Accessibility of Modified NK Fitness Landscapes

In this paper we present two modifications of traditional $NK$ fitness landscapes, the $\theta NK$ and $HNK$ models, and explore these modifications via accessibility and ruggedness. The $\theta NK$ model introduces a parameter $\theta$ to integrate local Rough Mount Fuji-type correlations in subgenotype contributions, simulating more biologically realistic correlated fitness effects. The $HNK$ model incorporates gene regulation effects by introducing a masking mechanism where certain loci modulate the expression of other loci, simulating effects observed in gene regulatory networks without modeling the full network. Through extensive simulations across a wide range of parameters ($N$, $K$, $\theta$, and $H$), we analyze the impact of these modifications on landscape accessibility and the number of local optima. We find that increasing $\theta$ or the number of masking loci ($H$) generally enhances accessibility, even in landscapes with many local optima, showing that ruggedness doesn't necessarily hinder evolutionary pathways. Additionally, distinct interaction patterns (blocked, adjacent, random) lead to different observations in accessibility and optimum structure. While more complex than traditional $NK$, we believe each model provides a new biologically relevant facet to fitness landscapes and provides insight into how genetic and regulatory structures influence the evolutionary potential of populations.