Crossover between neutral and strong-selection scaling of evolutionary path statistics on fitness landscapes

For monomorphic evolutionary dynamics on a genotype sequence space of length $L$ over an alphabet of $k$ alleles, the statistics of first-passage adaptive paths from the least-fit to the most-fit genotype obey neutral scaling ( $\bar\ell \sim k^L$ ) for $Ns \lesssim 1$ and strong-selection scaling ( $\bar\ell \sim L\log k$ , Poisson-like lengths) for $Ns \gtrsim 10^2$ , where $N$ is population size and $s$ the total selection coefficient across the landscape. Manhart and Morozov (2015), who established these limits, explicitly posed the nature of the crossover between them as future work; this problem asks what scaling functions of $Ns$ , $L$ , and $k$ control the transition, and how universal they are across landscape topographies and across evolutionary versus finite-temperature physical dynamics. The crossover between these two regimes has not been characterized.

Background

In models of molecular evolution, genotypes are represented as sequences of length $L$ over an alphabet of $k$ alleles, giving a sequence space of $k^L$ states; for proteins, realistic values (binding interfaces of tens of residues, $k = 20$ amino acids) make this space so large that exact computations are feasible only on coarse-grained sequences with reduced length and alphabet. Manhart and Morozov (2015) asked how coarse-graining affects the statistics of adaptive evolutionary paths. They consider a monomorphic population (all individuals share one genotype) evolving by single substitutions: a mutant with selection coefficient $s' = F(\sigma')/F(\sigma) - 1$ fixes with the diffusion-theory probability $\varphi(s') = (1 - e^{-2s'})/(1 - e^{-2Ns'})$ , where $N$ is the population size and the mutant's own selection coefficient enters both exponentials. When $N|s'| \gg 1$ , deleterious mutations never fix and the dynamics reduces to the 'strong-selection weak-mutation' (SSWM) or adaptive-walk limit, which is equivalent to zero-temperature Metropolis Monte Carlo dynamics with energy replaced by negative fitness; when $Ns \ll 1$ , evolution is nearly neutral. The regime is controlled by $Ns$ , where $s$ is the total selection coefficient between the least- and most-fit genotypes. The object of study is the ensemble of first-passage paths from the least-fit genotype to the global fitness maximum, summarized by the path-length distribution $\rho(\ell)$ , its mean $\bar\ell$ and variance $\ell_{var}$ , and the path entropy $S_{path} = -\sum_\phi \Pi[\phi]\log \Pi[\phi]$ , where $\Pi[\phi]$ is the probability of path $\phi$ ; $S_{path}$ measures the predictability of evolution, vanishing when only a single path is accessible. On a non-epistatic Mount Fuji landscape (fitness $F(\sigma) = f^{\sum_j (j-1)n_j(\sigma)}$ , with $n_j(\sigma)$ the number of copies of allele $A_j$ , so fitness increases monotonically toward a single peak), two limiting scalings are known: for $Ns \le 1$ the neutral results $\bar\ell \sim k^L$ and $\ell_{var} \sim k^{2L}$ hold, as for random walks on finite graphs, meaning the population explores essentially the whole sequence space before first reaching the peak; for $Ns \approx 100$ the greedy limit gives $\bar\ell \approx L H_{k-1} \sim L \log k$ (where $H_n$ is the $n$ -th harmonic number, since each site independently climbs its $k$ alleles) with Poisson-like length distributions. At intermediate strength ( $Ns \approx 10$ ), the numerically observed statistics are, in the authors' words, 'not simple functions of sequence space size'. The same framework was applied to a biophysical protein model in which proteins evolve new binding interactions while maintaining folding stability, with additive per-site Gaussian contributions to folding and binding energies; that landscape exhibits magnitude epistasis (mutational effects amplify or dampen depending on background) and limited sign epistasis (a mutation's beneficial or deleterious character depends on background), strongest when folding and binding energies are marginal, which causes quantitative deviations from the Mount Fuji scaling and is the main obstacle to extrapolating the small- $L$ , small- $k$ fits to realistic alphabets. Related crossover phenomena studied elsewhere vary complementary axes: Park, Szendro, Neidhart and Krug (2015) found a phase transition in adaptive-walk length with the steepness of the fitness gradient on rough-Mount-Fuji landscapes (a tunable-ruggedness generalization of the Mount Fuji construction analyzed by Neidhart, Szendro and Krug, 2014) at fixed zero-temperature dynamics, while Jain and Krug (2007) analyzed the crossover between stochastic, mutation-limited regimes (including clonal interference, the competition between concurrently segregating beneficial lineages) and deterministic quasispecies regimes of asexual populations (large populations in which mutation and selection balance deterministically), which concerns polymorphic population dynamics rather than monomorphic first-passage path statistics. The exact transfer-matrix computation of path statistics was generalized to random walks on arbitrary networks with memory by Manhart, Kion-Crosby and Morozov (2015). What no work has characterized is the crossover between the neutral and strong-selection regimes at fixed monomorphic substitution dynamics.

Problem Statement

For monomorphic substitution dynamics with diffusion-theory fixation probabilities on a genotype sequence space of length $L$ over an alphabet of $k$ alleles, characterize the crossover between the neutral and strong-selection regimes of first-passage adaptive path statistics from the least-fit to the most-fit genotype. Specifically: determine the scaling functions of the selection parameter $Ns$ (with $s$ the total selection coefficient across the landscape), $L$ , and $k$ that interpolate the mean path length $\bar\ell$ , its variance $\ell_{var}$ , and the path entropy $S_{path}$ between the neutral scaling $\bar\ell \sim k^L$ , $\ell_{var} \sim k^{2L}$ (valid for $Ns \lesssim 1$ ) and the strong-selection scaling $\bar\ell \sim L H_{k-1} \sim L \log k$ with Poisson-like length distributions (established for $Ns \gtrsim 10^2$ ), including the location and width of the crossover regime as $L$ and $k$ grow at fixed total selection coefficient $s$ (the coarse-graining convention of Manhart and Morozov (2015): since $s = f^{L(k-1)} - 1$ on a Mount Fuji landscape, the per-mutation fitness benefit renormalizes as $f = (1+s)^{1/(L(k-1))}$ when $L$ and $k$ vary); and establish how universal these crossover functions are across landscape topographies (non-epistatic Mount Fuji landscapes and biophysical protein landscapes with magnitude and sign epistasis) and across dynamics (Wright-Fisher substitution dynamics versus finite-temperature Metropolis dynamics, where the crossover is expected at temperatures comparable to the largest energy differences on the landscape). On each landscape class, the path ensemble is defined by the same first-passage process from the least-fit genotype to the global fitness maximum under the diffusion-theory fixation probabilities stated above. The interpolation functions for $\bar\ell$ , $\ell_{var}$ , and $S_{path}$ (including the location and width of the crossover regime as $L$ and $k$ grow at fixed total selection coefficient $s$ ) must be characterized on the non-epistatic Mount Fuji landscape under Wright-Fisher substitution dynamics, and the crossover functions must then be compared across the two landscape classes (non-epistatic Mount Fuji and the epistatic biophysical protein landscape) and across the two dynamics (Wright-Fisher substitution and finite-temperature Metropolis). It is open whether these crossover functions are shared across the two landscape classes and the two dynamics; the outcome may be positive (the functions are shared across both landscape classes and both dynamics), negative (they are not shared), or partial (they are shared for some statistic, landscape class, or dynamics and not for others), and may depend on the statistic, the landscape topography, or the dynamics.

References

Michael Manhart and Alexandre V. Morozov, "Scaling properties of evolutionary paths in a biophysical model of protein adaptation", Physical Biology 12, 045001 (2015). DOI: 10.1088/1478-3975/12/4/045001; arXiv:1410.1493; https://doi.org/10.1088/1478-3975/12/4/045001
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