# Random evolutionary dynamics driven by fitness and house-of-cards mutations. Sampling formulae

## TL;DR

This paper analyzes the impact of house-of-cards mutations on evolutionary dynamics, deriving sampling formulas and Ewens sampling distributions for multi-allelic Wright-Fisher models under weak selection and mutation regimes.

## Contribution

It introduces new sampling formulae for evolutionary models with house-of-cards mutations and extends Ewens sampling formulas to these complex scenarios.

## Key findings

- Derived normalizing partition functions for invariant distributions.
- Established generalized Ewens sampling formulas for house-of-cards mutation models.
- Addressed sampling issues in infinite-alleles weak limits.

## 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 Shahsha-hani 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.