The peaks and geometry of fitness landscapes

Abstract

Fitness landscapes are central in the theory of adaptation. Recent work compares global and local properties of fitness landscapes. It has been shown that multi-peaked fitness landscapes have a local property called reciprocal sign epistasis interactions. The converse is not true. We show that no condition phrased in terms of reciprocal sign epistasis interactions only, implies multiple peaks. We give a sufficient condition for multiple peaks phrased in terms of two-way interactions. This result is surprising since it has been claimed that no sufficient local condition for multiple peaks exist. We show that our result cannot be generalized to sufficient conditions for three or more peaks. Our proof depends on fitness graphs, where nodes represent genotypes and where arrows point toward more fit genotypes. We also use fitness graphs in order to give a new brief proof of the equivalent characterizations of fitness landscapes lacking genetic constraints on accessible mutational trajectories. We compare a recent geometric classification of fitness landscape based on triangulations of polytopes with qualitative aspects of gene interactions. One observation is that fitness graphs provide information not contained in the geometric classification. We argue that a qualitative perspective may help relating theory of fitness landscapes and empirical observations.

Topological features of rugged fitness landscapes in sequence space

“…Such asymmetrical systems in many dimensions are difficult to characterize theoretically or empirically. Thus, for the sake of simplicity, fitness landscapes are often assumed to have binary fitness with vertices corresponding to high or low fitness [39,40] (but see [41,42] for visualizations of nonbinary fitness landscapes). In that case, the vertices corresponding to low fitness are disconnected from all of their neighbors and the resulting fitness landscape corresponds to a subgraph comprising only vertices with high fitness connected by edges to neighboring vertices with high fitness (Figure 1).…”

The impact of high‐order epistasis in the within‐host fitness of a positive‐sense plant RNA virus

“…A necessary yet not sufficient condition for ruggedness in fitness landscapes is the existence of reciprocal sign epistasis (Poelwijk et al ., ; Crona et al ., ). In our previous study on epistasis among pairs of random deleterious mutations in TEV genome, we found a significant enrichment for this type of interaction (Lalić & Elena, ).…”

“…In the case of sign epistasis, the sign of the fitness effect of a mutation depends on the genetic background, such that only a fraction of the total paths to the optimum are selectively accessible, that is contain only steps that confer a fitness increase. Reciprocal sign epistasis, in which two mutations are individually deleterious but jointly advantageous, is a necessary but not sufficient condition for rugged landscape with multiple local optima (Poelwijk et al ., ; Crona et al ., ). The ruggedness of fitness landscapes determines whether an evolving population will reach the global optimum or it is possible for it to get stuck into suboptimal fitness peaks (Whithlock et al ., ; Weinreich, ; Poelwijk et al ., , ; Kvitek & Sherlock, ).…”

Recombination and peak jumping

“…Roughly, the nodes represent genotypes, and each arrow points toward the more fit genotype. A fitness landscape is smooth if it can be represented by a fitness graph where all arrows point up, or equivalently, if any shortest walk to the peak is an uphill walk [16, 17]. …”