f(u) (4.3.1) ∂t | {z } |x−y| Here u(x, t) is the cell density at position x and time t. D, η, and R0 are positive constants. A detailed description of which can be found in [5]. 4.3.2 Model mechanisms and derivation Inspired by the equation (4.3.1), let us consider a cross-section of an axon (as shown in Fig. 4.4). In this case microtubules, neurofilaments and organelles can all be regarded as particles in a two dimensional domain Ω. In [168], Ω is taken to be a circular domain with fixed radius R0, but in the derivation part, it can be a general one. The cross-sectional segregation of microtubules and neurofilaments occur in a time scale of hours, whereas axonal swelling occurs in a days, which is in a much larger time scale. In this model, we focus on the polymer segregation phenomena and ignore axonal swelling, thus we take the domain Ω to be fixed and assume that the total numbers of microtubules and neurofilaments in Ω are conserved over time. This reflects the fact that the addition and removal of neurofilaments in Ω due to their longitudinal transport cancel with each other on the time scale of polymer seg- regation. Furthermore, the volume fraction of organelles is very small compared to microtubules and neurofilaments in most places of an axon [120, 65], thus we do not track the organelle density 83 explicitly, but instead include their effect to the distribution of microtubules and neurofilaments implicitly. To derive a continuum model, both microtubules and neurofilaments are represented as discrete entities. We further denote nM , nN and nO to be the total number of microtubules, neurofilaments M and membranous organelles respectively in a cross-section of axon, xi , 1 ≤ i ≤ M to be N the position of i-th microtubule and xj , 1 ≤ j ≤ N the position of j-th neurofilament, and O xk , 1 ≤ k ≤ O the position of k-th neurofilament. Then the dynamics of microtubules and neurofilaments, and organelles are governed be the following system of stochastic differential equations dxM = FM /µ dt + σ dWM , 1 ≤ i ≤ n , i i M M i M dxN = FN /µ dt + σ dWN , 1 ≤ j ≤ n , (4.3.2) j j N N j N O O O dxk = Fk /µOdt + σOdWk , 1 ≤ k ≤ nO. M where µM,N,O are the drag coefficients of microtubules, neurofilaments and organelles, Wi are standard Brownian motions with σM,N,O being their corresponding magnitude, and Fα is the sum γ of all applied forces on that particle. Explicitly, Fα can be decomposed to be γ X X Fα = Aα,β + Rα,β , γ ∈ {M,N,O} (4.3.3) β β | {z } | {z } Attractive forces Repulsive forces Mechanism 1: Organelles interact with microtubules with an indirect way Organelles can interact with multiple motors microtubules simultaneously due to their large size, they can readily interact with multiple microtubules even if those microtubules are not close to each other[126, 76]. As they move along multiple microtubules they can pull nearby micro- 84 tubules closer to each other, similar to a“zippe” [174]. This mechanism provides a means of indirect interaction between microtubules, which can be simplified as attractive forces between microtubule pairs within an interaction distance (see Figure 4.6 for illustration). A B MT MT Org Figure 4.6: Simplifying assumption of the model. (A) An illustration of indirect interaction between two microtubules through a common organelle in 3D. As the organelle move along multiple microtubules, it pulls them towards each other in cross-section. MT: microtubule; Org: organelle. (B) Model simplifica- tion: replacing the indirect microtubule-microtubule (MT-MT) interaction by direct MT-MT interaction. Remark 4.3.1. This over-simplified mechanism will be used in derivation of the macroscopic model, which plays an important part in the nonlocal interaction term (4.3.4). Mechanism 2: microtubule and neurofilament cross-sectional movements are driven by fluxes We denote u(x, t) and v(x, t), where x ∈ Ω and t ∈ R+, as the cross-sectional densities of microtubules and neurofilaments respectively, measured as numbers of particles per unit area. 85 Under the above simplifications the equations for u and v can be written in the following form ut = −∇ · Ju, vt = −∇ · Jv, where Ju and Jv are the fluxes of u and v in Ω. The movement of microtubules in Ω is dominated by elastic spring forces acted by molecular motors that cross-bridge microtubules and engaged cargoes, volume exclusive forces acted by other particles, as well as Brownian motion. Hence we decompose the flux Ju into A P Ju = Ju + Ju , A P where Ju is an active flux due to microtubule-cargo binding and Ju is a passive flux due to volume exclusion and diffusion. Mechanism 3: Active fluxes are modeled by nonlocal spring-like interactions Movement of microtubules due to binding with various cargoes occurs as a result of the spring forces of the bridging kinesin and dynein motors. Both organelles and neurofilaments are moved along microtubules either anterogradely or retrogradely by molecular motors. We represent the attractive force on microtubules at position x by nearby microtubules as Z u Fu = k1 G1(y − x)u(y, t)dy, (4.3.4) Ω where k1G1(y − x) is the average force acted by microtubules at position y, k1 is proportional to the spring constant of the motor-cargo crossbridges between microtubules pairs and the average time fraction that the pair remains bound. 86 We assume that the force between an engaged microtubule pair is a linear spring force, and the average time fraction that the pair remains engaged is a decreasing function of their distance, in particular, proportional to (1 − |x|/r1) for |x| < r1, where x is the relative position vector and r1 is the maximum interaction distance governed by the length of the molecular motors and the size of the cargo. Under these assumptions we have |x| G1(x) = x 1 − χ|x| We represent the attractive force on microtubules at position x by nearby neurofilaments in a similar way, Z v Fu = k2 G2(y − x)v(y, t)dy, Ω with |x| G2(x) = x 1 − χ|x| Here k2 is proportional to the spring constant of the motor cross-bridges between microtubules and neurofilaments, and r2 is given by the maximum length of the molecular motors. A Because the movement is viscous-dominated, the active flux Ju can be written as A Ju = uFu/µ1, u v where Fu = Fu + Fu, and µ1 is the drag coefficient of a microtubule in the lateral direction. p We describe the passive flux Ju as p Ju = −D1∇u − η1u∇(u + v). The first term is based on Fick’s law with an effective diffusion coefficient D1. The second term, which is proportional to the gradient of the total density of the two populations, models the 87 volume exclusion effect. We note that the form (4.3.2) has been formally derived for interact- ing particle systems with repulsive potentials [112, 17] and has been used in modeling volume exclusion in the context of cell population dynamics [107,5]. 4.3.3 The model equations In summary, the general equations for the microtubule and neuralfilament densities take the following matrix form 1 ut = D1∆u + ∇ · η1u∇(u + v) − ∇ · uFu(u, v) , (4.3.7) µ1 with Z Z Fu(u, v) = k1 G1(y − x)u(y, t)dy + k2 G2(y − x)v(y, t)dy. (4.3.8) Ω Ω Similarly, the equation for the neurofilament density can be written as 1 vt = D2∆v + ∇ · η2v∇(u + v) − ∇ · vFv(u) , (4.3.9) µ2 with Z Fv(u) = k2 G2(y − x)u(y, t)dy. (4.3.10) Ω Here µ2 and D2 are the drag and diffusion coefficients of neurofilaments respectively, and η2 gives a measure of the repulsive forces for volume exclusion. 4.4 MATHEMATICAL RESULTS 4.4.1 Parameter estimation and nondimensionalization The baseline parameters of the model were estimated using experimental data and biophysical properties of microtubules and neurofilaments. Table 4.1 summarizes all the parameter values in 88 dimensional and non-dimensional forms. The effective diffusion coefficients, D1 and D2, and the drag coefficients, µ1 and µ2, for mi- crotubules and neurofilaments were taken from [168]. Specifically, the random movement of microtubules and neurofilaments is assumed to be a result of the Brownian motion of these parti- cles in the axonal cross-section and Di were calculated using the Einstein relation Di = kBT/µi, where kB is the Boltzmann constant and T is the absolute temperature. The drag coefficients µi were estimated by treating these polymers as thin elastic rods. The parameters η1 and η2 were estimated using the following method. Neurofilaments are polymer brushes that have highly-charged long sidearms projecting outward from the filament backbone. Neurofilament sidearms interact with other particles via long-range electrodynamic and entropic repulsive forces [144, 145]. Microtubule associated proteins have a similar volume- exclusive effect [135, 19, 106, 81]. Denote the repulsion acting on a particle at x by a particle at y be R(y − x). As in [168], when the repulsion of all the particle pairs have the same form with magnitude 0(r0/r − 1), r ≤ r0 R = 0, r > r0 where 0 = 0.5 pN is a force prefactor, r0 = 135 nm is the maximum repelling radius, and r = |y − x|. To estimate η1, we consider the situation that all the particles only interact through pairwise repulsion. Thus total force acting on the microtubule at position x is R R(y−x)(u(y)+ v(y))dy −µ1v, where v is the convective velocity of that particle. By neglecting inertia, we have the convective velocity Z v = R(y − x)(u(y) + v(y))dy/µ1. 89 Using the approximation u(y) ≈ u(x) + ∇u(x) · (y − x) and similarly for v(y), we obtain Z 3 π0r0 v ≈ R(y − x) · (y − x)dy/µ1 · ∇(u(x) + v(x)) = ∇ u(x) + v(x) . 6µ1 3 3 This gives η1 = π0r0/(6µ1). Similar arguments lead to η2 = π0r0/(6µ2). An equivalent method to obtain ηi is to use formulas derived in [17]. The force R(y − x) corresponds to the repulsion potential V (x) = 0(−r0 log |x| + |x| + r0 log r0 − r0) if |x| < r0 and 0 otherwise. The R parameters η1 and η2 are given by ηi = 2 V (x)dx/µi. R The lower bound for the MT-MT interaction radius r1 can be estimated as the sum of two microtubule radii (25 nm), the length of two motor proteins projected in the axonal cross-section. We consider the cross-bridging effect as a two-step process: first, one motor on the organelle surface bind to one microtubule, this brings the organelle close to the microtubule, with a distance about 17 nm; second, another motor on the organelle surface catches another microtubule within the full length of a motor (80 nm). Therefore the lower bound of r2 is approximately 122 nm. The upper bound for r1 should be the lower bound plus the diameter of an organelle, which ranges from 50 nm to 280 nm and above [141], as the two microtubules can be located at the opposite sides of the organelle. We note that in this model the organelles are only included implicitly, which means that their volume is neglected. Therefore the value of r2 should be taken to be between the lower bound and the upper bound. The MT-NF interaction radius r2 is estimated to be 100 nm, which is roughly the length of one molecular motor (80 nm) plus the radii of one microtubule (12.5 nm) and one neurofilament (5 nm) [77, 25]. The parameters k1 and k2 are estimated in the following way. We estimate k1 to be the product 90 of the spring constant for the motor-cargo cross-bridge, estimated to be 0.9 pN/nm in [168], and a small fraction α. The factor α depends on the flux rate of the organelles, the binding and unbinding rates of motors and microtubules, and the relative position of microtubules and the bridging organelle. Under normal conditions, at most 1/10 of microtubules are within the capturing distance of organelles. The binding and unbinding rate of microtubules and organelles were estimated to be both 2/s. So the time fraction that each microtubule is bound to the cargo is 1/2. Two microtubules may not always appear at the opposite side of the organelle, and the force between them thus need to be projected to the line that pass their centers. With all these considerations, we estimate α to range between 1/200 - 1/40. We estimate k2 as the product of the spring constant for the MT-NF bond and the time fraction for them to remain engaged. It has been estimated that in normal developing axons a neuro- filament spend about 3% of its time moving along microtubules, and the spring constant for the MT-NF bond was estimated to be 0.18pN/nm in [168]. Thus for normal axons, we have k2 = 0.18pN/nm × 3%. Reduced k2 represents impairment of neurofilament transport in the model. For nondimensionalization, we choose the length scale L0 = 0.2µm which is comparable to r1. The MT-NF segregation occurs on a time scale of hours, thus we choose the time scale T0 = 1h. The average density of microtubules and neurofilaments are 115 and 18 per µm2. Thus we choose 2 2 the density scale U0 = 0.4 particle per L0, which is equivalent to U0 = 10 particles per µm . The force that a single molecular motor can exert on a cargo is 5-6 pN [147]. Thus we choose the force scale to be F0 = 1 pN. We introduce the following nondimensionalized variables t x u v t0 = , x0 = , u0 = , v0 = . T0 L0 U0 U0 91 Table 4.1: Parameter values of (4.3.7) - (4.3.10) Parameter Description Dimensional value & Refs Nondim. −6 2 D1 MT diffusion coefficient 8.02 × 10 µm /s [168] 0.7218 −5 2 D2 NF diffusion coefficient 5.59 × 10 µm /s [168] 5.031 −2 µ1 MT drag coefficient 512 pN · s/µm [168] 2.844 × 10 −3 µ2 NF drag coefficient 73.5 pN · s/µm [168] 4.0833 × 10 −6 4 η1 param. for volume exclusion 1.258 × 10 µm /s per particle. 1.132 See text for estimation method. −6 4 η2 param. for volume exclusion 8.764 × 10 µm /s per particle. 7.887 See text for estimation method. r1 MT-MT interaction radius 120 – 400 nm [141, 77]. See text 0.6 – 2 for estimation method. r2 MT-NF interaction radius 100 nm [77]. See text for estima- 0.5 tion method. k1 force prefactor for MT-MT 0.9 pN/nm×α. α depends on the 72α (0.9 if interaction organelle flux, between 1/40 and α = 1/80) 1/200 k2 force prefactor for MT-NF 0.18 pN/nm × β, β = 0.03 in the 14.4β (0.432 interaction normal case and 0 in disease. or 0) 92 Plugging these new variables into (4.3.8)-(4.3.10), we get 3 0 T0 0 T0U0 0 L0 0 ri 0 U0L0 Di = 2 Di, ηi = 2 ηi, µi = µi, ri = , ki = ki, i = 1, 2, L0 L0 F0T0 L0 F0 and dropping the primes, the model equations have the same form as (4.3.7) and (4.3.9). 4.4.2 Linear stability analysis Linear stability 1D As an organelle moves along multiple microtubules, it pulls them closer together and overall causes microtubule clustering. In contrast, a neurofilament mainly moves along a single micro- tubule, stops frequently and thus neurofilaments disperse microtubules apart. In normal axons, a balance of these two competing processes leads to a more-or-less uniform distribution of micro- tubules and neurofilaments in cross-section. If neurofilament transport is impaired, e.g., by IDPN, neurofilaments cannot effectively separate microtubules apart, leading to microtubule clustering. In this section, we investigate this speculation systematically using a linear stability analysis of the model described above. To minimize the boundary effect and for simplicity of analysis, we consider a 1D situation with Ω = [0,L] and couple the model equations (4.3.8)-(4.3.10) with periodic boundary conditions. According to classical differential equations theory, the overall stability or lack of stability near a stationary point can be classified based on the eigenvalues of the system that has been linearized around that particular stationary point. By knowing the stability of a system near its stationary point, we can gain insight into the general behavior of the system for points close to the stationary point. In practice, the linearization for linear stability analysis involves building a first 93 order linear approximation of the system around the known quantity of the system?s stationary point. Let u0 and v0 be the average densities of microtubules and neurofilaments, then the uniform steady state of the model is (u0, v0). Let u(x, t) = u0 +u ¯(x, t) and v(x, t) = v0 +v ¯(x, t) where u¯ and v¯ represent the deviation of the solution from the uniform states. Substituting these into the governing equations in 1D, and neglecting the nonlinear terms in u¯ and v¯, we obtain the linearized model 1 ¯ u¯t = D1u¯xx + η1u0(¯u +v ¯)xx − u0 Fu(¯u, v¯) in [0,L] µ x 1 (4.4.1) 1 ¯ v¯t = D2v¯xx + η2v0(¯u +v ¯)xx − v0 Fv(¯u) in [0,L]. µ2 x where Z Z ¯ Fu(¯u, v¯) = k1 G1(y − x)¯u(y, t)dy + k2 G2(y − x)¯v(y, t)dy. Ω Ω (4.4.2) Z ¯ ¯ Fv(¯u) = k2 G2(y − x)¯u(y, t)dy. Ω To obtain the above system we used the fact that the nonlocal integrals (4.3.8) and (4.3.10) are zero if u and v are constant. We next take the Fourier expansion with respect to x X iwqx X iwqx u¯ = φq(t)e , v¯ = ψq(t)e , (4.4.3) q6=0 q6=0 where wq = 2qπ/L and q is an integer. Simple calculations lead to X 0 iwqx X 0 iwqx u¯t = φq(t)e , v¯t = ψq(t)e , q6=0 q6=0 X 2 iwqx X 2 iwqx u¯xx = − φq(t)wq e , v¯xx = − ψq(t)wq e , q6=0 q6=0 94 and Z X ˆ iwqx G1(y − x)¯u(y, t)dy = φq(t)G1,q e , q6=0 Z X ˆ iwqx G2(y − x)¯u(y, t)dy = φq(t)G2,q e , q6=0 Z X ˆ iwqx G2(y − x)¯v(y, t)dy = ψq(t)G2,q e , q6=0 where Z Z ˆ iwqy ˆ iwqy G1,q = G1(y)e dy, G2,q = G2(y)e dy, Using (4.3.5) and (4.3.6) we have Z ˆ iwqx Gj,q = Gj(x)e dx Z rj = 2i x(1 − |x|/rj) sin(wqx)dx 0 2 = 2irj H(wqrj), where j = 1, 2 and 4 sin2(z/2) − z sin(z) H(z) = . z3 We note that H(z) is an odd function that decreases when z > 0 and for reference we plotted it for z > 0 in Figure 4.7. 12 10 8 6 H(z) 4 2 0 0 1 2 3 4 5 z Figure 4.7: A plot of the H(z). Note that it’s an odd function, which makes the computation easier. 95 Plugging (4.4.3) into (4.4.1) and using the above calculations, we arrive at a system of differ- ential equations for φq and ψq for each q 6= 0, 0 φ φ q q = Aq , (4.4.4) ψq ψq where Aq = −(D + η u )w2 + 2k u /µ · w r2H(w r ) −η u w2 + 2k u /µ · w r2H(w r ) 1 1 0 q 1 0 1 q 1 q 1 1 0 q 2 0 1 q 2 q 2 . 2 2 2 −η2v0wq + 2k2v0/µ2 · wqr2H(wqr2) −(D2 + η2v0)wq We denote the maximum real part of the eigenvalues of Aq by Λq. MT-NF segregation occurs if the uniform steady state (u0, v0) becomes unstable, in which case there exists at least one q for which Λq is positive. This will not occur without considering the active binding between microtubules through organelles and between microtubules and neurofilaments, i.e, k1 = k2 = 0. Indeed, in this case Aq reduces to −(D + η u )w2 −η u w2 1 1 0 q 1 0 q Aq = . 2 2 −η2v0wq −(D2 + η2v0)wq It is easy to see that tr(A) < 0 and det(A) > 0, thus both eigenvalues have negative real part and the uniform steady state is stable. We considered the situation with normal MT-MT binding through organelles and variable de- grees of MT-NF binding (k2): k2 normal for healthy axons, and k2 reduced or 0 for IDPN-treated axons. Figure 4.8 shows that with normal or slightly increased k2, one has Λq < 0 for all q 6= 0, 96 i.e., the uniform steady state is stable and the two polymer populations form a mixture. As k2 decreases to half of its value or 0, Λq becomes positive for the first few modes with small q, and this means that the uniform steady state is unstable. 2 k2 × 2 1.5 k2 × 1 k2 × 0.5 1 k2 × 0 ) λ 0.5 0 -0.5 Max Re( -1 -1.5 -2 0 1 2 3 4 5 ω q Figure 4.8: Decreasing MT-NF interaction destabilize the uniform steady states. The x-axis is the wave number wq, and the y-axis is Λq, the maximum real part of the eigenvalues of the matrixAq. Parameters used: r1 =; α =; β =. The parameter k2 is changed to 2, 1, 0.5, 0 fold of the value in Table 4.1 for the blue, red, yellow, purple curves respectively. All other parameters used are the same as in Table 4.1. The circles represent the discrete modes with q = ±1, ±2, ±3,... and L = 10. To further understand how the densities of microtubules and neurofilaments evolve given k2 = 0, we simulated the model with periodic boundary conditions and a small perturbation away from the uniform steady state as the initial condition, u(x, 0) = u0 + u1(x), v(x, 0) = v0 + v1(x), where u1(x) and v1(x) are independent random numbers with a uniform distribution in [−1, 1]. Figure ?? demonstrates that microtubules and neurofilaments gradually relocate into different regions of the domain and separate almost completely from each other within a few hours. If k2 97 is restored to its normal value, the two polymers gradually intersperse. 4.4.3 A simplified model in 2D Now we investigate microtubule-clustering phenomena in two dimensional case. Stimulated by 1D case, we carry out corresponding linear stability analysis on a 2D square Ω = [0,L]×[0,L] with periodic boundary condition on both arguments. Thus the equation system reduces to ∂u 1 h i = D ∆u + ∇ · (η u∇(u + v)) − ∇ · uK (u, v) in Ω ∂t 1 1 µ u 1 (4.4.5) ∂v 1 h i = D2∆v + ∇ · (η2v∇(u + v)) − ∇ · vKv(u, v) in Ω. ∂t µ2 where Z Ku(u, v) = k1 Gr1 y − x u(y, t)dy Ω Z (4.4.6) + k2 Gr2 y − x v(y, t)dy, Ω Z Kv(u, v) = k2 Gr2 y − x u(y, t)dy. Ω |x| Grj (x) = x 1 − χ|x| u(x1, 0, t) = u(x1, L, t), v(x1, 0, t) = v(x1, L, t), x1 ∈ [0,L] (4.4.8) u(0, x2, t) = u(L, x2, t), v(0, x2, t) = v(L, x2, t), x2 ∈ [0,L] (4.4.9) Linear stability 2D Next we seek to linearize equations (4.4.5). Assume u0 and v0 to be the average densities of microtubules and neurofilaments respectively, similar to 1D case, we may expect the uniform 98 steady state of above system to be (u, v) = (u0, v0). We can therefore write u(x1, x2, t) = u0 +u ¯(x1, x2, t) and v(x1, x2, t) = v0 +v ¯(x1, x2, t), where u¯ and v¯ represent the deviation of the densities from uniform steady state. Applying the above notations into (4.4.5), neglecting nonlinear (higher than first order) terms and we obtain the linearized model h i ∂u¯ 1 ¯ = D1∆¯u + η1u0∆(¯u +v ¯) − u0∇ · Ku(¯u, v¯) in Ω (4.4.10) ∂t µ1 h i ∂v¯ 1 ¯ = D2∆¯v + η2v0∆(¯u +v ¯) − v0∇ · Kv(¯u) in Ω. (4.4.11) ∂t µ2 where ¯ R R Ku(¯u, v¯) = k1 Ω Gr1 y − x u¯(y, t)dy + k2 Ω Gr2 y − x v¯(y, t)dy, (4.4.12) ¯ R Kv(¯u, v¯) = k2 Ω Gr2 y − x u¯(y, t)dy. (4.4.13) Since we have assumed periodic boundary conditions on both arguments, we can express u¯ and v¯ into Fourier series X iwpx1 iwqx2 X iwpx1 iwqx2 u¯ = φp,q(t)e e , v¯ = ψp,q(t)e e . (4.4.14) p,q6=0 p,q6=0 here wp = 2pπ/L, wq = 2qπ/L for any nonzero integers p and q, φp,q(t) and ψp,q(t) are smooth functions. Corresponding differential terms can hence be writen as P 0 iwpx1 iwqx2 X 0 iwpx1 iwqx2 u¯t = p,q6=0 φp,q(t)e e , v¯t = ψp,q(t)e e , p,q6=0 P iwpx1 iwqx2 2 2 X iwpx1 iwqx2 2 2 ∆¯u = − p,q6=0 φp,q(t)e e (wq + wp), ∆¯v = − ψp,q(t)e e (wq + wp). p,q6=0 Plugging (4.4.14) into (4.4.10) and (4.4.11). Since {eiwpx1 eiwqx2 , (p, q) 6= (0, 0)} forms a basis in L2(Ω), for each pair of integer (p, q), we have 0 (p,q) (p,q) φp,q a a φp,q φp,q 11 12 p,q = := A · (4.4.15) (p,q) (p,q) ψp,q a21 a22 ψp,q ψp,q 99 p Figure 4.9: Max eigenvalue of A with different k1 and k2 values in 1D case. where (p,q) 2 2 u0k1 ˆ(1) ˆ(2) a11 = −(D1 + η1u0)(wp + wq ) − i (Gp,q,1wp + Gp,q,1wq) µ1 (p,q) 2 2 u0k2 ˆ(1) ˆ(2) a12 = −η1u0(wp + wq ) − i (Gp,q,2wp + Gp,q,2wq) µ1 (p,q) 2 2 v0k2 ˆ(1) ˆ(2) a21 = −η2v0(wp + wq ) − i (Gp,q,2wp + Gp,q,2wq) µ2 (p,q) 2 2 a22 = −(D2 + η2v0)(wp + wq ) Z p 2 2 ˆ(1) x1 + x2 Gp,q,j = i x1 1 − sin(wpx1)cos(wqx2)dx1dx2, |x| Z p 2 2 ˆ(2) x1 + x2 Gp,q,j = i x2 1 − cos(wpx1)sin(wqx2)dx1dx2, |x| We get the contour graph of k1 and k2 in Fig. 4.10. 100 p,q Figure 4.10: Max eigenvalue of A with different k1 and k2 values in 2D case. 4.4.4 An investigation to circular domain model Model equations in polar coordinates To simulate the model in a circular domain, we transform (4.4.5) into polar coordinates. We assume x and y have polar coordinates (r, θ) and (s, α). For simplicity we use u = u(r, θ, t) and v = v(r, θ, t) to denote these functions in polar coordinates. Using the formulas 1 ∇f = f e + f e , (4.4.16) r r r θ θ 1 1 ∇ · (fe + ge ) = (rf) + g , (4.4.17) r θ r r r θ 101 after some computation, we obtain the equations in polar coordinates ∂u 1 1 u = ∂r D1rur + η1ru(ur + vr) − ruKr [u, v] ∂t r µ1 1 1 1 1 u + ∂θ D1 uθ + η1 u(uθ + vθ) − uKθ [u, v] , (4.4.18) r r r µ1 ∂v 1 1 v = ∂r D2rvr + η2rv(ur + vr) − rvKr [u] ∂t r µ2 1 1 1 1 v + ∂θ D2 vθ + η2 v(uθ + vθ) − vKθ [u] . (4.4.19) r r r µ2 where + + Z θ1 Z min{r1 ,R0} u Kr [u, v] = k1 (r − s cos(α − θ))f1(α)u(s, α, t)s dsdα − − θ1 max{r1 ,−R0} + + Z θ2 Z min{r2 ,R0} +k2 (r − s cos(α − θ))f2(α)v(s, α, t)s dsdα (4.4.20) − − θ2 max{r2 ,−R0} + + Z θ2 Z min{r2 ,R0} v Kθ [u] = k2 (−s sin(α − θ))f2(α)u(s, α, t)s dsdα, − − θ2 max{r2 ,−R0} |r2 + s2 − 2rs cos (α − θ)| fj(α) = (1 − ), j = 1, 2. rj u v with similar expressions for Kθ [u, v],Kr [u], and ± −1 θi = θ ± sin (ri/r) if ri < r, and θ ± π/2 otherwise, q ± 2 2 2 ri = r cos(θ − α) ± ri − r sin (θ − α). with boundary conditions 1 u −D1∂ru − η1u∂r(u + v) + uK [u, v] · er = 0, r = R0 (4.4.21) µ1 R0,θ 1 v −D2∂rv − η2v∂r(u + v) + vK [u] · er = 0, r = R0 (4.4.22) µ2 R0,θ u(r, θ, t) = u(r, θ + 2π, t), v(r, θ, t) = v(r, θ + 2π, t), 0 ≤ r ≤ R0 (4.4.23) 102 Radially symmetric case In the case when the solution does not depend on θ, we must have uθ = uθθ = vθ = vθθ = 0, so for simplicity we can denote u(r, t) := u(r, θ, t) and v(r, t) := v(r, θ, t). As the non-local integral terms should be the same regardless of the choice of θ if r is fixed, we can consider the non-local terms when θ = 0, in which case es · er = cos α, es · eθ = sin α. Thus our equations are simplified into ∂u 1 1 = D1(urr + ur) + η1 ru(ur + vr) ∂t r r r θ+ min{r+,R } 1 1 h Z 1 Z 1 0 − ∂r ruk1 g1(α)u(s, t)s dsdα µ r − − 1 θ1 max{r1 ,−R0} θ+ min{r+,R } Z 2 Z 2 0 i +ruk2 g2(α)v(s, t)s dsdα , (4.4.24) − − θ2 max{r2 ,−R0} ∂v 1 1 = D2(vrr + vr) + η2 rv(ur + vr) ∂t r r r " θ+ min{r+,R } #! 1 1 Z 2 Z 2 0 − ∂r rvk2 g2(α)u(s, t)s dsdα , (4.4.25) µ r − − 2 θ2 max{r2 ,−R0} |r − s cos α| gj(α) = (r − s cos α)(1 − ), j = 1, 2. (4.4.26) rj with boundary conditions 1 R0,0 −D1∂ru − η1u∂r(u + v) + uK (u, v) · er = 0, r = R0 (4.4.27) µ1 u 1 R0,0 −D2∂rv − η2v∂r(u + v) + vK (u, v) · er = 0, r = R0 (4.4.28) µ2 v ∂ru = ∂rv = 0, r = 0, (4.4.29) The additional boundary condition (4.4.29) appears because of radially symmetry assumption. Numerical method for 2D case is more or less subtle compared to 1D, because of the right hand 103 side nonlocal advection terms. Since the integrals are taken over circles, a polar-coordination form of the model is more preferable. To begin with, we continue our simulations from section (4.4.4), in which we only consider radially symmetric case with no flux boundary conditions. Since u and v equations are independent of argument θ, numerical simulations for radially symmetric case are essentially same to 1D case. Simulation results are displayed in Fig. 4.13. As expected, strong segregation takes place when k2 = 0, and segregation behavior gets weaker as k2 becomes larger. Note that because of no flux boundary condition, the result does not completely matches that of Figure 4.9. See AppendixA for detailed numerical methods. 4.5 NUMERICAL SIMULATIONS AND BIOLOGICAL IMPLICATIONS 4.5.1 Simulations in 1D We begin with 1D model. In 1D, the model (4.3.7)-(4.3.10) reduce to 1 ut = D1uxx + (η1u(ux + vx))x − uKu(u, v) in [0,L] µ x 1 (4.5.1) 1 vt = D2vxx + (η2v(ux + vx))x − vKv(u) in [0,L]. µ2 x u(0, t) = u(L, t), v(0, t) = v(0, t), t ∈ R+ (4.5.2) if we impose model (4.5.1) with periodic boundary condition, or 1 D u + η u(u + v ) − uK (u, v) = 0 1 x 1 x x µ u 1 (4.5.3) 1 D2vx + η2v(ux + vx) − vKv(u) = 0, x = 0 or L. µ2 which is the no-flux boundary condition. 104 Periodic boundary condition We consider periodic boundary condition (4.5.2) first. Typically, periodic condition can be imposed when the simulation domain is large enough and boundary condition is less important. We applied second order Euler’s method, please refer to AppendixA for detailed numerical methods. Simulations results are shown in Figure 4.11. Figure 4.11: Numerical simulations of microtubule and neurofilament distribution in 24 hours in 1D with periodic condition. (a) and (b): Microtubule and neurofilament distribution in normal case; (c) and (d): Microtubule and neurofilament distribution in IDPN injected case. No-flux boundary condition We consider no-flux boundary condition (4.5.3) first. Discretization scheme is similar to (4.5.3), and results are shown in Figure 4.12. In the case of no-flux boundary condition, we see a concentration valley of microtubule, and concentration peak of neurofilament near the bound- 105 Figure 4.12: Numerical simulations of microtubule and neurofilament distribution in 24 hours in 1D with noflux condition. (a) and (b): Microtubule and neurofilament distribution in normal case; (c) and (d): Microtubule and neurofilament distribution in IDPN injected case. aries. This indicates a better match with experimental data, compared to periodic boundary con- dition case. However, there maybe more dedicated biological reasons involved other than no-flux near axon membrane. For example, actin-binding proteins (e.g. Abp1 involved in membrane trafficking) can affect the behaviors of actin filaments, and may thereby affect microtubule and neurofilament concentration [39, 159]. 4.5.2 The impact of domain size Recent biological experiments had indicated that for larger axons, more neurofilament clusters are observed [25, 172]. In previous simulations, we kept L = 1µm (L = 5 for dimensionless unit). In this subsection, we explore how domain size can affect MT-NF phase separation patterns. We impose no-flux boundary condition, keep k1 = 0.6, k2 = 0.432 (IDPN injected case) and 106 Figure 4.13: Microtubule and neurofilament distribution in 80 hours in 2D with k1 = 0.6 and different values of k2. (a) and (d): k2 = 0.864; (b) and (e): k2 = 0.432; (c) and (f): k2 = 0 choose L to be L = 5, 10, 15, and the simulations are shown in Fig. 4.14 and Fig. 4.15 below. (a) L = 1µm (b) L = 2µm (c) L = 3µm Figure 4.14: Comparison of microtubule-neurofilament distribution patterns computed in 50 hours in 1D with different sizes. Simulation implies that larger domain size can increase the number of distribution peaks of microtubule, forming more clusters as we can observe in experiments. 107 (a) L = 1µm (b) L = 2µm (c) L = 3µm Figure 4.15: Surface plot of Fig. 4.14 From the simulation indicated above, as domain size increases, we expect microtubules to form more clusters, while neurofilament are excluded away from these microtubule clusters for IDPN injected cells. So larger axons are not necessarily more defensive to IDPN-like toxics, because neurofilaments are totally separated away by all microtubule clusters, even though there can be more microtubules in larger axons. 4.5.3 MT-NF phase separation is reversible due to IDPN washout Although MT-NF phase separation can happen even for large axons, the phenomena is re- ported to be reversible [65, 120] by IDPN washout in hour. In the following simulation, we will reproduce the remarkable penomena after washing out. By simulating “washing out”, we simply take MT-NF interaction rate k2 from 0 back to 0.432 at t = 30 hours. We still simulate for different domain sizes, i.e. L = 1µm, 2µm, 3µm (L = 5, 10, 15 for non-dimensional units) respectively. The results are shown in Fig. and Fig. 108 (a) L = 1µm (b) L = 2µm (c) L = 3µm Figure 4.16: Microtubule-neurofilament distribution patterns computed in 50 hours in 1D, IDPN washout takes place at t = 30 hours. Distribution inhomogeneity of microtubules and neurofilaments are dimin- ished almost immediately. (a) L = 1µm (b) L = 2µm (c) L = 3µm Figure 4.17: Plot of Dirichlet energy of Fig. 4.16. Dirichlet energy is a measure of inhomogeneity of a function, a sharp drop of Dirichlet energy is indicated by the simulation, regardless of the domain size. 109 The numerical simulations indicated that, the non-homogeneous steady-state loses its stability almost immediately (compared with 1 hour). While biological experiments showed that MT-NF separation is only reversed several hours after IDPN washout, we guess that the neuron may take hours before fully recover from the washout effect. 4.5.4 Nonlinear diffusion is important for homogeneity, but not for MT-NF phase separa- tion Although what we are most interested in are nonlocal interaction terms, the effects of non- linear diffusion terms (arisen from excluded-volume effect) of model (4.3.7)-(4.3.10) are also non-neglectable. In fact, if we only have linear diffusion, then (4.4.5) becomes 2 2q 2 u0r1 2q 2qr1 u0 2q 2 2qr2 −D1( ) + 2k1 H( ) 2k2 r H( ) πL µ1 πL πL µ1 πL 2 πL Aq = . 2 v0r2 2q 2qr2 2q 2 2k2 H( ) −D2( ) µ2 πL πL πL whose maximum real part of eigenvalues is larger than 0 as shown in , destabilizing the homoge- neous steady state. Fig. 4.19 and Figure 4.20 display the numerical result. Numerical simulations indicate that when nonlinear diffusion terms are dropped, distribution inhomogeneity are even stronger. But it is important to notice that there is still a significant difference between this case and IDPN-injected case – microtubule and neurofilament distribution are now synchronized. Although strong microtubule clusters are formed, NO MT-NF phase separation is observed. Therefore, we expect that although nonlinear diffusion is important for forming microtubule and neurofilament clusters, nonlocal interactions are the core for MT-NF phase separation. 110 (a) 1d casd (b) 2d casd Figure 4.18: Linear stability for linear-diffusion-only cases in 1d and 2d, periodic boundary conditions are imposed. Computation shows that in linear-diffusion-only cases, MT-NF distribution are almost surely inhomogeneous for any k1, k2 > 0. Compare these figures with Fig. 4.9 and Fig. 4.10 (a) L = 1µm (b) L = 2µm (c) L = 3µm Figure 4.19: Microtubule-neurofilament distribution patterns computed in 20 hours in 1D, without non- linear diffusion terms. Even if MT-NF interaction is not blocked (k2 = 0.432), distribution inhomogeneity follows in only around 3 hours. Extending the bifurcation diagram Another interesting feature of the model (4.3.7)-(4.3.10) is that, it admits instability of the homogenous steady-state in a different way that a Turing’s system produces. This intuition was 111 (a) L = 1µm (b) L = 2µm (c) L = 3µm Figure 4.20: Plot of Dirichlet energy of Fig. 4.19. Without nonlinear diffusion terms, distribution inho- mogeneity of microtubules and neurofilaments seem to be even stronger than IDPN-injected cases. formed by continually increase the k2 value in the bifurcation diagram Fig. 4.9, leading to a extended contour plot as shown in Fig. 4.21. So when we increase the MT-NF interaction rate to a sufficiently large value, inhomogeneity happens again. The following Fig. 4.22 shows the simulation result when k1 = 0.432 and k2 = 6. The simulation result can be explained in the following biological point of view: if neurofilament bind onto microtubule too frequently, then microtubule will turn to pull neurofilament altogether, violating homogeneity in healthy neurons. 4.6 SUMMARY AND DISCUSSION In this Chapter, we developed a novel nonlocal continuum model modeling cross-sectional distribution of microtubules and neurofilaments in axons. Following the modeling philosophy in [168], the model describes microtubules and neurofilaments as interacting particles in an axonal 112 Figure 4.21: An extended bifurcation diagram for Fig. 4.9. (a) Heat plot. (b) Surface plot. Figure 4.22: Simulation result for large MT-NF interaction in T = 40 hours. Although inhomogeneity happens, microtubule and neurofilament distributions are still synchronized, no phase separation happens. 113 cross-section, while the effect of membranous organelles are simplified through direct MT-MT interaction. The model incorporates detailed descriptions of key molecular processes that occur within seconds, including the binding and unbinding process of neurofilament onto microtubule, through a nonlocal interaction term. Repulsive forces and diffusive motions of microtubule and neurofilament are taking into effect, addressing the MT-NF phase separation phenomena that occur on a time scale of hours to days. Except for [168], other mathematical models of the axonal transport of neurofilaments and organelles have been developed previously to describe the longitudinal distribution of cargoes along axons, addressing on slow axonal transport [16, 27, 83], fast axonal transport [15, 27] and However, those models were in 1D and did not consider the spatial arrangement and mechanical interactions of the cargoes and tracks in the radial dimension which are essential in understanding the segregation of microtubules and neurofilaments as well as the subsequent axonal swelling in neurological diseases. 114 CHAPTER 5 HOMOTOPY TRACKING METHOD – FINDING MULTIPLE PATTENS OF A REACTION-DIFFUSION SYSTEM Pattern formation can be modeled by reaction-diffusion systems (RDS). It is remarkable that even with simple nonlinear terms and choices of one or two model parameters, Turing’s system can produce fruitful patterns (steady-state patterns and dynamic patterns). Because of their non- linearities, however, classical PDE analytical methods are quite limited for RDSs. One technical reason that analytical methods failed to work is that they are often not able to capture multiple solution of a RDS. Therefore, numerical methods are highly appreciated for studying such kind of problems. One powerful tool of calculating multiple solutions of a RDS is so-called numerical algebraic geometry [142,4], which was coined by Dr. Andrew Sommese in 1990s. The main goal of numerical algebraic geometry is to find all the solutions of a polynomial system. On the other hand, if we consider a general reaction-diffusion system ∂u = D∆u + F (u, p) (5.0.1) ∂t where u ∈ Rn represents concentration of n types of molecules or chemical species, p ∈ Rk represents k parameters, D ∈ Rn×n represents the constant matrix of diffusion coefficients, and F is a function of u, which denotes reactions and interactions among different species. Then we can transform (5.0.1) into a polynomial system in a certain proper way. The organization of this Chapter is as follows. Section 5.1 will be the motivation of this 115 chapter, it explains why studying multiple patterns of a RDS is important. Section 5.2 will be a basic introduction to numerical algebraic geometry, especially the homotopy tracking method and a new method called bootstrapping method, which efficiently reduces the computational cost of solving a large polynomial system. Section 5.3 gives explicit examples on the Gray-Scott system (5.1.1) explaining how the bootstrapping method works. Section 5.3 studies the stability of each solution got from Section 5.4 use both linear (local) stability test and nonlinear stability test. Section 5.5 is a proposed work, the aim is to draw a bifurcation diagram of (5.1.1) in determining how the number of stable patterns changes as we vary the model parameters. Finally, Section 5.6 is the summary and discussion section. 5.1 MOTIVATION – THE GRAY-SCOTT MODEL Our start point is the Gray-Scott model, first appeared in 1990s [123, 91]. If we denote A to be the density function of chemical A (typically an activator), S the density function of chemical S (typically a substrate), Gray-Scott system then takes the following form: ∂A 2 = DA∆A + SA − (µ + ρ)A, ∂t (5.1.1) ∂S = D ∆S − SA2 + ρ(1 − S). ∂t S whose corresponding chemical reaction is: µ ρ S + 2A →1 3A, A → C, A, S → drain. and f(1 − S) is the feeding rate of S, (µ + ρ)A is the reaction rate of A → C. Equations of C concentration doesn’t appear here, therefore the equation has at least two parameters of interest, 116 ρ (the “feeding” rate), and µ (the “killing” rate) respectively. Numerical simulations indicated that various different patterns can be produced by (5.1.1) if we choose different values of ρ and µ. In the following section of numerical simulation, we choose: DA = 0.1,DA = 1, ρ = 0.0367, µ = 0.0649. Based on visual effect of these patterns, Pearson classified the patterns formed by (5.1.1) into 17 types, each type was designated by a Greek letter [123, 91]. However, since Pearson’s clas- sification is completely based on comparing different chemical density distributions produced by computer simulations, the work lacks insight into the PDE (5.1.1) itself. From mathematical point of view, all these patterns are either steady-state solutions (static patterns) or oscillating solutions (dynamic patterns) and can be approximated by numerical simulations. Figure 5.1: Classification of static patterns developed in (5.1.1). The picture is adapted from [123], reprinted with permission under AAAS copyright. 117 5.1.1 Numerical simulations To have a better feeling of how different solutions can be formed from the simple systems, we carry out a 2D simulation of (5.1.1) described as follows. We simulate the system on a 64 × 64 mesh grid, choose three different sets of ρ and µ values while keeping DA, DS and initial conditions, with periodic boundary condition imposed. The simulations are summarized in Fig. 5.2 below. (a) Cell division pattern (b) Coral like pattern (c) Temporal periodic pattern Figure 5.2: Numerical simulation of 2D Gray-Scott system with different µ and ρ values when DS = 1,DA = 0.5. (a): ρ = 0.0367, µ = 0.0649; (b): ρ = 0.061, µ = 0.062 and (c): ρ = 0.018, µ = 0.051. Pattern (c) is different from (a) and (b) in the sense that it’s dynamic, while both (a) and (b) are steady- states. 118 Fig. 5.3 displays total concentration of chemical A corresponding to the three different patterns in 5.2 evolving with time, under different parameter sets. (a) Coral like pattern (b) Cell division pattern (c) Temporal periodic pattern Figure 5.3: Concentration of chemical A. As a result, the solutions of (5.1.1) can be very complicated, even if we only simply vary two parameters! From theoretical point of view, due to the author’s knowledge, there is no general method to efficiently study the multiple solution behaviors of (5.1.1), and no generals 119 guarantee on stability of these solutions (non-homogeneous steady-state solutions). Nevertheless, These two issues can be efficiently studied by using computational methods, and will be further addressed in the following sections. 5.1.2 Multiple patterns arising with the same parameters As a matter of fact, even the same set of parameters yields different steady-state solutions (static patterns). For example, we follow the setup in the last section and choose ρ = 0.061, fµ = 0.062 (i.e. coral like patterns) and simulate for a long time, but start with different initial condi- tions, we can get different steady-state patterns: Fig. 5.4 is an example of static patterns of model (5.1.1) with exactly the same parameter set, this figure reflected the non-uniqueness of general RDSs. The multiple patterns in 5.4 are computed using the following algorithm. Although it works for our case, the algorithm does always guarantee a steady-state solution in general. Fig. 5.4 implies that the time-independent counterpart of (5.1.1) 2 DA∆A + SA − (µ + ρ)A = 0, (5.1.3) 2 DS∆S − SA + ρ(1 − S) = 0. which is a second order elliptic equation, has different solutions when periodic boundary con- dition is imposed (note that these solutions do not have constant difference with each other). Therefore, looking for all possible solutions of (5.1.3) and study their stability are appreciated, and they are the major motivation for the remainder of this note. 120 (a) D2 symmetric patterns (b) D4 symmetric patterns (c) Non-symmetric pat- terns Figure 5.4: Various steady-state patterns formed an RDS with identical parameters. 5.2 BASICS ON NUMERICAL ALGEBRAIC GEOMETRY 5.2.1 Solving polynomial systems The main goal of numerical algebraic geometry is to solve polynomial systems [4, 71]: f (x) 0 1 f (x) 0 2 = (5.2.1) . . . . . . fN (x) 0 where fi(x) is a polynomial of degree di, 1 ≤ i ≤ N, x = (x1, x2, ..., xM ). From geometric point of view, if solutions of (5.2.1) consists of discrete points, we call it a 121 zero dimmensional problem; if on the other hand, solutions of (5.2.1) has a compontent which is a variaty of at least one dimension, we call it a positive dimmensional problem. For (5.1.1), since we may expect there are only finite many steaty state solutions, we require it to be a zero dimmensional problem. We hence introduce the following definition: Definition 5.2.1. If (5.2.1) is a zero dimmensional problem, we call it well-defined. From a computational point of view, one may ask how many solutions does (5.2.1) have. let us consider a single polynomial first. We all know that if f(x) is a non-zero polynomial of degree n, then it has at most n roots in complex domain (of course in real domain, we expect less number of solutions). In this case, we can naturally think of the (real) roots of a polynomial f(x) as the points where the graph of y = f(x) crosses the x-axis. If we go back to the polynomial system (5.2.1), then we can keep the idea of “intersection point”, with each polynomial representing a (algebraic) curve. Figure 5.5: Polynomials can be regarded as algebraic curves, their roots are the intersections of these curves. Depending on the computational method, different upper bounds of solutions are given [4], they are respectively total degree, two-homogeneous, equation by equation and more. In the following sections, we use both total degree and two-homogeneous as upper bounds of the size 122 of initial system (start system). 5.2.2 Homotopy tracking method Before solving (5.1.1), we illustrate how to compute the steady-states of the general equation (5.0.1) directly by solving the following system of equations D∆u + F (u, p) = 0, (5.2.2) with no-flux boundary conditions. The general polynomial system is given by discretizing the differential equation system in (5.1.1) (e.g., the finite difference method or the finite element method). Homotopy methods are the main tool we use to solve the resulting polynomial equations [10,9]. In this section, we briefly describe the homotopy method to solve the differential equation system (5.2.1), which is re-written in the following form through a spatial discretization: fh(U, p) = DL(U) + F (U, p) = 0, (5.2.3) where h is the step size, U ∈ RnN is the numerical solutions of u, N is the number of grid points, Ui+1+Ui−1−2Ui L is a discretized Laplace operator (e.g., L(U) = h2 through a second order central difference scheme on the uniform grid), and fh(U, p) denotes the discretized polynomial system. Define our homotopy function as follows: H(U, t) = fh(U, p)(1 − t) + γt(U + F (U, p)) = 0, (5.2.4) where γ is a randomly chosen complex number. U + F (U, p) = 0 is our start system, and can be solved for each Ui, i = 1, ··· ,N. The set of these N-tuples (U1,..., UN ) is the solution set of the start system. The polynomial systems (5.2.3) is solved by tracking t from 1 to 0. 123 We consider a general homotopy H(U, t) = 0, where U consists of the variables and t ∈ [0, 1] is the path tracking parameter. When t = 1, we have known solutions to H(U, 1) = 0. At t = 0, we recover the original system that we want to solve. The problem of getting the solutions of the target system now reduces to tracking solutions of H(U, t) = 0 from t = 1 where we know solutions to t = 0. The numerical method used in path tracking from t = 1 to t = 0 arises from solving the Davidenko differential equation: dH(U(t), t) ∂H(U(t), t) dU(t) ∂H(U(t), t) = + = 0. (5.2.5) dt ∂U dt ∂t Fig. 5.6 shows how homotopy tracking method works, and Fig. 5.7 shows the general idea of the prediction-correction used in solving (5.2.5). Figure 5.6: A graphical explanation of the homotopy tracking method. Each initial guess (on the right hand side) lead to a true solution to the original system (on the left hand side) by tracking paths. In particular, path tracking reduces to solving initial value problems numerically with the start points being the initial conditions. Since we also have an equation which vanishes along the path, namely H(U, t) = 0, predictor/corrector methods, such as Euler’s predictor and Newton’s 124 Figure 5.7: A graphical explanation of the prediction-correction method. The correction (e.g. Newton’s method) step make path tracking process more accurate compared to using prediction (e.g. forward Euler’s method) along. corrector, are used in practice to solve these initial value problems. Additionally, the predictor- corrector methods are combined with adaptive step-size and adaptive precision algorithms [10,9] to provide eligibility and efficiency. The basic numerical idea for a path tracking algorithm is as follows. If the initial prediction is not adequate, the corrector fails and the algorithm responds by shortening the step-size to try again. For a small enough step and a high enough precision, the prediction/correction cycle must succeed and the tracker advances along the path. Moreover, for too large a step-size, the predicted point can be far enough from the path that the rules set the precision too high that the algorithm fails before a decrease in step-size is considered. So we employ adaptive path tracker [10,9] that adaptively changes the step-size and precision simultaneously. This adaptive path tracker increases the security of adaptive precision path tracking while simultaneously reducing the computational cost. 125 5.2.3 Bootstrapping method let us go back to the polynomial system (5.2.2). Since it is a discretization of an (elliptic) PDE, one could expect that it is of a high dimensional system if accuracy is required. The high dimen- sion of the system arises a huge difficulty, because computational time grows exponentially as the number of grid is included. Besides, loss of accuracy is another reason that high dimensional systems are not appreciated. To better illustrate the computational difficulty, we use the numerical algebraic geometry soft- ware Bertini [8] to solve 1d version of (5.1.1) on a late 2016 model of MacBook Pro (with 2 GHz Intel Core i5 CPU). The experiment is summarized in table 5.1. To overcome the difficulty produced by Bertini, we propose Bootstrapping algorithm in this section. This algorithm is essentially a multigrid method, which provides a way to ‘transfer’ a coarse grid solution onto a finer grid by means of spline interpolation, and therefore increasing the accuracy of the solution in an efficient and fast manner. Details of the algorithm The basic idea of the bootstrapping approach is that the polynomial systems resulting from discretizing on the whole domain, and each subdomain is structurally the same and only differ- ing by values of parameters. In short, this means that coefficient-parameter continuation can be repeatedly used to solve such systems. Although not guaranteed to find all solutions, the method has reliably supplied solutions not obtained any other way [10]. Specifically, the discretized poly- nomial system is completely solved on a coarse grid, fH (U, p) = 0. Then, to obtain solutions for a finer grid, we first solve subsystems on each subdomain. The solutions are obtained by using 126 homotopy continuation to build from the solutions of the subdomains. Solutions to even finer grids can be obtained by iterating this bootstrapping approach (see [10] for more detail). The algorithm for bootstrapping method is shown in Algorithm3 . Algorithm 3: Algorithm for bootstrapping method Input : the stepsize of coarse grid H, and the stepsize of the grids h on appropriate subdomains. • solve the discretized system on coarse grid fH (U, p) = 0 by using the homotopy method; • on each subdomain, set the solution as Dirichlet boundary condition, and solve the discretized system fh(U, p) = 0; • on the whole domain, obtain a hybrid discretized system fh,H (U, p) = 0 by previous two steps; solutions to fh(U, p) = 0 are computed from solutions of fh,H (U, p) = 0 using the homotopy H(U, p, t) = (1 − t)fh(U, p) + tfh,H (U, p), 5.3 MULTIPLE STEADY-STATES OF GRAY-SCOTT SYSTEM 5.3.1 Introduction To illustrate how homotopy tracking and bootstrapping methods can be applied to hunt for multiple steady-state solutions of a reaction-diffusion system, we consider 1D example of Gray- Scott system (5.1.1). 5.3.2 Discretization in 1D Note: this section will be put into appendix in the future. 127 The first order implicit Euler discretization of (5.1.1) will be A − 2A + A D k+1 k k−1 ∆t + S A2 − (µ + ρ)A = 0, A h2 k k k Sk+1 − 2Sk + Sk−1 2 DS 2 ∆t − SkAk + ρ(1 − Sk) = 0, for 1 ≤ k ≤ N − 1 h (5.3.1) 3A0 = 4A1 − A2, 3S0 = 4S1 − S2, 3AN = 4AN−1 − AN−2, 3SN = 4SN−1 − SN−2. If we consider the steady-state problem of (5.1.1) in 1D with no flux boundary condition, then it becomes a elliptic system 2 0 = DA∆A + SA − (µ + ρ)A, (5.3.2) 2 0 = DS∆S − SA + ρ(1 − S). We discretize (5.3.2) by second order central difference for interior points, and one-sided sec- ond order scheme for boundary points, then we arrive at a discretized version of (5.3.2): A − 2A + A D k+1 k k−1 + S A2 − (µ + ρ)A = 0, A h2 k k k Sk+1 − 2Sk + Sk−1 2 DS 2 − SkAk + ρ(1 − Sk) = 0, for 1 ≤ k ≤ N − 1 h (5.3.3) 3A0 = 4A1 − A2, 3S0 = 4S1 − S2, 3AN = 4AN−1 − AN−2, 3SN = 4SN−1 − SN−2. where the domain of interest ([a, b] in this case) is equally divided by N + 1 grid points, and h = (b − a)/N. Therefore, we are reduced to solve a 2(N + 1) number of algebraic equation system. Since there are 2(N − 1) equations of degree 3, we expect there are 32(N−1) independent solutions of (5.3.3)[8, 70]. 128 5.3.3 Solving the 1D system via Homotopy tracking Since the discretized system is a system of algebraic equations, we can therefore solve the above problem by using numerical algebraic geometry. The key method is homotopy tracking, which is already described in Section 5.3.4 Multigrid refining Although from computational point of view, solving the equation system (5.1.1) on a coarse grid is much more efficient, the accuracy is a potential issue, as seen in Fig. 5.8. Therefore, we need to interpolate the original solutions on a coarse grid onto a finer grid in a proper way. One popular choice for interpolation method is by using the third-order spline functions. The following results shows the 1-step refined solutions as the number of subdomains doubled. Up to now, we are yet not sure if all the solutions we got on N = 10 subdomains correspond to independent accurate solutions, so we continue on our refining process. Fig. 5.10 shows more refined solutions as the number of subdomains get doubled at each step. It is worth to notice that when the mesh get refined, total number of solution also decreases because numerical errors are reduced in finer meshes, and thereby different coarse-grid solutions may turn out to be an identical fine-grid solution. Total number of solutions versus number of grid points is summarized in Table 5.2. As we can see, when the mesh grid number is greater than 40, the number of solutions will remain to be 11. All these solutions are independent accurate steady-state solutions of (5.1.1) as 129 (a) (b) Figure 5.8: A test of refined results when DA = 0.1,DS = 1, µ = 1, ρ = 0.01. (a) We solve (5.1.1) on a coarse grid when there are 5 subdomains (therefore 8 algebraic equations of 3rd order, 4 algebraic equation of first order), we find 31 solutions in total. (b) After 1-step refining, the number of solutions reduces to 11, because due to numerical error, some of the solutions on the coarse grid coinside to be the identical true solution. 130 we desire. Figure 5.9: Refined solutions when DA = 0.1,DS = 1, µ = 1, ρ = 0.01. The system starts from 5 to 160 subdomains (i.e. 6 to 161 grids), each time the number of subdomains get doubled, new grids are generated by the center points of the subdomains in previous step. 5.4 STABILITY OF THE STEADY-STATES In this section, we study the stability of each solution got in the last section. 131 Figure 5.10: Refined solutions when DA = 0.1,DS = 1, µ = 0.062, ρ = 0.055. The system starts from 5 to 160 subdomains (i.e. 6 to 161 grids), each time the number of subdomains get doubled, new grids are generated by the center points of the subdomains in previous step. 132 5.4.1 Linear stability Linear stability is the standard way to study the asymptotic behavior of a steaty state solution of a nonlinear dynamical system. Consider a steady-state solution (A0(x),S0(x)), we write A(x, t) A0(x) A1(x, t) 2 = + + O( ) (5.4.1) S(x, t) S0(x) S1(x, t) plugging (5.4.1) into (5.1.1) and collecting O() term, we get ∂A1 2 = DA∆A1 + S1A0 + 2S0A0A1 − (µ + ρ)A1, ∂t (5.4.2) ∂S 1 = D ∆S − S A2 − 2S A A − ρS . ∂t S 1 1 0 0 0 1 1 using matrix notation, (5.4.2) becomes A (x, t) A (x, t) d 1 1 = L (5.4.3) dt S1(x, t) S1(x, t) where D 0 2S A − (µ + ρ) A2 A 0 0 0 L(·) := ∆(·) + (·) 2 0 DS −2S0A0 −A0 − ρ Therefore, studying the stability of (5.1.1) is to calculate the eigenvalues of L, that is, when the largest eigenvalue of L (evaluated at some steady-state solution) is negative, then the correspond- ing some steady-state solution of (5.1.1) is linearly stable. Using matlab, we are able to find 4 solutions that are stable, while the other 7 unstable. 5.4.2 Nonlinear stability While linear stability can only reflect the local asymptotic behavior of a steady-state solution, we do not know its global stability behaviors. In such cases, nonlinear stability may be able to capture more information of the global (at least, nonlocal) asymptotic behavior. 133 Figure 5.11: Linear stability test of (5.1.1). 134 To carry out nonlinear stability test, we proceed by giving a small random perturbation onto the steady-state, and then use the perturbed steady-state as initial condition of the corresponding N-dimensional ODE system, solve the ODE system up to some positive time Tend. By comparing the perturbed steady-state with the solution at time Tend, we can see global stability of that steady- state. The result of nonlinear stability test is shown in Figure. 5.12 below. Figure 5.12: Linear stability test of (5.1.1). Notice that since small perturbation is random, it can cause problems especially when dimen- sion N is large. In fact, to carry out a full version of nonlinear stability test, one need to perturb the steady-state along independent directions. That means we need to solve N ODE systems with N variables, which would be extremely expensive for large Ns! Therefore, a combination of linear and nonlinear stability test is more favored from our perspective. 135 5.5 PARAMETER REGIME – PROPOSED WORK The parameter regimes of all patterns for (5.1.1) in a one-dimensional domain is classified through performing Algorithm4 . 5.5.1 Homotopy tracking Since there are two different parameters in (5.1.1), ρ and µ respectively, we are able to draw the two-parameter bifurcation diagram in the following algorithm: Algorithm 4: Algorithm for one-parameter tracking Input : Coarse gird H, finer grid h, variable ρ, RHS function with first order terms fl(U, ρ). • solve the equation flH (U, ρ) = 0 by using an additional linear constraint; • on each subdomain, set the solution as Dirichlet boundary condition, and solve the discretized system flh(U, ρ) = 0; • extract ρ values solved from last step, sort these values and use (5.2.4) to track ρ. As ρ varies, so does µ, and therefore the outcome contain boundary curves in ρ − µ plane. These curves correspond to bifurcation points of (5.1.1). As for an example, we rewrite the system (5.1.1) here for convenience ∂A 2 = DA∆A + SA − (µ + ρ)A, ∂t (5.5.1) ∂S = D ∆S − SA2 + ρ(1 − S). ∂t S Remember that in Section 5.3.4, we were able to refine the coarse-grid solution of (5.5.1) onto refined grids and thereby bring all the solutions to be smooth enough in a computational efficient way. 136 5.6 SUMMARY AND DISCUSSION In this Chapter, we studied the computational method to find multiple stable steady-states for reaction-diffusion systems, with specific application to the Gray-Scott model (5.1.1). First, we combined a difference discretization of (5.1.1) and numerical algebraic geometric methods to solve the boundary value problem at the steady-state of the system, and obtain seven steady- state solutions. Second, in order to reduce computational cost, we combined numerical algebraic geometric method with multigrid method and works much more efficiently than purely using homotopy tracking method. Last, we analyzed the stability of each solution using both the eigen- value analysis (i.e. linear stability test) and nonlinear stability test, and find that seven out of the eleven steady-state solutions are stable. The polynomial numerical algebraic geometric method is a powerful approach which enables us to find as many steady-state solutions as we can, and gives guidance on finding multiple stable and biological meaningful steady-state solutions. Homotopy tracking method is a powerful tool but (a) it starts from a large initial system which grows exponentially on the size of the spatial grid and (b) each path may reach at most one steady-state solution, it is necessary to reduce the computational cost in an efficient way. The algorithm discussed in this Chapter is one solution. Since multiple steady-states arising in a reaction-diffusion system has many meaningful bio- logical implications, e.g. multiple skin patterns of zebrafish [69], numerical algebraic geometry can perform as a powerful tool in studying biological-related reaction-diffusion systems. Another potential application of numerical algebraic geometry in an evolutionary equation/system is that, it provides a natural way to study the bifurcation behavior of that equation/system by continuous varying the parameters and counting how the number of steady-states changes. This approach 137 provides a numerical solution to PDE bifurcation theory and extends the traditional numerical continuation method in ODE bifurcation theory. This will be included in my proposed work. 138 Algorithm 2: Algorithm for getting static patterns (steady-state solutions) Input : Positive integer N, RHS function of (5.1.1), tolerence δ, initial end time T . Step 1: Choose a initial condition A0, S0; Step 2: Use finite difference method to discretize the right hand side of (5.1.1) into N equations, resulting in an ODE system: dA dS 1 = f (A, S), 1 = g (A, S) dt 1 dt 1 ... (5.1.2) dA dS N = f (A, S), N = g (A, S) dt N dt N denote F(C) := (f1, ..., fN , g1, ...gN ); Step 3: For a large enough end time T , solve (5.1.2) by the method of line, and the result will be a approximation of a stable steady-state of (5.1.1), denoted by C; Step 4: Evaluate e := F(C). if 1. |e| > δ, then go back to Step 3 and assign T ← 2T ; 2. |e| ≤ δ then print C, go back to Step 1 and change initial conditions to get more static patterns; 139 N Total degree path # Two-homogenous path # 5 81 (= 34) 321 8 2,187 (= 37) 48,639 10 19,683 (= 39) 1,462,563 20 1,162,261,467 (= 319) – Table 5.1: Computational complexity versus subinterval number N. 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(A.1.4) Ω A.1.1 Numerical sheme in 1D We use the following discretization scheme of (A.1.1)-(A.1.2) inside the domain [0,L]. We iL discretize [0,L] with N + 1 equidistant points 0 ≡ x0 < x1 < x2 < ... < xN ≡ L, i.e. xi = N for i = 0, 1, ..., N. We approximate u(xi, t), v(xi, t) by Ui(t),Vi(t) Denote ui ≡ ui(t) = u(xi, t), vi ≡ vi(t) = v(xi, t) and ∆x = xi+1 − xi, then a second order difference scheme for RHS of (??) will be U − 2U + U η hU + U U 0(t) = D i+1 i i−1 + 1 i+1 i (U + V − U − V ) i 1 (∆x)2 (∆x)2 2 i+1 i+1 i i Ui + Ui−1 i 1 Ui+1KU,i+1 − Ui−1Ku,i−1 − (Ui + Vi − Ui−1 − Vi−1) − (A.1.5) 2 µ1 2∆x 160 V − 2V + V η hV + V V 0(t) = D i+1 i i−1 + 2 i+1 i (U + V − U − V ) i 2 (∆x)2 (∆x)2 2 i+1 i+1 i i Vi + Vi−1 i 1 Vi+1Kv,i+1 − Vi−1Kv,i−1 − (Ui + Vi − Ui−1 − Vi−1) − (A.1.6) 2 µ2 2∆x i = 1, 2, ..., N − 1 where Ku,i ≈ Ku(u, v)(xi, t), Kv,i ≈ Kv(u)(xi, t). Next we consider the flux, Ju,Jv corresponding to u and v equation respectively, which are defined in the following: 1 Ju(x, t) = D1ux + η1u(ux + vx) − Ku(u, v), (A.1.7) µ1 1 Jv(x, t) = D2vx + η2v(ux + vx) − vKv(u). (A.1.8) µ2 We then use half-grid discretization for the flux terms and approximate them by central difference method. Thus the half-grid flux terms will have the form 1 (i + 2 )L Ui+1 − Ui Ju( , t) ≈ Ju,i+ 1 := D1 N 2 ∆x (Ui+1 + Ui) (Ui+1 + Ui+1 − Vi − Vi) 1 Ui+1Ku,i+1 + UiKu,i +η1 − , 2 ∆x µ1 2 1 (i + 2 )L Vi+1 − Vi Jv( , t) ≈ Jv,i+ 1 := D2 N 2 ∆x (Vi+1 + Vi) (Ui+1 + Ui+1 − Vi − Vi) 1 Vi+1Kv,i+1 + uiKv,i +η2 − , i = 0, 1, ..., N − 1. 2 ∆x µ2 2 and therefore, (A.1.5) and (A.1.6) can be rewritten as Ju,i+ 1 − Ju,i− 1 U 0(t) = 2 2 , (A.1.9) i ∆x Jv,i+ 1 − Jv,i− 1 V 0(t) = 2 2 , i = 0, 1, ..., N. (A.1.10) i ∆x We wish to use the same discretization for i = 0 and i = N, and thus an approximation of ”ghost flux” J 1 ,J 1 ,J 1 and J 1 should be established. To do that, enforce conservation of u,− 2 v,− 2 u,N+ 2 v,N+ 2 161 mass of the numerical method, that is we wish to have N−1 1 X (U 0 + U 0 ) + U 0 = 0, (A.1.11) 2 0 N i i=1 N−1 1 X (V 0 + V 0 ) + V 0 = 0. (A.1.12) 2 0 N i i=1 plugging (A.1.9) and (A.1.10) into (A.1.11) and (A.1.12), we obtain 1 1 (Ju,− 1 + Ju, 1 ) − (Ju,N− 1 + Ju,N+ 1 ) = 0 (A.1.13) 2 2 2 2 2 2 1 1 (Jv,− 1 + Jv, 1 ) − (Jv,N− 1 + Jv,N+ 1 ) = 0 (A.1.14) 2 2 2 2 2 2 so we can assign J 1 = −J 1 ,J 1 = −J 1 , (A.1.15) u,− 2 u, 2 u,N− 2 u,N+ 2 J 1 = −J 1 ,J 1 = −J 1 . (A.1.16) v,− 2 v, 2 v,N− 2 v,N+ 2 A.2 FINITE VOLUME SCHEME The key feature of finite volume method is that, it considers flux terms on the boundaries of each cell (the computation domain is separated into small cells, called volume), and it’s therefore a natural choice for solving evolution equations with no-flux boundary condition. Since Equation (A.1.1)-(A.1.2) can be written into gradient forms, we can also use finite volume method to solve the integro-PDE system. As a counterpart to finite difference scheme, we only consider 2d case of (A.1.1)-(A.1.2) as an 162 illustrative example. The model equations in 2D have the form ∂u 1 1 u = ∂r D1rur + η1ru(ur + vr) − ruKr [u, v] ∂t r µ1 1 1 1 1 u + ∂θ D1 uθ + η1 u(uθ + vθ) − uKθ [u, v] , (A.2.1) r r r µ1 ∂v 1 1 v = ∂r D2rvr + η2rv(ur + vr) − rvKr [u] ∂t r µ2 1 1 1 1 v + ∂θ D2 vθ + η2 v(uθ + vθ) − vKθ [u] . (A.2.2) r r r µ2 where + + Z θ1 Z min{r1 ,R0} u Kr [u, v] = k1 h1(r, s, α, θ)u(s, α, t)s dsdα − − θ1 max{r1 ,−R0} + + Z θ2 Z min{r2 ,R0} +k2 h1(r, s, α, θ)v(s, α, t)s dsdα, (A.2.3) − − θ2 max{r2 ,−R0} + + Z θ2 Z min{r2 ,R0} v Kθ [u] = k2 h2(r, s, α, θ)u(s, α, t)s dsdα, − − θ2 max{r2 ,−R0} |r2 + s2 − 2rs cos (α − θ)| h1(r, s, α, θ) := (r − s cos(α − θ))(1 − ), r1 |r2 + s2 − 2rs cos (α − θ)| h2(r, s, α, θ) := (−s sin(α − θ))(1 − ). r2 u v with similar expressions for Kθ [u, v],Kr [u], and ± −1 θi = θ ± sin (ri/r) if ri < r, and θ ± π/2 otherwise, q ± 2 2 2 ri = r cos(θ − α) ± ri − r sin (θ − α). with boundary conditions 1 u −D1∂ru − η1u∂r(u + v) + uK [u, v] · er = 0, r = R0 (A.2.4) µ1 R0,θ 1 v −D2∂rv − η2v∂r(u + v) + vK [u] · er = 0, r = R0 (A.2.5) µ2 R0,θ u(r, θ, t) = u(r, θ + 2π, t), v(r, θ, t) = v(r, θ + 2π, t), 0 ≤ r ≤ R0 (A.2.6) 163 Next, we define the flux terms, Fu, Fv as u u,r u,θ F (r, θ, t) = F (r, θ, t)er + F (r, θ, t)eθ, (A.2.7) v v,r v,θ F (r, θ, t) = F (r, θ, t)er + F (r, θ, t)eθ. (A.2.8) where u,r 1 u F (r, θ, t) = D1∂ru + η1u∂r(u + v) − uKr,θ[u, v], µ1 u,θ 1 1 u F (r, θ, t) = D1∂θu + η1u∂θ(u + v) − uKr,θ[u, v], r µ1 v,r 1 v F (r, θ, t) = D2∂rv + η2v∂r(u + v) − vKr,θ[u], µ2 v,θ 1 1 v F (r, θ, t) = D2∂θv + η2v∂θ(u + v) − vKr,θ[u]. r µ2 Then (A.2.1) and (A.2.2) can be rewritten as u,r u,θ v,r v,θ r∂tu = ∂r(rF ) + ∂θF , r∂tv = ∂r(rF ) + ∂θF . (A.2.9) A.2.1 Radially symmetric case In the case of radially symmetric case, we use the following discretization inside the domain. We discretize [0,R0] with N + 1 equidistant points 0 ≡ r0 < r1 < r2 < ... < rN ≡ L, i.e. iR0 ri = N for i = 0, 1, ..., N. We approximate u(ri, t) and v(ri, t) by Ui(t) and Vi(t). Denote ui ≡ ui(t) := u(ri, t), vi ≡ vi(t) := v(ri, t) and ∆r = R0/N. The numerical method we use is based on the above conservative form, r 1 F 1 − r 1 F 1 0 1 i+ 2 u,i+ 2 i− 2 u,i− 2 Ui (t) = , (A.2.10) ri ∆r r 1 F 1 − r 1 F 1 0 1 i+ 2 v,i+ 2 i− 2 v,i− 2 Vi (t) = , i = 1, 2, ..., N − 1. (A.2.11) ri ∆r 164 where Ui+1 − Ui (Ui+1 + Ui) (Ui+1 + Ui+1 − Vi − Vi) Fu,i+ 1 := D1 + η1 2 ∆r 2 ∆r 1 Ui+1Ku,i+1 + UiKu,i − ≈ Fu(ri+ 1 , t), µ1 2 2 Vi+1 − Vi (Vi+1 + Vi) (Ui+1 + Ui+1 − Vi − Vi) Fv,i+ 1 := (D2 + η2 2 ∆r 2 ∆r 1 Vi+1Kv,i+1 + UiKv,i − ≈ Fv(ri+ 1 , t), µ2 2 2 and Ku,i, Kv,i are numerical approximations of the integrals at ri. Based on the no-flux boundary conditions of the model, we enforce the following conservation law for the numerical scheme, 0 2 N−1 ! π∆r X π(rN + rN− 1 )∆r U + 2πr ∆rU + 2 U = 0, (A.2.12) 4 0 i i 2 N i=1 0 2 N−1 ! π∆r X π(rN + rN− 1 )∆r V + 2πr ∆rV + 2 V = 0. (A.2.13) 4 0 i i 2 N i=1 which is equivalent to N−1 !0 ∆r X rN + rN− 1 U + r U + 2 U = 0, (A.2.14) 8 0 i i 4 N i=1 N−1 !0 ∆r X rN + rN− 1 V + r V + 2 V = 0. (A.2.15) 8 0 i i 4 N i=1 Plugging (A.2.10) and (A.2.11) into (A.2.14) and (A.2.15), we obtain 0 2 U = 8r 1 F 1 /∆r , (A.2.16) 0 2 u, 2 0 2 V = 8r 1 F 1 /∆r , (A.2.17) 0 2 v, 2 0 U = −2r 1 F 1 /(r 1 ∆r), (A.2.18) N N− 2 u,N− 2 N− 4 0 V = −2r 1 F 1 /(r 1 ∆r). (A.2.19) N N− 2 v,N− 2 N− 4 165 A.2.2 Square domain and periodic boundary condition case Similar to 1D case, we use the following discretization inside the domain. We discretize (1) (2) (1) (2) (1) (2) [0,L] × [0,L] with (N + 1) × (N + 1) equidistant points (x0 , x0 ), (x0 , x1 ), ..., (xN , xN ) (1) (1) (1) (1) (2) (2) (2) (2) with 0 ≡ x0 < x1 < x2 < ... < xN ≡ L, 0 ≡ x0 < x1 < x2 < ... < xN ≡ L, i.e. (1) (2) iL xi = xj = N for i = 0, 1, ..., N. We approximate u(xi, yj, t) and v(xi, yj, t) by Uij(t) and Vij(t). Denote uij ≡ uij(t) := u(xi, yj, t), vij ≡ vij(t) := v(xi, yj, t) and ∆x ∆y = L/N. The numerical method we use is based on the above conservative form, u u u u Fi+ 1 ,j − Fi− 1 ,j Fi,j+ 1 − Fi,j− 1 U 0 (t) = 2 2 + 2 2 , (A.2.20) ij ∆x ∆y v v v v Fi+ 1 ,j − Fi− 1 ,j Fi,j+ 1 − Fi,j− 1 V 0 (t) = 2 2 + 2 2 , i = 1, 2, ..., N − 1. (A.2.21) ij ∆x ∆y where u Ui+1,j − Uij (Ui+1,j + Uij) (Ui+1,j + Ui+1,j − Vi,j − Vi,j) Fi+ 1 ,j := D1 + η1 2 ∆x 2 ∆x u u 1 Ui+1,jK + UijK − i+1,j ij ≈ (Fu(x(1) , x(2), t))(1), i+ 1 j µ1 2 2 v Vi+1,j − Vij (Vi+1,j + Vij) (Ui+1,j + Ui+1,j − Vij − Vij) Fi+ 1 ,j := (D2 + η2 2 ∆x 2 ∆x v v 1 Vi+1,jK + UiK − i+1,j ij ≈ (Fv(x(1) , x(2), t))(1), i+ 1 j µ2 2 2 u v and Ku,i, Kv,i are numerical approximations of the integrals at ri. F 1 ,F 1 can be defined i,j+ 2 i,j+ 2 in a similar way. 166 A.2.3 Circular domain with noflux boundary condition case In this case we consider model (A.2.1)-(A.2.2). We define the flux terms, Fu, Fv as u u,r u,θ F (r, θ, t) = F (r, θ, t)er + F (r, θ, t)eθ, (A.2.22) v v,r v,θ F (r, θ, t) = F (r, θ, t)er + F (r, θ, t)eθ. (A.2.23) where u,r 1 u F (r, θ, t) = D1∂ru + η1u∂r(u + v) − uKr,θ[u, v], µ1 u,θ 1 1 u F (r, θ, t) = D1∂θu + η1u∂θ(u + v) − uKr,θ[u, v], r µ1 v,r 1 v F (r, θ, t) = D2∂rv + η2v∂r(u + v) − uKr,θ[u], µ2 v,θ 1 1 v F (r, θ, t) = D2∂θv + η2v∂θ(u + v) − uKr,θ[u]. r µ2 Then (A.2.1)-(A.2.2) can be rewritten as u,r u,θ v,r v,θ r∂tu = ∂r(rF ) + ∂θF , r∂tv = ∂r(rF ) + ∂θF . (A.2.24) We use the following discretization inside the domain. We discretize [0,R] × [0, 2π] into M × N boxes of equal size, and (r 1 , θ 1 ), (r 1 , θ 1 ), ..., (r 1 , θ 1 ) being the centers of the 2 2 2 1+ 2 M+ 2 N+ 2 boxes. Here 0 ≡ r0 < r 1 < r1 < r 1 < ... < r 1 < rM ≡ L, 0 ≡ θ0 < θ 1 < θ1 < 2 1+ 2 M− 2 2 iR 2jπ θ 1 < ... < θ 1 < θN ≡ 2π, i.e. ri = , θj = for i = 0, 1, ..., M and j = 0, 1, ..., N. We 1+ 2 N− 2 M N approximate u(ri, θj, t) and v(ri, θj, t) by Uij(t) and Vij(t). Denote uij ≡ uij(t) := u(ri, θj, t), vij ≡ vij(t) := v(ri, θj, t) and ∆r = R/M, ∆θ = 2π/N. 167 The numerical method we use is based on the above conservative form, u,r u,r u,θ u,θ ri+1F 1 − riF 1 F 1 − F 1 0 i+1,j+ 2 i,j+ 2 i+ 2 ,j+1 i+ 2 ,j ri+ 1 Ui+ 1 ,j+ 1 (t) = + , (A.2.25) 2 2 2 ∆r ∆θ v,r v,r v,θ v,θ ri+1F 1 − riF 1 F 1 − F 1 0 i+1,j+ 2 i,j+ 2 i+ 2 ,j+1 i+ 2 ,j ri+ 1 Vi+ 1 ,j+ 1 (t) = + , (A.2.26) 2 2 2 ∆r ∆θ i = 0, 1, 2, ..., M − 1, j = 0, 1, ..., N − 1. where Ui+ 1 ,j+ 1 − Ui− 1 ,j+ 1 F u,r := D 2 2 2 2 i,j+ 1 1 2 ∆r (Ui+ 1 ,j+ 1 + Ui− 1 ,j+ 1 ) (Ui+ 1 ,j+ 1 + Vi+ 1 ,j+ 1 − Ui− 1 ,j+ 1 − Vi− 1 ,j+ 1 ) +η 2 2 2 2 2 2 2 2 2 2 2 2 1 2 ∆r u u Ui+ 1 ,j+ 1 K 1 1 + Ui− 1 ,j+ 1 K 1 1 1 2 2 i+ 2 ,j+ 2 2 2 i− 2 ,j+ 2 u,r − ≈ F (ri, θj+ 1 , t), µ1 2 2 1 Ui+ 1 ,j+ 1 − Ui+ 1 ,j− 1 F u,θ := D 2 2 2 2 i+ 1 ,j 1 2 r 1 ∆θ i+ 2 (U 1 1 + U 1 1 ) (U 1 1 + V 1 1 − U 1 1 − V 1 1 ) 1 i+ 2 ,j+ 2 i+ 2 ,j− 2 i+ 2 ,j+ 2 i+ 2 ,j+ 2 i+ 2 ,j− 2 i+ 2 ,j− 2 +η1 r 1 2 ∆θ i+ 2 u u Ui+ 1 ,j+ 1 K 1 1 + Ui+ 1 ,j− 1 K 1 1 1 2 2 i+ 2 ,j+ 2 2 2 i+ 2 ,j− 2 u,θ − ≈ F (ri+ 1 , θj, t). µ1 2 2 u v u v v,r v,θ and Ki,j, Ki,j are numerical approximations of Kr,θ, Kr,θ at ri and θj. F 1 ,F 1 can be i+ 2 ,j i,j+ 2 defined in a similar way. u,θ Next we need to determine flux terms at boundaries, i.e F 1 , i = 1, 2, ..., M − 1 and i,N+ 2 u,θ u,θ u,θ u,θ F0,j ,FM,j, j = 0, 1, ..., N − 1 (note that we can define Fi,N ≡ Fi,0 ). Using periodicity on u,θ u,θ u,θ v,θ v,θ v,θ θ, we can simply define F 1 = F 1 ≈ F (ri, θ 1 ), F 1 = F 1 ≈ F (ri, θ 1 ). In case i,N+ 2 i, 2 2 i,N+ 2 i, 2 2 u,θ of r-flux terms, we assign F 1 = 0 for all j = 1, 2, ..., N − 1 in concordance with noflux M,j+ 2 u,θ boundary condition, and F 1 = 0 because radius is equal to zero at origin. 0,j+ 2 168 Figure A.1: Visual interpretation of notations. A.2.4 Calculating integral terms u Now let’s consider the integral terms. To illustrate our algorithm we only calculate Kr,θ, and similar methods apply to other integral terms. u Suppose we want to calculate K with the circular integral domain Ωij centering at r 1 , θ 1 , r,θ i+ 2 j+ 2 we approximate Ku by Ku ≈ Ku . We use following interpolation: Denote r 1 ,θ 1 r 1 ,θ 1 i+ 1 ,j+ 1 i+ 2 j+ 2 i+ 2 j+ 2 2 2 = {(k, l)|(r 1 , θ 1 ) ∈ Ωij, k = 0, ..., M, l = 0, ..., N − 1} be the index set of grid points that S l+ 2 l+ 2 u lie in Ωij, and therefore K 1 1 is given by i+ 2 ,j+ 2 u X K 1 1 = ri+ 1 − rk+ 1 cos(θl+ 1 − θj+ 1 ) i+ 2 ,j+ 2 2 2 2 2 (k,l)∈S 2 2 r 1 + r 1 − 2r 1 r 1 cos(θ 1 − θ 1 ) i+ k+ i+ 2 k+ 2 l+ 2 j+ 2 × 1 − 2 2 r1 ×U 1 1 r 1 ∆r∆θ i+ 2 ,j+ 2 k+ 2 169 The algorithm is summarized as follows: Algorithm S 5: Algorithm for the non-local integration Input : Step size hx, hy, the radius r and the integrant f(x, y) Compute the number of grid points of the local circle on the mesh M = r , and y hy residual ry = r − Myhy. for n = −My : My do n p 2 2 • Compute the intersection of the circle and the line y = nhy, x = r − (nhy) ; • Compute the number of grid points on x direction M n = xn , and residual x hx n n n rx = x − Mx hx; • Compute the integration of f(x, yn) on x-direction, gn; • Compute the integration of g(y) on y-direction. end 170