On the existence of accessible paths in various models of fitness landscapes

Abstract

We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the $n$-dimensional binary hypercube, for some $n$, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concern is with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node $\mathbf{v}^{0}$ to the all-ones node $\mathbf{v}^{1}$ tends respectively to $0$, $1$ and $1$, as $n$ tends to infinity. A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of $\mathbf{v} ^{0}$ is set to some $\alpha=\alpha_{n}\in[0,1]$. We prove that there is a very sharp threshold at $\alpha_{n}=\frac{\ln n}{n}$ for the existence of accessible paths from $\mathbf{v}^{0}$ to $\mathbf{v}^{1}$. As a corollary we prove significant concentration, for $\alpha$ below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from $\mathbf{v}^{0}$ to $\mathbf{v}^{1}$ existing tends to $1$ provided the drift parameter $\theta=\theta_{n}$ satisfies $n\theta_{n}\rightarrow\infty$, and for any fitness distribution which is continuous on its support and whose support is connected.