MATHEMATICAL TOPICS IN POPULATION GENETICS

By C. HAZAO

CONTENTS PART I BIOLOGICAL NETWORKS AND GRAPH THEORY INTRODUCTION TO GRAPH THEORY DE?NITIONS AND EXAMPLES SPECTRAL GRAPH THEORY CENTRALITY MEASURES GEOMETRIC CENTRALITIES CLOSENESS PATH-BASED CENTRALITIES SPECTRAL MEASURES AXIOMS FOR CENTRALITY THE SIZE AXIOM THE DENSITY AXIOM THE SCORE-MONOTONICITY AXIOM BIOLOGICAL NETWORKS NETWORKS: THE REPRESENTATION OF A SYSTEM AT THE BASIS OF SYSTEMS BIOLOGY BIOCHEMICAL NETWORKS METABOLIC NETWORKS PROTEIN–PROTEIN INTERACTION NETWORKS GENETIC REGULATORY NETWORKS NEURAL NETWORKS PHYLOGENETIC NETWORKS SIGNALLING NETWORKS ECOLOGICAL NETWORKS CHALLENGES IN COMPUTATIONAL NETWORK BIOLOGY NETWORK INFERENCE FOR DRUG DISCOVERY HOW NETWORK BIOLOGY HELPS DRUG DISCOVERY COMPUTATIONAL METHODS CLASSI?ER-BASED METHODS REVERSE ENGINEERING METHODS INTEGRATING STATIC AND DYNAMIC DATA: A PROMISING VENUE PART II CALCULUS AND CHEMICAL REACTIONS AN INTRODUCTION TO DIFFERENTIAL AND INTEGRAL CALCULUS DERIVATIVE OF A REAL FUNCTION EXAMPLES OF DERIVATIVES GEOMETRIC INTERPRETATION OF THE DERIVATIVE THE ALGEBRA OF DERIVATIVES DE?NITION OF INTEGRAL RELATION BETWEEN INTEGRAL AND DERIVATIVE METHODS OF INTEGRATION INTEGRATION BY PARTS INTEGRATION BY SUBSTITUTION INTEGRATION BY PARTIAL FRACTION DECOMPOSITION THE REVERSE CHAIN RULE USING COMBINATIONS OF METHODS ORDINARY DIFFERENTIAL EQUATIONS FIRST-ORDER LINEAR EQUATIONS INITIAL VALUE PROBLEMS PARTIAL DIFFERENTIAL EQUATIONS DISCRETIZATION OF DIFFERENTIAL EQUATIONS THE IMPLICIT OR BACKWARD EULER METHOD THE RUNGE–KUTTA METHOD SYSTEMS OF DIFFERENTIAL EQUATIONS MODELLING CHEMICAL REACTIONS MODELLING IN SYSTEMS BIOLOGY THE DIFFERENT TYPES OF MATHEMATICAL MODELS CHEMICAL KINETICS: FROM DIAGRAMS TO MATHEMATICAL EQUATIONS KINETICS OF CHEMICAL REACTIONS THE LAW OF MASS ACTION EXAMPLE 1: THE LOTKA–VOLTERRA SYSTEM EXAMPLE 2: THE MICHAELIS–MENTEN REACTIONS CONSERVATION LAWS MARKOV PROCESSES THE MASTER EQUATION THE CHEMICAL MASTER EQUATION MOLECULAR APPROACH TO CHEMICAL KINETICS REACTIONS ARE COLLISIONS REACTION RATE ZEROTH-, FIRST-, AND SECOND-ORDER REACTIONS HIGHER-ORDER REACTIONS FUNDAMENTAL HYPOTHESIS OF STOCHASTIC CHEMICAL KINETICS THE REACTION PROBABILITY DENSITY FUNCTION THE STOCHASTIC SIMULATION ALGORITHMS DIRECT METHOD FIRST REACTION METHOD NEXT REACTION METHOD SPATIO-TEMPORAL SIMULATION ALGORITHMS ORDINARY DIFFERENTIAL EQUATION STOCHASTIC MODELS: THE LANGEVIN EQUATION HYBRID ALGORITHMS REACTION–DIFFUSION SYSTEMS THE PHYSICS OF REACTION–DIFFUSION SYSTEMS DIFFUSION OF NON-CHARGED MOLECULES INTRINSIC VISCOSITY AND FRICTIONAL COEF?CIENT CALCULATED SECOND VIRIAL COEF?CIENT ALGORITHM AND DATA STRUCTURES DRUG RELEASE THE HIGUCHI MODEL SYSTEMS WITH DIFFERENT GEOMETRIES THE POWER-LAW MODEL WHAT DRUG DISSOLUTION IS THE DIFFUSION LAYER MODEL (NOYES AND WHITNEY) THE WEIBULL FUNCTION IN DISSOLUTION INHOMOGENEOUS CONDITIONS DRUG DISSOLUTION IS A STOCHASTIC PROCESS THE INTER-FACIAL BARRIER MODEL COMPARTMENTAL MODEL PART III LINEAR ALGEBRA AND MODELLING LINEAR ALGEBRA BACKGROUND MATRICES INTRODUCTION SPECIAL MATRICES OPERATION ON MATRICES TRANSPOSITION AND SYMMETRIES LINEAR SYSTEMS SPECIAL LINEAR SYSTEMS GENERAL LINEAR SYSTEMS THE GAUSSIAN ELIMINATION METHOD GAUSSIAN ELIMINATION FOR RECTANGULAR SYSTEMS CONSISTENCY OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS NONHOMOGENEOUS LINEAR SYSTEMS LEAST-SQUARES PROBLEMS PERMUTATIONS AND DETERMINANTS EIGENVALUE PROBLEMS COMPUTING THE EIGENVALUES AND THE EIGENVECTORS REGRESSION REGRESSION AS A GEOMETRIC PROBLEM STANDARD ERROR ON REGRESSION COEF?CIENTS REGRESSION VIA MAXIMUM-LIKELIHOOD ESTIMATION REGRESSION DIAGNOSTIC HOW TO ASSESS THE GOODNESS OF THE MODEL OTHER TYPES OF REGRESSION CASE STUDY 1: REGRESSION ANALYSIS OF SWEAT SECRETION VOLUMES IN CYSTIC FIBROSIS PATIENTS THE EXPERIMENTS THE MULTILINEAR MODEL RESULTS NONLINEAR REGRESSION CASE STUDY 2: INFERENCE OF KINETIC RATE CONSTANTS PARAMETER SPACE RESTRICTION VARIANCE OF THE ESTIMATED PARAMETERS CARDIAC ELECTROPHYSIOLOGY THE BIDOMAIN MODEL ADAPTIVE ALGORITHMS ITERATIVE METHODS FOR LINEAR SYSTEMS KRYLOV SUBSPACE METHODS PARALLEL IMPLEMENTATION REFERENCES