Exact path-integral representation of the Wright-Fisher model with mutation and selection
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929424526475264 |
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| author | Waxman, David |
| author_facet | Waxman, David |
| contents | The Wright-Fisher model describes a biological population containing a finite number of individuals. In this work we consider a Wright-Fisher model for a randomly mating population, where selection and mutation act at an unlinked locus. The selection acting has a general form, and the locus may have two or more alleles. We determine an exact representation of the time dependent transition probability of such a model in terms of a path integral. Path integrals were introduced in physics and mathematics, and have found numerous applications in different fields, where a probability distribution, or closely related object, is represented as a 'sum' of contributions over all paths or trajectories between two points. Path integrals provide alternative calculational routes to problems, and may be a source of new intuition and suggest new approximations. For the case of two alleles, we relate the exact Wright-Fisher path-integral result to the path-integral form of the transition density under the diffusion approximation. We determine properties of the Wright-Fisher transition probability for multiple alleles. We show how, in the absence of mutation, the Wright-Fisher transition probability incorporates phenomena such as fixation and loss. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12548 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact path-integral representation of the Wright-Fisher model with mutation and selection Waxman, David Populations and Evolution Statistical Mechanics The Wright-Fisher model describes a biological population containing a finite number of individuals. In this work we consider a Wright-Fisher model for a randomly mating population, where selection and mutation act at an unlinked locus. The selection acting has a general form, and the locus may have two or more alleles. We determine an exact representation of the time dependent transition probability of such a model in terms of a path integral. Path integrals were introduced in physics and mathematics, and have found numerous applications in different fields, where a probability distribution, or closely related object, is represented as a 'sum' of contributions over all paths or trajectories between two points. Path integrals provide alternative calculational routes to problems, and may be a source of new intuition and suggest new approximations. For the case of two alleles, we relate the exact Wright-Fisher path-integral result to the path-integral form of the transition density under the diffusion approximation. We determine properties of the Wright-Fisher transition probability for multiple alleles. We show how, in the absence of mutation, the Wright-Fisher transition probability incorporates phenomena such as fixation and loss. |
| title | Exact path-integral representation of the Wright-Fisher model with mutation and selection |
| topic | Populations and Evolution Statistical Mechanics |
| url | https://arxiv.org/abs/2407.12548 |