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URI [dataset] UNIVERSITY OF SOUTHAMPTON FACULTY OF SOCIAL SCIENCES Mathematical Sciences Progress in the Mathematical Modelling of Developmental Processes in Biological Systems, since Publication of On Growth and Form by D’Arcy W. Thompson by Brian H. Bonney Thesis for the degree of Master of Philosophy 08 August 2019 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF SOCIAL SCIENCES Mathematical Sciences Thesis for the degree of Master of Philosophy PROGRESS IN THE MATHEMATICAL MODELLING OF DEVELOPMENTAL PROCESSES IN BIOLOGICAL SYSTEMS, SINCE PUBLICATION OF ON GROWTH AND FORM BY D’ARCY W. THOMPSON by Brian H. Bonney In his renowned book On Growth and Form, first published in 1917, D’Arcy Wentworth Thomp- son stressed the significance of physical laws and mechanisms in determining the development of form and pattern in living organisms. This was at variance with the view held by the majority of his contemporaries in Biology, that natural selection and evolution were the primary factors involved in driving these processes. This disparity of views was understandable in Thompson’s day, given that most biologists of the time were (at least implicitly) vitalists. Now, given a general acceptance that the phenomena of life are subject to the laws of physics and chemistry, Thompson’s ideas are seen as complementary to the processes of evolution by natural selection. Yet, Thompson’s viewpoint still finds itself marginal to present day thinking in developmental biology, in that his ideas emphasise the significance of physical, more than biochemical and genetic processes. In particular On Growth and Form highlights the role of minimisation principles, and of formal symmetries of one type or another in developmental phenomena. Although these are manifestly of great significance to a full understanding of developmental systems, such principles are still not central to the attention of experimentalists in mainstream research. I conclude that Thompson, in the early 20th Century, could not quantify biology, because neither enough biology, nor enough of the underlying physics and chemistry, were yet understood. Even if they had been, the mathematical tools available at the time were inadequate. The real revolution has come with the advent of high performance computing, enabling the complexities with which Thompson aspired to grapple, to be studied meaningfully. A Note on Some Conventions Used in the Writing For economy of space, paraphrasing of quotes from D.W.Thompson and other authors cited will be the norm henceforward. Any changes in wording, or deleted parts of text will be indicated respectively within [square brackets] and with ... ellipses ... like so. Unless otherwise stated, any italicised emphases within quotations are my own, and not the cited authors’. All citations from On Growth and Form are from the first edition (1917), unless otherwise stated. If a particular citation used can only be found in the second edition (1942), then this will be made clear. On Growth and Form in the general sense is referred to as ‘OGF’, and the first and second editions are referred to respectively as ‘OGF(1)’ and ‘OGF(2)’. Passages in bold are there to point out links between successive Chapters and/or sections and subsections within chapters. Short statements in bold/italic are there to cross-reference material, usually by section or page numbers, within or between chapters. Any words or phrases shown in italic are technical terms, usually appearing at the first occurrence of the term in the text. Table of Contents Title Page i Abstract iii Table of Contents v List of Figures and Tables ix Declaration of Authorship xiii Acknowledgements xv 1 Introduction 1 1.1 A Brief Introduction to Thompson’s Life and Career ............ 2 1.2 Thompson’s Programme as Enshrined in ‘On Growth and Form’ ..... 4 1.3 Thompson’s Key Messages from ‘On Growth and Form’ .......... 5 1.4 Some Physico-mathematical Principles Highlighted by Thompson in ‘On Growth and Form’ ............................... 16 1.4.1 Force as an Explanatory Principle in the Determination of Form . 17 1.4.2 The Principle of Similitude, and the Effects of Scale ........ 18 1.4.3 Conservation and Minimisation Principles .............. 19 1.4.4 Symmetry and Symmetry-breaking .................. 21 1.4.5 The Second Law of Thermodynamics - evolution and entropy as analogues ................................ 22 2 Descriptive Mathematical Biology 25 2.1 Structuralism versus Functionalism ...................... 26 2.1.1 Structuralism .............................. 27 v vi Table of Contents 2.1.2 Functionalism .............................. 29 2.1.3 Dichotomisation versus Unification .................. 31 2.2 Growth as a Precisely Defined Concept .................... 32 2.3 Allometry, and the Theory of Transformations ............... 35 3 Empirically-based Theories of Organismic Development 39 3.1 How Thompson Interprets Form and Function Using Structural Mechanics 40 3.1.1 Adaptation of Bone to Stresses and Strains ............. 41 3.1.2 Adaptation of Bone to Shearing Stresses ............... 44 3.2 Attempts to Apply Thompson’s Theory of Transformations ........ 47 3.3 The Place of ‘On Growth and Form’ in the Contemporary and Subsequent History of The oretical Biology ........................ 50 3.3.1 Some Reviews of Both Editions of On Growth and Form ...... 50 3.3.2 The Place of Joseph Needham in Twentieth Century Biology ... 52 3.3.3 Thompson’s Contribution in the Pre-molecular Era of Developmen- tal Biology ............................... 53 3.4 How Empirically-based Models in Developmental Biology Came to Involve Genes as Developmental Agents ........................ 56 3.5 How Gene-based Models of Development Came to Incorporate the Concepts of Molecular Biology .............................. 58 4 Cybernetics and Systems Theory 61 4.1 Framing Problems of Developmental Biology in Systems Theoretic (or Cy- bernetic) Terms ................................. 63 4.2 Characterising Systems Theoretic/Cybernetic Models, and their Relation- ship to Mathematical Models ......................... 65 4.2.1 Effective Dynamic Systems and Models ............... 65 4.2.2 Feedback-loops and Self-regulation .................. 66 4.2.3 Two Significant Early Examples of Effective Models in Developmen- tal Systems Theory ........................... 68 4.3 Concepts of Information, Pattern and Complexity ............. 69 4.3.1 The Information Theory of Shannon and Weaver .......... 70 4.3.2 Elaborating the Ideas of Information Theory ............ 71 4.3.3 Patterns and their Representation as Networks, in the Modelling of Living Systems ............................. 73 4.4 Self-organisation ................................ 76 5 Finite State Models of Development 77 5.1 Some Early Philosophical Difficulties ..................... 78 5.1.1 Preformation versus Epigenesis .................... 79 5.1.2 Self-reproduction versus Development ................ 80 Table of Contents vii 5.1.3 The Interrelationships of Parts, and the Significance of Connected- ness in Developmental Modelling ................... 81 5.2 Growing Automata Nets ............................ 84 5.2.1 Growth versus Development ...................... 84 5.2.2 Constructing Growing Net Automata, with Examples ....... 86 5.2.3 Simple Examples of Developmental Algorithms ........... 87 5.3 Lindenmayer Systems, Lindenmayer Languages, and Genetic Grammars . 91 5.3.1 How L-systems Relate to Chapters 2 , 3 and 4 above ........ 92 5.3.2 Some General Aspects of Formal Languages ............. 93 5.3.3 Construction of L-Systems as ‘Grammars for Development’ .... 93 5.3.4 Generalising the Applicability of L-systems ............. 95 5.3.5 Cellular Automata ........................... 96 6 Complexity Theory and Complex Networks 99 6.1 Dynamical Systems and Nonlinear Mathematics ............... 101 6.1.1 Some Discussion of the Nomenclature of Complexity Theory ... 102 6.1.2 Self-organisation and Emergence in Dynamic Systems ....... 105 6.1.3 A Mathematical Interpretation of Complexity ............ 106 6.1.4 Relating Abstract Complexity Theory to the Modelling of Real World Systems ................................. 114 6.1.5 Spontaneous Emergence of Order in Developmental Systems .... 119 6.1.6 Ilya Prigogine’s Theory of Dissipative Structures .......... 120 6.2 Dynamics on Complex Networks ....................... 124 6.2.1 Some Historical Links Between Network Modelling and the Ideas of D’Arcy Thompson ........................... 124 6.2.2 Modelling the Dynamical Behaviours of Gene Regulatory Networks (GRNs) ................................

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