We propose to investigate a number of problems in probability theory that arise from genetics and ecology. We will extend our analysis of the impact of the fixation of beneficial mutations, develop analytical results to help quantify the effects of Hill-Robertson interference, which occurs when a second advantageous mutation occurs before the first one reaches fixation and study diffusion approximations for the related problem of codon usage bias. We will obtain rigorous results for the sequentially Markovian coalescent, which has recently been introduced as an approximation for genealogies with recombination, investigate the nature of regulatory sequence evolution in Drosophila and other organisms with large effective population size; study the subfunctionalization explanation of the persistence of gene duplicates, and consider random Boolean networks, in order to gain insights into gene networks. Inspired by ecology, we will use spatial models to understand conditions needed for the emergence and maintenance of cooperation between individuals, and we will investigate changes that occur when the parameters governing the interactions in predator-prey and epidemic models evolve. The last decade has seen the generation of incredible amounts of DNA sequence data. However, having the text of the instruction manual that controls life is just the beginning, we need to understand how the expression of genes is controlled and how genes interact in biological networks. These interactions cannot be directly observed so one needs to develop mathematical models to test hypotheses, and to infer how genetic systems evolved to operate as they do. In particular, our research will investigate (i) how rapidly changes can occur due to regulatory sequence evolution, which is commonly quoted explanation for many of the differences between humans and chimpanzees, (ii) whether or not the subfunctionalization hypothesis provides a satisfactory explanation of the paradox of the preservation of gene duplicates, and (iii) to improve our understanding the patterns of variations caused by adaptive evolution. These are just three examples of many at the interface between biology and mathematics. An important aspect of our proposal is to train graduate students to work in this exciting area.
