Greedy adaptive walks on a correlated fitness landscape

TL;DR

This paper analyzes greedy adaptive walks on a correlated fitness landscape combining random and deterministic components, deriving explicit and asymptotic results for walk length depending on landscape parameters and initial conditions.

Contribution

It provides explicit formulas and asymptotic analysis for the distribution of walk lengths on a correlated fitness landscape with a tunable gradient, extending understanding beyond uncorrelated and additive models.

Findings

Walk length varies non-monotonically with fitness gradient for certain starting points.

Explicit distribution formulas are derived for Gumbel-distributed fitness components.

Asymptotic results show non-trivial limits for large sequence length when scaling the gradient.

Abstract

We study adaptation of a haploid asexual population on a fitness landscape defined over binary genotype sequences of length $L$. We consider greedy adaptive walks in which the population moves to the fittest among all single mutant neighbors of the current genotype until a local fitness maximum is reached. The landscape is of the rough mount Fuji type, which means that the fitness value assigned to a sequence is the sum of a random and a deterministic component. The random components are independent and identically distributed random variables, and the deterministic component varies linearly with the distance to a reference sequence. The deterministic fitness gradient $c$ is a parameter that interpolates between the limits of an uncorrelated random landscape ($c = 0$) and an effectively additive landscape ($c \to \infty$). When the random fitness component is chosen from the Gumbel distribution, explicit expressions for the distribution of the number of steps taken by the greedy walk are obtained, and it is shown that the walk length varies non-monotonically with the strength of the fitness gradient when the starting point is sufficiently close to the reference sequence. Asymptotic results for general distributions of the random fitness component are obtained using extreme value theory, and it is found that the walk length attains a non-trivial limit for $L \to \infty$, different from its values for $c=0$ and $c = \infty$, if $c$ is scaled with $L$ in an appropriate combination.