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Mathematical Models in Developmental Biology

Jerome K. Percus and Stephen Childress
American Mathematical Society/Courant Institute of Mathematical Sciences
Publication Date: 
Number of Pages: 
Courant Lecture Notes 26
[Reviewed by
William J. Satzer
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Developmental biology is the study of the development of multicellular animals from a single egg cell. The history of mathematical investigations of the subject goes back at least as far as Alan Turing and his work on morphogenesis and pattern formation. Then René Thom’s Structural Stability and Morphogenesis appeared in the 1970s. It offered a new and intriguing path of exploration and inspired a good deal of work by others. Some of that work was highly speculative and poorly conceived. Largely on account of that, the whole geometrical-topological approach to the subject fell into disrepute.

The authors of this book taught a graduate class on mathematical models of developmental biology at the Courant Institute in 1977–1978 and prepared lecture notes. The current book preserves much of the content of the original lectures and updates them with notes that incorporate work from the ensuing three decades. Part of what the authors do is to remind readers that some very good work had been discarded along with the less reputable.

The authors note that there have been rapid advances in our understanding of developmental processes at the molecular level and even more dramatic changes in our understanding of the biochemical basis of pattern formation. At the same time, they note, there is new appreciation for the value of mathematical modeling at an intermediate level between the molecular instructions for a developmental event and the results of that event.

Our best current understanding is that the information used by a developing organism is present in an unstructured form at the beginning of development. The problem is then to explain how the highly complex spatial structure of the animal arises from something that has minimal structure. The book explores this question from a variety of perspectives — local and global, deterministic and stochastic. It begins with a discussion of how the egg cell cleaves to form the hollow ball called the blastula, and how a cavity — the blastocoel — grows inside the blastula.

From there the authors move on to catastrophe theory and its methods of energy minimization, unfolding of singularities, and control parameters. The idea that a small set of control parameters associated with the elementary catastrophes might have biological significance has not yet been realized but it continues to intrigue investigators. Other topics treated here are pattern formation and its modeling by reaction-diffusion systems, the adhesion of cells to one another, cell proliferation and the movement of cells and cell-aggregates. The mathematical tools that dominate throughout are partial differential equations, though some deep questions of geometry and topology lurk in the background.

This book is not meant as an introduction, and indeed seems aimed mostly at specialists. It is best suited to those who already have at least a modest knowledge of developmental biology.

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Bill Satzer ( 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.