  Speciation is fundamental to the huge diversity of life on Earth. Evidence
suggests reproductive isolation arises most commonly in allopatry with a higher
speciation rate in small populations. Current theory does not address this
dependence in the important weak mutation regime. Here, we examine a
biophysical model of speciation based on the binding of a protein transcription
factor to a DNA binding site, and how their independent co-evolution, in a
stabilizing landscape, of two allopatric lineages leads to incompatibilities.
Our results give a new prediction for the monomorphic regime of evolution,
consistent with data, that smaller populations should develop incompatibilities
more quickly. This arises as: 1) smaller populations having a greater initial
drift load, as there are more sequences that bind poorly than well, so fewer
substitutions are needed to reach incompatible regions of phenotype space; 2)
slower divergence when the population size is larger than the inverse of
discrete differences in fitness. Further, we find longer sequences develop
incompatibilities more quickly at small population sizes, but more slowly at
large population sizes. The biophysical model thus represents a robust
mechanism of rapid reproductive isolation for small populations and large
sequences, that does not require peak-shifts or positive selection.
