Biological interactions between multiple genetic mutations

Abstract

The aim of this project is to provide mathematical results for the effect of selection and environmental changes on the distribution of mutations in DNA sequences. This project is divided in two parts. The first part is devoted to the stationary solutions of the Kolmogorov equations for the distribution of mutation frequencies in a population. We generalize the well-known diffusion equations for a single mutation to pairs of mutations, and we give an explicit stationary solution in the neutral case. In the second part, the evolution of the frequencies of pairs of mutations is studied in the case of a sudden environmental change. In particular, we assume that the mutations are evolving neutrally (therefore their distribution follows the expression found in the first part) until a change occurs and one of the mutations becomes selected. We derive the final distribution of frequencies for this scenario.