MODELLING CHEMOTACTIC MOTION OF CELLS IN BIOLOGICAL TISSUE WITH APPLICATIONS TO EMBRYOGENESIS THESIS SUBMITTED IN ACCORDANCE WITH THE REQUIREMENTS OF THE UNIVERSITY OF LIVERPOOL FOR THE DEGREE OF DOCTOR OF PHILOSOPHY BY NIGEL CLIFFORD HARRISON SEPTEMBER 2012 TABLE OF CONTENTS General Introduction ................................................................................................................................. 5 Motivation ................................................................................................................................................... 5 Thesis Outline ............................................................................................................................................. 6 Chapter 1 Background Review ..................................................................................................................... 8 1.1 Background Review ......................................................................................................................... 8 1.1.1 Developmental Biology .......................................................................................................... 8 1.1.2 Mechanisms Of Cell Migration ........................................................................................... 13 1.2 Mathematical Modelling in Developmental Biology ................................................................ 17 1.2.1 Introduction ........................................................................................................................... 17 1.2.2 Modeling Morphogen Concentrations and Pattern Formation ..................................... 18 1.2.3 Chemotaxis ............................................................................................................................. 21 1.2.4 Numerical Methods .............................................................................................................. 25 1.3 The Cellular Potts Model .............................................................................................................. 29 1.3.1 The Ising Model .................................................................................................................... 29 1.3.2 The Potts (Clock) model ...................................................................................................... 33 1.3.3 The Extended Large-q Potts Model ................................................................................... 34 1.3.4 The Monte Carlo Method. ................................................................................................... 35 1.3.5 The Cellular Potts Model ..................................................................................................... 39 1.3.6 Implementation Of The CPM............................................................................................. 42 Chapter 2 1D Continuous Models for Chemotactically Moving Cells ............................................... 43 2.1 Introduction .................................................................................................................................... 43 2.2 Homogenous Model of a Migrating Group Of Cells ............................................................... 44 2.2.1 Concentration profile of Internally Produced Chemotactic Agent ............................... 44 2.2.2 Motion due to chemotaxis ................................................................................................... 47 2.2.3 Existence of Travelling Solutions ....................................................................................... 50 2.2.4 Concentration profile of an Externally Produced Chemotactic Agent ........................ 53 2.2.5 Motion Due To Chemotaxis ............................................................................................... 54 2.3 Model for the Heterogeneous Migrating Group ....................................................................... 55 2 | P a g e 2.3.1 Concentration profile of an Internally Produced Chemotactic Agent ......................... 55 2.3.2 Motion Due To Chemotaxis ............................................................................................... 57 2.3.3 Concentration profile of an Externally Produced Chemotactic Agent ........................ 60 2.3.4 Motion Due To Chemotaxis. .............................................................................................. 60 2.3.5 Generalisation of the Heterogeneous Model .................................................................... 63 2.4 Chapter Summary ........................................................................................................................... 65 Chapter 3 2D Modeling of a Migrating Group of Cells ......................................................................... 67 3.1 Introduction .................................................................................................................................... 67 3.2 2D Continuous Model of Homogenous Migrating Group ..................................................... 67 3.2.1 2D Model of a Migrating Group ........................................................................................ 68 3.2.2 2D Polar Coordinate Representation of 1D Model ........................................................ 69 3.2.3 Solutions to the 2D Polar System ....................................................................................... 70 3.2.4 Comparison between 1D and 2D Profiles ........................................................................ 71 3.2.5 Chemotactic Motion of a Circular Group: Numerical Implementation ...................... 73 3.2.6 Travelling Solution of a 2D Migrating Group. ................................................................. 74 3.3 Preliminaries Of CPM Models Of Group Migration: .............................................................. 75 3.4 CPM Homogenous Model of a Migrating Group .................................................................... 77 3.4.1 Concentration Profiles for an Internally Produced Chemotactic Agent ...................... 77 3.4.2 Motion Due To Chemotaxis On The CPM. ..................................................................... 78 3.4.3 Group Motion for an Internally Produced Chemotactic Agent .................................... 79 3.4.4 Group Motion for an Externally Produced Chemotactic Agent ................................... 85 3.5 Heterogeneous Model of a Migrating Group ............................................................................ 87 3.5.1 Group Migration for an Internally Produced Chemotactic Agent ................................ 88 3.5.2 Group Migration for an Externally Produced Chemotactic Agent ............................... 90 3.6 Chapter Summary ........................................................................................................................... 90 Chapter 4 Coordination of Cell Differentiation and Migration In Mathematical Models of Embryonic Axis Extension ...................................................................................................................................... 92 4.1 Abstract ............................................................................................................................................ 93 4.2 Introduction .................................................................................................................................... 93 4.3 Results .............................................................................................................................................. 98 3 | P a g e 4.3.1 Concentration profiles in the continuous one-dimensional model ............................... 98 4.3.2 Self-regulation of the size of the DoT via negative feedback ........................................ 99 4.3.3 Size regulation of the FGF8 domain of transcription ................................................... 101 4.3.4 Maintenance of the migrating DoT size in GGHM ...................................................... 103 4.3.5 Promotion of cell migration by a caudal morphogen .................................................... 104 4.3.6 Chemotactic mechanism for the DoT migration ........................................................... 105 4.3.7 Chemo-repulsion in GGHM ............................................................................................. 107 4.3.8 Experimental study of regulative properties of the FGF8 DoT ................................. 109 4.4 Discussion...................................................................................................................................... 111 4.5 Materials and Methods ................................................................................................................ 114 4.5.1 One-dimensional continuous model ................................................................................ 114 Chapter 5 Discussion and Conclusions ................................................................................................... 124 5.1 Summary Of Thesis ..................................................................................................................... 124 5.2 Discussion...................................................................................................................................... 125 5.3 Conclusion ....................................................................................................................................