Universality classes of interaction structures for NK fitness landscapes

TL;DR

This paper analyzes how different interaction structures in NK fitness landscapes influence the number of local maxima and accessible mutational pathways, revealing two universality classes with distinct asymptotic behaviors.

Contribution

It introduces a unified framework for analyzing the growth rate of local maxima and identifies two universality classes of interaction structures with different asymptotic properties.

Findings

Number of local maxima grows exponentially with sequence length for certain structures.

Probability of accessible paths decreases exponentially with sequence length.

Two distinct universality classes of interaction structures are identified.

Abstract

Kauffman's NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of $L$ binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of $k \le L$ loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of $L$, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large $k$. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in $L$ for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the NK interaction structures on the dynamics of evolution using adaptive walk models.