Stochastic Chemical Reaction Systems in Biology

Preface
Acknowledgments
Contents
Acronyms
Chapter 1 Introduction
1.1 Why and What is a Mathematical Science
1.2 Mechanistic Models Constitute Understanding
1.2.1 The Role of Natural Laws and Scientific Hypotheses
1.2.2 Dynamics and Causality
1.3 What Do We Know About Biological Cells?
1.3.1 Basics of Chemical Reactions
1.3.2 Cells: The Proteins, DNA and RNA Within
1.3.3 The Central Dogma of Molecular Biology
1.3.4 Cellular Regulation
1.3.5 Cell Biology As a System of Biochemical Reactions
1.3.6 Chemical Reactions in Terms of Molecular Processes and Beyond
Part I Essentials of Deterministic and Stochastic Chemical Kinetics
Chapter 2 Kinetic Rate Equations and the Law of Mass Action
2.1 Reaction Kinetic Equation: Conservation of Molecules and the Law of Mass Action
2.2 Forward and Backward Reaction Fluxes J+ and J-, Chemical Potential and Boltzmann’s Law
2.3 General Chemical Kinetic System in a Continuously Stirred Vessel
2.4 The Principle of Detailed Balance for Chemical Reaction
2.5 Equilibrium and Kinetics of Closed Chemical Reaction Systems
2.5.1 An Example
2.5.2 Shear’s Theorem for General Mass-action Kinetics
2.5.3 Complex Balanced Chemical Reaction Networks
2.5.3.1 Lyapunov Function for Complex Balanced Kinetics
2.5.3.2 Null Space of a Stoichiometric Matrix
2.5.4 Equilibrium Gibbs Function for General Rate Laws
2.6 Nonlinear Chemical Kinetics: Bistability and the Limit Cycle
2.6.1 Schlögl System and Chemical Bistability
2.6.2 The Schnakenberg System and Chemical Oscillation
Chapter 3 Probability Distribution and Stochastic Processes
3.1 Basics of Probability Theory
3.1.1 Probability Space and Random Variables
3.1.2 Probability Distribution Functions— Mass and Density
3.1.3 Expected Value, Variance, and Moments
3.1.4 Function of An RV and Change of Variables
3.1.4.1 Cdf and Pdf of a Function of An RV
3.1.4.2 Expected Value of a Function of An RV
3.1.5 Popular Discrete and Continuous RVs and Their Distributions
3.1.6 A Pair of RVs and Their Independence
3.1.7 Functions of Two RVs
X
Y:
Z
X,Y), W=
X,Y)
Z
W:
Z
X,Y), W= X,
3.2 Discrete-time, Discrete-state Markov Chains
3.2.1 Transition Probability Matrix
3.2.2 Invariant Probability Distribution
3.3 Continuous-time, Discrete-state Markov Processes
3.3.1 Poisson Process
3.3.1.1 Waiting-time Distribution
3.3.1.2 Uniform Distribution and Poisson Process
3.3.2 Continuous-time, Discrete-state Markov Process
3.3.3 Exact Algorithm For Generating a Markov Trajectory
3.3.4 Mean First Passage Time
3.3.5 Stochastic Trajectory In Terms of Poisson Processes
3.3.6 The Birth-death Process
3.3.6.1 The Pure Birth Process
3.3.6.2 The Birth-death Process
3.3.6.3 Birth-death Process and Chemical Reactions
3.4 Continuous-time, Real-valued Diffusion Process
3.5 The Theory of Entropy and Entropy Production
Chapter 4 Large Deviations and Kramers’ rate formula
4.1 The Law of Large Numbers and the Central Limit Theorem
4.1.1 Statistical Estimation of Expectation and Variance
4.1.2 Central Limit Theorem For Independent and Identically Distributed RVs
4.2 Laplace’s Method for Integrals
4.3 Legendre–Fenchel Transformation and LDT of sample mean
4.4 LDT of Empirical Distribution and Contraction
4.5 Large Deviations for Stationary Diffusion on a Circle and λ-Surgery
4.5.1 Caricature of a Turning Point: An Example
4.5.2 Saddle-node Bifurcation of Diffusion On a Circle
4.6 Kramers’ Rate Formula
Chapter 5 The Probabilistic Basis of Chemical Kinetics
5.1 Probability and Stochastic Processes — A New Language for Population Dynamics
5.2 Radioactive Decay and Exponential Time
5.2.1 Basic Formulation
5.2.2 Memoryless Property
5.2.3 From Binomial Distribution to Poisson Distribution
5.2.4 Minimum of Independent Exponential Distributed Random Variables
5.3 Known Mechanisms that Yield Memoryless Time
5.3.1 Minimal Time of a Set of Nonexponential I.I.D. Random Times
5.3.2 Khinchin’s Theorem
5.3.3 Kramers’ Theory and Saddle-crossing As a Rare Event
5.4 Stochastic Population Growth
5.5 Theory of Chemical and Biochemical Reaction Systems
5.5.1 Delbr¨üuck–Gillespie Process (DGP)
5.5.2 Integral Representations With Random Time Change
5.5.2.1 Poisson Process
5.5.2.2 Random Time-Changed Poisson Representation
5.5.3 Single-population System
Part II From Chemical Kinematics to Mesoscopic and Macroscopic Thermodynamics
Chapter 6 Mesoscopic Thermodynamics of Markov Processes
6.1 Entropy Balance Equation
6.2 Detailed Balance and Reversibility
6.2.1 Free Energy
6.3 Free Energy Balance Equation
6.4 First and Second Mesoscopic Thermodynamic Laws of Markov Processes
6.4.1 Application to Population Kinetic Processes
Chapter 7 Emergent Macroscopic Chemical
Thermodynamics
7.1 Kurtz’s Law of Large Numbers, WKB Ansatz and Hamilton–Jacobi Equation
7.1.1 Stationary Solution to the HJE As a Lyapunov Function
7.1.2 Mass-action Kinetics With Complex Balance
7.2 Macroscopic Nonequilibrium Thermodynamic Formalism
7.3 Kinetics with Detailed Balance and Chemical Equilibrium
7.3.1 Law of Mass Action and Ideal Solution
7.4 Mathematical Theory of the Macroscopic NET
7.4.1 Kurtz’s Theorem
7.4.2 Large-Deviation Principle
7.4.3 Diffusion Approximation
7.4.4 Macroscopic Limits
7.5 Fluctuation-Dissipation Theorem
Chapter 8 Phase Transition and Mesoscopic Nonlinear Bistability
8.1 A Canonical Description in Terms of a Double-Well φss(x)
8.2 Nonlinear Stochastic Dynamics and Landau’s Theory
8.2.1 Ordinary Differential Equation With Bistability
8.2.2 Stochastic Differential Equation With Phase Transition
8.3 Symmetry Breaking, Nonlinear Catastrophe and Phase Transition
8.4 Hierarchical Organization: Different Levels and Time Scales
8.4.1 Symmetry Breaking As the Underpinning of Stochastic Population Kinetics
Part III Stochastic Kinetics of Biochemical Systems and Processes
Chapter 9 Classic Enzyme Kinetics — The Michaelis–Menten and Briggs–Haldane Theories
9.1 Enzyme: Protein as a Catalyst
9.2 Product Formation Rate and Double-Reciprocal Relation
9.3 Michaelis–Menten (MM) Theory
9.3.1 Michaelis–Menten Kinetic Equation
9.3.2 Singular Perturbation Problem: Examples
9.3.3 Singular Perturbation Theory: Outer and Inner Solutions and Their Matching
9.3.4 MM kinetics, Saturation and Bimolecular Reactions
9.4 Briggs–Haldane Reversible Enzymatic Reaction
9.5 Allosteric Cooperativity
9.5.1 Combinatorics of Identical and Independent Subunits
9.5.2 Mathematical Definition of Molecular Cooperativity
Chapter 10 Single-Molecule Enzymology and Driven Biochemical Kinetics with Chemostat
10.1 Single-Molecule Irreversible Michaelis–Menten Enzyme Kinetics
10.1.1 Product Arrival As a Renewal Process
10.1.2 Mean Waiting Time and Steady-state Flux
10.2 Single-molecule Reversible Michaelis–Menten Enzyme Kinetics
10.2.1 Mean Waiting Cycle Times
10.2.2 Stepping Probabilities
10.2.3 Haldane Equality and Its Generalization
10.2.4 Fluctuation Theorems
10.3 Modifier Activation-inhibition Switching in Enzyme Kinetics
10.3.1 Case Study of a Simple Example
10.3.2 Modifier With a More General Mechanism— An In-depth Study
10.4 Fluctuating Enzymes and Dynamic Cooperativity
10.4.1 A Simple Model for Fluctuating Enzymes
10.4.2 Mathematical Method For Analyzing Dynamic Cooperativity
10.5 Kinetic Proofreading and Specificity Amplification
10.5.1 Minimal Error Rate Predicted By the Hopfield–Ninio Model
10.5.2 Absolute Thermodynamic Limit on the Error Rate With Finite Available Free Energy.
Chapter 11 Stochastic Linear Reaction Kinetic Systems
11.1 Stochastic Two-state Unimolecular Reaction
11.1.1 Approaching a Stationary Distribution
11.1.2 Identical, Independent Molecules With Unimolecular Reactions
11.1.3 Autocorrelation and Autocovariance Functions
11.1.4 Statistical Analysis of Stochastic Trajectories
11.2 A Three-state Unimolecular Reaction
11.2.1 Diffusion Approximation
11.3 Stochastic Theory for Closed Unimolecular Reaction Networks
11.4 Linear Reaction Systems with Open Boundaries
11.4.1 A Simple Chemical System With Synthesis and Degradation
11.4.1.1 Expected Value and Variance
11.4.1.2 Time-dependent Solution to (11.45)
11.4.2 K-state Open Linear Reaction System
11.4.3 Correlation Functions of Open Systems
11.4.4 An Example
11.5 Grand Canonical Markov Model
11.5.1 Equilibrium Grand Canonical Ensemble
11.5.2 Open Boundaries, Potential Function and Reaction Conductance
Chapter 12 Nonlinear Stochastic Reaction Systems with Simple Examples
12.1 Bimolecular Nonlinear Reaction
12.2 One-Dimensional Birth-Death Processes of a Single Species
12.2.1 Landscape For Stochastic Population Kinetics
12.2.2 More Discussions On Diffusion Approximation
12.3 Bimolecular association-dissociation reaction
12.4 Simple Bifurcations in One-Dimensional Nonlinear Dynamics
12.4.1 Saddle-node Bifurcation
12.4.2 Transcritical Bifurcation
12.4.3 Pitch-fork Bifurcation
12.5 Keizer’s Paradox and Time Scale Separation
12.5.1 Mean Time to Extinction
12.6 Schlögl’s Model and Nonlinear Bistability
12.7 General Theory of Nonlinear Reaction Systems
12.7.1 Deterministic Chemical Kinetics
12.7.2 Chemical Master Equation (CME)
12.7.3 Integral Representation With Random Time Change
12.8 Stochastic Bistability
Chapter 13 Kinetics of the Central Dogma of Molecular Cell Biology
13.1 The Simplest Model of the Central Dogma
13.2 Translational Burst
13.3 Transcriptional Burst and a Two-state Model
13.4 Self-Regulating Genes in a Cell: the Simplest Model
13.5 Fluctuating-Rate Model and Landscape Function of Self-Regulating Genes
13.5.1 Kinetics In the Intermediate Regime
13.5.2 Global Landscapes and Phenotypic Switching
13.5.3 Transition Rate Formula For Phenotypic Switching
Chapter 14 Stochastic Macromolecular Mechanics and Mechanochemistry
14.1 Mechanics and Chemistry: Two Complementary Representations
14.2 Potential of Mean Force
14.3 Chemical Model of Molecular Motors
14.3.1 Chemical Model of Single Molecular Motors
14.3.2 Chemomechanical Energy Transduction
14.3.3 Efficiency of a Molecular Motor
14.4 Polymerization Motors
14.4.1 Polymerization With Nucleation
14.4.2 Polymerization Against a Force
14.5 Energy, Force and Stochastic Macromolecular Mechanics
14.5.1 Chemical Transformation As Motion In a Potential Force
14.5.2 Single Receptor-ligand Dissociation Under Force
Part IV Epilogue: Beyond Chemical Reaction
Kinetics
Chapter 15 Nonequilibrium Landscape Theory, Attractor-State Switching, and Autonomous Differentiation
15.1 Emergent Global Landscape for Nonequilibrium Steady State
15.1.1 The Emergent Global Nonequilibrium Landscape
15.2 Bifurcation and the Far-from-Equilibrium State of a System
15.2.1 From Near-equilibrium to Far-from-equilibrium Through Bifurcation
15.2.2 Thermodynamic vs. Kinetic Branches of Nonequilibrium States of Matter
15.2.3 The Biochemistry of Energy-driven Bifurcation
15.3 Physics of Complexity Meets Waddington
15.3.1 Self-organization and Differentiation
15.4 Irreversible Living Process on a Landscape
Chapter 16 Nonlinear Stochastic Dynamics of Complex Individuals and Systems: New Paradigm and Syntheses
16.1 Fundamental Theorem of Natural Selection and Price’s Equation
16.1.1 The Mathematical Ideas of the FTNS and LDT
16.1.2 The FTNS and Price’s Equation
16.1.3 Fundamental Mechanics of Natural Selection
16.2 Classical Nonequilibrium Thermodynamics (NET)
16.2.1 Macroscopic NET of Continuous Media
16.2.2 Mesoscopic NET With Fluctuating Variables
16.2.3 Stochastic Liouville Dynamics
16.3 The Theory of Complex Systems Biology
16.4 The Theory of Replicator Dynamics
16.5 Outlooks
References
Index