Why Is Mathematical Biology So Hard?, Volume 51, Number 3

Although there is a long history of the applications of mathematics to biology, only recently has mathematical biology become an accepted branch of applied mathematics. Undergraduates are doing research projects and graduate students are writing Ph.D. dissertations in mathematical biology, and departments are trying to hire them. But what should the Ph.D. training consist of? How should departments judge work in mathematical biology? Such policy questions are always important and controversial, but they are particularly difficult here because mathematical biology is very different from the traditional applications of mathematics in physics. I'll begin by discussing the nature of the field itself and then return to the policy questions. Where's Newton's Law? The phenomena that mathematical biology seeks to understand and predict are very rich and diverse and not derived from a few simple principles. Consider, in comparison, classical mechanics and continuum mechanics. Newton's Law of Motion is not just a central explanatory principle; it also gives an immediate way to write down equations governing the important variables in a real or hypothetical physical situation. Since the Navier-Stokes equations express Newton's Law for fluids, they are fundamental and have embedded in them both the fundamental principle and the complexity of the fluid phenomena that we see. Thus a pure mathematician who proves a theorem about the Navier-Stokes equations and an applied mathematician who develops new numerical tools knows that he or she has really contributed something. Alas, there are no such central fundamental principles in biology. There are principles of course—some would say dogmas—such as " evolution by natural selection " , " no inheritance of acquired characteristics " , or " DNA → RNA → proteins ". But these are not trans-latable into mathematical equations or other structures without hosts of additional facts and assumptions that are context-dependent. This means that mathematical biology is very unsatisfying for pure mathematicians, who usually are interested in discovering fundamental and universal structural relationships. It also means that there is no " mathematics of biology " in the same way that ordinary differential equations is the mathematics of classical mechanics and partial differential equations is the mathematics of continuum mechanics. Diverse, yet special. Because of evolution, biological systems are exceptionally diverse, complex, and special at the same time, and this presents several difficulties to a mathematician. The first is choosing what to work on. There's too much biology ! How do changes in the …