Adaptation to DNA damage, an asymptotic approach for a cooperative non-local system

TL;DR

This paper investigates a two-population cooperative system modeling DNA damage adaptation, deriving a constrained Hamilton-Jacobi equation as the mutation rate approaches zero, supported by numerical simulations.

Contribution

It introduces a novel asymptotic analysis of a cooperative integro-differential system, linking it to a constrained Hamilton-Jacobi equation with biological applications.

Findings

Limit of rare mutations described by a constrained Hamilton-Jacobi equation.

Eigenvalue of a matrix determines the system's behavior.

Numerical simulations illustrate theoretical and biological insights.

Abstract

Following previous works about integro-differential equations of parabolic type modelling the Darwinian evolution of a population, we study a two-population system in the cooperative case. First, we provide a theoretical study of the limit of rare mutations and we prove that the limit is described by a constrained Hamilton-Jacobi equation. This equation is given by an eigenvalue of a matrix which accounts for the diffusion parameters and the coefficients of the system. Then, we focus on a particular application: the understanding of a phenomenon called Adaptation to DNA damage. In this framework, we provide several numerical simulations to illustrate our theoretical results and investigate mathematical and biological questions.