{"title":"The Accumulation of Beneficial Mutations and Convergence to a Poisson Process","authors":"Nantawat Udomchatpitak, Jason Schweinsberg","doi":"arxiv-2407.01999","DOIUrl":null,"url":null,"abstract":"We consider a model of a population with fixed size $N$, which is subjected\nto an unlimited supply of beneficial mutations at a constant rate $\\mu_N$.\nIndividuals with $k$ beneficial mutations have the fitness $(1+s_N)^k$. Each\nindividual dies at rate 1 and is replaced by a random individual chosen with\nprobability proportional to its fitness. We show that when $\\mu_N \\ll 1/(N \\log\nN)$ and $N^{-\\eta} \\ll s_N \\ll 1$ for some $\\eta < 1$, large numbers of\nbeneficial mutations are present in the population at the same time, competing\nagainst each other, yet the fixation times of beneficial mutations, after a\ntime scaling, converge to the times of a Poisson process.","PeriodicalId":501044,"journal":{"name":"arXiv - QuanBio - Populations and Evolution","volume":"178 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - QuanBio - Populations and Evolution","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.01999","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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 $\mu_N$.
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 $\mu_N \ll 1/(N \log
N)$ and $N^{-\eta} \ll s_N \ll 1$ for some $\eta < 1$, large numbers of
beneficial mutations are present in the population at the same time, competing
against each other, yet the fixation times of beneficial mutations, after a
time scaling, converge to the times of a Poisson process.