This book discusses the mathematical simulation of biological systems, with a focus on the modeling of gene expression, gene regulatory networks and stem cell regeneration. The diffusion of morphogens is addressed by introducing various reaction-diffusion equations based on different hypotheses concerning the process of morphogen gradient formation. The robustness of steady-state gradients is also covered through boundary value problems.
The introduction gives an overview of the relevant biological concepts (cells, DNA, organism development) and provides the requisite mathematical preliminaries on continuous dynamics and stochastic modeling. A basic understanding of calculus is assumed.
The techniques described in this book encompass a wide range of mechanisms, from molecular behavior to population dynamics, and the inclusion of recent developments in the literature together with first-hand results make it an ideal reference for both new students and experienced researchers in the field of systems biology and applied mathematics.
Author(s): Jinzhi Lei
Series: Lecture Notes on Mathematical Modelling in the Life Sciences
Publisher: Springer
Year: 2021
Language: English
Pages: 320
City: Singapore
Preface
References
Contents
1 Biological Background—Information, Energy, and Matter
1.1 Introduction
1.2 Building Blocks of Life—Cells
1.2.1 Cells
1.2.2 Prokaryotic Cells
1.2.3 Eukaryotic Cells
1.3 Storage of Information—DNA Sequences
1.3.1 DNA and the Genome
1.3.2 Genes
1.4 Epigenetic Information
1.4.1 Nucleosome and Histone Modification
1.4.2 DNA Methylation
1.5 Expression of Genetic Information—The Central Dogma
1.5.1 DNA Replication and Genome Inheritance
1.5.2 Transcription and mRNA
1.5.3 Translation and the Genetic Code
1.6 Variation of the Information
1.7 Cellular Energy Exchangers—Mitochondria
1.8 Organism Development—From Genotype to Phenotype
1.9 Summary
References
2 Mathematical Preliminary–Continuous Dynamics
2.1 Introduction
2.2 Ordinary Differential Equations
2.2.1 First-Order Differential Equations
2.2.2 Second-Order Differential Equations
2.3 Delay Differential Equations
2.3.1 Delay Differential Equations
2.3.2 Stability Analysis of Delay Differential Equations
2.4 Stochastic Differential Equations
2.4.1 Stochastic Differential Equations and Stochastic Integral
2.4.2 Itô Formula
2.4.3 Fokker-Planck Equation
2.5 Reaction-Diffusion Equations
2.5.1 One-Dimensional Conservation Law
2.5.2 Various Forms of Flux
2.5.3 Random Walk and the Diffusion Equation
2.5.4 Initial and Boundary Conditions
2.5.5 High Dimensional Conservation Laws
2.6 Discrete Dynamical Systems
2.7 Multi-timescale Analysis
2.7.1 Quasi-Equilibrium Assumption
2.7.2 Perturbation Method
2.8 Michaelis-Menten Function and Hill Function
2.8.1 Michaelis-Menten Function
2.8.2 Hill Function
2.8.3 Hill-Type Function Transcription Rate
2.9 Summary
References
3 Mathematical Preliminary–Stochastic Modeling
3.1 Introduction
3.2 Biochemical Reaction Systems
3.3 Mathematical Formulations–Intrinsic Noise
3.3.1 Chemical Master Equation
3.3.2 Fokker-Planck Equation
3.3.3 Reaction Rate Equation
3.3.4 Chemical Langevin Equation
3.3.5 Discussions of the Chemical Langevin Equation
3.4 Mathematical Formulations–Fluctuations in Kinetic Parameters
3.4.1 Reaction Rate as a Random Process
3.4.2 Reaction Rate Equation
3.4.3 Chemical Langevin Equation
3.5 Stochastic Simulations
3.5.1 Stochastic Simulation Algorithm
3.5.2 Tau-Leaping Algorithm
3.5.3 Other Simulation Methods
3.6 Summary
References
4 Stochastic Modeling of Gene Expression
4.1 Introduction
4.2 Gene Expression in Prokaryotes
4.3 Stochasticity in Prokaryotic Gene Expression
4.3.1 Intrinsic Noise
4.3.2 Extrinsic Noise
4.4 Gene Expression in Eukaryotes
4.5 Noise in Eukaryotic Gene Expression
4.5.1 Stochastic Simulation
4.5.2 Analytic mRNA Distribution
4.6 Epigenetic Regulation of Transcription
4.6.1 Stochastic Modeling of Histone Modification and Transcriptional Regulation
4.6.2 Stochastic Modeling of DNA Methylation
4.7 A Unified Model of Gene Expression with Epigenetic Modification
4.8 Summary
References
5 Mathematical Models for Gene Regulatory Network Dynamics
5.1 Introduction
5.2 Gene Regulation
5.3 Positive Feedback and Bistability
5.3.1 Lac Operon
5.3.2 Mathematical Formulation
5.3.3 Steady State Analysis
5.4 Noise Perturbation and Cell State Switches
5.4.1 Cell State Switches in the Lac Operon
5.4.2 Lysis-Lysogeny Decision of Bacteriophage λ
5.4.3 Genetic Toggle Switch Induced by Intrinsic Noise
5.5 Negative Genetic Circuit and Oscillatory Dynamics
5.5.1 Atkinson Oscillator
5.5.2 Noise Resistance in Genetic Oscillators
5.5.3 Time-Delayed Negative Feedback
5.5.4 Circadian Rhythms
5.6 Summary
References
Dynamical Modeling of Stem Cell Regeneration
6.1 Introduction
6.2 The G0 Cell Cycle Model
6.2.1 Age-Structured Model
6.2.2 Delay Differential Equation Model
6.2.3 Formulation of the Proliferation Rate
6.2.4 Stability of the G0 Cell Cycle Model
6.2.5 Hematopoietic Stem Cell Regeneration
6.3 Heterogeneous Stem Cell Population Dynamics
6.3.1 Mathematical Formulation
6.3.2 Mathematical Model of Cell Lineages
6.3.3 A One-Dimensional Variable to Represent Transcriptome Heterogeneity
6.4 Mathematical Models of Dynamic Hematological Disease
6.4.1 Dynamic Hematological Disease
6.4.2 Mathematical Models of Hematopoiesis
6.4.3 Origins of Dynamic Hematological Diseases
6.4.4 Neutrophil Dynamics in Response to Chemotherapy and G-CSF
6.5 Summary
References
7 Mathematical Models of Morphogen Gradients and Growth Control
7.1 Introduction
7.2 Morphogen-Mediated Pattern Formation
7.3 Mathematical Models of Morphogen Gradient Formation
7.3.1 Ligand Diffusion in Extracellular Space
7.3.2 Self-enhanced Ligand Degradation
7.3.3 Non-receptor-Mediated Ligand Transport
7.3.4 Receptor-Mediated Transcytosis
7.3.5 Other Models
7.4 Robustness of Morphogen Gradients
7.4.1 Definition of Robustness
7.4.2 Robustness with Respect to Perturbations in Morphogen Production
7.5 Morphogen-Mediated Growth Control
7.6 Summary
References
Index
