Evolutionary accessibility of mutational pathways

Abstract

Functional effects of different mutations are known to combine to the total effect in highly nontrivial ways. For the trait under evolutionary selection ('fitness'), measured values over all possible combinations of a set of mutations yield a fitness landscape that determines which mutational states can be reached from a given initial genotype. Understanding the accessibility properties of fitness landscapes is conceptually important in answering questions about the predictability and repeatability of evolutionary adaptation. Here we theoretically investigate accessibility of the globally optimal state on a wide variety of model landscapes, including landscapes with tunable ruggedness as well as neutral 'holey' landscapes. We define a mutational pathway to be accessible if it contains the minimal number of mutations required to reach the target genotype, and if fitness increases in each mutational step. Under this definition accessibility is high, in the sense that at least one accessible pathway exists with a substantial probability that approaches unity as the dimensionality of the fitness landscape (set by the number of mutational loci) becomes large. At the same time the number of alternative accessible pathways grows without bounds. We test the model predictions against an empirical 8-locus fitness landscape obtained for the filamentous fungus Aspergillus niger. By analyzing subgraphs of the full landscape containing different subsets of mutations, we are able to probe the mutational distance scale in the empirical data. The predicted effect of high accessibility is supported by the empirical data and is very robust, which we argue reflects the generic topology of sequence spaces. Together with the restrictive assumptions that lie in our definition of accessibility, this implies that the globally optimal configuration should be accessible to genome wide evolution, but the repeatability of evolutionary trajectories is limited owing to the presence of a large number of alternative mutational pathways.

Figure 1. Graphical representation of three fitness landscapes of size extracted from the empirical 8-locus fitness data set for A. niger.

The presence/absence of a given mutation is indicated by 1/0. Arrows point towards higher fitness, local maxima are enlarged and underlined, and colors mark basins of attraction of maxima under a greedy (steepest ascent) adaptive walk. (A) All combinations of mutations argH12, pyrA5, leuA1, oliC2. This landscape has a single fitness maximum (the wildtype), but only 9 out of 4! = 24 paths from {1111} to {0000} are accessible. (B) Mutations argH12, pyrA5, leuA1, pheA1. This landscape has three maxima and no accessible path. (C) Mutations fwnA1, leuA1, oliC2, crnB12. The landscape has four maxima and 2 accessible paths.

Figure 2. Accessibility of mutational pathways in the House-of-Cards model.

Main figure shows the distribution of the number of accessible paths for three different sequence lengths in the HoC model in semi-logarithmic scales. The value of is an outlier, indicating that a large fraction of landscapes have no accessible paths at all. This is a typical feature of rugged fitness landscapes of moderate dimensionality , see Figures S4 and S5. Inset shows as function of for the HoC model. The top curve makes no assumptions about the antipodal sequence, while the bottom curve assumes it to be the global fitness minimum. Note the decline in the bottom curve.

Figure 3. Accessibility in fitness landscape models with tunable ruggedness.

(A) Behavior of in the RMF model as function of the correlation parameter . Inset shows normalized rescaled curves, all taking their maximum at . This implies that increases monotonically only for . (B) Probability for the model as a function of at fixed (main figure) and fixed (inset), respectively.

Figure 4. Comparison of models to empirical data.

(A) Mean number of accessible paths for HoC, RMF and models compared to the empirical A. niger data. With the exception of the HoC model, all curves show an increase of with . Both RMF (inset) and (main plot) models can be fit to the empirical data. Error bars on the empirical data represent standard deviations obtained from the resampling analysis. (B) Cumulative probability of the number of accessible paths as observed in the empirical fitness landscape compared to (main plot) and RMF (inset) model. Error bars represent the standard deviation estimated by the resampling method.