Distinguishing the effects of epistasis and pleiotropy using a variant of the NK model

Pleiotropy and epistasis are central to understanding how genes are expressed. Kauffman’s NK model is used ubiquitously to investigate gene expression in evolution and other contexts; it is widely understood to reflect the results of epistasis, but it is less often used to study pleiotropy. In this paper we introduce the NEP model, a variant of the NK model which allows epistasis and pleiotropy to be studied individually. We apply our methods to global and local optima and adaptive walks, and elucidate new insights into Kauffman’s complexity catastrophe. Introduction Pleiotropy, which refers to a single locus affecting more than one trait, and epistasis, several loci collectively affecting a single trait, have long been recognized to be fundamental to our understanding of gene expression (Tyler et al., 2009). Epistasis is widely encountered in humans (Moore, 2003) and other organisms (Remold and Lenski, 2004; Bonhoeffer et al., 2004), as is pleiotropy (Ostrowski et al., 2005; Wagner et al., 2008; Scarcelli et al., 2007). Epistasis and pleiotropy are also seen to play a key role in evolution (Phillips, 2008; Fenster et al., 1997). Thus it is important to form a clear picture of the mechanisms of epistasis and pleiotropy and their effects on phenotypes. Kauffman’s NK model (Kauffman and Levin, 1987), a computational model of genomes in fitness landscapes, has been widely used to investigate properties of fitness spaces (Kauffman, 1993; Weinberger and Stadler, 1993; Macken and Perelson, 1989; Orr, 2005) and evolution thereon (Østman et al., 2010). A number of variants of the NK model have also been studied, such as the infinite-allele variant (Welch and Waxman, 2005) and the block model (Perelson and Macken, 1995); the NK model and its variants have been shown to be applicable to a variety of biological phenomena (Macken and Perelson, 1989; Perelson and Macken, 1995; Kauffman and Weinberger, 1989; Orr, 2006). Epistasis and pleiotropy can be tuned in the NK model, but they always vary in tandem, which makes it difficult to study the two effects separately. In this paper we describe the NEP model, a variant of the NK model in which epistasis and pleiotropy can be tuned independently. Methods Models The NK Model The NK model comprises a population of genomes, each of which consists of N loci, with A alleles at each locus. The model defines one trait for each locus; the locus interacts epistatically with K other loci in determining that trait. The fitness of a genome is calculated by averaging the fitness contributions of all of the traits. Each trait is represented by a (K + 1)-dimensional table, with the length along each dimension equal to A, where the values in the table are stochastically chosen from a uniform distribution. The fitness contribution for each trait is selected from the table by choosing the row corresponding to the allele of the base locus, the column corresponding to the allele of the next locus, etc. Choosing which other loci interact with a given locus can either be done deterministically, by having each locus interact with the succeeding K loci (where the genome is assumed to be circular), or stochastically, by choosing K other loci at random. Because the NK model contains a trait/table for each of the N loci, there are N traits/tables in total. In the rest of the paper we will refer to traits and tables interchangeably. Fig. 1 gives an example genome with its associated tables; the top part of this figure refers to the NK model, and the bottom part contains tables that are added for the NEP model. In this example N = 4 and A = 2, so each genome contains 4 loci with 2 alleles each; the interaction degree, K, is 1. The horizontal bar in the middle of the diagram is the genome, and the 4 dots on the bar are the loci. The numbers 0, 1, 1, 0 along the genome are the alleles at each locus. In this example, each consecutive pair of loci interact in a trait, as do the outer two loci, for a total of four traits. Each trait is shown in the diagram as a two-dimensional lookup table above the genome, which is linked to its pair of loci by a pair of lines. The first pair of alleles is (0, 1), so the corresponding fitness contribution is the .71 shown in the