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Mathematics in Biology

by Meister, Lee, Portugues

| ISBN: 9780262049405 | Copyright 2025

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Overview

A concise but rigorous textbook for advanced undergraduate and graduate students across the biological sciences that provides a foundation for understanding the methods used in quantitative biology. 

Biology has turned into a quantitative science. The core problems in the life sciences today involve complex systems that require mathematical expression, yet most biologists are untrained in this dimension of the discipline. Bridging that gap, this practical textbook equips students to integrate advanced mathematical concepts with their biological education. Mathematics in Biology covers three broad subjects—linear algebra, probability and statistics, and dynamical systems—each treated at three levels: basic principles, advanced topics, and applications. Motivations and examples are drawn from diverse areas of study, while end-of-chapter exercises encourage creative applications. Based on nearly two decades of teaching at Harvard and Caltech, this rigorous but concise text provides an essential foundation for understanding the methods used in quantitative biology. 

•Proven in the classroom
•Suitable for advanced undergraduate and graduate students across the biological sciences
•Offers accompanying online materials including code and solved exercises

Expand/Collapse All
Cover (pg. Cover)
Contents (pg. v)
Introduction (pg. 1)
Notation (pg. 3)
Prerequisites (pg. 5)
1. Elements of Calculus (pg. 7)
1.1 Elementary Functions (pg. 7)
1.2 Differentiation (pg. 8)
1.3 Integration (pg. 13)
1.4 Differential Equations (pg. 15)
1.5 Multiple Variables (pg. 16)
1.6 Complex Numbers (pg. 19)
1.7 The Delta Function (pg. 21)
I. Linear Systems (pg. 23)
2. Basics of Linear Algebra (pg. 25)
2.1 Motivation (pg. 25)
2.2 Vector Space (pg. 28)
2.3 Basis Sets (pg. 29)
2.4 Linear Operators (pg. 31)
2.5 Matrix Algebra (pg. 36)
2.6 Change of Basis (pg. 40)
2.7 Scalar Product (pg. 43)
2.8 Special Matrix Properties (pg. 46)
2.9 Eigenvalues and Eigenvectors (pg. 46)
2.10 The Characteristic Equation (pg. 47)
2.11 Diagonalizing a Matrix (pg. 50)
3. Linear Systems (pg. 55)
3.1 Linear Systems Analysis (pg. 55)
3.2 Fourier Transforms (pg. 60)
4. Applications of Linear Systems (pg. 75)
4.1 Microscopy (pg. 75)
4.2 Crystal Analysis (pg. 82)
4.3 X-ray Scattering (pg. 86)
4.4 Detecting Periodicity (pg. 87)
4.5 Filtering (pg. 91)
4.6 Sampling (pg. 96)
4.7 Optimal Estimation (pg. 99)
5. Exercises (pg. 107)
II. Probability and Statistics (pg. 115)
6. Basics of Probability and Statistics (pg. 117)
6.1 Motivation (pg. 117)
6.2 Events and Probabilities (pg. 119)
6.3 Discrete Random Variables (pg. 121)
6.4 Continuous Random Variables (pg. 129)
6.5 Multiple Random Variables (pg. 135)
6.6 The Central Limit Theorem (pg. 142)
7. Inference and Statistical Testing (pg. 145)
7.1 Maximum Likelihood Estimation (pg. 145)
7.2 Bayesian Estimation (pg. 150)
7.3 Hypothesis Testing (pg. 152)
7.4 The z-test (pg. 153)
7.5 The t-test (pg. 156)
7.6 Goodness of Fit to a Distribution (pg. 160)
7.7 Nonparametric Tests (pg. 164)
7.8 Other Statistical Tests (pg. 165)
7.9 Linear Regression (pg. 165)
7.10 Bootstrapping (pg. 171)
8. Advanced Topics in Probability and Statistics (pg. 175)
8.1 Random Walks and Diffusion (pg. 175)
8.2 Random Time Series (pg. 186)
8.3 Hidden Markov Models (pg. 192)
8.4 Point Processes (pg. 196)
8.5 Dimensionality Reduction (pg. 201)
8.6 Information Theory (pg. 212)
9. Applications of Probability and Statistics (pg. 221)
9.1 Luria-Delbrück Revisited (pg. 221)
9.2 Signal Processing (pg. 226)
9.3 Population and Quantitative Genetics (pg. 230)
9.4 Vision at the Quantum Limit (pg. 236)
9.5 Neural Coding (pg. 238)
10. Exercises (pg. 245)
III. Nonlinear Dynamics (pg. 255)
11. Basics of Dynamical Systems (pg. 257)
11.1 Motivation (pg. 257)
11.2 What Is a Dynamical System? (pg. 257)
11.3 Flows, Fixed Points, and Bifurcations (pg. 261)
11.4 Dynamics in Two Dimensions (pg. 266)
11.5 Bifurcation Analysis of 2D Systems (pg. 273)
12. Advanced Topics in Nonlinear Dynamics (pg. 291)
12.1 Dynamics in Three or More Dimensions (pg. 291)
12.2 Chaos (pg. 291)
12.3 The Turing Model of Morphogenesis (pg. 292)
13. Applications of Nonlinear Dynamics (pg. 299)
13.1 Repressilator (pg. 299)
13.2 Fold Change Detection (pg. 303)
13.3 Bistability (pg. 306)
13.4 Turing Patterns (pg. 312)
13.5 Circadian Rhythms (pg. 314)
13.6 How Do Neurons Communicate? (pg. 318)
14. Exercises (pg. 325)
14.1 Further Exercises (pg. 327)
References (pg. 329)
Index (pg. 335)
Cover (pg. Cover)
Contents (pg. v)
Introduction (pg. 1)
Notation (pg. 3)
Prerequisites (pg. 5)
1. Elements of Calculus (pg. 7)
1.1 Elementary Functions (pg. 7)
1.2 Differentiation (pg. 8)
1.3 Integration (pg. 13)
1.4 Differential Equations (pg. 15)
1.5 Multiple Variables (pg. 16)
1.6 Complex Numbers (pg. 19)
1.7 The Delta Function (pg. 21)
I. Linear Systems (pg. 23)
2. Basics of Linear Algebra (pg. 25)
2.1 Motivation (pg. 25)
2.2 Vector Space (pg. 28)
2.3 Basis Sets (pg. 29)
2.4 Linear Operators (pg. 31)
2.5 Matrix Algebra (pg. 36)
2.6 Change of Basis (pg. 40)
2.7 Scalar Product (pg. 43)
2.8 Special Matrix Properties (pg. 46)
2.9 Eigenvalues and Eigenvectors (pg. 46)
2.10 The Characteristic Equation (pg. 47)
2.11 Diagonalizing a Matrix (pg. 50)
3. Linear Systems (pg. 55)
3.1 Linear Systems Analysis (pg. 55)
3.2 Fourier Transforms (pg. 60)
4. Applications of Linear Systems (pg. 75)
4.1 Microscopy (pg. 75)
4.2 Crystal Analysis (pg. 82)
4.3 X-ray Scattering (pg. 86)
4.4 Detecting Periodicity (pg. 87)
4.5 Filtering (pg. 91)
4.6 Sampling (pg. 96)
4.7 Optimal Estimation (pg. 99)
5. Exercises (pg. 107)
II. Probability and Statistics (pg. 115)
6. Basics of Probability and Statistics (pg. 117)
6.1 Motivation (pg. 117)
6.2 Events and Probabilities (pg. 119)
6.3 Discrete Random Variables (pg. 121)
6.4 Continuous Random Variables (pg. 129)
6.5 Multiple Random Variables (pg. 135)
6.6 The Central Limit Theorem (pg. 142)
7. Inference and Statistical Testing (pg. 145)
7.1 Maximum Likelihood Estimation (pg. 145)
7.2 Bayesian Estimation (pg. 150)
7.3 Hypothesis Testing (pg. 152)
7.4 The z-test (pg. 153)
7.5 The t-test (pg. 156)
7.6 Goodness of Fit to a Distribution (pg. 160)
7.7 Nonparametric Tests (pg. 164)
7.8 Other Statistical Tests (pg. 165)
7.9 Linear Regression (pg. 165)
7.10 Bootstrapping (pg. 171)
8. Advanced Topics in Probability and Statistics (pg. 175)
8.1 Random Walks and Diffusion (pg. 175)
8.2 Random Time Series (pg. 186)
8.3 Hidden Markov Models (pg. 192)
8.4 Point Processes (pg. 196)
8.5 Dimensionality Reduction (pg. 201)
8.6 Information Theory (pg. 212)
9. Applications of Probability and Statistics (pg. 221)
9.1 Luria-Delbrück Revisited (pg. 221)
9.2 Signal Processing (pg. 226)
9.3 Population and Quantitative Genetics (pg. 230)
9.4 Vision at the Quantum Limit (pg. 236)
9.5 Neural Coding (pg. 238)
10. Exercises (pg. 245)
III. Nonlinear Dynamics (pg. 255)
11. Basics of Dynamical Systems (pg. 257)
11.1 Motivation (pg. 257)
11.2 What Is a Dynamical System? (pg. 257)
11.3 Flows, Fixed Points, and Bifurcations (pg. 261)
11.4 Dynamics in Two Dimensions (pg. 266)
11.5 Bifurcation Analysis of 2D Systems (pg. 273)
12. Advanced Topics in Nonlinear Dynamics (pg. 291)
12.1 Dynamics in Three or More Dimensions (pg. 291)
12.2 Chaos (pg. 291)
12.3 The Turing Model of Morphogenesis (pg. 292)
13. Applications of Nonlinear Dynamics (pg. 299)
13.1 Repressilator (pg. 299)
13.2 Fold Change Detection (pg. 303)
13.3 Bistability (pg. 306)
13.4 Turing Patterns (pg. 312)
13.5 Circadian Rhythms (pg. 314)
13.6 How Do Neurons Communicate? (pg. 318)
14. Exercises (pg. 325)
14.1 Further Exercises (pg. 327)
References (pg. 329)
Index (pg. 335)

Markus Meister

Markus Meister is Anne P. and Benjamin F. Biaggini Professor of Biological Sciences at Caltech.

Kyu Hyun Lee

Kyu Hyun Lee is a postdoctoral fellow in neurobiology at the University of California, San Francisco.

Ruben Portugues

Ruben Portugues is Assistant Professor in the Institute of Neuroscience at the Technical University of Munich.
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