Predictability of evolutionary trajectories in fitness landscapes

Abstract

Experimental studies on enzyme evolution show that only a small fraction of all possible mutation trajectories are accessible to evolution. However, these experiments deal with individual enzymes and explore a tiny part of the fitness landscape. We report an exhaustive analysis of fitness landscapes constructed with an off-lattice model of protein folding where fitness is equated with robustness to misfolding. This model mimics the essential features of the interactions between amino acids, is consistent with the key paradigms of protein folding and reproduces the universal distribution of evolutionary rates among orthologous proteins. We introduce mean path divergence as a quantitative measure of the degree to which the starting and ending points determine the path of evolution in fitness landscapes. Global measures of landscape roughness are good predictors of path divergence in all studied landscapes: the mean path divergence is greater in smooth landscapes than in rough ones. The model-derived and experimental landscapes are significantly smoother than random landscapes and resemble additive landscapes perturbed with moderate amounts of noise; thus, these landscapes are substantially robust to mutation. The model landscapes show a deficit of suboptimal peaks even compared with noisy additive landscapes with similar overall roughness. We suggest that smoothness and the substantial deficit of peaks in the fitness landscapes of protein evolution are fundamental consequences of the physics of protein folding.

Figure 1. Fitness averaged over all points at a particular distance from the peak for folding landscapes, additive landscapes with the same three levels of multiplicative noise used in Fig. 6 and the sesquiterpene synthase landscape.

Figure 2. Deviation from additivity, monotonic paths and suboptimal peak suppression in folding and experimental landscapes.

(A) Deviation from additivity for the folding landscapes (larger symbols), their scrambled versions (smaller symbols) and the two experimental landscapes. Error bars show one standard deviation within the ensemble of permuted landscapes. (B) Fraction of monotonic paths to the main peak in folding, scrambled and experimental landscapes. (C) The number of peaks is vastly greater in scrambled landscapes than in folding or experimental landscapes (with the exception of the sesquiterpene synthase landscape).

Figure 3. The Z-scores of different characteristics of the original folding and experimental landscapes measured with respect to the ensembles of their randomly permuted counterparts.

Figure 4. Correlations between different quantitative characteristics of the folding landscapes.

Each panel quotes the Spearman rank correlation coefficient between the particular pair of characteristics.

Figure 5. Mean path divergence as a function of selection pressure, which is a product of and , for a folding landscape with 5936 nodes and 65 peaks.

Solid lines are labeled by the Hamming distance between the pairs of starting and ending points of the trajectory bundles over which the path divergence is averaged.

Figure 6. The dependence of the path divergence (top row) and the monotonic path fraction (bottom row) on the measures of landscape roughness.

The dots of different color correspond to noisy additive landscapes with differing amounts of multiplicative noise: low (red), two intermediate levels (green smaller than blue), and high (magenta). Yellow circles represent the folding landscapes, the cyan squares–the -lactamase landscape, and the red triangles–the sesquiterpene synthase landscape.

Figure 7. Mean path divergence in folding and experimental landscapes (larger symbols) landscapes, as well as their scrambled versions (smaller symbols) as a function of Hamming distance from the main peak.