The Accumulation of Beneficial Mutations and Convergence to a Poisson Process
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arXiv
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| Format: | Preprint |
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2024
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| author | Udomchatpitak, Nantawat Schweinsberg, Jason |
| author_facet | Udomchatpitak, Nantawat Schweinsberg, Jason |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Accumulation of Beneficial Mutations and Convergence to a Poisson Process Udomchatpitak, Nantawat Schweinsberg, Jason Probability Populations and Evolution 92D15 (Primary) 60J27, 60J80, 92D25 (Secondary) 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. |
| title | The Accumulation of Beneficial Mutations and Convergence to a Poisson Process |
| topic | Probability Populations and Evolution 92D15 (Primary) 60J27, 60J80, 92D25 (Secondary) |
| url | https://arxiv.org/abs/2407.01999 |