A stochastic model for the development of Bateson–Dobzhansky–Muller incompatibilities that incorporates protein interaction networks

Abstract

Speciation is characterized by the development of reproductive isolating barriers between diverging groups. Intrinsic post-zygotic barriers of the type envisioned by Bateson, Dobzhansky, and Muller are deleterious epistatic interactions among loci that reduce hybrid fitness, leading to reproductive isolation. The first formal population genetic model of the development of these barriers was published by Orr in 1995, and here we develop a more general model of this process by incorporating finite protein–protein interaction networks, which reduce the probability of deleterious interactions in vivo. Our model shows that the development of deleterious interactions is limited by the density of the protein–protein interaction network. We have confirmed our analytical predictions of the number of possible interactions given the number of allele substitutions by using simulations on the Saccharomyces cerevisiae protein–protein interaction network. These results allow us to define the rate at which deleterious interactions are expected to form, and hence the speciation rate, for any protein–protein interaction network.

The Pace of Hybrid Incompatibility Evolution in House Mice

“…It has helped investigators identify specific genes involved in hybrid incompatibilities (e.g., Sawamura and Yamamoto 1997;Bomblies et al 2007;Lee et al 2008;Seidel et al 2008;Bikard et al 2009;Mihola et al 2009), and genetic studies of hybrid dysfunction are often interpreted in light of the DM model even when experiments lack the ability to detect epistasis. This attention has inspired multiple theoretical advances in speciation genetics, including treating fitness landscapes (Gavrilets 2004), protein evolution (Kondrashov et al 2002), gene networks (Palmer and Feldman 2009;Livingstone et al 2012;Tulchinsky et al 2014), and developmental pathways (Porter and Johnson 2002;Johnson and Porter 2007) in the context of the DM model.…”

The probability of speciation on an interaction network with unequal substitution rates

“…Note: With random sampling, as in [5], the result is that $$Efalse[Xfalse]=true(\begin{array}{c}centerK\\ center2\end{array}true)\alpha$$ $$\approx \frac{K^2}{2}\alpha$$which is the case when r = 1. In [7], both r and α equal 1 and $X\equiv true(\begin{array}{c}centerK\\ center2\end{array}true)$.…”

“…The case of random sampling, as in [5], corresponds to a uniform distribution: f ( i ) = 1 / N for all i . Let D be the degree of a node chosen according to the f ( i ) and note that, as there are Nq ( i ) nodes of degree i , we get $$Pfalse(D=ifalse)=\text{italicNq}false(ifalse)ffalse(ifalse)$$and hence expected value $$Efalse[Dfalse]=\sum _i\text{italiciNq}false(ifalse)ffalse(ifalse)$$…”

“…In [5], we elaborated upon Orr’s model by recognizing that most nodes in a real biological network are connected to only a small number of other nodes, while very few nodes act as central hubs with a large number of interactions (see [2]). We modeled an interaction network as an undirected graph where nodes are loci and edges are existing interactions, each edge leading to a BDMI with probability p .…”

“…The sampling in [5] is done randomly, that is, every node is equally likely to be included in the sample of size K . Biologically, this translates into all alleles being equally likely to become substituted; however, as is evident from the literature on the subject, this assumption is questionable.…”