Dopamine (DA) is a neurotransmitter involved in attention, goal oriented behavior, movement, reward learning, and short term and working memory. For the past four decades, mathematical and computational modeling approaches have been useful in DA research, and although every modeling approach has limitations, a model is an efficient way to generate and explore hypotheses. This work develops a model of DA dynamics in a representative, single DA neuron by integrating previous experimental, theoretical and computational research. The model consists of three compartments: the cytosol, the vesicles, and the extracellular space and forms the basis of a new mathematical paradigm for examining the dynamics of DA synthesis, storage, release and reuptake. The model can be driven by action potentials generated by any model of excitable membrane potential or even from experimentally induced depolarization voltage recordings. Here the model is forced by a previously published model of the excitable membrane of a mesencephalic DA neuron in order to study the biochemical processes involved in extracellular DA production. After demonstrating that the model exhibits realistic dynamics resembling those observed experimentally, the model is used to examine the functional changes in presynaptic mechanisms due to application of cocaine. Sensitivity analysis and numerical studies that focus on various possible mechanisms for the inhibition of DAT by cocaine provide insight for the complex interactions involved in DA dynamics. In particular, comparing numerical results for a mixed inhibition mechanism to those for competitive, non-competitive and uncompetitive inhibition mechanisms reveals many behavioral similarities for these different types of inhibition that depend on inhibition parameters and levels of cocaine. Placing experimental results within this context of mixed inhibition provides a possible explanation for the conflicting views of uptake inhibition mechanisms found in experimental neuroscience literature.

Building mathematical models and examining the compatibility of their theoretical predictions with empirical data are important for our understanding of evolution. The rapidly increasing amounts of genomic data on polymorphisms greatly motivate evolutionary biologists to find targets of positive selection. Although intensive mathematical and statistical studies for characterizing signatures of positive selection have been conducted to identify targets of positive selection, relatively little is known about the effects of other evolutionary forces on signatures of positive selection. In this dissertation, I investigate the effects of various evolutionary factors, including purifying selection and population demography, on signatures of positive selection. Specifically, the effects on two highly used methods for detecting positive selection, one by Wright's Fst and its analogues and the other by footprints of genetic hitchhiking, are investigated. In Chapters 2 and 3, the effect of purifying selection on Fst is studied. The results show that purifying selection intensity greatly affects Fst by modulating allele frequencies across populations. The footprints of genetic hitchhiking in a geographically structured population are studied in Chapter 4. The results demonstrate that footprints of genetic hitchhiking are significantly influenced by geographic structure, which may help scientists to infer the origin and spread of the beneficial allele. In Chapter 5, the stochastic dynamics of a hitchhiking allele are studied using the diffusion process of genetic hitchhiking conditioned on the fixation of the beneficial allele. Explicit formulae for the conditioned two-locus diffusion process of genetic hitchhiking are derived and stochastic aspects of genetic hitchhiking are investigated. The results in this dissertation show that it is essential to model the interaction of neutral and selective forces for correct identification of the targets of positive selection.