Mathematics for the Life Sciences

909 Pages·2014·22.92 MB·English

by  Erin N. Bodine

Mathematics for the Life Sciences
Mathematics
for the Life Sciences

Erin N. Bodine
Suzanne Lenhart
Louis J. Gross
PRINCETON UNIVERSITY PRESS Princeton and Oxford
Copyright © 2014 by Princeton University Press Published by Princeton University Press, 41 William
Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire, OX20 1TW
press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Bodine,
Erin N.
Mathematics for the life sciences / Erin N. Bodine, Suzanne Lenhart, Louis J. Gross.
pages cm
Summary: “The life sciences deal with a vast array of problems at different spatial, temporal, and
organizational scales. The mathematics necessary to describe, model, and analyze these problems is
similarly diverse, incorporating quantitative techniques that are rarely taught in standard undergraduate
courses. This textbook provides an accessible introduction to these critical mathematical concepts, linking
them to biological observation and theory while also presenting the computational tools needed to address
problems not readily investigated using mathematics alone. Proven in the classroom and requiring only a
background in high school math, Mathematics for the Life Sciences doesn’t just focus on calculus as do
most other textbooks on the subject. It covers deterministic methods and those that incorporate uncertainty,
problems in discrete and continuous time, probability, graphing and data analysis, matrix modeling,
difference equations, differential equations, and much more. The book uses Matlab throughout, explaining
how to use it, write code, and connect models to data in examples chosen from across the life sciences. The
text provides undergraduate life science students with a succinct overview of major mathematical concepts
that are essential for modern biology, covers all the major quantitative concepts that national reports have
identified as the ideal components of an entry-level course for life science students, and provides good
background for the MCAT, which now includes data-based and statistical reasoning. The book explicitly
links data and math modeling, includes end-of-chapter homework problems, end-of-unit student projects,
select answers to homework problems, and provides an online supplement with Matlab m-files and an R
supplement. It prepares students to read with comprehension the growing quantitative literature across the
life sciences, gives an online answer key, solution guide, and illustration package (available to
professors)”—Provided by publisher.
Includes bibliographical references and index.
ISBN 978-0-691-15072-7 (hardback) 1. Biology—Mathematical models. 2. Mathematics. I. Lenhart,
Suzanne. II. Gross, Louis J. III. Title.
QH323.5.B63 2014
570.1’51—dc23
2014006493
British Library Cataloging-in-Publication Data is available This book has been composed in
Printed on acid-free paper. ∞
Printed in the United States of America 10 9 8 76 5 4 3 2 1
Dedication
This book is dedicated to Bryan and Vellie Barnes, Peter and Phillip Andreae,
and Marilyn Kallet and Heather Gross.
Contents
Preface
Acknowledgments
UNIT 1 Descriptive Statistics
CHAPTER 1 Basic Descriptive Statistics
1.1 Types of Biological Data
1.2 Summary of Descriptive Statistics of Data Sets
1.3 Matlab Skills
1.4 Exercises
CHAPTER 2 Visual Display of Data
2.1 Introduction
2.2 Frequency Distributions
2.3 Bar Charts and Histograms
2.4 Scatter Plots
2.5 Matlab Skills
2.6 Exercises
CHAPTER 3 Bivariate Data and Linear Regression
3.1 Introduction to Linear Regression
3.2 Bivariate Data
3.3 Linear Analysis of Data
3.4 Correlation
3.5 Matlab Skills
3.6 Exercises
CHAPTER 4 Exponential and Logarithmic Functions
4.1 Exponential and Logarithmic Functions in Biology
4.2 Review of Exponential and Logarithm Properties
4.3 Allometry
4.4 Rescaling Data: Log-Log and Semilog Graphs
4.5 Matlab Skills
4.6 Exercises
UNIT 1 Student Projects
UNIT 2 Discrete Time Modeling
CHAPTER 5 Sequences and Discrete Difference Equations
5.1 Sequences
5.2 Limit of a Sequence
5.3 Discrete Difference Equations
5.4 Geometric and Arithmetic Sequences
5.5 Linear Difference Equation with Constant Coefficients
5.6 Introduction to Pharmacokinetics
5.7 Matlab Skills
5.8 Exercises
CHAPTER 6 Vectors and Matrices
6.1 Vector Structure: Order Matters!
6.2 Vector Algebra
6.3 Dynamics: Vectors Changing over Time
6.4 Matlab Skills
6.5 Exercises
CHAPTER 7 Matrix Algebra
7.1 Matrix Arithmetic
7.2 Applications
7.3 Matlab Skills
7.4 Exercises
CHAPTER 8 Long-Term Dynamics or Equilibrium
8.1 Notion of an Equilibrium
8.2 Eigenvectors
8.3 Stability
8.4 Matlab Skills
8.5 Exercises
CHAPTER 9 Leslie Matrix Models and Eigenvalues
9.1 Leslie Matrix Models
9.2 Long-Term Growth Rate (Eigenvalues)
9.3 Long-Term Population Structure (Corresponding Eigenvectors)
9.4 Matlab Skills
9.5 Exercises
UNIT 2 Student Projects
UNIT 3 Probability
CHAPTER 10 Probability of Events
10.1 Sample Spaces and Events
10.2 Probability of an Event
10.3 Combinations and Permutations
10.4 Binomial Experiments
10.5 Matlab Skills
10.6 Exercises
CHAPTER 11 Probability of Compound Events
11.1 Compound Events
11.2 Finding the Probability of a Compound Event
11.3 Probability Viewed as Darts Tossed at a Dart Board
11.4 Matlab Skills
11.5 Exercises
CHAPTER 12 Conditional Probability
12.1 Conditional Probability
12.2 Independence
12.3 Matlab Skills
12.4 Exercises
CHAPTER 13 Sequential Events
13.1 Partition Theorem
13.2 Bayes’ Theorem
13.3 Exercises
CHAPTER 14 Population Genetics Models
14.1 Hardy-Weinberg Equilibrium
14.2 Hardy-Weinberg Selection Model
14.3 Exercises
UNIT 3 Student Projects
UNIT 4 Limits and Continuity
CHAPTER 15 Limits of Functions
15.1 Limit of a Function
15.2 Limit Properties
15.3 Matlab Skills
15.4 Exercises
CHAPTER 16 Limits of Continuous Functions
16.1 Right and Left Limits
16.2 Continuity
16.3 Intermediate Value Theorem
16.4 Matlab Skills
16.5 Exercises
UNIT 4 Student Projects
UNIT 5 Derivatives
CHAPTER 17 Rates of Change
17.1 Average Rate of Change
17.2 Estimating Rates of Change for Data
17.3 Velocity
17.4 Photosynthesis
17.5 Other Examples of Rates of Change
17.6 Definition of a Derivative at a Point
17.7 Matlab Skills
17.8 Exercises
CHAPTER 18 Derivatives of Functions
18.1 Concept of a Derivative
18.2 Limit Definition of a Derivative of a Function
18.3 Derivatives of Exponential Functions
18.4 Derivatives of Trigonometric Functions
18.5 Derivatives and Continuity
18.6 Derivatives of Logarithmic Functions
18.7 Matlab Skills
18.8 Exercises
CHAPTER 19 Computing Derivatives
19.1 Derivatives of Frequently Used Functions
19.2 The Chain Rule for the Composition of Functions
19.3 Quotient and Reciprocal Rules
19.4 Exponential Models
19.5 Higher Derivatives
19.6 Exercises
CHAPTER 20 Using Derivatives to Find Maxima and Minima
20.1 Maxima and Minima
20.2 First Derivative Test
20.3 Mean Value Theorem
20.4 Concavity
20.5 Optimization Problems
20.6 Matlab Skills
20.7 Exercises
UNIT 5 Student Projects
UNIT 6 Integration
CHAPTER 21 Estimating the Area under a Curve
21.1 The Area under a Curve
21.2 Increasing the Accuracy of the Area Estimation
21.3 Area below the Horizontal Axis
21.4 Matlab Skills
21.5 Exercises
CHAPTER 22 Antiderivatives and the Fundamental Theorem of Calculus
22.1 Definition of an Integral
22.2 Antiderivatives
22.3 Fundamental Theorem of Calculus
22.4 Antiderivatives and Integrals
22.5 Average Values
22.6 Matlab Skills
22.7 Exercises
CHAPTER 23 Methods of Integration
23.1 Substitution Method
23.2 Integration by Parts