Mathematical biology sounds, at first, like a contradiction in terms. Biologists are supposed to observe and describe living systems, not equations. Despite my love for both mathematics and biology, I felt a choice was needed and I chose biology. The face of biology is changing, however. Today, a lot of research is performed in front of a lab bench, seeing little more than DNA or protein extracts from an organism. And mathematics, especially statistics, has become an essential tool.
Most mathematical biology involves the interpretation of data. Are data statistically consistent with a particular hypothesis or model? As a particular example, models are used to reconstruct a phylogenetic tree (the "family tree" of a group of organisms) from DNA sequences. A phylogeny is often difficult or impossible to recreate from the fossil record; most ancestors have simply left no record of their existence. Yet traces of the ancestry of living organisms are recorded in their genetic codes. Organisms that are closely related are characterised by very similar DNA sequences, whereas distantly related organisms have few similarities. For example, humans and chimpanzees have nearly identical genetic codes (roughly 2% divergence), whereas humans and Old World monkeys are less similar (with about 7% divergence). Models, with varying degrees of complexity and realism, translate these DNA sequences into an estimate of the most likely phylogenetic tree.
Such models have been at the heart of several recent debates in biology, including whether humans are more closely related to chimpanzees or gorillas and whether the common ancestor of all human mitochondrial DNA (the "mitochondrial Eve") lived in Africa.
Yet models can be used not just to interpret data, but to tell us how the world might work. Darwin's theory of evolution, although essentially verbal, is a prime example. Given heritable differences in survival between organisms, we predict that the next generation will be different, composed of the progeny only of survivors and not of everyone. This verbal argument has been made mathematical under a broad range of different assumptions by the great evolutionary biologists of this century, including R. A. Fisher, S. Wright, and J.B.S. Haldane. In addition, M. Kimura was instrumental in showing that populations change even in the absence of selection, since even individuals with the same inherent fitness can, by chance, have different numbers of offspring.
It is not enough, however, to know that the fittest survive if we have no way of knowing who the fittest might be. The work of many theoretical biologists, myself included, involves predicting what traits confer a higher fitness and when. To develop one example, I have looked closely into the evolution of haploidy and diploidy. Adult humans are diploid; we carry two copies of every autosomal gene. Our eggs and sperm are haploid, with only one copy of every gene, but are short-lived. There are many organsisms, from every major group besides "higher" animals, that spend most of their life cycle as haploids not as diploids, however, including mosses, bread molds, and many green algae. I have constructed a series of models that include both haploid and diploid individuals in a population. In these models, many parameters are varied, including reproductive system, population size, mutation rate, and recombination rate. Assumptions about the action of selection are also varied - selection may act to weed out deleterious mutations or to promote the spread of favorable mutations; it may act among cell lines within an individual or only among individuals within a population. I look across this entire range of parameters to find out when haploid individuals fare better and when diploids do. Then, with these patterns in hand, we can look at the distribution of traits in the world and see which variable, or combination of variables, has the most predictive power.
A major task of theoretical evolutionary biology is to explain how the same mechanism, evolution, has led to such a vast array of extremely different organisms, why particular traits are favored in some organisms but are lost in others. Models are a way to signpost the direction of evolution. We can specify the conditions under which evolution can lead to haploidy, or parthenogenesis, or elaborate tail feathers, and we can determine when such evolution is impossible. By better specifying how populations can and cannot change, we will better understand how evolution has shaped the world around us and may better design experiments to test this understanding.
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