Analysis of epistatic interactions and fitness landscapes using a new geometric approach

Abstract

Background: Understanding interactions between mutations and how they affect fitness is a central problem in evolutionary biology that bears on such fundamental issues as the structure of fitness landscapes and the evolution of sex. To date, analyses of fitness landscapes have focused either on the overall directional curvature of the fitness landscape or on the distribution of pairwise interactions. In this paper, we propose and employ a new mathematical approach that allows a more complete description of multi-way interactions and provides new insights into the structure of fitness landscapes.

Results: We apply the mathematical theory of gene interactions developed by Beerenwinkel et al. to a fitness landscape for Escherichia coli obtained by Elena and Lenski. The genotypes were constructed by introducing nine mutations into a wild-type strain and constructing a restricted set of 27 double mutants. Despite the absence of mutants higher than second order, our analysis of this genotypic space points to previously unappreciated gene interactions, in addition to the standard pairwise epistasis. Our analysis confirms Elena and Lenski's inference that the fitness landscape is complex, so that an overall measure of curvature obscures a diversity of interaction types. We also demonstrate that some mutations contribute disproportionately to this complexity. In particular, some mutations are systematically better than others at mixing with other mutations. We also find a strong correlation between epistasis and the average fitness loss caused by deleterious mutations. In particular, the epistatic deviations from multiplicative expectations tend toward more positive values in the context of more deleterious mutations, emphasizing that pairwise epistasis is a local property of the fitness landscape. Finally, we determine the geometry of the fitness landscape, which reflects many of these biologically interesting features.

Conclusion: A full description of complex fitness landscapes requires more information than the average curvature or the distribution of independent pairwise interactions. We have proposed a mathematical approach that, in principle, allows a complete description and, in practice, can suggest new insights into the structure of real fitness landscapes. Our analysis emphasizes the value of non-independent genotypes for these inferences.

Figure 1

Three-dimensional genotopes. The genotope of the complete bi-allelic three-locus system with eight genotypes is the regular cube, depicted in (a). The three-locus systems that arise from data structured as in Table 1 are displayed in (b), (c), and (d). The genotope (b) lacks the triple mutant, genotope (c) lacks the triple and one double mutant, and genotope (d) contains only the wild-type and the single mutants.

Figure 2

Standard and non-standard gene interactions. Displayed are density estimates of gene interactions as measured by the 27 standard tests (solid curve), for example w·ar - a·r, and by the 216 non-standard tests (dashed curve), for example ar·bs - as·br. The raw data are shown below the density curves.

Figure 3

Epistasis correlates with relative fitness loss. For each standard test (filled black circles), for example w·ar - a·r, and each double-double test (open red circles), for example ar·bs - as·br, anchored with the fittest type on the left-hand side of the equation, the value of the test (epistasis) is plotted versus the average fitness decrement (Δ) associated with the two deleterious two-point mutations that define the genotypes on the right hand side of the equation. Epistasis tends toward more positive values in the context of more deleterious mutations. The significance of this correlation was robust to different assumptions about the independence of the data, as described in the text.

Figure 4

Mixing ability of mutations. For each focal pair of mutations (a, b), ..., (y, z), six tests of the sort a·br - b·ar were performed in order to test the relative mixing ability of component mutations a and b relative to a tester mutation r. A small amount of jitter has been added to the vertical coordinate of each point in order to facilitate visualization. Mixing abilities vary considerably between mutations, with the (y, z) comparison (last column) revealing the most significant difference in favour of y as the superior mixer to z.

Figure 5

The 16 shapes of fitness landscapes of genotope (b). In the graph, each vertex represents one of the 16 possible geometric shapes of a fitness landscape on the truncated three-locus system that corresponds to genotope (b) in Figure 1. The shapes are determined by a collection of different tests for gene interactions. Two shapes are connected by an edge if they differ only by the sign of a single test. The labelling of the vertices follows [1, Table 5.1].