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Theory in Biology without which, as we know, nothing makes sense in biology. In the 1960s, the Japanese mathematical biologist Motoo Theory is Kimura conceived the neutral available light theory of evolution, which assumes that the overwhelming majority of molecular mutations Martin A. Nowak do not affect the fitness of an individual. Today most methods Many people praise Gregor for reconstructing phylogenies Mendel as the founding father of from genetic data assume the genetics. Few, however, know that neutral theory is correct. In a as a student at the University of brilliant PhD thesis submitted in Vienna he excelled in 1964, Bill Hamilton formulated the Mathematics, but failed Botany theory behind ‘selfish genes’. In Sir Ronald Fisher, with Haldane and twice, which prevented him from the 1970s, John Maynard Smith Wright one of the founders of population pursuing an academic career. He brought game theory to biology: genetics. (Courtesy of Antony Barrington- Brown and the Science Photo Library.) entered a monastery and evolutionary game theory is a continued to do science. He found generic description of evolutionary the rules of genetic inheritance dynamics whenever fitness is not population dynamics of infectious because he had a clear constant, but depends on the agents and immune cells without mathematical hypothesis even to composition of the population. At the aid of mathematical models. the extent of ignoring ambiguous about the same time, Manfred Crucial questions, such as what results that did not fit his Eigen and Peter Schuster determines the virus load in HIV mathematical expectation. pioneered an information theoretic infection, how fast is the turnover Ronald Fisher considered approach to evolution with quasi- rate of HIV-infected cells, or what Mendel as a mathematician with species theory, while Josef explains the long and variable an interest in biology. Fisher, Hofbauer and Karl Sigmund asymptomatic period between HIV together with J.B.S. Haldane and studied rigorously the replicator infection and development of Sewall Wright, unified Mendelian equation, one of the most AIDS, cannot be answered by a genetics with Darwinian evolution important descriptions of purely verbal analysis of by formulating a precise evolutionary dynamics. experimental observations. mathematical description of Contributions such as these — It is sometimes stated that evolutionary dynamics. This work and many others — have made it biology is too complicated for was done in the 1930s and is the clear that any discourse of mathematical investigation, but very foundation of our current evolutionary dynamics ultimately this argument is more logical if understanding of evolution, requires exact mathematical reversed: surely only simple formulations. All evolutionary mechanisms can be understood biologists I have ever met without the use of mathematical appreciate mathematics and analysis. understand its role in scientific In cancer genetics, one of the research. most crucial questions concerns Evolution does not stand alone the role of genetic instability in as a biological discipline that tumor progression. Mutations in would be unthinkable without oncogenes or tumor suppressor theory. The work of Robert May, for example, has extended earlier studies in ecology and epidemiology to ensure that these subjects are also grounded in precise mathematical formalism. Clearly, the spread of epidemics or the dynamic interaction of species in an ecosystem cannot be understood without the help of mathematical methods. Notably, the most cited ecologist of our It is no coincidence that Gregor Mendel, time, David Tilman, is a the founder of genetics, trained as a theoretician. mathematician. A portrait painted after In immunology, molecular and J.B.S. Haldane ended his book ‘The Mendel’s death, preserved in the Augus- causes of evolution’ (1932) with the words tinian Abbey of St Thomas in Brno and cellular experiments have “The permeation of biology by mathemat- printed in Gregor Mendel: the First generated fascinating insights, but ics is only beginning”. (Courtesy of Klaus Geneticist (1996), Oxford University Press. nobody can argue about the Patau.) Magazine R407

requires the mathematics of chemical kinetics and thermodynamics. Protein folding and predictions of the secondary or tertiary structures of RNA is another vast and exciting field which demonstrates the fruitful interaction of mathematics and biology. Although mathematical biology has a long and impressive history, as I have argued here, the present time is particularly exciting. Advances in cellular and molecular biology, mostly in the Sewell Wright studied genetic variation in subdivided populations and invented context of the human genome the ‘fitness landscape’. (Courtesy of The project, have yielded extensive University of Chicago Press.) amounts of information that not only require mathematical and genes enhance the net growth computational analysis, but rate of the cell, whereas mutations represent invaluable resources for Lord May of Oxford, President of the Royal in genetic instability genes testing new theories. In genomics, Society, brought a clear mathematical line increase its mutation rate. mathematical techniques are into ecology and epidemiology. Chromosomal instability refers to needed to align sequences, the phenomenon that most cancer search for genes or characterize same amount of rigor. Insisting on cells have increased rates of the interactions of genetic control purely non-mathematical gaining or losing whole networks. Notably, Eric Lander, interpretation of empirical chromosomes. Do mutations that one of the leaders of the genomic observations limits the range of trigger genetic instability revolution, came top of his class questions that can be asked and accelerate cancer progression? at Princeton University in conclusions that can be reached. How long does it take to activate mathematics. Evolutionary biology For any scientific investigation, oncogenes or inactivate tumor can ask questions concerning the theory is available light. suppressor genes in populations organization of whole genomes, of dividing cells with and without the selection pressure acting on Background reading genetic instability? Both questions individual genes, the evolution of Anderson, R.M. and May, R.M. (1991). require mathematical analysis. chromosomes or the relationship Infectious Diseases of Humans, Oxford Univ Press. The very concept of a tumor between the complexity of Fisher, R.A. (1930). The Genetical suppressor gene was first organisms, the size of the genome Theory of Natural Selection, Oxford proposed by Al Knudson in 1971, and the number of genes. The Univ Press. based on a mathematical analysis data for evaluating exciting new Ewens, W.J. (2004). Mathematical Population Genetics, Springer. of incidence patterns of theories may already exist. Gillespie, J.H. (1991). The causes of retinoblastoma in children. Much Somebody only has to ask the molecular evolution, Oxford Univ earlier, at a time before the nature right questions. Press. of DNA mutation had been Students feel this excitement. Hamilton, W. D. (1996). Narrow roads of gene land. NY: Freeman. revealed, Richard Doll used Harvard undergraduates in Hofbauer, J. and Sigmund, K. (1988). mathematics and statistics to mathematics, physics or Evolutionary Games and Replicator prove that smoking causes cancer. computer science, for example, Dynamics, Cambridge University The Hodgkin–Huxley equation is are keen to participate in Press central to neuro-physiology and biological research projects. Kimura, M (1983). The neutral theory of molecular evolution, Cambridge has inspired much experimental Similarly, biology students now University Press. and theoretical work. Moreover, feel the need to take advanced May, R. M. (1972). Stability and cognitive functions of the brain classes in mathematics. The complexity of model ecosystems, cannot be understood choice of the students will define Princeton University Press. Maynard Smith, J. (1982). Evolution quantitatively without the the direction of biology. and the theory of games. formalism of computer science, All these examples show that Cambridge University Press. learning theory or neural biology is already permeated by Nowak, M. A. and May, R. M. (2000). networks. Human language, for essential theory in every corner. Virus Dynamics, Oxford University Press. example, was studied for This is not surprising, because thousands of years, but the mathematics is the most powerful tremendous progress of the last language of science that we have. Program for Evolutionary Dynamics, 50 years coincides with the Mathematical models provide a Department of Organismic and Evolutionary Biology, Department of insistence on mathematical rigor. logical link between assumption Mathematics, Harvard University, Understanding biophysical and conclusion. Verbal Cambridge, Massachusetts 02138, USA. principles of molecular motors interpretations do not have the E-mail: [email protected]