The ascent of the abundant: how mutational networks constrain evolution

Abstract

Evolution by natural selection is fundamentally shaped by the fitness landscapes in which it occurs. Yet fitness landscapes are vast and complex, and thus we know relatively little about the long-range constraints they impose on evolutionary dynamics. Here, we exhaustively survey the structural landscapes of RNA molecules of lengths 12 to 18 nucleotides, and develop a network model to describe the relationship between sequence and structure. We find that phenotype abundance--the number of genotypes producing a particular phenotype--varies in a predictable manner and critically influences evolutionary dynamics. A study of naturally occurring functional RNA molecules using a new structural statistic suggests that these molecules are biased toward abundant phenotypes. This supports an "ascent of the abundant" hypothesis, in which evolution yields abundant phenotypes even when they are not the most fit.

Figure 1. Contiguity statistic and thermostability percentiles for natural functional molecules from the Rfam database.

The blue circles represent percentiles calculated from consensus structures and individual sequences, respectively. The red squares represent percentiles for thermostability predictions of molecules folding into the wildtype structures. We used 239 families in which the consensus structure was relatively well conserved among the individual genotypes. The x-axis gives the fraction of random phenotypes that are predicted to be less abundant (or less thermostable) than the actual phenotype, based on a comparison to 500 randomized molecules. The functional taxonomy is determined by the Rfam database.

Figure 2. Phenotype abundance distributions for all fitness landscapes.

The graph shows the phenotype abundances (y-axis) for each phenotype, ranked in order of abundance (x-axis). The most common phenotype is rank 1, the second most common is rank 2, and so on. Also shown is the distribution of abundances for the 12-mer RNA landscape (at left), along with some representative structures from this landscape.

Figure 3. Simple mutational networks.

(A) a two-locus, two-allele network and (B) a more complex (hypothetical) mutational network. The lower networks show mutational connections among genotypes; vertices are unique genotypes and edges are point mutations. Colored edges represent neutral mutations, which connect genotypes with the same phenotype (color); black edges represent non-neutral mutations, which lead to a change in phenotype. The middle networks show mutational connections among phenotypes. The size of a phenotype vertex is proportional to the number of genotypes that produce it. Pairs of vertices are connected if there is at least one point mutation that converts one phenotype to the other. The top networks show possible fitness landscapes in which each phenotype is assigned a fitness value, indicated in grayscale.

Figure 4. Mutational connectivity among RNA phenotypes.

(Top) The Astatistic (described in text) indicates the likelihood that a given phenotype will arise through point mutation. Random mutations are more likely to hit upon larger neutral networks that smaller neutral networks (r ² = 0.886, P<2.2×10⁻¹⁶; calculated on log-transformed data). (Middle) The E statistic (described in text) indicates the likelihood of given phenotype will produce diverse alternative phenotypes upon mutation. Point mutations to sequences in large neutral networks are less likely to yield novelty than point mutations to sequences in small neutral networks (r ² = 0.265, P = 3.56×10⁻⁵). (Bottom) The B statistic (described in text) suggests that point mutations to abundant phenotypes create other abundant phenotypes (r ² = 0.559, P = 1.58×10⁻¹¹).

Figure 5. Network connectivity correlates with mutation frequency in the 12-mer fitness landscape.

The rates of mutation between phenotype i and phenotype j in simulations is nearly identical to the fraction of nonneutral mutations to i that produce j (f_ij ). The top pane depicts this correlation for an abundant phenotype (rank 2, 218567 sequences), whereas the bottom pane shows this for a small neutral network (rank 47, 800 sequences). The mean slope of the regression line (taken over all 52 of 59 neutral networks that arose in simulation) was r ² = 0.978 with 95% confidence interval [0.945, 1.011], which is statistically indistinguishable from one.

Figure 6. Stochastic evolutionary simulation in the 12-mer fitness landscape.

(A) The phenotype abundance of the target strongly affects the success of adaptation (r = 0.76, P = 2.2×10⁻⁴). (B) The phenotype abundance at the start of the simulation has no effect on the evolutionary outcome (r = −0.023, P = 0.17). We simulated adaptation over one million generations with a genomic mutation rate of U = 0.0003 and a constant population size of N = 1000.

Figure 7. Populations exploring the 12-mer fitness landscape.

(A) The number of appearances of a phenotype is strongly correlated with the abundance of that phenotype (r = 0.92, P = 2.2×10⁻¹⁶, calculated on logtransformed data). (B) The total number of time steps that a phenotype occurs in the evolving populations is positively correlated with its abundance (r = 0.75, P = 1.5×10⁻¹¹, calculated on logtransformed data).

Figure 8. Calculation of the contiguity statistic.

(A) Sample calculations of the contiguity statistic for three 18-mer shapes. (B) The contiguity statistic is strongly correlated with abundance for all lengths of RNA molecules studied; example shown is the 18-mer land-scape (r ² = 0.69, P<2.2×10⁻¹⁶). Minimal Gaussian noise was added to reduce granularity in the data.