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CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 114 Chapter 5â Calibrating the Clock: Using Stochastic Processes to Measure the Rate of Evolution Simon Tavaré University of Southern California Deoxyribonucleic acid (DNA) sequences record the history of life. Although DNA replication is remarkably accurate, mutations do occur at a small but nonnegligible rate, with the result that an individual's descendants begin to diverge in DNA sequence over time. By examining DNA sequences among different species or among different individuals within a single species, it is possible to reconstruct aspects of their evolutionary history. Such studies have been pursued with special interest in the human, where an unusual DNA sequence called the mitochondrial genome has been used to trace human migrations and human evolution. The author shows how mathematical tools from the theory of stochastic processes assist in calibrating the molecular clock inherent in DNA sequences. While DNA sequences are transmitted from parent to child with remarkable fidelity, mutations do occur at a small but nonnegligible rate, with the result that an individual's descendants begin to diverge in DNA sequence over time. Some mutations are deleterious and are eliminated by natural selection, but many are thought to be selectively neutral and thus accumulate at a roughly steady rateâproviding a molecular clock for measuring the time since two species or two individuals within a species shared a common ancestor. In this manner, it is possible to reconstruct an evolutionary tree and even estimate the times of key separation events.
CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 115 Different biological sequences within an organism may obey different clocks. The amino acid sequence of a protein encoded by a gene changes more slowly than the DNA sequence of the underlying gene because many amino acid changes may be selectively disadvantageous (because they disrupt function). On the other hand, a significant proportion of DNA changes may be selectively neutral because they create a synonymous codon (that is, one that specifies the same amino acid). Similarly, DNA regions within genes change at a slower rate than the DNA sequences located between genes. Accordingly, evolutionary studies of distant species are often carried out by examining amino acid sequences of proteins, while evolutionary comparisons among more closely related species are better done by examining DNA sequences within or between genes. To study evolution within a single species such as the human, it is often useful to study DNA sequences that change at especially rapid rates. The mitochondrial genome provides an ideal substrate for such studies. The mitochondrion is an organelle found in the cytoplasm of eukaryotic cells, whose primary role is to generate high- energy compounds that the cell uses to drive chemical reactions. Although the mitochondria use many proteins that are encoded by genes in the cell nucleus, each mitochondrion has its own small circular chromosome that encodes a few dozen genes essential for its function. In the human, the mitochondrial genome consists of 16,569 base pairs whose DNA sequence has been completely determined (Anderson et al., 1981). Human mitochondria are inherited only from the mother, and so their genealogy is considerably simpler to follow than for genes encoded in the nucleus (which are inherited from both parents and are subject to recombination between the two homologous copies in the cell). Conveniently for evolutionary studies, mitochondrial DNA (mtDNA) has an increased rate of nucleotide substitution compared to nuclear genes, owing to the presumed absence of certain DNA repair mechanisms. Moreover, the mitochondrial genome contains certain regions that are particularly tolerant of mutation, that is, appear to be subject to little selective pressure (Avise, 1986) and thus show a great deal of variation. In all, the mitochondrial genome may be evolving 10 times faster than the nuclear genome. For these reasons, molecular population geneticists have carried out many studies of the DNA sequences of mitochondrial variable regions in many human populations (Di Rienzo and Wilson, 1991; Horai and
CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 116 Hayasaka, 1990; Vigilant et al., 1989, 1991; Ward et al., 1991). Studies of mitochondrial sequences of different Native American tribes strongly suggest that there were multiple waves of colonization of North America by migrant groups from Asia, and even allow one to estimate the dates of these events (Schurr et al., 1990; Ward et al., 1991). Assuming a constant evolutionary rate, the pattern of mutations between diverse human groups has been used to argue (Cann et al., 1987) that the mitochondria of all living humans descended from a mother that lived in Africa some 200,000 years agoâthe so-called Eve hypothesis. Although the precise details of the hypothesis are disputed (Maddison, 1991; Nei, 1992; Templeton, 1992), the general power of the methodology is well accepted. (As an aside, the reader should note that the existence of a common ancestorâEve, so to speakâis a mathematical necessity in any branching process that satisfies very weak conditions. The biological controversies pertain to when and where Eve lived.) Each of these applications requires a knowledge of the rate at which mutations occur in an mtDNA sequence. Estimates of this rate have been obtained by comparing a single DNA sequence from each of several species whose times of divergence are presumed known. Divergence is calculated from the number of nucleotide differences between species (using methods that correct for the possibility of multiple mutations at a site), and rate estimates are obtained by dividing the amount of sequence divergence by the divergence time. For data taken from multiple individuals in a single population, one requires a model that takes account of the population genetic aspect of the sampling: individuals in the sample are correlated by their common ancestry. In this chapter, we describe the underlying stochastic structure of this ancestry and use the results to estimate substitution rates. We have chosen to focus on rate estimation to give the chapter a single theme. We are not interested per se in statistical aspects of tests for selective neutrality of DNA differences; rather, we assume neutrality for the data sets discussed as examples. The techniques described here should be regarded as illustrative of the theoretical and practical problems that arise in sequence analysis of samples from closely related individuals. The emphasis is on exploratory methods that might be used to summarize the structure of such samples.
