A Primer on Mathematical Models in Biology

This attractive introduction to mathematical biology arose from an unintended collaboration. Segel died while the book was in progress and his former student Edelstein-Keshet picked up his work, shaped, edited, re-arranged and added a good deal of material. Segel’s intention had been to write a text for master’s level students in biology using material he had collected and taught over several years at the Weizmann Institute. Edelstein-Keshet does a marvelous job of integrating his material and crafting a cohesive introductory textbook.

Segel wrote “My goal was not to teach biologists useful mathematics (although this occurs) but rather to show biologists by example how mathematical modeling can deepen their understanding of biology.” Although the book was developed originally for graduate students in biology, it would also appeal to mathematics students as long as they know or are willing to learn a little biology and chemistry. Key for students of either variety is seeing how modeling can lead to new insights in biology.

Following an introductory chapter on modeling, Chapter 2 jumps right into biochemical kinetics. Although the book includes a few other topics in mathematical biology, the core topics are molecular level processes (biochemistry with enzyme kinetics), excitable systems (largely about signal propagation in nerves), and biochemical circuits.

Chapters 3 through 7 make up a short course in basic ordinary differential equations and phase plane analysis. Especially valuable is a chapter that develops a model for the spread of infectious disease “from the ground up”. This is a well-known model, but the purpose here is to illustrate and exercise the tools developed in previous chapters: deriving the model, applying dimensional analysis, identifying the steady state, analyzing the qualitative behavior of solutions, running a simulation, doing stability and bifurcation analyses, and interpreting the results.

Chapters 8–11 include the core of Segel’s original planned textbook. The first two chapters discuss enzyme-mediated biochemical kinetics. A central idea here is to use a quasi steady state approximation that avoids analytical difficulties and offers more direct insight into the dynamics of the system. The next two chapters focus on neurophysiology, and in particular on the Hodgkin-Huxley and FitzHugh-Nagumo models. The treatment of the Hodgkin-Huxley equation is particularly good. It carefully steps through the development of the chemical and electrical aspects of the model, pulls all the pieces together and explains its importance in launching both the study of neurophysiology and the discovery of ion selective channels. Having read this, I feel that I can go back and read the original paper with new insight.

Chapter 12 pulls together many of the idea and techniques from the previous chapters as it addresses biochemical networks and biochemical “circuits”. A highlight here — a capstone really — is a pretty detailed model of the cell division cycle.

The book comes with a lot of good exercises and project suggestions, a collection of simulation programs (in XPP, but easily translated into MATLAB or something similar), a short review of basic electricity, and an excellent bibliography.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.