UNIVERSITY OF SOUTHAMPTON Mathematical Modelling of Lymphangiogenesis by Kenneth Y. Wertheim A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy in the Faculty of Engineering and the Environment Engineering Sciences, Bioengineering Sciences Research Group March 2017 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF ENGINEERING AND THE ENVIRONMENT ENGINEERING SCIENCES Doctor of Philosophy by Kenneth Y. Wertheim This thesis is concerned with lymphangiogenesis, the formation of new lymphatic vessels. The lymphatic system of a vertebrate is important for its homeostasis and immune responses to pathogens. Additionally, lymphangiogenesis is a means of cancer metastasis. This thesis is concerned with how the vascular endothelial growth factor C (VEGFC) may regulate lymphangiogenesis in the zebrafish embryo. We built a reaction-diffusion-convection model to describe the distribution of VEGFC in an idealised zebrafish trunk. We solved the model with the finite element method under a wide range of conditions. The results suggest that VEGFC can act as a morphogen for the progenitor cells of the lymphatic system: it induces their differentiation during lymphangiogenesis. However, it is unlikely to be a chemotactic factor which guides their migration. The abundance of collagen I in the trunk is the key regulator: it determines the dominant transport phenomenon and the extent of VEGFC-collagen I binding, thus affecting the distribution of VEGFC. The abundance of collagen I is in turn regulated by the matrix metalloproteinase 2 (MMP2). Then, we simplified the model and studied it by Turing pattern analysis. The results suggest that VEGFC can form Turing patterns in the zebrafish embryo. However, further studies are needed before we can use this patterning mechanism to explain lymphangiogenesis. Our conclusions about VEGFC can be tested experimentally. The demonstrated pat- terning mechanisms are not specific to VEGFC, MMP2, and collagen I; they can pattern other potential regulators of lymphangiogenesis, such as Cxcl12a; they can even be used to explain events other than lymphangiogenesis. Due to evolutionary conservation, they can be generalised to vertebrates other than the zebrafish too. The two mathematical models are new tools which will help in further studies about lymphangiogenesis and other biological phenomena too. Contents Nomenclature xxi Declaration Of Authorship xxix Acknowledgements xxxi 1 Lymphatic System1 1.1 Structure....................................2 1.1.1 Primary Lymphatic Vessels......................2 1.1.2 Secondary Lymphatic Vessels.....................3 1.2 Functions....................................7 1.3 Chapter Summary...............................7 2 Lymphatic Development9 2.1 Mouse...................................... 10 2.1.1 Transdifferentiation of Venous Endothelial Cells into Lymphatic Endothelial Cell Progenitors...................... 10 2.1.2 Sprouting of Lymphatic Endothelial Cell Progenitors........ 12 2.1.3 Lymphatic Vascular Remodelling and Maturation......... 13 2.1.4 Separation from Blood Vessels.................... 13 2.2 Zebrafish.................................... 14 2.2.1 Formation of Blood Vessels...................... 15 2.2.2 Transdifferentiation.......................... 17 2.2.3 Sprouting of Lymphatic Endothelial Cell Progenitors........ 17 2.2.4 Formation of Intersegmental Veins.................. 17 2.2.5 Formation of Parachordal Lymphangioblast String......... 18 2.2.6 Migration from Parachordal Lymphangioblast String........ 18 2.2.7 Functional Lymphatic System..................... 18 2.2.8 Further Development.......................... 19 2.2.9 Differences from the Mouse...................... 19 2.3 Research Questions and Hypotheses..................... 19 2.4 Chapter Summary............................... 21 3 Interstitial Flow 23 3.1 Morphogenetic Functions........................... 23 3.2 Idealised Geometry............................... 25 3.3 Model Building................................. 28 3.4 Parametrisation................................. 30 v vi CONTENTS 3.5 Chapter Summary............................... 31 4 Coupled Phenomena 33 4.1 Extracellular Matrix Remodelling....................... 33 4.2 VEGFC Proteolytic Processing........................ 36 4.3 Model Building................................. 36 4.3.1 Reaction-Diffusion-Convection Equations.............. 36 4.3.2 Diffusion Terms............................. 37 4.3.3 Other Equations............................ 38 4.3.4 Reaction Terms............................. 39 4.3.5 Boundary and Initial Conditions................... 40 4.3.6 Specific Hydraulic Conductivity.................... 42 4.4 Parametrisation................................. 43 4.4.1 Transport Parameters......................... 43 4.4.2 Kinetic Parameters........................... 44 4.5 Nondimensionalisation............................. 46 4.5.1 Interstitial Flow Component..................... 50 4.5.2 Reaction-Diffusion-Convection Equation and its Simplified Forms. 51 4.5.3 Reaction Terms............................. 53 4.5.4 Model Features............................. 53 4.6 Simplification.................................. 55 4.6.1 Leaky Blood Vessels.......................... 55 4.6.2 Specific Hydraulic Conductivity.................... 56 4.6.3 Diffusion Term............................. 56 4.7 Chapter Summary............................... 58 5 Computer Simulations 59 5.1 COMSOL Multiphysics Settings........................ 59 5.2 Simulation Results............................... 61 5.2.1 Diffusion and Sequestration Act Together.............. 62 5.2.2 MMP2 Acts Uniformly when Diffusion Dominates......... 66 5.2.3 Convection and Asymmetry...................... 70 5.2.4 Channelisation............................. 74 5.2.5 Concentration Gradients Vanish Without Sequestration...... 77 5.3 Discussion.................................... 79 5.3.1 VEGFC................................. 79 5.3.2 Collagen I and MMP2......................... 80 5.3.3 MT1-MMP and TIMP2........................ 81 5.3.4 Answers to our Research Questions.................. 81 5.3.5 Generalisations and Extensions.................... 81 5.4 Chapter Summary............................... 82 6 Turing Patterns 85 6.1 Biological Pattern Formation......................... 85 6.1.1 Alan Turing and Lewis Wolpert.................... 86 6.2 Secondary System............................... 87 6.2.1 Differences from the Primary System................. 88 CONTENTS vii 6.2.2 Parametrisation and Nondimensionalisation............. 91 6.3 Turing Pattern Analysis............................ 93 6.3.1 Homogeneous Steady State...................... 93 6.3.2 Homogeneous Perturbation...................... 94 6.3.3 Heterogeneous Perturbation...................... 95 6.3.4 Dispersion Relation.......................... 97 6.3.5 Turing Space: Parametric Distributions............... 98 6.3.6 Turing Space: Dispersion Relations.................. 101 6.3.7 Turing Space: Bifurcation....................... 102 6.4 Computer Simulations............................. 108 6.4.1 COMSOL Multiphysics Settings................... 108 6.4.2 One Dimension............................. 110 6.4.3 Two Dimensions............................ 115 6.5 Discussion.................................... 118 6.5.1 Context................................. 118 6.5.2 Turing's Mechanism.......................... 118 6.5.3 VEGFC................................. 120 6.5.4 Generalisations............................. 120 6.5.5 Future Work.............................. 121 6.6 Chapter Summary............................... 122 7 Thesis Summary 123 7.1 Motivations................................... 123 7.2 Questions.................................... 123 7.3 Answers..................................... 124 7.3.1 Between 36 and 48 HPF: VEGFC.................. 124 7.3.2 Between 36 and 48 HPF: MMP2 and Collagen I.......... 124 7.3.3 After 120 HPF: VEGFC........................ 125 7.4 Implications................................... 126 7.4.1 Primary System............................ 126 7.4.2 Secondary System........................... 126 7.5 Extensions.................................... 127 7.5.1 VEGFC................................. 127 7.5.2 Primary System............................ 127 7.5.3 Secondary System........................... 128 7.5.4 Tertiary System............................ 128 7.6 Chapter Summary............................... 128 A Morphogens 131 B Extracellular Matrix Components 135 C Activation of VEGFC by Proteolytic Processing 137 D Primary System 141 D.1 Model Equations in the Interstitial Space Domain............. 141 D.2 Model Equations in the LEC Domain.................... 142 D.3 Boundary Conditions.............................. 143 viii CONTENTS D.4 Initial Conditions................................ 144 D.5 Scales and Parameters............................. 145 E Numerical Solution of Partial Differential Equations 149 E.1 Finite Difference Method........................... 150 E.2 Finite Element Method............................ 151 E.3 Finite Volume Method............................. 155 E.4 Spectral Method................................ 155 E.5 Consistency, Convergence, Stability...................... 157 F Secondary System 159 F.1 Model Equations................................ 159 F.2 Boundary Conditions.............................. 160 F.3 Initial Conditions................................ 160 F.4 Scales and Parameters............................. 161 G Python Programs 163 G.1 Turing Space Identification.........................