Biology in Time and Space:
A Partial Differential Equation Approach
James P. Keener
University of Utah
26 December, 2015
2
Contents
I
The Text
9
1 Introduction
11
2 Conservation
13
2.1
The Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Examples of Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3 The Diffusion Equation - Derivations
17
3.1
Discrete Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.2
A Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.3
The Cable Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.4
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4 Realizations of a diffusion process
25
4.1
Following a single particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.2
Other features of Brownian particle motion - Escape times and splitting probabilities . . . . .
26
4.2.1
Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.3
Following Several Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.4
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
5 Solutions of the Diffusion Equation
33
3
4
CONTENTS
5.1
On the Infinite Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
5.2
On the semi-infinite line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
5.3
With Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5.4
Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5.5
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
5.5.1
Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
5.5.2
Forward Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.5.3
Backward Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.5.4
Crank-Nickolson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.6
Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.7
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
6 Diffusion and Reaction
6.1
41
Birth-Death with Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
6.1.1
Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
6.2
Facilitated Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
6.3
Facilitated Diffusion in Muscle Respiration
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.4
Growth with a Carrying capacity - Fisher’s Equation . . . . . . . . . . . . . . . . . . . . . . .
47
6.4.1
On a Bounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
6.5
Resource Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
6.6
Spread of Rabies - SIR with Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
6.7
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
7 The Bistable Equation - Part I: Derivations
63
7.1
Spruce Budworm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
7.2
Wolbachia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
5
CONTENTS
7.3
Nerve Axons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
7.4
Calcium Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
7.5
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
8 The Bistable Equation - Part II: Analysis
71
8.1
Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
8.2
Threshold Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
8.3
Propagation Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
8.3.1
Regions of Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
8.3.2
Discrete Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
8.3.3
Discrete Release Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
8.4
9 Advection and Reaction
87
9.1
Simple Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
9.2
Transport across a Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
9.3
Oxygen exchange between capillaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
9.4
Structured Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
9.5
Red Blood Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
9.6
Protein Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
9.6.1
Muscle Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
9.6.2
Sliding Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9.7
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.7.1
Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.7.2
The Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.7.3
Method of Lines; Upwinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6
CONTENTS
9.8
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10 Advection with diffusion
111
10.1 Axonal transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
10.2 Transport with Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.3 Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
11 Chemotaxis
117
12 Waves and Vibrations
121
12.1 Blood Pressure Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
12.2 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
12.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
13 Spatial Patterns
13.1 The Turing Mechanism
125
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
13.2 Diffusion-Driven Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
13.2.1 Tiger Bush Stripes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
13.3 Cell Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
13.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
14 Dispersal-Renewal Theory
133
14.1 Invasive Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
14.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
15 Collective Behavior
137
15.1 Quorum Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
CONTENTS
7
15.2 cAmp Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
15.3 Flocking Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
16 Appendix: Background Material
139
16.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
16.1.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
16.1.2 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
16.1.3 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
16.2 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
16.2.1 First Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
16.2.2 Systems of first order equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
16.2.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
16.2.4 Modeling Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
16.3 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
16.3.1 Decay Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
16.3.2 Several Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
16.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
17 Appendix: Matlab Codes
157
17.1 A Matlab Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
17.2 Particle Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
17.3 Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
17.4 Discrete Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
17.5 Motion of a Brownian Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
17.6 Diffusion Equation solution via Crank Nicolson Method . . . . . . . . . . . . . . . . . . . . . 164
17.7 Fisher Equation and Bistable Equation via Crank-Nicolson . . . . . . . . . . . . . . . . . . . 166
8
CONTENTS
17.8 Diffusion Equation with growth/decay solution via Crank Nicolson . . . . . . . . . . . . . . . 167
17.9 Solution of Delay differential equation using the Method of Lines (MOL) . . . . . . . . . . . . 169
17.10Solution of the Ornstein-Uhlenbeck equation
. . . . . . . . . . . . . . . . . . . . . . . . . . . 169
17.11The Bisection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
18 Appendix: Physical Constants
173
18.1 Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
18.2 Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
II
Solutions
177
19 Selected Solutions to Chapter 3
1
20 Selected Solutions to Chapter 4
3
21 Selected Solutions to Chapter 5
5
22 Selected Solutions to Chapter 6
7
23 Selected Solutions for Chapter 9
9
24 Selected Solutions to Chapter 13
15
25 Selected Solutions to Chapter 14
17
Part I
The Text
9
Chapter 1
Introduction
Mathematical biology is, broadly speaking, the study of how biological objects move and interact. The
interaction part of biology is often studied using dynamical systems including discrete time maps and ordinary
differential equations. Indeed, there is a lot of insight that can be gained from the study of these dynamical
descriptions of biological processes. And, there are many excellent books that present this point of view.
However, this perspective is limited when it comes to understanding real biological situations, because
biological objects are essentially never homogeneously distributed in space, or well-mixed, even in a test
tube or on a Petrie dish, let alone in the human body or on a mountain slope. In fact, it is a primary feature
of biological objects that there are spatial differences and correspondingly movement of objects from one
place to another. So, whether one is studying the spread of an infectious disease in Africa, the spread of
an invasive species in the agricultural regions of California, or the movement of an action potential along a
nerve axon, it is crucially important to include the effects of spatial differences.
This book is about the dynamics of biological objects in time and space. Consequently, it is about partial
differential equations. It is intended for an undergraduate audience, and no previous background in partial
differential equations is required. In fact, in the Appendices, you will find summaries of the background that
is needed from calculus, from ordinary differential equations, and from probability theory and stochastic
processes, because these are used a lot. Also, this material is presented from a heavily computational
perspective, using Matlab to compute solutions and make plots. So, in the appendix you will find a primer
for Matlab as well as Matlab codes for many of the calculations done in this text. The hope is that these
codes can be readily modified to be used for Exercises, and that this will facilitate the process of learning
how to understand partial differential equations.
The content of the book is organized pedagogically around mathematical material. However, all of the
models described in this book are derived with a specific biological question in mind. Give a summary of
the topics covered.
11