Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology (1995)

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CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 148 where s = 7. Assuming that πC and πT are given by their observed frequencies, there is just the single parameter θ to be estimated. Preliminary simulation results give the maximum likelihood estimate of θ at about = 17 . This corresponds to a per site C → T rate of α = 1.14, and a per site T → C rate of β = 1.28. These rates are about 50 times higher than those based on the analysis in the section on the K-allele models above using all 201 sites. Of course, this set of sites was chosen essentially because of the high mutation rates in the region and so should represent an extreme estimate of the rates in the whole molecule. Nonetheless, the results do point to the lack of homogeneity in substitution rates in this molecule. For other approaches to the modeling of hypervariable sites, see Lundstrom et al. (1992b). Discussion The emphasis in this chapter has been the discussion of inference techniques for the coalescent, a natural model for the analysis of samples taken from large populations. An interesting development in the mathematical theory has been the study of measure-valued diffusions initiated by Fleming and Viot (1979). This is a generalization of the "usual" diffusions so prevalent in the classical theory of population genetics, described for example in Ewens (1979, 1990) and Tavaré (1984). A comprehensive discussion of the Fleming-Viot process appears in Ethier and Kurtz (1993), where the probabilistic structure of a broad range of examples, such as multiple loci with recombination, infinitely many alleles with selection, multigene families, and migration models, are discussed in some detail. Perhaps the most important aspect of the theory that has seen rather little theoretical treatment thus far is the area that might loosely be called variable population size processes, and their inference. These issues are becoming more important in the analysis and interpretation of human mitochondrial sequence data. Two recent articles in this area are Slatkin