The aim of this book is to provide a well-structured and coherent overview of existing mathematical modeling approaches for biochemical reaction systems, investigating relations between both the conventional models and several types of deterministic-stochastic hybrid model recombinations. Another main objective is to illustrate and compare diverse numerical simulation schemes and their computational effort. Unlike related works, this book presents a broad scope in its applications, from offering a detailed introduction to hybrid approaches for the case of multiple population scales to discussing the setting of time-scale separation resulting from widely varying firing rates of reaction channels. Additionally, it also addresses modeling approaches for non well-mixed reaction-diffusion dynamics, including deterministic and stochastic PDEs and spatiotemporal master equations. Finally, by translating and incorporating complex theory to a level accessible to non-mathematicians, this book effectively bridges the gap between mathematical research in computational biology and its practical use in biological, biochemical, and biomedical systems.
Author(s): Stefanie Winkelmann, Christof Schutte
Series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials
Publisher: Springer
Year: 2021
Language: English
Pages: 283
City: Cham
Introduction
Contents
Abbreviations
List of Figures
1 Well-Mixed Stochastic Reaction Kinetics
1.1 The Chemical Reaction Network
1.2 The Reaction Jump Process
1.2.1 Path-Wise Representation in Terms of Poisson Processes
1.2.2 The Chemical Master Equation
Existence of Solutions of the CME
1.2.3 Moment Equations
1.3 Computation of Expectations
1.3.1 Direct Approach: Solving the CME
Linear Reaction Networks
Other Special Systems
Finite State Space
State Space Truncation
Galerkin Projection Methods
1.3.2 Indirect Approach: Stochastic Simulation
Modifications of the SSA
Approximation by τ-Leaping
Binomial τ-Leaping
Other Approaches to Accelerate SSA
1.4 Stochastic Versus Deterministic Modeling
2 Population Scaling
2.1 Uniform Scaling: From Stochastic Jumps to DeterministicDynamics
2.1.1 The Reaction Rate Equation
Scaling of Reaction Propensities
The Rescaled Process
The Limit Process
Approximation of First-Order Moments for Finite V
2.1.2 The Chemical Langevin Equation
Approximation of First- and Second-Order Moments for Finite V
2.1.3 Chemical Fokker-Planck and Liouville Equations
Evolution of Expectation Values
System Size Expansion
Path-Wise Simulation and Relation to τ-Leaping
2.2 Hybrid Models for Multiple Population Scales
2.2.1 Piecewise-Deterministic Reaction Processes
Convergence
Numerical Simulation of Piecewise-Deterministic Reaction Processes
2.2.2 Piecewise Chemical Langevin Equation
Numerical Simulation of Hybrid Diffusion Processes
2.2.3 Three-Level Hybrid Process
2.2.4 Hybrid Master Equation
Numerical Effort
2.3 Application: Autorepressive Genetic System
2.3.1 The Biochemical Model
Hybrid Master Equation
2.3.2 Simulation Results and Model Comparison
Sample Paths
Long-Term Averages and Protein Bursts
Temporal Evolution
2.3.3 Small Volume Failure
2.3.4 Three-Level Dynamics
2.4 Dynamic Partitioning
2.4.1 Joint Equation for Time-Dependent Partitioning
2.4.2 Rules for Partitioning
Numerical Simulation of the Adaptive Hybrid Process
Error Analysis
3 Temporal Scaling
3.1 Separation in Terms of Reaction Extents
3.1.1 Alternative Master Equation in Terms of ReactionExtents
3.1.2 Slow-Scale RME
3.1.3 Fast-Scale RME
3.1.4 Approximative Solutions
3.2 Separation in Terms of Observables
3.2.1 Multiscale Observables
3.2.2 Separation of Fast and Slow Scales
3.2.3 Nested Stochastic Simulation Algorithm
Relation to the Deterministic Quasi-Equilibrium Assumption
3.3 Combination with Population Scaling
4 Spatial Scaling
4.1 Particle-Based Reaction-Diffusion
4.1.1 Modeling Diffusive Motion
Interacting Forces and Crowding
Molecular Dynamics
4.1.2 Diffusion-Controlled Bimolecular Reactions
4.1.3 Mathematical Model for Multi-article Systems
Fock Space
Single-Particle and Two-Particle Operators
Two-Species Fock Space
Creation and Annihilation Operators
4.1.4 Particle-Based Stochastic Simulation
4.2 Reaction-Diffusion PDEs
4.2.1 Deterministic Reaction-Diffusion PDEs
RREs in Compartments
4.2.2 Stochastic Reaction-Diffusion PDEs
Formal Derivation
Solving SPDEs
CLEs in Compartments
4.3 Compartment CMEs
4.3.1 The Reaction-Diffusion Jump Process
4.3.2 Lattice-Based Discretization: RDME
The Microscopic Limit
Limit of Fast Diffusion/Relation to CME
Reaction-Diffusion in Crowded Media
Numerical Simulation and Complexity
4.3.3 Metastable Dynamics: ST-CME
Markov State Modeling
Spatial Coarsening
Estimation of Jump Rates λij
4.4 Hybrid Reaction-Diffusion Approaches
4.5 Application
Modeling for the RDME
Summary and Outlook
Mathematical Background
A.1 Markov Jump Processes
A.1.1 Setting the Scene
A.1.2 Infinitesimal Generator and Master Equation
Evolution Equations
Operator Notation
Stationary Distribution and Steady State
A.1.3 Jump Times, Embedded Markov Chain,and Simulation
A.1.4 Poisson Processes
A.2 Diffusion Process and Random Walk
A.2.1 Stochastic Differential Equations
A.2.2 Diffusion in a Bounded Domain
Discretization in Space
A.3 Convergence of Stochastic Processes
A.3.1 Skorokhod Space
Extension to DX[0,∞)
A.3.2 Types of Path-Wise Convergence
Convergence in Distribution
Convergence in Probability
Convergence Almost Surely
A.3.3 Large Population Limit
A.4 Hybrid Stochastic Processes
A.4.1 Piecewise-Deterministic Markov Processes
A.4.2 Convergence of the Partially Scaled Reaction Jump Process to a PDMP
A.4.3 Hybrid Diffusion Processes
A.5 Multiscale Asymptotics of the CME
A.5.1 Large Population Limit
Evolution of Expectation Values
Concentration of the Initial Distribution
Solution Formula for the Generalized Liouville Equation
A.5.2 Multiple Population Scales
Partial Population Scaling via Multiscale Expansion
Partial Population Scaling via Splitting Approach
A.5.3 Multiple Time Scales
Method of Multiple Time Scales
The Fast Process Is Geometrically Ergodic
The Fast Process Is Increasing But Bounded
INDEX
