*Fully integrated book offers refreshingly modern explanations of the biochemical and biological sides of things*

The book herein reviewed, the second edition of *Mathematical Physiology I: Cellular Physiology*, is an interdisciplinary text. The authors state in the preface that their major goals were to expand the discussion of many models and principles from the first edition, and to provide pointers to recent works in as many areas as possible. They have succeeded remarkably well in the first goal and did as well as they could given space constraints on the second. The really informative reading is in the preface to the first edition, however.

Keener and Sneyd, the authors, start by positing that physiology is the area of biology in which mathematics has played the greatest historical role. I would argue that genetics is also up there in the running but, over the course of the last 150 years, they are probably correct. The life sciences were long known as a descriptive discipline and the integration of mathematics to provide the details of functionality moved many of the sub-disciplines into the Twentieth Century and beyond.

I chose to review this new volume as a sneaky way to introduce my life science readers to even more mathematics. As will become quickly apparent to any life scientist who peruses a copy, I may have overreached a wee bit! The authors are quick to point out that this is neither a math nor a physiology text but a fully integrated book of how mathematics is applied to many areas of cellular biochemistry and biology.

In reading this preface, I was struck by the authors’ candor in stating that their text is “*…necessarily brief, incomplete, and outdated (even before it was written)…*,” a concept probably foreign to mathematicians but easily grasped by the life scientists. In hoping that the physiologists would be interested in theoretical approaches to their subject of study, they may be correct, although the breadth and depth of the mathematics used will bury most. It will be far easier for the mathematicians to grasp a new and interesting application for their subject matter. In any case, I believe that the authors are entirely correct in asserting that each discipline has much to offer the other.

Readers are advised by the authors to look elsewhere for background, so those using the book need to already know some mathematics and biology. And the math background needs to be extensive beyond the experience of most life scientists. For example, in the preface for the first edition, the authors state that the mathematical background should include “*…a solid understanding of differential equations, including phase plane analysis and stability theory…basic bifurcation theory, linear transform theory (Fourier and Laplace transforms), linear systems theory, complex variable techniques…and some understanding of partial differential equations and their numerical simulation*.” Indeed, I’m still not sure what a slow manifold is, and the stochastic processes and mean field equations got awfully thick very quickly! And, as the students of mathematics can attest, there are a lot of “now we solve to obtain…” with many intermediate steps noticeably absent.

Some mathematical background is presented in chapter appendices but at a level not easily assimilated by the non-mathematician. Also omitted are solutions to any of the chapter exercises. There is easily enough here to teach the full-year course in mathematical physiology that they suggest.

As to the contents, the following subjects are included:

• biochemical reactions

• cellular homeostasis

• membrane ion channels

• passive electrical flow in neurons

• excitability

• wave propagation in excitable systems

• calcium dynamics

• intercellular communications

• neuroendocrine cells

• regulation of cell function

Omitted are areas such as the immune system, axonal transport, pharmacokinetics, compartmental modeling, wound healing and tumor modeling. Some of these are covered in the second volume dealing with systems physiology, however.

Lest the reader come away with a negative impression, let’s briefly review the good points (which are really good). The authors’ sense of humor is evident even in the introduction and sprinkled throughout the text. (*The illustration and caption of the Giant Squid on page 197 is but a single example.*) Perhaps the best features, however, are the lucid and refreshingly modern explanations of many of the biochemical processes that underlie the biology. As just one example, their explanation of the law of mass action is complete with qualifiers as to why it is not truly a law, where it cannot be used, facts such as the scarcity of trimolecular reactions and the lack of knowledge on many elementary biochemical reaction steps. Similar gems are found under thermodynamics and enzyme kinetics. Also, the brief section on detailed balance is most welcome, as many biology students may equate it with simple equilibrium. These insights occur throughout the text and represent helps in understanding which I would have loved to have seen in most of my introductory chemistry texts.

In considering the level of difficulty in most of the mathematics presented in this volume, it would have been very helpful if the authors had provided some assistance. This would include

• more backgrounders/reviews in specific basic areas such as calculus and differential equations

• solutions to many of the exercises (none are provided)

• far more complete derivations in the text

Unfortunately, this would no doubt greatly increase the page count and, thus, the price. Although the math here may be onerous to the life scientists, I would strongly recommend obtaining a copy from the library on the strength of the many fine explanations of the biochemical and biological sides of things.

**Availability**

Mathematical Physiology I: Cellular Physiology (2nd Ed.) by James Keener and James Sneyd. Springer Science+Business Media LLC. 547 pp. (2009), hardcover, $69.95.

*John Wass is a statistician based in Chicago, IL. He may be reached at editor@ScientificComputing.com.*