On the Fisher-Wright problem and on signal peptide cleavage site recognition
In this talk, we discuss two new results, the first of which was obtained in joint work with Samuel Buss. The first part of the talk concerns the classical Fisher-Wright problem from population genetics about the expected time for gene allele fixation or extinction in a population under the random mating hypothesis. In the 1930's, Fisher pioneered the use of the linear diffusion or "Fokker-Planck" equation in estimating expected time for fixation or extinction. The asymptotic correctness of the diffusion equation was not justified until much later, in work of G. Watterson and especially of W. Ewens. If follows from an analysis of the solution of the diffusion equation, together with the work of Ewens, that expected time for gene allele fixation or extinction is linear in population size. In this part of the talk, we give an entirely new, self-contained proof of this result, which relies on Markov chain analysis instead of the diffusion equation. We provide a fast algorithm for computing the expected fixation/extinction time, and give some sample output even for situations which might not be covered by the diffusion equation. In contrast to the theoretical contribution of the first part, the second part of the talk is rather applied and concerns a comparative study of many variants of the position-dependent weight matrix algorithm for detection of signal peptide cleavage site recognition.