Repair-Mediated Path Filtering of Mutation Load in Multilocus Homeostatic Populations

Abstract

We study a nonlinear multilocus mutation--selection--repair system for an asexual population maintained at exact homeostasis by state-dependent reproductive feedback. Each of $n$ identical biallelic damage loci mutates with probability $U$ at reproduction and is repaired at rate $\nu$, with per-locus reproduction and mortality factors $\zeta_\rho<1$ and $\zeta_\gamma>1$. Permutation symmetry and feedback through damage-class frequencies reduce the $2^n$-state dynamics exactly to an $(n+1)$-class ODE system on a forward-invariant simplex. Without repair, the nonlinear equilibrium is obtained in closed form throughout the branch on which the undamaged class persists. The microstate distribution factorizes as a Bernoulli product measure, the damage number is binomially distributed, and the branch terminates at $U_c=1-\zeta_\rho/\zeta_\gamma$, independently of $n$. Under $n\to\infty$, $U\to0$, and $nU\to\Theta$, the damage number converges to a Poisson distribution with mean $\Theta/(\zeta_\gamma-\zeta_\rho)$. With repair, we prove existence and local asymptotic stability of an analytic rare-mutation equilibrium branch and derive a recursion for the factor $E_k$ suppressing the leading mutation load of class $k$. Its source weights follow a conditional $\mathrm{Binomial}(k,\zeta_\rho/\zeta_\gamma)$ law independent of $n$ for every $n\ge k$. The resulting suppression lies between the product $\prod_{j=1}^k S_j$, corresponding to sequential accumulation through every intermediate repair filter, and the terminal filter $S_k$ alone, corresponding to direct multilocus jumps that bypass them, where $S_j=\lambda_j/(\lambda_j+j\nu)$ quantifies the balance between effective demographic depletion and repair. Repair also generates positive leading-order association among loci, breaking the no-repair product structure. Numerical integration of the full nonlinear system confirms the asymptotic predictions.