Jürgen Jost is a prolific author in various fields of advanced analysis and he has written an interesting book based on graduate courses given as part of a joint program between Leipzig University and the Max Planck Institute for Mathematics in the Sciences. The range of topics represented in these lectures is quite large — from cellular reaction kinetics and biological pattern formation to neuronal systems to phylogenetic tree reconstructions and more. This book sets itself the task of highlighting the use of graph theory, stochastic methods and partial differential equations in current biology.
It will come as a surprise to almost no one that the science of biology has undergone the most change of all the sciences in the past 50 years. The mathematical issues generated by scads of genetic information are still under debate, but one should probably expect discrete mathematics to play a large role. In particular, graph theory serves as a good way to model some of the connectivity that occurs between genes and the way they express themselves. The first part of this book is devoted to developing discrete methods to tackle some of the abiding questions in biological genetics. Within only a few pages, Jost defines many basic graph concepts (spanning trees, bipartite graphs and so on) but makes greatest use of the graph Laplacian and its associated eigenvalues. The mathematics is well-developed (if somewhat telegraphic in form) but the “introductory” review of DNA and RNA coding and the ways in which polypeptides can form oligomotives was entirely too brief for someone without a fair amount of biology expertise to follow. Even a few pictures could have helped the reader to understand the combinatorial ordering that connects these things. The graph theory, on the other hand, is quite elegant and the use of the Pólya-Cheeger constant to estimate the spectrum of the graph Laplacian was extremely instructive.
The next chapters follow more of the traditional uses of calculus-based mathematics to biology. Branching process and Poisson distributions are applied to neural coding and generating functions of random graphs are discussed briefly. The third portion of the book discusses pattern formation, diffusion processes, and random walks. Traveling wave solutions to nonlinear diffusion PDEs are used to talk about reaction-diffusion systems and a final section brings together the Wright-Fisher model and its diffusion approximation. Throughout all of this there is a nice interplay between micro-random phenomena and large scale deterministic models of population drift.
This book is ideal for someone with some mathematical expertise who wishes to put it to use at the current boundaries of biological exploration. The reader must have some background in ordinary and partial differential equations as well as combinatorial graph theory and linear algebra to make sense of the rapid-fire derivations (even though the presentation stresses its accessibility). I found myself wishing that there was a more introductory explanation of the molecular genetics concepts. This would have helped tremendously to create an appreciation of the deep applications of mathematics to the biological research world. Perhaps this will be forthcoming in the companion book “Biology and Mathematics” that the author has promised to write. Anyone with an interest in the cross-fertilization of these fields will look forward to this with great excitement!
Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.