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

Abstract

We present rigorous mathematical analyses of a number of wellknown 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 v 0 to the all-ones node 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 v 0 is set to some α = αn ∈ [0, 1]. We prove that there is a very sharp threshold at αn = ln n n for the existence of accessible paths from v 0 to v 1 . As a corollary we prove significant concentration, for α 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 v 0 to v 1 existing tends to 1 provided the drift parameter θ = θn satisfies nθn → ∞, and for any fitness distribution which is continuous on its support and whose support is connected.

Phase transition in random adaptive walks on correlated fitness landscapes

“…For this reason we here use a slight generalization of the Kimura formula, which reads 7Tf(s) = 1 -e ks. (35) For k -» oo we thus recover the case of the RAW studied in the previous sections, whereas for 7->0we obtain the Haldane-type fixation dynamics that is usually considered in the SSWM literature [4][5][6][7][19][20][21], As before, we consider the limit of infinite L. In this case, we expect that effectively all possible values of the random fit ness components should appear with their appropriate weights. Since the fixation probability of a beneficial mutation with random component y in the uphill direction is t c f{y -x + c), where x is the random component of the current genotype, we find the recursion relation for Q/(y) as [19,20,22] Qi+\(y)…”

“…The m utational pathways follow ed by the RAW are monotonically increasing in fitness, and a num ber o f papers have explored the conditions for the existence o f such selectively accessible paths [ 12,34,35]. In particular, in [35] it was proven that accessible paths to the reference sequence Cr exist in the RM F with a probability approaching unity for L -> oo and any c > 0.…”

“…(37) for the fixation probability Eq. (35) and an exponential distribution fix) = e~x/a, w hich gives *a + 2 Z i+ \ ~ Zi -a-------c . (38) A a + 1 T his equation is consistent w ith a diverging solution for c < a(Xa + 2)/(A.a + 1), and we conclude that the transition point is c* = a(Xa + 2)/(Xa + 1).…”

Adaptation in Tunably Rugged Fitness Landscapes: The Rough Mount Fuji Model

“…The probability for the existence of fitness-monotonic pathways in the RMF model has been investigated previously for the case when the paths end at the global fitness maximum , and it has been shown that such paths exist with unit probability for large L and any c . 0 (Hegarty and Martinsson 2014). It is not clear whether this result applies in the present setting, however, because the probability that the global maximum coincides with the reference sequence vanishes for large L (see Equation A29).…”

“…Using an approach similar to that of the present work, an explicit expression for the expected number of accessible pathways in the RMF model with Gumbeldistributed randomness can be derived (Franke et al 2010. Subsequently Hegarty and Martinsson (2014) presented a rigorous proof that accessible pathways exist with unit probability for large L in the RMF model for any c . 0, independent of the distribution of the random fitness component.…”