A coarse-grained biophysical model of sequence evolution and the population size dependence of the speciation rate

Bhavin S Khatri,Richard A Goldstein

doi:10.1016/j.jtbi.2015.04.027

Abstract

Speciation is fundamental to understanding the huge diversity of life on Earth. Although still controversial, empirical evidence suggests that the rate of speciation is larger for smaller populations. Here, we explore a biophysical model of speciation by developing a simple coarse-grained theory of transcription factor-DNA binding and how their co-evolution in two geographically isolated lineages leads to incompatibilities. To develop a tractable analytical theory, we derive a Smoluchowski equation for the dynamics of binding energy evolution that accounts for the fact that natural selection acts on phenotypes, but variation arises from mutations in sequences; the Smoluchowski equation includes selection due to both gradients in fitness and gradients in sequence entropy, which is the logarithm of the number of sequences that correspond to a particular binding energy. This simple consideration predicts that smaller populations develop incompatibilities more quickly in the weak mutation regime; this trend arises as sequence entropy poises smaller populations closer to incompatible regions of phenotype space. These results suggest a generic coarse-grained approach to evolutionary stochastic dynamics, allowing realistic modelling at the phenotypic level.

Highlights

Speciation underlies the diversity of life on Earth today
We examine the process of how incompatibilities arise in allopatry for a biophysical model of a transcription factor binding to DNA by developing a coarse-grained model of how the transcription factor protein and DNA sequences co-evolve within a stochastic dynamics framework
We aim to use this stochastic dynamics to study speciation for a co-evolving pair of loci representing the binding of a transcription factor (TF) to a region of DNA corresponding to the TF binding site (TFBS)

Summary

Introduction

Speciation underlies the diversity of life on Earth today. Yet the detailed genetic mechanisms by which distinct species arise are still largely not understood. Darwin (1859), despite the title of his magnus opus, struggled to understand how natural selection could give rise to hybrid inviability or infertility. Sequence-level simulations of protein–DNA binding similar to the model we examine here, showed in the intermediate to strong mutation regime, that hybrid fitness decayed more rapidly for smaller populations (Tulchinsky et al, 2014); the underlying mechanism or growth of DMIs was not explored. Despite these results in the strong mutation regime, many traits involved in speciation are found to be monogenic or oligogenic (involving only one or a few loci) (Orr, 2001) and so are expected to arise in monomorphic populations in the weak mutation regime. We consider two populations evolving independently from a common ancestor, and consider the viability of reproductive crosses between these populations

A Smoluchowski equation for evolutionary stochastic dynamics

Coarse grained biophysical model of protein–DNA binding

Results l

Discussion and conclusion

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