The evolution of postzygotic isolation: accumulating Dobzhansky-Muller incompatibilities

Hybrid sterility and inviability often result from the accumulation of substitutions that, while functional on their normal genetic backgrounds, cause a loss of fitness when brought together in hybrids. Previous theory has shown that such Dobzhansky-Muller incompatibilities should accumulate at least as fast as the square of the number of substitutions separating two species, the so-called snowball effect. Here we explicitly describe the stochastic accumulation of these incompatibilities as a function of time. The accumulation of these incompatibilities involves three levels of stochasticity: (1) the number of substitutions separating two allopatric lineages at a given time; (2) the number of incompatibilities resulting from these substitutions; and (3) the fitness effects of individual incompatibilities. Previous analyses ignored the stochasticity of molecular evolution (level 1) as well as that due to the variable effects of incompatibilities (level 3). Here we approximate the full stochastic process characterizing the accumulation of hybrid incompatibilities between pairs of loci. We derive the distribution of the number of incompatibilities as a function of divergence time between allopatric taxa as well as the distribution of waiting times to speciation by postzygotic isolation. We provide simple approximations for the mean and variance of these waiting times. These results let us estimate. albeit crudely, the probability, p, that two diverged sites from different species will contribute to hybrid sterility or inviability. Our analyses of data from Drosophila and Bombina suggest that p is generally very small, on the order of 10(-6) or less.