The Influence of Higher-Order Epistasis on Biological Fitness Landscape Topography

Abstract

The effect of a mutation on the organism often depends on what other mutations are already present in its genome. Geneticists refer to such mutational interactions as epistasis. Pairwise epistatic effects have been recognized for over a century, and their evolutionary implications have received theoretical attention for nearly as long. However, pairwise epistatic interactions themselves can vary with genomic background. This is called higher-order epistasis, and its consequences for evolution are much less well understood. Here, we assess the influence that higher-order epistasis has on the topography of 16 published, biological fitness landscapes. We find that on average, their effects on fitness landscape declines with order, and suggest that notable exceptions to this trend may deserve experimental scrutiny. We conclude by highlighting opportunities for further theoretical and experimental work dissecting the influence that epistasis of all orders has on fitness landscape topography and on the efficiency of evolution by natural selection.

Keywords: Fitness landscapes topography; Higher-order epistasis; NK landscape; Natural selection; Sequence space combinatorics.

Fig. 1

Analytic pipeline, illustrated with data from Palmer et al. [45]. a For each dataset, published fitness data (or a suitable proxy, written $\overrightarrow{W}$) were first converted to the corresponding epistatic terms ($\overrightarrow{E}$ using the Fourier–Walsh transformation (Eq. 1). b Explanatory power of a succession of models using only the m largest epistatic terms in absolute value ($\overrightarrow{W_{\mathrm{best}}^{\left(m\right)}}$) were compared with the published data. For given value of m, these models provably have the greatest explanatory power (smallest residual variance) of any model with exactly m parameters (Appendix). The symbols plotted represent the epistatic order (Sect. 1.2) of each successive parameter added to the model. c Rank correlation coefficient $({\tau }_{\mathrm{b}})$ between the empirical sequence of epistatic orders and those of our naïve expectation (Eq. 2) were computed. In cases where experimental variance was reported, these sequences were truncated as soon as the remaining model variance was less than the experimental variance. For the data shown, that truncation occurred after the $55\mathrm{th}$ epistatic term. Finally, statistical significance was assessed by a permutation test that asked whether the observed sequence of epistatic orders was significantly different than random. For the data shown (red arrow), the observed value of ${\tau }_{\mathrm{b}}$ (0.1921) was smaller than only the 3639 largest of ${10}^5$ values obtained by the permutation test, yielding $P=0.03639$

Fig. 2

Distribution of uncorrected $\mathit{P}$ values among 16 empirical datasets. Under any null model, P values are expected to be uniformly distributed (black bars; note both axes are log-transformed). Instead observed P values (grey bars) are sharply skewed toward small values ($G=143.77$, $P_{\mathrm{d}.\mathrm{f}.=5}\ll 0.01$)