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arxiv: 2605.03046 · v1 · submitted 2026-05-04 · 🧬 q-bio.PE · cond-mat.dis-nn

Recognition: unknown

Epistatic strength, modularity, and locus heterogeneity shape the number of local optima in fitness landscapes

Alejandro Castro Cabrera, Alejandro Lage-Castellanos, Guillaume Achaz, Joachim Krug, Luca Ferretti, Mahan Ghafari

Pith reviewed 2026-05-08 01:54 UTC · model grok-4.3

classification 🧬 q-bio.PE cond-mat.dis-nn
keywords fitness landscapesepistasislocal optimaGaussian random fieldsreciprocal sign epistasismodularitylocus heterogeneityevolutionary trajectories
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The pith

The expected number of local optima in unstructured fitness landscapes is fixed by the correlation of fitness effects.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when fitness landscapes are modeled as isotropic Gaussian random fields, a single local statistic—the correlation of fitness effects—sets the expected number of peaks. This correlation directly tracks the prevalence of reciprocal sign epistasis and therefore controls how often an evolving population will be trapped on a suboptimal peak rather than reaching the global maximum. The same overall strength of epistasis can produce very different peak counts once interactions are allowed to cluster into modules or to concentrate on only a few loci. Clustering modestly raises the number of optima, while concentrating epistasis on a small subset of loci collapses it. These relations supply a concrete baseline for the range of ruggedness expected in typical unstructured landscapes.

Core claim

For a broad class of unstructured fitness landscapes modeled as isotropic Gaussian random fields, the expected number of local optima is determined by a single local measure of epistasis: the correlation of fitness effects. This measure links peak density directly to the amount of reciprocal sign epistasis. When epistatic interactions are structured by clustering within blocks of loci, the number of local optima increases slightly; strong heterogeneity that restricts epistasis to a small subset of loci causes the number of peaks to collapse. The results reconcile the central role of reciprocal sign epistasis with the observation that landscapes sharing similar overall epistasis can still be

What carries the argument

The correlation of fitness effects, which quantifies how the fitness impact of a mutation at one locus depends on the state at another and serves as the sole parameter controlling peak density in the Gaussian random-field model.

If this is right

  • Stronger negative correlation between fitness effects produces more local optima and therefore more trapping of adaptive walks.
  • Clustering epistatic interactions into modular blocks of loci modestly increases the expected number of peaks relative to the unstructured baseline.
  • Locus heterogeneity that confines epistasis to a small fraction of sites sharply reduces the number of local optima.
  • The model supplies a quantitative baseline for peak numbers in typical unstructured landscapes once the correlation of fitness effects is measured.
  • Landscapes with comparable overall epistasis can still differ widely in ruggedness depending on how the interactions are distributed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Measuring correlation of fitness effects in laboratory or natural populations could predict evolutionary accessibility without exhaustive mapping of the full landscape.
  • Genomes with high modularity may therefore impose more trapping than random-interaction models suggest.
  • Experiments that systematically vary the degree of locus heterogeneity while holding overall epistasis fixed could test whether peak counts collapse as predicted.

Load-bearing premise

Real fitness landscapes can be usefully approximated by isotropic Gaussian random fields in which the correlation of fitness effects fully captures the epistatic structure that matters for counting local optima.

What would settle it

An empirical fitness landscape whose measured correlation of fitness effects predicts a certain number of local optima but whose actual enumeration of peaks deviates substantially from that prediction.

Figures

Figures reproduced from arXiv: 2605.03046 by Alejandro Castro Cabrera, Alejandro Lage-Castellanos, Guillaume Achaz, Joachim Krug, Luca Ferretti, Mahan Ghafari.

Figure 1
Figure 1. Figure 1: An illustration of the argument that local epistasis, not roughness, view at source ↗
Figure 2
Figure 2. Figure 2: Correlation between the number of peaks/sinks and several mea view at source ↗
Figure 3
Figure 3. Figure 3: Expected number of local optima for unstructured landscapes view at source ↗
Figure 4
Figure 4. Figure 4: Expected number of local optima for unstructured landscapes with view at source ↗
Figure 5
Figure 5. Figure 5: Exponential scaling with L of the density of peaks, πpeaks ∼ e λL. Exact results for λ are shown in view at source ↗
Figure 6
Figure 6. Figure 6: Number of maxima versus measures of local epistasis in experi view at source ↗
Figure 7
Figure 7. Figure 7: Expected number of local optima for landscapes with clustered view at source ↗
Figure 8
Figure 8. Figure 8: Exponential scaling with L of the density of peaks, πpeaks ∼ e λL for clustered interactions. for k ≫ 1. For finite values of heterogeneity h, the number of optima can increase or decrease compared to unstructured landscapes ( view at source ↗
Figure 9
Figure 9. Figure 9: Expected number of local optima for landscapes with heterogene view at source ↗
Figure 10
Figure 10. Figure 10: Exponential scaling with L of the density of peaks, πpeaks ∼ e λL for landscapes with heterogeneity between loci. There are landscapes with peculiar epistatic structures that are even more extreme in this respect, such as NK models with star structure [Hwang et al., 2018] where the number of peaks remains finite for L → ∞. 3.4.3 Full model: The formula for the expected number of peaks with both clustering… view at source ↗
Figure 11
Figure 11. Figure 11: Exponential scaling with L of the density of peaks, πpeaks ∼ e λL for landscapes with both clustering of interactions and heterogeneity between loci. and the exponential dependence of the density of peaks on L is πpeaks ∼ e λL with λ = − h 1 + h log A + β 1 + h log Z ∞ −∞ dyφ(y)H(y) 1/β (29) and it is shown in view at source ↗
read the original abstract

Fitness landscapes provide a quantitative framework for understanding how natural selection shapes evolutionary trajectories. A central feature of these landscapes is their number of local optima, which determines whether fitness-increasing evolution can proceed towards a global optimum or become trapped on suboptimal peaks. Although multiple peaks are known to require reciprocal sign epistasis, the quantitative relationship between epistasis and number of peaks remains incompletely understood. Here, we show that for a broad class of unstructured fitness landscapes, i.e. isotropic Gaussian random fields, the expected number of local optima is determined by a single local measure of epistasis: the correlation of fitness effects. This provides a baseline prediction for the number of peaks in typical unstructured landscapes and links peak density directly to the amount of reciprocal sign epistasis. This baseline changes when epistatic interactions are structured. We show that clustering interactions within blocks of loci slightly increases the number of local optima. In contrast, strong heterogeneity between loci, where only a small subset of loci participate in epistatic interactions, causes the number of peaks to collapse. These results show that the number of local optima is governed not only by the overall strength of epistasis, but also by how epistatic interactions are distributed across the genotype space. Our framework therefore reconciles the central role of reciprocal sign epistasis with the observation that landscapes with similar amounts of epistasis can differ substantially in ruggedness, and provides a guide to the range of peak numbers expected in typical landscapes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that for isotropic Gaussian random field fitness landscapes on the hypercube, the expected number of local optima equals 2^L times the negative orthant probability of the L-dimensional MVN whose components are the mutational fitness effects, which are exchangeable and equicorrelated with off-diagonal correlation rho determined solely by the covariance at Hamming distances 0, 1 and 2. It further shows that clustering epistatic interactions into modular blocks slightly raises this count while strong locus heterogeneity (only a small subset of loci participating in epistasis) causes the count to collapse, thereby linking peak density to both the overall strength and the spatial distribution of epistasis.

Significance. If the derivations hold, the result supplies a clean, parameter-light baseline for the number of peaks in unstructured landscapes and demonstrates how interaction structure modulates ruggedness even at fixed overall epistasis. The use of linearity of expectation on local-optimality indicators together with the MVN orthant probability is a methodological strength that makes the unstructured case fully determined by a single local statistic.

major comments (2)
  1. [Main derivation] Main derivation (unstructured case): the claim that only the correlation of fitness effects enters the peak count is correct for isotropic GRFs, but the manuscript should explicitly derive the value of rho from the underlying covariance function evaluated at distances 0, 1 and 2 and state the precise isotropy assumption required for the deltas to remain exchangeable.
  2. [Structured landscapes] Structured landscapes section: the statements that modularity 'slightly increases' and heterogeneity 'causes collapse' of the peak count need to be accompanied by the exact block sizes, participation fractions, and whether these results follow from an analytic extension of the MVN construction or from Monte Carlo sampling; without these details the quantitative claims cannot be verified.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'reciprocal sign epistasis' should be tied explicitly to the sign pattern of the vector of mutational effects delta_i(x) in the main text.
  2. Notation: define the correlation of fitness effects (rho) in a dedicated equation early in the methods so that later references to it are unambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive suggestions, which will improve the clarity of the derivations and results. We address each major comment below and will incorporate the requested details in the revised manuscript.

read point-by-point responses

  1. Referee: Main derivation (unstructured case): the claim that only the correlation of fitness effects enters the peak count is correct for isotropic GRFs, but the manuscript should explicitly derive the value of rho from the underlying covariance function evaluated at distances 0, 1 and 2 and state the precise isotropy assumption required for the deltas to remain exchangeable.

    Authors: We agree that an explicit derivation will strengthen the presentation. In the revised manuscript we will add a dedicated paragraph deriving ρ directly from the covariance function evaluated at Hamming distances 0, 1 and 2, and we will state the isotropy assumption (invariance under permutation of loci together with dependence only on Hamming distance) that guarantees exchangeability of the mutational fitness-effect vector. This makes transparent why the orthant probability, and hence the expected number of local optima, depends only on this single local statistic. revision: yes

  2. Referee: Structured landscapes section: the statements that modularity 'slightly increases' and heterogeneity 'causes collapse' of the peak count need to be accompanied by the exact block sizes, participation fractions, and whether these results follow from an analytic extension of the MVN construction or from Monte Carlo sampling; without these details the quantitative claims cannot be verified.

    Authors: We accept that the current text is insufficiently precise on these points. The modularity and heterogeneity results are obtained by Monte Carlo sampling of the corresponding multivariate normal distributions (not by analytic extension of the unstructured MVN formula). In the revision we will report the exact block sizes (e.g., equal-sized modules of k loci), the participation fractions (e.g., fraction of loci that participate in epistatic interactions), the number of Monte Carlo replicates, and the sampling procedure used to estimate the orthant probabilities. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives that, for isotropic Gaussian random fields on the hypercube, the expected number of local optima equals 2^L times the negative orthant probability of an L-dimensional MVN whose off-diagonal correlation rho is fixed by the field's covariance at Hamming distances 0, 1 and 2. This follows directly from linearity of expectation applied to the indicator that a genotype is a local optimum (sign pattern of its mutational effects) and the exchangeability of the jointly Gaussian effects; no parameters are fitted to data, no self-citation chain is load-bearing for the central reduction, and the correlation of fitness effects is an input statistic computed from the covariance rather than a quantity defined in terms of the peak count itself. The result is therefore a genuine simplification, not a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the modeling choice of isotropic Gaussian random fields and the definition of the correlation-of-fitness-effects statistic; no new entities are postulated.

free parameters (1)
  • correlation of fitness effects
    Local epistasis statistic that the derivation shows determines expected peak number; treated as an input measure rather than a fitted constant.
axioms (1)
  • domain assumption Fitness landscapes belong to the class of isotropic Gaussian random fields
    Invoked to obtain the baseline prediction for unstructured landscapes.

pith-pipeline@v0.9.0 · 5584 in / 1158 out tokens · 63620 ms · 2026-05-08T01:54:47.694874+00:00 · methodology

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Reference graph

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