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Evolutionary accessibility of mutational pathways
Jasper Franke, Alexander Klözer, J. Arjan G. M. de Visser, Joachim Krug
TL;DR
The paper investigates whether epistasis makes global fitness optima inaccessible and studies this question using shortest, monotonically increasing paths on model and empirical fitness landscapes. It finds that accessibility remains high while alternative accessible pathways proliferate, although the analysis does not determine which paths populations will actually discover.
Problem
The paper asks whether epistasis makes the global fitness optimum selectively inaccessible, a question difficult to resolve because complete genome-wide fitness landscapes remain elusive.
Method
The authors analyze shortest monotonically increasing paths from antipodal genotypes to global optima across model landscapes and an empirical Aspergillus niger landscape.
Results
The number of accessible pathways grows much faster with landscape dimensionality than per-path inaccessibility, while model behavior is also examined across HoC, RMF, and holey landscapes.
Takeaways & Limitations
Under the paper’s accessibility definition, global optima remain broadly accessible and alternative pathways proliferate, with the empirical Aspergillus niger analysis supporting this pattern.
Takeaways & Limitations
The analysis addresses whether accessible pathways exist, not the probability that a population will find a particular pathway under specific population dynamics.
Abstract
from arXiv · showhide
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 pathwayexists 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 bound. We test the model predictions against an empirical 8-locus fitness landscape obtained for the filamentous fungus \textit{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 very robust, which we argue to reflect the generic topology of sequence spaces.
Introduction
The paper asks whether epistasis makes global fitness optima inaccessible and defines accessibility through shortest, monotonically increasing mutational paths. It analyzes this question using fitness-landscape models and empirical landscapes, while focusing on whether paths exist rather than whether populations discover them.
- Motivation: Epistatic interactions can make fitness landscapes rugged because mutations may combine to produce effects unlike their individual effects.Such interactions include combinations that are advantageous or deleterious despite little individual effect.
- Motivation: Existing experiments typically examine small, carefully selected mutation sets, whereas natural selection acts across broader genomic variation.The largest landscapes described in the supplied passage involve five mutations.
- Research question: The paper asks whether epistasis makes the global fitness optimum selectively inaccessible, a question motivated by the difficulty of exploring whole fitness landscapes experimentally.The authors frame theoretical analysis as necessary while genome-wide landscape exploration remains elusive.
- Research question: Fisher’s pathway-proliferation view and Wright’s local-maximum view make opposing predictions about accessibility as genotype-space dimensionality increases.The paper argues that both intuitions hold qualitatively, but Fisher’s scenario prevails under its quantitative accessibility definition.
- Framework: Genotypes are represented as binary sequences forming an L-dimensional hypercube, and accessible paths are shortest paths whose fitness increases at every step.Two states differing at l loci have l! shortest paths, corresponding to mutation orders.
- Framework: The analysis focuses on paths from the antipodal genotype to the global maximum, which are longest direct paths and therefore provide a lower limit on accessibility of typical paths.The mean path length from a randomly chosen genotype to the global maximum is L/2.
- Framework: The study evaluates both whether at least one accessible path exists and how many accessible paths occur, linking the latter to evolutionary repeatability.Different pathways may be chosen in replicate experiments depending on population dynamics.
Results
Across correlated, epistatic, and holey landscape models, accessibility generally increases with dimensionality, whereas the uncorrelated House of Cards model is the exception. Empirical Aspergillus niger subgraphs support correlated and epistatic model predictions, although their sizes do not yet reveal the predicted eventual decline in the probability of no accessible path.
- Rough Mount Fuji model: ⟨nL⟩ grows like L^2 for large L at constant θ, showing that even slight fitness correlations alter the large-L behavior relative to HoC.The RMF model reduces to HoC when θ = 0.
- Rough Mount Fuji model: For any θ > 0 in the RMF model, the probability of finding at least one accessible path is predicted to increase for large L, unlike the HoC model.Simulations suggest pL(0) decreases for large L and most likely approaches zero.
- LK model: In the LK model, pL(0) decreases monotonically when K is fixed, but behaves non-monotonically when the interacting-locus fraction K/L is fixed.When L − K is fixed, behavior resembles the HoC limit and pL(0) increases with L.
- Holey landscapes: For holey landscapes, the no-path probability is conjectured to vanish at large L for any p > 0, while the expected number of accessible paths grows without bounds.The probability of a specific path is p^L, but L! possible paths grow faster than p^L declines.
- Comparison with empirical data: In A. niger subgraphs, the average number of accessible paths increases with mutational distance m, ruling out uncorrelated fitness values and matching RMF and LK predictions.The data are consistent with RMF θ ≈ 0.25 and, for even m, LK with L = m and K = m/2.
- Comparison with empirical data: For m = 4, p4(0) is approximately 0.5, and p_m(0) increases through m = 6, so the models’ eventual accessibility increase is not yet visible empirically.This agrees with model estimates placing the maximum in pL(0) at or beyond six loci.
Discussion
Across diverse landscape models, accessibility generally increases with dimensionality: accessible paths become likely and numerous, although pathway counts fluctuate substantially across landscapes. The empirical A. niger data support this broad pattern, while the analysis remains constrained by selection-regime assumptions and sampling biases.
- General model predictions: Except for uncorrelated fitness landscapes, all models predict increasing accessibility with dimensionality and an unbounded expected number of accessible paths.The probability of at least one accessible path is conjectured to approach unity as L becomes large, while many alternative paths limit trajectory repeatability.
- General model predictions: The accessibility pattern is attributed to path combinatorics: exponentially decreasing per-path accessibility is overwhelmed by approximately L! possible shortest paths.The estimate is conservative because the analysis excludes valley crossing by double mutations and considers only shortest paths.
- Empirical variation: Accessibility fluctuates strongly across epistatic landscapes, with some landscapes lacking any accessible path while accessible landscapes often contain many.For 70 m = 4 A. niger subgraphs, half had no accessible path; among the remainder, the mean was 4 paths and two had 10.
- Empirical variation: Empirical analyses of collectively beneficial mutations are biased toward landscapes with larger accessibility and should be compared using distributions conditioned on n ≥1.Whether landscapes composed of beneficial or deleterious mutations share these topographical properties requires further empirical studies.
- The A. niger landscape: The A. niger analysis rules out completely uncorrelated fitness and estimates substantial epistasis, including K ≈ L/2 in the RMF/LK model analysis.The study also reports high intergenic sign epistasis relative to several intragenic examples, though comparisons are confounded by differences in mutation selection.
- Scope and limitations: The study measures whether accessible pathways exist, not the probability that evolution will follow a particular pathway under a specified population scenario.That probability depends on population parameters, especially mutation supply Nu; larger populations introduce additional effects that are difficult to assess.
- Scope and limitations: Increasing mutation supply can make adaptation more deterministic while increasing trapping at local maxima, and it can also enable fitness-valley crossing through multiple mutations.The interplay between landscape structure and population parameters is left for future work.
Materials and Methods
The study combines numerical analyses of model landscapes with an empirical eight-locus Aspergillus niger landscape. It estimates accessibility while accounting for missing strains, lethal genotypes, and measurement uncertainty.
- Numerical simulations: Fitness values were assigned to all 2^L genotypes under HoC, RMF, or LK ensembles, and accessible paths were counted by depth-first backtracking.The search began at the antipodal genotype and proceeded toward the global maximum, backtracking after local maxima.
- Analytic model calculations: For θ ≪ 1, π_L still decays factorially as L increases, although higher-order terms in θ alter this behavior.The HoC case θ = 0 differs from the general θ > 0 case.
- Analytic model calculations: For the Gumbel distribution, π_L decays exponentially as π_L ∼ (1 − e^−c)^L, a behavior expected for most choices of f(x).This result follows because the denominator approaches a constant for large L.
- Empirical dataset: The empirical dataset comprised 186 isolated haploid segregants from 256 possible genotypes, with fitness measured in replicate radial-growth assays.The eight marker mutations occupied one position on each of the eight chromosomes.
- Empirical data analysis: The analysis examined m-locus subgraphs for 2 ≤ m ≤ 6 and compared viable-subgraph and lethal-path predictions with observed accessibility.Viable subgraphs contained no non-viable strains, while lethal-path calculations estimated accessibility attributable solely to lethal states.
- Error analysis: Measurement uncertainty was assessed by resampling viable-genotype fitnesses as normal variables with s_0 ≈ 0.03 and averaging statistics over 10^5 landscapes.Non-viable genotypes retained fitness zero, and reported error bars came from the resampled ensemble.
Tables
Table 1 summarizes empirical subgraphs by size, viability, lethal-path accessibility, mean accessible paths, and the probability of no accessible path.
- Lethal-genotype effects: The table also reports accessible paths remaining after excluding paths containing lethal genotypes, alongside model predictions in brackets.These values isolate the accessibility reduction attributable to non-viable genotypes.
- Accessibility outcomes: Mean accessible paths and the probability of no accessible path are reported as the final two columns for each subgraph size.These measures summarize the empirical accessibility of the A. niger subgraph ensembles.
Figures
The figures compare accessibility across empirical and model fitness landscapes, showing how ruggedness, dimensionality, and model structure affect accessible mutational paths.
- Empirical landscapes: Figure 1 shows empirical landscapes with varied topographies, including one with 9 of 24 accessible paths, one with none, and one with 2.The landscapes contain one, three, and four maxima, respectively.
- House-of-Cards model: Figure 2 shows that House-of-Cards landscapes frequently have no accessible paths at moderate sequence lengths.The zero-path probability is shown separately for unconstrained antipodal sequences and antipodes fixed as global minima.
- Tunable ruggedness: Figure 3 examines the zero-path probability in RMF and LK landscapes as functions of ruggedness parameters and sequence length.The RMF inset rescales data by θL(L −1), while LK results distinguish fixed K/L from fixed K.
- Empirical comparison: Figure 4 compares model predictions with Aspergillus niger data using mean and cumulative distributions of accessible-path counts.Except for HoC, the models show increasing mean accessibility with L, and RMF and LK can be fitted to the empirical data.