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In this thesis, consisting of four papers, we develop probabilistic models of molecular evolution to advance our conceptual understanding of how evolutionary processes jointly shape sequence evolution and genetic variation across different time scales.  

As a first direction, we introduce a flexible framework to study the effects of nonstationary dynamics of various evolutionary processes on allele frequency trajectories. We obtain nonequilibrium allele frequency spectra within a Poisson random field model and derive measures of evolutionary processes over different time scales. In paper I, we consider a demographic nonequilibrium in form of a change in population size, and demonstrate that the selection-drift relationship after the change in population size deviates substantially from the equilibrium balance. This deviation is sensitive to the chosen combination of measures. In paper II, we examine how temporal dynamics of recombination hotspots can be inferred from measures of GC-biased gene conversion, and show that a combination of measures across different time scales reveals whether a recombination hotspot has formed or eroded, and indicates the relative age of the change.

As a second direction, in paper III we present a mutation-selection-drift model of sequence evolution that explicitly integrates both population genetic and phylogenetic modeling approaches and the corresponding time scales. Allele frequency trajectories at a locus are described by the path of a hybrid jump-diffusion process, with selection coefficients based on a fitness landscape. Within this framework, we present rigorous arguments that directional selection, in comparison to neutral evolution, reduces the magnitude of genetic variation. In paper IV, we apply the mutation-selection-drift model to codon sequence evolution within the context of speciation, during which polymorphisms contain essential information. By employing the link to the underlying fitness landscape and introducing a Poisson formulation of the model, we express divergence between two species, both on a common fitness landscape and on divergent fitness landscapes, with the aim to investigate differences between divergence due to genetic drift and divergent selection.

Altogether, in addition to augmenting conceptual understanding of sequence evolution, our analytical results provide valuable implications for the interpretation of empirical observations and form a basis for refined methodological development.