M ATHEMATICS IN B IOLOGY tions provide models of the same general introductory course practical access to con- of computation will provide students with form. Although the different systems have ceptual tools that are much more sophisti- tools that will serve them well in all of their ECTION important special features (e.g., the conserva- cated than those currently taught in the scientific careers thereafter. S tion laws), surely we would like to commu- standard first-year mathematics courses. Our proposal for an integrated introduc- nicate the more general idea that dynamics Although real mastery over these ideas will tory education for quantitatively oriented bi- are described by differential equations and require continuing reinforcement through- ologists really is an experiment in a more encourage students to discover the applicabil- out the undergraduate curriculum (as is cur- general problem: science education in the PECIAL ity of this approach to the dynamics of more rently done for physical science students), a modern world. This is a problem whose so- S complex biological systems through well- unified introduction can empower the stu- lution will require collaborations among designed laboratory exercises. In a similar dents to explore ideas far beyond what is scientists who now reside in quite different spirit, statistical physics and kinetic theory currently accessible to them. departments and cultures; enthusiastic as we provide probabilistic models of the world, but A final and, in the context of biology, are, we also are cognizant of the difficulties Mendelian genetics is also a probabilistic possibly the most important synergy derives that will no doubt arise. On the other hand, model and an understanding of probability is from the judicious use of nonstandard exam- the necessary collaborations among the fac- at the heart of all practical data analysis. ples for basic principles and methods of phys- ulty from several disciplines may well set a Today, not only can we integrate subjects ics and chemistry. For example, it makes wonderful example for students. that share common mathematical structures, sense, in modern times, to introduce students To conclude, we believe there is a great we can also integrate these abstract structures to the idea of molecular motion and thermo- opportunity to construct a unified, mathemat- with their practical implementation through dynamics in solution rather than focusing ically sophisticated introduction to physics computation. If the students are taught to only on the world of ideal gases. With afford- and chemistry, which draws on examples program and to use simple algorithms and if able modern instrumentation, students can from biology wherever possible. Such a they learn to use high-level languages (e.g., observe and record Brownian motion in a course would provide a coherent introduction Matlab or Mathematica), they can visualize microscope, for example, and satisfy them- to quantitative thinking about the natural and verify for themselves the mathematical selves quantitatively how this motion derives world, and invite all students, including biol- ideas and thereby become comfortable with from invisible molecules bouncing around in ogists of the future, to partake of the grand those they find less intuitive or more abstract. the solution and even how many such mole- tradition, which flows from Galileo’s vision. In statistics, for example, it is possible to cules there must be. This hands-on approach begin by applying simulation and bootstrap has the advantage that the phenomena (and of algorithms (e.g., for finding P values). By course the underlying principles) are directly References and Notes starting in this way, students will more easily and obviously relevant to research in biology. 1. Committee on Undergraduate Biology Education to come to appreciate parametric methods and In a similar vein, much of basic combinato- Prepare Research Scientists for the 21st Century, Board on Life Sciences, Division of Earth and Life closed-form solutions and learn to understand rics, probability theory, and statistics can be Sciences, National Research Council, BIO 2010: and to use them appropriately. presented in tandem with basic genetics, re- Transforming Undergraduate Education for Future Re- We believe that integrating mathemat- sulting a substantial saving in overall time search Biologists (National Academies Press, Wash- ington, DC, 2003). ics, computation, and the scientific context when compared with separate courses in dif- 2. C. P. Snow, The Two Cultures and the Scientific Rev- for these ideas will allow students in an ferent departments. Again, the concurrent use olution (Cambridge Univ. Press, New York, 1959). VIEWPOINT Uses and Abuses of Mathematics in Biology Robert M. May In the physical sciences, mathematical theory and experimental investigation the particulate nature of inheritance were have always marched together. Mathematics has been less intrusive in the life contemporary with Darwin, and his pub- sciences, possibly because they have until recently been largely descriptive, lished work accessible to Darwin. Fisher lacking the invariance principles and fundamental natural constants of physics. and others have suggested that Fleeming Increasingly in recent decades, however, mathematics has become pervasive in Jenkin’s fundamental and intractable ob- biology, taking many different forms: statistics in experimental design; pattern jections to The Origin of Species could seeking in bioinformatics; models in evolution, ecology, and epidemiology; and have been resolved by Darwin or one of his much else. I offer an opinionated overview of such uses—and abuses. colleagues, if only they had grasped the mathematical significance of Mendel’s Darwin once wrote “I have deeply regretted have solved one of Darwin’s major prob- results (1). But half a century elapsed that I did not proceed far enough at least to lems. In his day, it was thought that in- before Hardy and Weinberg (H-W) re- understand something of the great leading heritance “blended” maternal and paternal solved the difficulties by proving that par- principles of mathematics; for men thus characteristics. However, as pointed out to ticulate inheritance preserved variation endowed seem to have an extra sense.” Darwin by the engineer Fleeming Jenkin within populations (2). With the benefit of hindsight, we can see and others, with blending inheritance it is Today, the H-W Law stands as a kind of how much an “extra sense” could indeed virtually impossible to preserve the natural Newton’s First Law (bodies remain in their variation within populations that is both state of rest or uniform motion in a straight Zoology Department, Oxford University, Oxford OX1 observed and essential to his theory of how line, except insofar as acted upon by external 3PS, UK. evolution works. Mendel’s observations on forces) for evolution: Gene frequencies in a 790 6 FEBRUARY 2004 VOL 303 SCIENCE www.sciencemag.org M ATHEMATICS IN B IOLOGY S PECIAL population do not alter from generation to exploration—and sometimes understanding— earlier stages. Various conjectures about under- generation in the absence of migration, selec- we could not have dreamed of 50 years ago. lying mechanisms can be made explicit in tion, statistical fluctuation, mutation, etc. Consider the role played by applications of mathematical terms, and the consequences can Subsequent advances in population genetics, mathematics in sequencing the human and oth- be explored and tested against the observed S led by Fisher, Haldane, and Wright, helped er genomes. This adventure began with the patterns. In this general way, we can, in effect, ECTION make the neo-Darwinian Revolution in the recognition of the doubly helical structure of explore possible worlds. Some hard-nosed ex- early 20th century. Current work on the one DNA and its implications, an oft-told tale in perimentalists may deride such exploration of hand provides illuminating metaphors for ex- which classical mathematical physics played a imaginary worlds. And such derision may have ploring current evolutionary problems, par- central role. Brilliant biochemical advances, al- some justification when the exploration is in ticularly in molecular evolution, whilst on the lowing the 3 billion-letter-long human se- vaguely verbal terms. The virtue of mathemat- other hand having important applications in quence to be cut up into manageable fragments, ics in such a context is that it forces clarity and plant and animal breeding programs (Fig. 1). were a crucial next step. The actual reassem- precision upon the conjecture, thus enabling Before I embark on chauvinistic elaboration bling of the sequence fragments, to obtain a meaningful comparison between the conse- of other uses of mathematics in biology today, final human genome sequence, drew on both quences of basic assumptions and the empirical it is well to reflect on the varied encounters with huge computational power and complex soft- facts. Here mathematics is seen in its quintes- mathematics that today’s nascent biological re- ware, itself involving new mathematics. The sence: no more, but no less, than a way of searcher is likely to have as undergraduate and sequence information, however, represents only thinking clearly. earlier. First encounters are usually with A point that arguably deserves more the simpler aspects of “pure mathemat- emphasis than it usually gets is that, in ics” such as numbers, algebra, elemen- such exploration of mathematical mod-