Journal of Statistical Physics

, Volume 168, Issue 1, pp 15–42 | Cite as

Random Evolutionary Dynamics Driven by Fitness and House-of-Cards Mutations: Sampling Formulae

Article
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Abstract

We first revisit the multi-allelic mutation-fitness balance problem, especially when mutations obey a house of cards condition, where the discrete-time deterministic evolutionary dynamics of the allelic frequencies derives from a Shahshahani potential. We then consider multi-allelic Wright–Fisher stochastic models whose deviation to neutrality is from the Shahshahani mutation/selection potential. We next focus on the weak selection, weak mutation cases and, making use of a Gamma calculus, we compute the normalizing partition functions of the invariant probability densities appearing in their Wright–Fisher diffusive approximations. Using these results, generalized Ewens sampling formulae (ESF) from the equilibrium distributions are derived. We start treating the ESF in the mixed mutation/selection potential case and then we restrict ourselves to the ESF in the simpler house-of-cards mutations only situation. We also address some issues concerning sampling problems from infinitely-many alleles weak limits.

Keywords

Evolutionary genetics Fitness landscape House-of-cards mutations Shahshahani gradient Wright–Fisher random genetic drift Gamma calculus Generalized Ewens sampling formulae 
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Notes

Acknowledgements

T. Huillet acknowledges partial support both from the “Chaire Modélisation mathématique et biodiversité” and the labex MME-DII Center of Excellence (Modèles mathématiques et économiques de la dynamique, de l’incertitude et des interactions, ANR-11-LABX-0023-01 project).

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 et Université de Cergy-PontoiseCergy-PontoiseFrance

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