Nantawat Udomchatpitak

arXiv2024

The Accumulation of Beneficial Mutations and Convergence to a Poisson Process

We consider a model of a population with fixed size $N$, which is subjected to an unlimited supply of beneficial mutations at a constant rate $μ_N$. Individuals with $k$ beneficial mutations have the fitness $(1+s_N)^k$. Each individual dies at rate 1 and is replaced by a random individual chosen with probability proportional to its fitness. We show that when $μ_N \ll 1/(N \log N)$ and $N^{-η} \ll s_N \ll 1$ for some $η< 1$, the fixation times of beneficial mutations, after a time scaling, converge to the times of a Poisson process, even though for some choices of $s_N$ and $μ_N$ satisfying these conditions, there will sometimes be multiple beneficial mutations with distinct origins in the population, competing against each other.

math.PR↗

arXiv2019

The Effect of Recombination on the Speed of Evolution

It has been a puzzling question why some organisms reproduce sexually. Fisher and Muller hypothesized that reproducing by sex can speed up the evolution. They explained that in the sexual reproduction, recombination can combine beneficial alleles that lie on different chromosomes, which speeds up the time that those beneficial alleles spread to the entire population. We consider a population model of fixed size $N$, in which we will focus on two loci on a chromosome. Each allele at each locus can mutate into a beneficial allele at rate $μ_N$. The individuals with 0, 1, and 2 beneficial alleles die at rates $1, 1-s_N$ and $1-2s_N$ respectively. When an individual dies, with probability $1-r_N$, the new individual inherits both alleles from one parent, chosen at random from the population, while with probability $r_N$, recombination occurs, and the new individual receives its two alleles from different parents. Under certain assumptions on the parameters $N, μ_N, s_N$ and $r_N$, we obtain an asymptotic approximation for the time that both beneficial alleles spread to the entire population. When the recombination probability is small, we show that recombination does not speed up the time that the two beneficial alleles spread to the entire population, while when the recombination probability is large, we show that recombination decreases the time, which agrees with Fisher-Muller hypothesis, and confirms the advantage of reproducing by sex.

math.PR↗