In general, the evolutionary and genetic processes involved in changing DNA sequences in populations are not well understood. Within this dissertation several problems in the field are studied involving statistical estimation of molecular evolutionary and genetic phenomena, hypothesis testing of underlying population genetic models, and the construction of new quantitative models that exhibit testable properties which can be compared to molecular data. With a few exceptions, this work is theoretical in nature; and for the sake of precision, the tool of mathematics is employed ad libitum. In CHAPTER I, a mathematical model for haploid population genetics is discussed. The sampling-with-replacement assumption used to beget subsequent generations in the standard model of population genetics is rejected in favor of a model that uses a sampling without-replacement assumption for observational reasons concerning the lifecycles of many haploid organisms. Analytic properties of this model are presented. Significant differences in transition probabilities and higher moments of the allele frequency differentiate the two models. CHAPTERS II and III comprise a united investigation into the population genetic causes behind substitution variability. A new method for measuring temporal changes in the dispersion index is presented in CHAPTER II. Analytic results for three dispersion estimators are derived under neutrality. This motivates an inquiry into the causes of substitution variation patterns seen in 80 protein-coding mammalian genes in CHAPTER III. A stochastic model consistent with the mammalian DNA data is proposed—one in which weak selection coefficients very slightly fluctuate in time according to a diffusion process within a potential-well.
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