Table Of ContentAndreas Deutsch •Sabine Dormann
Cellular Automaton Modeling
of Biological Pattern
Formation
Characterization, Examples, and Analysis
Second Edition
Foreword by Philip K. Maini
Fellow of the Royal Society London
AndreasDeutsch SabineDormann
CentreforInformationServices FBMathematik/Informatik
andHighPerformanceComputing Universita¨tOsnabru¨ck
TechnischeUniversita¨tDresden Osnabru¨ck,Germany
Dresden,Germany
ISSN2164-3679 ISSN2164-3725 (electronic)
ModelingandSimulationinScience,EngineeringandTechnology
ISBN978-1-4899-7978-0 ISBN978-1-4899-7980-3 (eBook)
DOI10.1007/978-1-4899-7980-3
LibraryofCongressControlNumber:2017941183
©SpringerScience+BusinessMediaNewYork2005,2017
Foreword to the
Second Edition
The enormous advances over the past 10 years in molecular and cell biology
(most notably imaging) have given us unprecedented data on behaviour at the
single cell and tissue level. Consequently, more and more experimentalists and
theoreticians are developing mathematical and computational models at this
level and, with ever-enhancing computing power, these models are becoming
increasinglydiverseandsophisticated. Therearenowmanydifferentapproaches
to setting up models for spatio-temporal phenomena, ranging from the classical
mathematicalpartialdifferentialequationapproachtothecomputationallyvery
efficient cellular automaton models. To someone entering the field (and even to
the expert) this diversity can be bewildering and confusing. Which approach
should I use? Are various approaches related in some way?
In light of the above, this new edition of Cellular Automaton Modeling of Bio-
logical Pattern Formation couldnotbemoretimely. Toreflectthevastamount
of new research in this area over the past 10 years after the appearance of the
first edition, the authors have produced a book that not only greatly expands
chapters from the previous edition, but also includes new chapters on cell mi-
gration, tissue development, and tumour growth/invasion to reflect important
applications of cellular automata. They have also updated the suggested re-
search projects listed at the end of several chapters. In doing so, the authors
have produced the most comprehensive text on modelling spatial patterning
that I have seen. It brings together diverse modelling approaches (determin-
istic, stochastic, cell-based), explains each in detail, compares and contrasts
them, and applies them to cutting-edge research in biology and medicine. Ex-
amples include single and collective cell migration, adhesive cell interactions,
alignment and swarming, pigment pattern formation, tissue growth and de-
velopment, Turing pattern formation, excitable systems, tumour invasion, and
angiogenesis. The book elucidates key principles of cell interactions, as well
as concepts such as self-organisation, stability, and bifurcation. It begins with
a thought-provoking and entertaining history of theories of pattern formation
going back to the ancient Greeks and, through 14 chapters, brings the reader
right up to date with the present thinking on biological patterning processes
and disease dynamics. Notably, an accompanying cellular automaton simulator
is now available for readers to perform their own simulations on many of the
models covered in the text. QR codes are included within the figures for easy
access to the simulator.
The book is accessible to theoreticians at all levels. The beginning graduate
student will learn the basic theory behind the modelling method she/he wishes
to adopt, while the experienced researcher will see where her/his approach sits
in the wider context of models in this field. Many researchers are restricted
by their background to the particular framework they use, and it may not be
the most appropriate one for the problem at hand. This book, by putting
in one place such a breadth of modelling approaches, and benefiting from the
wealth of experience and insights acquired by the authors, who have worked in
this field all their careers, will facilitate the choice of appropriate methodology.
Therefore, not only does the book serve as an educational and training text, it
willalsoadvanceresearchinsignificantwaysaswecontinuetoaddressthemost
important and challenging problems facing scientific researchers this century.
Centre for Mathematical Biology Philip K. Maini
University of Oxford Fellow of the Royal Society London
Oxford, United Kingdom
August 2016
Foreword to the
First Edition
Therecentdramaticadvancesinbiotechnologyhaveledtoanexplosionofdata
in the life sciences at the molecular level as well as more detailed observation
and characterisation at the cellular and tissue levels. It is now absolutely clear
that one needs a theoretical framework in which to place this data to gain from
itasmuchinformationaspossible. Mathematicalandcomputationalmodelling
approaches are the obvious way to do this. Heeding lessons from the physical
sciences, one might expect that all areas in the life sciences would be actively
pursuing quantitative methods to consolidate the vast bodies of data that ex-
ist and to integrate rapidly accumulating new information. Remarkably, with
a few notable exceptions, quite the contrary situation exists. However, things
are now beginning to change and there is the sense that we are at the begin-
ning of an exciting new era of research in which the novel problems posed by
biologists will challenge the mathematicians and computer scientists, who, in
turn, will use their tools to inform the experimentalists, who will verify model
predictions. Only through such a tight interaction between disciplines will we
have the opportunity to solve many of the major problems in the life sciences.
Onesuchproblem,centraltodevelopmentalbiology,istheunderstandingofhow
various processes interact to produce spatio-temporal patterns in the embryo.
From an apparently almost homogeneous mass of dividing cells in the very
early stages of development emerge the vast and sometimes spectacular array
of patterns and structures observed in animals. The mechanisms underlying
the coordination required for cells to produce patterns on a spatial scale much
larger than a single cell are still largely a mystery, despite a huge amount of
experimental and theoretical research. There is positional information inherent
inoocytes,whichmustguidepattern,butcellswhicharecompletelydissociated
and randomly mixed can recombine to form periodic spatial structures. This
leads to the intriguing possibility that at least some aspects of spatio-temporal
patterning in the embryo arise from the process of self-organisation. Spatial
patternsalsoariseviaself-organisationinotherpopulationsofindividuals,such
as the swarming behaviour of bacteria, and in chemical systems, so that it is a
widespread phenomenon.
Modelling in this area takes many forms, depending on the spatio-temporal
scale and detail one wishes (or is able) to capture. At one extreme are coupled
systems of ordinary differential equations, in which one assumes that the sys-
tem is well stirred so that all spatial information is lost and all individuals (for
example, molecules) are assumed to have identical states. At the other extreme
are cellular automaton models in which each element may represent an individ-
ual (or a collection of individuals) with assigned characteristics (for example,
age) that can vary from one individual to the next. This approach allows for
population behaviour to evolve in response to individual-level interactions. In
hybrid cellular automata, one can model intracellular phenomena by ordinary
differential equations, while global signalling may be modelled by partial dif-
ferential equations. In this way, one can begin to address the crucial issue of
modelling at different scales. There are many modelling levels between these
extremes and each one has its own strengths and weaknesses.
AndreasDeutschandSabineDormannbringtobearonthissubjectadepthand
breadth of experience that few can match. In this book, they present many dif-
ferent modelling approaches and show the appropriate conditions under which
each can be used. After an introduction to pattern formation in general, this
book develops the cellular automaton approach and shows how, under certain
conditions, one can take the continuum limit, leading to the classical partial
differential equation models. Along the way, many interesting pattern forma-
tion applications are presented. Simple rules are suggested for various elemen-
tary cellular interactions and it is demonstrated how spatio-temporal pattern
formation in corresponding automaton models can be analysed. In addition,
suggestions for future research projects are included. It is also shown that the
model framework developed can be used more generally to tackle problems in
other areas, such as tumour growth, one of the most rapidly growing areas in
mathematical biology at the present time. The accompanying website allows
the reader to perform online simulations of some of the models presented.
This book, aimed at undergraduates and graduate students as well as experi-
enced researchers in mathematical biology, is very timely and ranges from the
classical approaches right up to present-day research applications. For the ex-
perimentalist,thebookmayserveasanintroductiontomathematicalmodelling
topics, while thetheoretician will particularly profit fromthedescription ofkey
problems in the context of biological pattern formation. The book provides
the perfect background for researchers wishing to pursue the goal of multi-scale
modellinginthelifesciences,perhapsoneofthemostchallengingandimportant
tasks facing researchers this century.
Centre for Mathematical Biology Philip K. Maini
University of Oxford Fellow of the Royal Society London
Oxford, United Kingdom
August 2003
Contents
Foreword to the Second Edition vii
Foreword to the First Edition ix
I General Principles and Models of Pattern Formation 1
1 Introduction and Outline 3
2 On the Origin of Patterns 13
2.1 Space, Time, and Mathematics . . . . . . . . . . . . . . . . . . 14
2.1.1 Static and Dynamic World Conceptions in Ancient
Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Scholasticism . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.3 The Deterministic World of Classical Mechanics . . . . 20
2.1.4 Discovering the History of Time . . . . . . . . . . . . . 26
2.1.5 From Equilibrium to Self-Organizing Systems . . . . . . 29
2.2 Principles of Biological Pattern Formation . . . . . . . . . . . . 35
2.2.1 Preformation and Epigenesis . . . . . . . . . . . . . . . 35
2.2.2 Ontogeny and Phylogeny . . . . . . . . . . . . . . . . . 37
2.2.3 On Organizers and Embryonic Regulation . . . . . . . . 39
2.2.4 Molecular and Genetic Analysis . . . . . . . . . . . . . . 40
2.2.5 Self-Assembly . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.6 Physical Analogues . . . . . . . . . . . . . . . . . . . . . 43
2.2.7 On Gradients and Chemical Morphogens . . . . . . . . 44
2.2.8 Self-Organization and Morphogenesis. . . . . . . . . . . 46
2.2.9 Cell-Cell Interactions. . . . . . . . . . . . . . . . . . . . 46
3 Modeling Biological Pattern Formation 49
3.1 The Art of Modeling . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 How to Choose the Appropriate Model . . . . . . . . . . . . . . 51
3.2.1 Model Perspectives . . . . . . . . . . . . . . . . . . . . . 56
3.2.2 From Individual Behavior to Population Dynamics . . . 58
II Cellular Automaton Modeling 63
4 Cellular Automata 65
4.1 Biological Roots . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Biological Applications . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Cellular Automaton Definition . . . . . . . . . . . . . . . . . . 76
4.3.1 Lattice Geometry . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 State Space . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.3 Neighborhood of Interaction . . . . . . . . . . . . . . . . 78
4.3.4 System Dynamics. . . . . . . . . . . . . . . . . . . . . . 80
4.4 Analysis and Characterization . . . . . . . . . . . . . . . . . . . 87
4.4.1 Chapman-Kolmogorov Equation . . . . . . . . . . . . . 90
4.4.2 Cellular Automaton Mean-Field Equations . . . . . . . 93
4.4.3 Linear Stability Analysis. . . . . . . . . . . . . . . . . . 100
4.5 Partial Differential Equations . . . . . . . . . . . . . . . . . . . 107
4.6 Further Research Projects . . . . . . . . . . . . . . . . . . . . . 109
III Applications 113
5 Random Movement 115
5.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Discrete Random Walk and Diffusion . . . . . . . . . . . . . . . 117
5.3 Probabilistic Cellular Automaton Models . . . . . . . . . . . . 121
5.3.1 Random Walk Rule According to Toffoli
and Margolus . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3.2 Asynchronous Updating . . . . . . . . . . . . . . . . . . 125
5.4 Lattice-Gas Cellular Automaton Models . . . . . . . . . . . . . 126
5.4.1 Stability Analysis. . . . . . . . . . . . . . . . . . . . . . 130
5.4.2 Checkerboard Artifact . . . . . . . . . . . . . . . . . . . 133
5.5 Diffusion-Limited Aggregation. . . . . . . . . . . . . . . . . . . 137
5.6 Further Research Projects . . . . . . . . . . . . . . . . . . . . . 139
6 Cell Migration 141
6.1 Cell Migration Modes . . . . . . . . . . . . . . . . . . . . . . . 142
6.2 Mathematical Models of Cell Migration . . . . . . . . . . . . . 144
6.3 Migration in Static Environment . . . . . . . . . . . . . . . . . 145
6.3.1 LGCA Model for Haptotactic Cell Migration . . . . . . 145
6.3.2 LGCA Model for Contact Guidance . . . . . . . . . . . 147
6.3.3 Analysis of the LGCA Models. . . . . . . . . . . . . . . 149
