Geometry of fitness landscapes: Peaks, shapes and universal positive epistasis

Abstract

Darwinian evolution is driven by random mutations, genetic recombination (gene shuffling) and selection that favors genotypes with high fitness. For systems where each genotype can be represented as a bitstring of length L, an overview of possible evolutionary trajectories is provided by the L-cube graph with nodes labeled by genotypes and edges directed toward the genotype with higher fitness. Peaks (sinks in the graphs) are important since a population can get stranded at a suboptimal peak. The fitness landscape is defined by the fitness values of all genotypes in the system. Some notion of curvature is necessary for a more complete analysis of the landscapes, including the effect of recombination. The shape approach uses triangulations (shapes) induced by fitness landscapes. The main topic for this work is the interplay between peak patterns and shapes. Because of constraints on the shapes for L = 3 imposed by peaks there are in total 25 possible combinations of peak patterns and shapes. Similar constraints exist for higher L. Specifically, we show that the constraints induced by the staircase triangulation can be formulated as a condition of universal positive epistasis, an order relation on the fitness effects of arbitrary sets of mutations that respects the inclusion relation between the corresponding genetic backgrounds. We apply the concept to a large protein fitness landscapes for an immunoglobulin-binding protein expressed in Streptococcal bacteria.

Occurrences of reciprocal sign epistasis in single- and multi-peaked theoretical fitness landscapes

“…Definitions This paper will consider genetic fitness landscapes abstractly as Boolean lattices under the assumptions laid out by Crona et al in their 2013 paper [3]. Crona and their collaborators have a long history of work in the area including [4][5][6][7]. Restricting ourselves to haploid genomes, we will call the physical site of each gene under investigation a locus.…”