This thesis presents a statistical mechanical analysis of different formulations of quasispecies theory of molecular evolution. These theories, characterized by two different families of models, the Crow-Kimura and the Eigen model, constitute a microscopie description of evolution. These models are most often used for RNA viruses, where a phase transition is predicted, in agreement with experiments, between an organized or quasispecies phase, and a disordered non-selective phase when the mutation rate exceeds a critical value.
The methods of statistical mechanics, in particular field-theoretic methods, are employed to obtain analytic solutions to four problems relevant to biological interest. The first chapter presents the study of evolution under a multiple-peak fitness landscape, with biological applications in the study of the proliferation of viruses or cancer under the control of drugs or the immune system. The second chapter studies the effect of incorporating different forms of horizontal gene transfer and two-parent recombination to the classical formulation of quasispecies models. As an example, we study the effect of the sign of epistasis of the fitness landscape on the advantage or disadvantage of recombination for the mean fitness. The third chapter considers the relaxation of the purine/pyrimidine assumption in the classical formulation of the models, by formulating and solving the parallel and Eigen models in the context of a four-letter alphabet. The fourth and final chapter studies finite population effects, both in the presence and in the absence of horizontal gene transfer.