A Mathematical Introduction to Biology: Understanding Biological Systems with Computational Models
Description
This book is perfect for a one-semester course called "Introduction to Mathematical Biology," which is a 4-credit winter course. It provides a well-organized and complete learning experience, making it great for both students and teachers in mathematical biology. Discover the dynamic intricacies of metabolic, signaling, and genetic network motifs with "A Mathematical Introduction to Biology: Understanding Biological Systems with Computational Models." This book elegantly intertwines mathematics and life sciences to reveal the roles of feedback and feedforward patterns embedded within network structures, uncovering the mechanisms that govern living systems. Customised for both students and instructors, it offers an engaging and accessible approach to bridging the gap between mathematics and biology.
Spanning 19 comprehensive chapters, this book covers essential topics such as constructing differential equation models based on the principles of mass-action kinetics and Michaelis-Menten kinetics. It delves deeply into the feedback and feedforward mechanisms that form the backbone of signaling, metabolic, and gene regulatory network motifs. By applying dynamical systems theory to adaptation models, MAPK signaling pathways, the cell division cycle, and circadian rhythms, this resource becomes indispensable for understanding biological complexity.
Problems, examples, and downloadable instructor resources—including programmable MATLAB codes and lecture slide decks for each chapter—make it an ideal companion for both independent study and classroom instruction. This book aims to the undergraduate and postgraduate students who are studying computational or mathematical biology courses and professionals working in the biotechnology and pharmaceutical industries who may find the foundational knowledge useful.
Table of Contents
Preface About the Book About the Author Acknowledgment Introduction to Cell Biology and Biochemistry Introduction Signalling Pathways in a Cell Time Scales in Cellular Signalling DNA, Promoters, Enhancers, and Silencers Transcriptional Unit in a Cell Translational Unit in a Cell Post-Translation Modification Metabolic Pathways Homoeostasis and Adaptation Motifs Modules ODEs are the Most Preferred Way of Modelling Biological Systems Biochemical Modelling and Dynamical Systems Theory Noise in Biological Systems Role of Mathematical Models in Biology Problems of Modelling Biological Systems Broad Steps to Follow to Model the Biological Systems Basic Requirements to Carry Out Mathematical Modelling in Biology Summary References 1 Basic Calculus and Algebra 1.1 Introduction 1.2 Types of Function 1.2.1 Power Function 1.2.2 Linear Function 1.2.3 Parabolic Function 1.2.4 Cubic Function 1.3 Exponential Function 1.3.1 Logarithm 1.3.2 Polynomial Function, Maxima, Minima, and the Inflexion Point 1.3.3 Extremum: Maxima, Minima, and an Inflection Point 1.3.4 Extrema of the Ratio of Polynomials 1.4 Rational Functions 1.4.1 Vertical and Horizontal Asymptotes 1.4.2 Hyperbolic and Sigmoidal Function 1.5 Zeros of a Function 1.5.1 Quadratic 1.5.2 Higher Order: Cubic Equation 1.6 Composite Functions 1.7 Important Integrals Summary Practice Zone References 2 Mass Action Kinetics, Species Conservation Relationships, Steady State, and Equilibrium Approximations 2.1 Mass Action Kinetics 2.2 Mass Action Kinetics 2.2.1 Examples 2.3 Species Conservation Relationship 2.4 Stoichiometric Matrix 2.4.1 Brief Concepts from Linear Algebra to Identify Species Conservation Relationship 2.4.2 Linearly Dependent and Independent Vectors in a Matrix 2.4.3 Echelon and Row-Reduced Echelon Form 2.4.4 Rank of a Matrix 2.4.5 Gaussian Elimination 2.5 Species Conservation Relationship from the Gaussian Elimination 2.5.1 Conservation of Enzyme in MM Kinetics 2.6 Time Scales and Approximations 2.6.1 Steady-State Approximation (SSA) 2.7 Rapid-Equilibrium Approximation 2.7.1 Quasi-Steady-State Approximation (QSSA) of the Above Equation 2.8 Characteristic Time Scales Summary Practice Zone References 3 Time-Scale Analysis of Enzyme Kinetics: Quasi-Steady-State, Rapid-Equilibrium, and Total Quasi-Steady-State Approximations 3.1 Enzyme Kinetics 3.2 Disparate Time Scales in ODE Models 3.3 Enzyme Kinetics and the Quasi-Steady-State Approximation (QSSA): Example of Time-Scale Separation 3.3.1 Assumptions 3.4 Validity of QSSA Over the Parameter Range: Time Scales in MM Kinetics 3.5 Total Quasi-Steady-State Approximation (tQSSA): When the Enzyme Exceeds the Substrate 3.6 Rapid-Equilibrium Approximation: Steps Involved in the Derivation of Enzyme Kinetic Equation 3.6.1 Rapid Approximation of Competitive Enzyme–Inhibition Dynamics 3.7 Allostery and Cooperativity 3.8 Modelling Cooperativity and Sigmoidal Dynamics: Hill’s Equation 3.8.1 Parameters Affecting Threshold and Switching Response in Hill’s Equation 3.9 Receptor–Ligand Interactions 3.9.1 kon, koff , and KD Values Summary Practice Zone References 4 Linear and Nonlinear First-Order Differential Equations 4.1 Introduction 4.1.1 Linear First-Order Differential Equation 4.1.2 The Steady State Is dx/dt = 0 4.1.3 Linear ODE with a Constant Input p 4.1.4 Linear, Non-Autonomous ODE: Time-Dependent Input S(t) 4.2 Example: S(t) = e−t 4.3 Nonlinear First-Order Differential Equations 4.3.1 Ricatti’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.2 Bernoulli’s ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.4 Quantitative Analysis of 1D Linear and Nonlinear Autonomous Differential Equations 4.5 Linear Stability Analysis 4.5.1 Example 1 4.5.2 Example 2 4.6 Qualitative Analysis of 1D-ODE: Slope Fields Summary Practice Zone References 5 Rate-Balance Plot, Threshold, and Ultrasensitivity 5.1 Introduction 5.2 Rate-Balance Plot of Phosphorylation–Dephosphorylation Reaction with Mass Action Kinetics 5.3 Derivation of Goldbeter–Koshland Function for the PdP Reaction with Enzyme Kinetics 5.3.1 Rate-Balance Plot of GK Function: Zero-Order Ultrasensitivity 5.4 Quantifying Ultrasensitivity 5.4.1 Three-Tiered Signalling Cascade: Ultrasensitivity in Michaelis–Menten’s or Hill’s Equation Type of Models 5.5 Ultrasensitivity by Molecular Titration 5.5.1 Example: Molecular Titration in Mitotic Trigger: Entry from Interphase to Mitotic Phase 5.6 Example: Rate-Balance Plot for Reciprocal Regulation in Cell Cycle: Robustness of Mitotic Trigger Summary Practice Zone References 6 Second-Order Linear and Nonlinear Differential Equations 6.1 Second-Order Linear Differential Equations 6.2 System of Linear Second-Order Differential Equation in a Matrix Form 6.3 Behaviour of the Solutions 6.3.1 Example 1: Decaying Solution–Stable Node 6.3.2 Plotting the Eigenvectors and the Trajectories for Different Initial Conditions 6.3.3 Example 2: Growing Solution–Unstable Node 6.3.4 Example 3: Eigenvalues have Opposite Sign: Growing–Saddle 6.3.5 Example 4: Eigenvalues are Complex and Oscillating 6.4 Trace–Determinant Method to Determine the Stability and the Dynamics of the Equilibrium Point 6.5 Mapping the Parameter Dynamics Based on the Trace–Determinant Equations 6.5.1 Example 1 6.5.2 Example 2 6.6 Nonlinear 2D-ODEs 6.7 Hyperbolic Equilibrium Point and Hartman–Grobman Theorem 6.8 Linearisation of Nonlinear ODE 6.8.1 Example 1 6.8.2 Example 2 6.9 Higher-Dimensional ODEs 6.9.1 Routh–Hurwitz Criteria 6.9.2 Determination of Stability from the Hurwitz Matrix 6.9.3 Example 1 Summary Practice Zone References 7 Phase-Plane Analysis of Two-Dimensional Linear and Nonlinear Differential Equations 7.1 Phase Plane, Flow, Trajectories, and Portraits 7.2 Nullclines 7.3 General Points to Construct the Nullclines 7.4 Examples of Dynamics Through Phase-Plane Construction 7.4.1 Example 1: Stable Node 7.4.2 Example 2: Saddle 7.5 Coupled Linear ODEs 7.5.1 Example a = 2; b = 1; c = −1; d = 2 7.6 Phase-Plane Analysis of the 2D-Nonlinear ODEs 7.6.1 Example 1: Lotka–Volterra Model 7.6.2 Example 2: Bistability Summary Practice Zone References 8 Scaling and Dimensionless Analysis of Differential Equations 8.1 Introduction 8.1.1 General Idea About Scaling 8.2 Examples 8.2.1 Dimensionless Logistic Equation 8.2.2 Variations of the Logistic Equation 8.3 Scaling and Dimensionless Analysis of Michaelis–Menten Equation 8.3.1 Short and Long Characteristic Time Scales of t0 8.4 Example-1: Dimensionless Analysis of Enzymatic Nonlinear Product Formation with Degradation 8.5 Example-2: Dimensionless Analysis of Substrate Inhibition by an Enzyme 8.5.1 Scaling and Dimensionless Analysis 8.6 Scaling and Dimensionless Analysis Problems from Gene Regulatory Networks 8.6.1 Scaling and Dimensionless Analysis of 2D-Model of GRN Summary Practice Zone References 9 Bifurcation, Dynamical Analysis of Nonlinear Differential Equations, and Chemical Reaction Network Theory 9.1 Introduction 9.2 Bifurcation of the 1D Nonlinear ODE System 9.2.1 Transcritical Bifurcation 9.2.2 1D Directional Field Plot of yy 9.2.3 Bifurcation Diagram of the Transcritical Bifurcation with Etot as Bifurcation Parameter 9.2.4 Determination of Steady-State Stability 9.2.5 Conditions for a Nonlinear ODE Model to Exhibit Transcritical Bifurcation 9.3 Saddle–Node Bifurcation of 1D Nonlinear Model 9.4 Conditions for Saddle–Node Bifurcation 9.5 Determination of the Steady State and Its Stability 9.5.1 Conditions for Schlögl Model to Exhibit Saddle–Node Bifurcation 9.5.2 Information from the Saddle–Node Bifurcation Diagram 9.6 Pitchfork Bifurcation 9.7 Chemical Reaction Network Theory (CRNT) 9.7.1 Structural Properties of Chemical Reaction Networks 9.8 Deficiency One and Advance Deficiency Algorithms: Use of the CRNT Toolbox 9.8.1 Example 1: Deficiency Zero Network Using CRNT Toolbox 9.8.2 Example 2: Deficiency One Network Summary Practice Zone References 10 Oscillations in Cell Biology 10.1 Introduction to Biological Oscillations 10.2 Limit Cycle Oscillations and Types 10.3 Conditions for Oscillations 10.4 Hopf Bifurcation 10.5 Poincare–Bendixson Theorem and Other Methods 10.6 Example: Illustration of Conditions for Selkov’s Two-Variable Oscillatory Models 10.7 Higgins’ Multiple Template Circuits to Generate Oscillations in 2D-ODE 10.7.1 Example: Derivation of the Rate Equation Using Rapid-Equilibrium Approximation for One Specific Template Motif Backward Activation 10.7.2 Example 1: Regulation of Glycolytic Oscillatory Motif by Backward Activation 10.8 Three-Variable Model for Oscillations: Goodwin Model, Secant, and Routh–Hurwitz Conditions 10.9 Secant Condition 10.10 Routh–Hurwitz Condition and Linear Stability Analysis 10.11 Hopf Bifurcation Through Numerical Methods: Oscillations via Supercritical HB 10.12 Oscillations via Subcritical Hopf Bifurcation 10.13 Role of Positive Feedback in the Negative Feedback Loop in Sustained Oscillations: Fine-Tuning Frequency with a Near-Constant Amplitude Summary Practice Zone References 11 Network Motifs: Modelling of Positive and Negative Feedback Loops, and Identifying Their Dynamics 11.1 Introduction 11.2 Feedback Loops: Important Recurrent Motifs in Biological Networks 11.3 Converting Abstract Biological Circuits into a Meaningful Mechanism 11.3.1 PdP Reaction with MM Kinetics 11.4 Template of Combining Two Case I Circuits Appropriately to Generate Perfect Adaptive Response 11.5 Template of Combining Case I Linear Circuit with Nonlinear Case II PdP Circuits to Generate Bistability 11.6 Coupling Bistable System to Activator–Inhibitor and Substrate Depletion Systems: Generation of Oscillations 11.7 Template for Gene Regulatory Networks: Hill’s Equation 11.7.1 Reducing 2D to 1D ODE: Composite Function for Rate-Balance Plot 11.8 Example: Building a Minimal Model of G1-S Phase of Cell Using the Hill’s and MM Equations as Templates Summary Practice Zone References 12 Feedforward Loops: Templates for Modelling Biological Systems and Identifying Their Dynamics 12.1 Introduction 12.2 Threshold and Induction Time 12.2.1 Adding an Intermediate Step in the Conversion 12.2.2 Threshold from an Autocatalytic Reaction 12.3 Feedforward Loops 12.4 Modelling Feedforward Loops 12.4.1 C1-FFL–AND Logic 12.4.2 C1-FFL–OR Logic 12.5 Incoherent Feedback Loop and its Properties 12.5.1 Biphasic Dynamics in Incoherent FFLs 12.6 Example of the Role of Coherent and Incoherent Feedforward Motifs in Cell Cycle Regulation 12.6.1 Feedforward Loops in Cell Cycle 12.6.2 Mathematical Model of Eukaryotic Cell Cycle with Only Feedforward Loops 12.6.3 Dynamics of Cell Cycle with Only Feedforward Motifs 12.7 Long Signalling Cascade with Feedforward and Feedback Motifs 12.7.1 Case I: Positive Feedforward Cascade 12.7.2 Case II: Long Positive Feedback Loop Motif: A Permanent Response 12.7.3 Case III: Negative Feedforward Cascade and Positive Feedback Loop Motif Summary Practice Zone References 13 Signalling Motifs and Networks: Templates for Modelling and Identifying the Dynamics 13.1 Introduction 13.2 Signal Amplification in the Three-Tiered Mono-Phosphorylation– Dephosphorylation Cascade 13.3 Modelling the Dynamics of Transient Responses in the Signalling Pathway 13.4 Increased Output Sensitivity and Conversion of Graded Inputs to Switch-Like Outputs in the Linear Signalling Cascade 13.5 Ultrasensitivity in the Cascade Reaction: Huang–Ferrell Model of Mitogen- Activated Protein Kinase (MAPK) Signalling Pathway 13.5.1 Comparison of MAPK Signalling Output Response with Hill’s Equation for Ultrasensitivity 13.6 Bistability and Oscillations in Three-Tiered and One-Tiered MAPK Pathway 13.6.1 Markevich et al.’s Bistable Model of Single-Tiered Dual PdP Reactions 13.7 Death Receptor–Mediated Apoptosis 13.7.1 Law of Mass Action Kinetics and Mathematical Model for the Apoptosis Network Summary Practice Zone References 14 Signalling Motifs in Adaptation, Homoeostasis, and Two-Component Systems 14.1 Adaptation in Bacterial Chemotaxis 14.2 Molecular Mechanism of Perfect Adaptation 14.2.1 BL Model of Chemotaxis 14.2.2 Two-State Barkai–Leibler Model for Chemotaxis 14.3 Examples of Perfect and Near-Perfect Adaptation 14.4 Sensitivity and Precision Characterise Different Types of Adaptive Response 14.5 A Simple Example of Robust, Perfect Adaptation 14.6 Negative Feedback Circuit without a Buffer: Near-Perfect Adaptation 14.7 Negative Feedback Circuit with a Buffer: Robust, Perfect Adaptation 14.8 State-Dependent Activation and Inactivation for Perfect Adaptation 14.9 Integral Feedback Control as a Mechanism for Perfect Adaptation 14.10 Dynamics of Bacterial Two-Component System 14.10.1 Igoshin’s Model for Bistability in EnvZ/OmpR System Summary Practice Zone References 15 Circadian Oscillations 15.1 Circadian Oscillations 15.2 Terminologies 15.2.1 Free Running Period and Endogenous Oscillations 15.2.2 What Is a Circadian Time (CT)? 15.2.3 What Is a Zeitgeber Time (ZT)? 15.2.4 What Is an Entrainment? 15.2.5 What Is a Phase–Response Curve (PRC)? 15.2.6 What Characteristic Features a Circadian Mathematical Model Should Capture? 15.3 Circadian Models: General Ideas 15.4 Mathematical Model of the Circadian Rhythms of Neurospora crassa 15.5 Mathematical Model of Mammalian Circadian Rhythms: KF Model for Circadian Rhythms 15.6 Circadian Model for Cyanobacteria: A Case of Autocatalysis 15.6.1 Chemical Steps in Positive Autocatalysis Reaction 15.6.2 Two-Parameter Bifurcation Diagram in the k3 − k6 Plane 15.7 Temperature Compensation 15.7.1 Temperature Compensation in Brusselator, the Chemical Oscillator 15.8 Example of Temperature Compensation in Circadian Rhythms of Neurospora crassa 15.8.1 FRQ Stability and Temperature Compensation 15.8.2 Temperature Compensation in FRQ Mutants 15.8.3 Loss of Temperature Compensation in the Mutants Summary Practice Zone References 16 Cell Division Cycle 16.1 Introduction 16.2 Molecular Controls in Cell Cycle: Cyclin-Dependent Kinases (CDKs) and Cyclins 16.2.1 Three Modes Broadly Control Cyclins 16.3 Mitotic Oscillations 16.4 Model-1: Norel and Agur’s Model of Embryonic Cell Cycle Oscillations 16.5 Model-2: Goldbeter Model of Embryonic Cell Cycle Oscillations 16.6 Model-3: Tyson’s Model of for MPF Oscillations 16.7 General Principles of Cell Division Cycle and Dynamical Systems Theory 16.7.1 Detour: Writing a GK Function 16.8 A Two-Variable Toy Model of Eukaryotic Cell Division Cycle 16.9 Adding Controls-1: Transition from S-G2-M → G1 Phases – Role of APC Auxiliary Protein Cdc20 in Allowing G1 Control and Exit from Mitosis 16.10 Adding Controls-2: Transition from G1 → S-G2-M Phases – Role of Starter Kinase (SK) and Cyclin Kinase Inhibitor 16.11 Can the Novak and Tyson Model Undergo Oscillations without a Checkpoint? 16.12 Activator–Inhibitor Models of Cell Cycle Checkpoint Dynamics 16.12.1 SIMM: Substrate-Inhibitor Multiply Modified 16.13 SIMM∗ MOTIF 16.14 Sic as Inhibitor Checkpoint and CycB as Activator: G1-S Checkpoint Engagement Summary Practice Zone References 17 Probability 17.1 Introduction 17.1.1 Examples 17.2 Density and Distribution Function 17.3 Probability Distributions 17.3.1 Probability Mass Function 17.3.2 Bernoulli and Binomial Discrete Distributions 17.3.3 Expectation and Variance of a Discrete Distribution 17.3.4 Geometric Distribution 17.3.5 Expectation and Variance of a Geometric Distribution 17.4 Poisson Distribution 17.4.1 Mean and Variance of the Poisson Distribution 17.5 Exponential Distribution and Inter-Arrival Time 17.5.1 Expectation and Variance of the Exponential Distribution 17.5.2 Example: Waiting Time Between a Chemical Reaction 17.6 Uniform Distribution 17.7 Gaussian from Poisson Distribution: Both Distributions Have the Same Mean and Variance ???? 17.7.1 Properties of a Gaussian/Normal Distribution 17.7.2 Other Important Properties of a Normal Random Variable 17.7.3 Property 1: The Sum of Two Independent Normal Random Variables Is Again a Normal Random Variable 17.7.4 Property 2: The Sum of More Than Two Mutually Independent Normal Random Variables Is a Normal Random Variable 17.7.5 Property 3: Linear Transformation of a Normal Random Variable 17.8 Numerical Simulation of Random Variables from Distributions 17.8.1 Generation of Bernoulli’s Random Variable 17.9 Generation of Exponential Random Variable from Exponential Distribution 17.10 Simulation of a Poisson Random Variable Summary Practice Zone References 18 Stochastic Simulations 18.1 Introduction 18.2 Internal and External Noise 18.3 Some Basics 18.4 Chemical Master Equation 18.5 Stochastic Simulation Algorithm: Gillespie’s Method 18.5.1 Software Codes 18.6 Example 1: Protein Production and Loss 18.6.1 Example 2: Stochastic Simulation of Schlögl Model: Bistability 18.6.2 Example 3: Triggering of Oscillations by Noise 18.7 Chemical Langevin Equation 18.8 Modeling the Stochastic Dynamics of MM and Hill’s Equation of Gene Regulatory Networks 18.8.1 Stochastic Simulation of MM Kinetics: Quasi-Steady State Approximation 18.9 Example: Molecular or Internal Noise in Circadian Oscillations Summary Practice Zone References Index |