This book helps the reader make use of the mathematical models of biological phenomena starting from the basics of programming and computer simulation. Computer simulations based on a mathematical model enable us to find a novel biological mechanism and predict an unknown biological phenomenon. Mathematical biology could further expand the progress of modern life sciences. Although many biologists are interested in mathematical biology, they do not have experience in mathematics and computer science. An educational course that combines biology, mathematics, and computer science is very rare to date. Published books for mathematical biology usually explain the theories of established mathematical models, but they do not provide a practical explanation for how to solve the differential equations included in the models, or to establish such a model that fits with a phenomenon of interest.
MATLAB is an ideal programming platform for the beginners of computer science. This book starts from the very basics about how to write a programming code for MATLAB (or Octave), explains how to solve ordinary and partial differential equations, and how to apply mathematical models to various biological phenomena such as diabetes, infectious diseases, and heartbeats. Some of them are original models, newly developed for this book. Because MATLAB codes are embedded and explained throughout the book, it will be easy to catch up with the text. In the final chapter, the book focuses on the mathematical model of the proneural wave, a phenomenon that guarantees the sequential differentiation of neurons in the brain. This model was published as a paper from the author’s lab (Sato et al., PNAS 113, E5153, 2016), and was intensively explained in the book chapter “Notch Signaling in Embryology and Cancer”, published by Springer in 2020.
This book provides the reader who has a biological background with invaluable opportunities to learn and practice mathematical biology.
Author(s): Makoto Sato
Series: Theoretical Biology
Publisher: Springer
Year: 2023
Language: English
Pages: 210
City: Singapore
Preface
Contents
Chapter 1: Preparation
1.1 What Is Programming?
1.2 Installation
1.3 Trying to Use MATLAB as a Functional Calculator
1.3.1 Calculating in the Command Window
1.3.2 Calculating Functions
1.3.3 Calculating Inverse Functions
1.4 Variables
1.4.1 What Are Variables?
1.4.2 Names of Variables
1.5 Vectors
1.5.1 What Are Vectors?
1.5.2 Manipulating Elements of Vectors
1.6 Drawing a Graph Using a Vector
1.6.1 Using plot Function
1.6.2 Applications of plot Function
1.7 Matrices
1.7.1 Defining a Matrix
1.7.2 The Size of a Matrix
1.8 Calculating Matrices
1.8.1 Sum and Product of Matrices
1.8.2 Hadamard Product
Chapter 2: Introduction to MATLAB Programming
2.1 Scripting
2.1.1 Creating a Script
2.1.2 Creating a Script to Draw a Graph
2.2 Circles and Spirals
2.2.1 Using Polar Coordinates to Draw Circles and Spirals
2.3 for Statement
2.3.1 Creating a Loop with for Statement
2.3.2 Nesting for Statements
2.4 Converting Numbers to Integers
2.5 Characterization of Numbers
2.6 Mixed Number and Character Sentences
2.7 while Statement
2.8 Rock-Paper-Scissors and if/switch Statements
2.8.1 Conditional Judgment by if Statement
2.8.2 Conditional Judgment by switch Statement
2.9 Functions
2.9.1 Functions for Judging Rock-Paper-Scissors
2.9.2 Placing a Function in a Script
2.10 Handling 2D Images
2.10.1 Image Display with imagesc
2.10.2 Creating 2D Patterns
2.11 Animation
2.11.1 Animation of plot
2.11.2 Animation of imagesc
2.11.3 Saving a Movie File
2.12 Gradient Rings
2.12.1 Drawing a Gradient Ring
2.12.2 Rotating a Gradient Ring
Chapter 3: Simulating Time Variations in Life Phenomena
3.1 Synthesis and Degradation of Proteins
3.1.1 Euler Method for Solving Differential Equations
3.1.2 Comparison with Exact Solutions
3.2 Mathematical Model of Hematopoietic Stem Cells
3.2.1 Mathematical Model of Hematopoietic Stem Cells: Part 1
3.2.2 Mathematical Model of Hematopoietic Stem Cells: Part 2
3.2.3 Logistic Equation
3.3 Mathematical Model of Glucose Metabolism
3.4 Mathematical Model of Infectious Disease
3.4.1 SIR Model
3.4.2 Comparing the Results of Multiple Parameters
3.4.3 Introducing the Basic Reproduction Number, R0
3.5 Lotka-Volterra Model
3.5.1 Numerical Calculation Using the Euler Method
3.5.2 Numerical Calculation Using ode45
Chapter 4: Simulating Temporal and Spatial Changes in Biological Phenomena
4.1 Time Variation of Spatial Pattern Formation by Diffusive Materials
4.2 Intuitive Explanation of Diffusion Equation
4.3 Calculation of the Diffusion Equation
4.3.1 Numerical Algorithm for Diffusion
4.3.2 Calculating Diffusion Using for Statement
4.3.3 Calculating Diffusion Using del2
4.3.4 Fast Calculation of Diffusion Using Matrices
4.4 Pattern Formation by Morphogens
4.5 Pattern Formation in Fly Wing
4.5.1 Calculating the Diffusion of Dpp
4.5.2 Expression of Target Genes of Dpp
4.6 Pattern Formation in Fly Leg
4.6.1 Regulation of Dll expression by Dpp and Wg
4.6.2 Effects of Activated Dpp Receptors
4.7 Turing Model
4.7.1 Reaction-Diffusion Equation Consisting of Activators and Inhibitors
4.7.2 Setting an Upper Bound on Concentration
4.7.3 Setting an Upper Bound on Concentration Without Using if Statement
4.7.4 Turing Pattern
4.8 Simulation of Heartbeats
4.8.1 FitzHugh-Nagumo Equation
4.8.2 Adding Diffusion to FitzHugh-Nagumo Equation
4.8.3 Numerical Simulation of Arrhythmia
4.8.4 Mechanism of Heartbeat
4.9 Mathematical Model of the Wave of Differentiation
4.9.1 The Proneural Wave
4.9.2 Two-Variable Model Consisting of EGF and AS-C
4.9.3 A Four-Variable Model with Notch and Delta
4.9.4 Termination of the Wave in EGF Mutants
4.9.5 Mechanism of Wave Acceleration in Notch Mutant
4.9.6 Why not a Salt-and-Pepper Pattern?
Untitled
Postscript
MATLAB Commands and Functions
References
Index
