The spatio-temporal dynamics of interacting genetic incompatibilities. Part I: The case of stacked underdominant clines

TL;DR

This paper models the interaction of two genetic incompatibilities in a spatial population, deriving equations to describe their dynamics and revealing conditions for stacked clines and traveling wave propagation.

Contribution

It introduces a mathematical framework for understanding how two interacting genetic incompatibilities behave spatially and temporally, including traveling wave solutions.

Findings

Stacked clines can remain stable and propagate as traveling waves.

Recombination slows down the wave speed but does not prevent propagation.

A quasi-linkage equilibrium approximation simplifies the complex dynamics.

Abstract

We explore the interaction between two genetic incompatibilities (underdominant loci in diploid organisms) in a population occupying a one-dimensional space. We derive a system of partial differential equations describing the dynamics of allele frequencies and linkage disequilibrium between the two loci, and use a quasi-linkage equilibrium approximation in order to reduce the number of variables. We investigate the solutions of this system and demonstrate the existence of a solution in which the two clines in allele frequency remain stacked together. In the case of asymmetric incompatibilities (i.e. when one homozygote is favored over the other at each locus), these stacked clines propagate in the form of a traveling wave. We obtain an approximation for the speed of this wave which, in particular, is decreased by recombination between the two loci but is always larger than the speed of "one cline alone".