Mathematical Modeling for Genes to Collective Cell Dynamics

Preface
Contents
1 Transcription Dynamics: Cellular Automaton Model of Polymerase Dynamics for Eukaryotes
	1.1 Brief Review of Transcription
	1.2 Experimental Result
	1.3 Cellular Automaton and Traffic Flow Model
		1.3.1 What Is Cellular Automaton?
		1.3.2 Traffic Flow Cellular Automaton
	1.4 Cellular Automaton Model of Transcription Dynamics
	References
2 Angiogenesis: Dynamics of Endothelial Cells in Sprouting and Bifurcation
	2.1 Angiogenesis: A Biological Overview
		2.1.1 Introduction
		2.1.2 Mechanisms Involved in Cell Migration
		2.1.3 Endothelial Cell Behaviors During Angiogenesis
		2.1.4 Cell-to-Cell Junction Proteins as a Modulator of Angiogenesis
		2.1.5 Clinical Significance of Angiogenesis
	2.2 Modeling by Differential and Difference Equations
		2.2.1 Continuous Model and Discrete Model
		2.2.2 Newtonian Equation of Motion
		2.2.3 Diffusion Equation
	2.3 Mathematical Model for the Dynamics of Endothelial Cells in Angiogenesis
		2.3.1 Introduction
		2.3.2 Discrete Model for EC Dynamics
		2.3.3 Cell-Mixing and Scaling Behavior of EC Dynamics
		2.3.4 Pattern Formation of the Model with Elongation and Bifurcation of Neogenetic Vessels
		2.3.5 A Continuous Model for Angiogenesis
		2.3.6 Exact Solutions of the Continuous Model in Case of Constant VEGF Concentration
		2.3.7 Numerical Simulations in the Presence of VEGF Concentration Gradient
		2.3.8 Concluding Remarks
	2.4 Two-Dimensional Pattern Formation with Ellipses
		2.4.1 Two-Dimensional Extension of the Model
		2.4.2 Pattern Formation and Order Parameter
	References
3 Synchronization and Fluctuation of Cardiac Muscle Cells
	3.1 Introduction
	3.2 The Stochastic Phase Models for the Cardiomyocyte Beating
		3.2.1 Some Preliminaries for the Stochastic Phase Model
			3.2.1.1 The Phase Model
			3.2.1.2 The Brownian Motion and White Noise
			3.2.1.3 Itô\'s Integral
			3.2.1.4 Itô\'s Formula
		3.2.2 The Phase Model for an Isolated Cardiomyocyte
		3.2.3 The Phase Model for Two Coupled Cardiomyocytes
		3.2.4 The CV of Synchronization
		3.2.5 The Phase Model for N-cardiomyocytes Network
	3.3 Experimental Approach
		3.3.1 On-Chip Cellomics Technology: Reconstructive Understanding of the Community Effect in Cardiomyocytes
		3.3.2 Photothermal Etching on Agarose Layer for Cell Network Formation Control
		3.3.3 Community Effect of Cells for Their Synchronization (1): Two-Cell Model
		3.3.4 Community Effect of Cells for Their Synchronization (2): Cell Number Dependence
		3.3.5 Community Effect of Cells on Their Synchronization (3): Mixture of Different Types of Cells
		3.3.6 Summary of Experimental Results
		3.3.7 Ability and Limitation of Constructed Experimental Approach
	3.4 Numerical Approach to Synchronization of Cardiomyocytes
		3.4.1 Comparison of the Mathematical Modeling with Experimental Results and Numerical Simulations
			3.4.1.1 Mathematical Modeling for Synchronization of Cardiomyocytes
			3.4.1.2 Numerical Simulation Method
			3.4.1.3 Comparison of the Model with Experimental Results of Two Cardiomyocytes
			3.4.1.4 Comparison with the Kuramoto Model
		3.4.2 Numerical Experiments
			3.4.2.1 Size and Configuration Dependence on Fluctuation of the System
			3.4.2.2 Dependence of Cell Properties and Numbers on Fluctuation of the System
		3.4.3 Discussion
	3.5 Summary
	References
4 Statistical Analysis of Cellular Directional Movement: Application for Research of Single Cell Movement
	4.1 Single Cell Movement
		4.1.1 Persistent Random Walk
		4.1.2 Mean Squared Displacement (MSD) and the Fürth Formula
	4.2 Circular Statistics
		4.2.1 Raw Data Plot
		4.2.2 Histograms
		4.2.3 Summary Statistics
		4.2.4 Probability Models
	4.3 Application for Single Cell Movement
	References
5 Protein Structures
	5.1 Introduction
	5.2 Lattice Polymer Models
		5.2.1 Self-avoiding Walk
		5.2.2 HP Model
		5.2.3 An Extension of HP Model: Including Coulomb Force
	5.3 Fatgraph Model
	References