arxiv: 2604.22611 · v1 · submitted 2026-04-24 · 🧬 q-bio.PE · cond-mat.dis-nn

Recognition: unknown

Simple sign epistasis and evolutionary detours in fitness landscapes

Alejandro Castro, Alejandro Lage-Castellanos, Alisa Sergeeva, Guillaume Achaz, Joachim Krug, Luca Ferretti, Mahan Ghafari, Paolo Ribeca, Ruben Gustavo Paccosi, Sebastian Matuszewski, Vitaly Belik

Pith reviewed 2026-05-08 08:48 UTC · model grok-4.3

classification 🧬 q-bio.PE cond-mat.dis-nn

keywords sign epistasisfitness landscapesevolutionary detoursback-mutationsepistatic interactionsweak epistasisevolutionary trajectories

0 comments

The pith

Simple sign epistasis forces evolutionary detours that include back-mutations to reach fitness peaks.

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

In fitness landscapes, the effect of a mutation can depend on the presence of other mutations. Simple sign epistasis occurs when the fitness impact of one mutation flips from beneficial to deleterious (or the reverse) depending on whether a second mutation is already present. The paper proves that this interaction creates indirect paths to higher peaks, paths that must reverse an earlier mutation before progressing. In real experimental landscapes that show only weak overall epistasis, simple sign epistasis appears far more often than reciprocal sign epistasis. The pattern matches predictions from most theoretical models of fitness landscapes.

Core claim

The presence of simple sign epistasis is associated with evolutionary detours, that is, indirect longer fitness-increasing paths to fitness peaks that include back-mutations. In experimentally resolved weakly epistatic landscapes, simple sign epistasis occurs much more frequently than reciprocal sign epistasis. This result is consistent with theoretical predictions derived for most landscape models, with the exception of the block model and of landscapes dominated by pairwise allelic incompatibilities such as RNA stability landscapes. The results indicate that detours represent a general feature of evolutionary trajectories in weakly epistatic landscapes.

What carries the argument

Simple sign epistasis, the case in which only one of two mutations changes the sign of its fitness effect depending on the presence of the other, which imposes conditional dependencies that block direct mutational routes and require reversals.

If this is right

Evolutionary trajectories to higher fitness must often include reversals of prior mutations, making paths longer than the shortest mutational distance.
Direct routes between genotypes can be inaccessible, requiring exploration of additional genetic backgrounds before ascent.
Simple sign epistasis is expected to predominate over reciprocal sign epistasis in weakly epistatic regimes, rendering detours a common outcome.
Most theoretical landscape models predict the observed excess of simple sign epistasis, except in special cases such as block models or RNA stability landscapes.

Where Pith is reading between the lines

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

Accounting for possible back-mutation steps may be necessary to predict the order and timing of mutations observed in microbial evolution experiments.
The prevalence of detours could reduce the predictability of which genetic backgrounds will ultimately fix in adapting populations.
Similar detour dynamics may appear in other weakly epistatic systems such as protein sequence evolution where sign changes have been documented.

Load-bearing premise

That the experimentally resolved landscapes are representative of weakly epistatic regimes and that the theoretical models capture the dominant features of real fitness landscapes without strong selection biases or measurement artifacts.

What would settle it

An experimentally measured weakly epistatic landscape in which the frequency of simple sign epistasis is not substantially higher than reciprocal sign epistasis and in which evolutionary trajectories to peaks lack back-mutations would falsify the claimed association.

Figures

Figures reproduced from arXiv: 2604.22611 by Alejandro Castro, Alejandro Lage-Castellanos, Alisa Sergeeva, Guillaume Achaz, Joachim Krug, Luca Ferretti, Mahan Ghafari, Paolo Ribeca, Ruben Gustavo Paccosi, Sebastian Matuszewski, Vitaly Belik.

**Figure 1.** Figure 1: Above: landscapes with 2 loci and 2 alleles per locus with different types of epistasis. Below: the corresponding fitness graphs. Arrows in the graph point the direction in of the higher fitness, not the sequence of mutations. Mutations go from genotype g (bottom left) to g[1,2] (up right) either by applying allele flips ∆1 (blue, horizontal) and then ∆2 (yellow vertical), or in reverse order ∆2 first, the… view at source ↗

**Figure 2.** Figure 2: Comparison between theory and simulations for the fractions view at source ↗

**Figure 3.** Figure 3: Comparison between theory and simulations for the fractions view at source ↗

**Figure 4.** Figure 4: (a) Empirical fraction of RSE motifs versus all motifs involving sign epistasis, i.e. ϕrs ϕss+ϕrs , computed for empirical (sub)landscapes of different size L = 3, 4, 5 as a function of the estimated epistasis 1 − γ; the moving average of ϕrs ϕss+ϕrs is shown by the dashed line. (b) Comparison of SSE and RSE for empirical landscapes with theoretical predictions for unstructured Gaussian landscapes. The poi… view at source ↗

**Figure 5.** Figure 5: Theoretical predictions for the fractions view at source ↗

**Figure 6.** Figure 6: Theoretical predictions for the fractions view at source ↗

**Figure 7.** Figure 7: Statistical analysis of the computational RNA landscape reconstructed by ViennaRNA for the view at source ↗

read the original abstract

In epistatic fitness landscapes, the fitness effect of a mutation depends on the genetic background and may even switch between deleterious and beneficial depending on the presence of another mutation. Epistatic interactions may cause both mutations to change the sign of each other's fitness effects (reciprocal sign epistasis) or only one mutation to do so (simple sign epistasis). Both these forms of epistasis influence evolutionary trajectories. While reciprocal sign epistasis has been associated with multi-peaked landscapes and their ruggedness, the role and relative frequency of simple sign epistasis in fitness landscapes have not been systematically investigated. Here, we prove that the presence of simple sign epistasis is associated with evolutionary detours, i.e., indirect, longer fitness-increasing paths to fitness peaks that include back-mutations. We also show that in experimentally resolved, weakly epistatic landscapes, simple sign epistasis occurs much more frequently than reciprocal sign epistasis. This result is consistent with the theoretical predictions we derive for most landscape models, with the exception of the block model and of landscapes dominated by pairwise allelic incompatibilities, such as RNA stability landscapes. Our results suggest that detours represent a general feature of evolutionary trajectories in weakly epistatic 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.

Desk Editor's Note private letter to a colleague

Simple sign epistasis links to detours via a clean proof and appears far more common than reciprocal sign epistasis in the experimental cases, but the frequency claim needs checks on dataset selection.

read the letter

This paper shows that simple sign epistasis produces evolutionary detours—longer paths to peaks that require back-mutations—and that this interaction type occurs much more often than reciprocal sign epistasis in the weakly epistatic landscapes they examined. The proof part is the clearest advance. They derive the association directly from standard landscape models and show it holds except in specific cases like the block model or RNA stability landscapes. That gives a straightforward theoretical connection without extra assumptions. The frequency comparison in real data is also new enough to be useful, as it moves the discussion from theory alone to a systematic check against experiments. The work is direct and stays within the models and datasets it uses. The main soft spot is the empirical frequency result. It depends on the selected experimental landscapes being representative of weakly epistatic regimes. If those landscapes were resolved because fitness effects were detectable or if sign flips are sensitive to measurement thresholds, the reported difference could be narrower in a broader sample. The paper notes the exceptions but does not test how often real data fall into the block or incompatibility regimes. This is for readers who study fitness landscapes, adaptation in microbes or viruses, and how epistasis shapes trajectories. It adds a usable distinction between epistasis types without claiming to explain all cases. The thinking is clear and the math stays grounded in the models. It deserves a serious referee to confirm the proof steps and to probe whether the data selection affects the frequency claim. I would send it for peer review with that focus.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the presence of simple sign epistasis in fitness landscapes is associated with evolutionary detours—indirect, longer fitness-increasing paths to peaks that include back-mutations. It further reports that simple sign epistasis occurs much more frequently than reciprocal sign epistasis in experimentally resolved, weakly epistatic landscapes, a pattern consistent with theoretical predictions derived for most landscape models (with noted exceptions for the block model and RNA stability landscapes dominated by pairwise allelic incompatibilities).

Significance. If the proof is gap-free and the empirical frequency result is robust to data selection, this establishes detours as a general feature of trajectories in weakly epistatic regimes rather than a rare phenomenon tied only to reciprocal sign epistasis and ruggedness. The work derives parameter-free theoretical predictions across standard models and contrasts them with empirical data, providing a falsifiable link between a common epistatic interaction and path structure that could guide future experimental tests in microbial and viral systems.

major comments (2)

[results on experimental landscapes] Empirical frequency claim (results section on experimental landscapes): the assertion that simple sign epistasis occurs 'much more frequently' than reciprocal sign epistasis rests on a small set of resolved landscapes whose selection criteria, measurement-error thresholds for sign flips, and exclusion rules are not fully specified; without these, it is impossible to rule out that the reported difference is an artifact of detection biases rather than a general feature.
[discussion of model exceptions] Theoretical consistency statement (discussion of model exceptions): while the block model and RNA stability landscapes are correctly identified as exceptions, the manuscript does not quantify the fraction of real data that fall into these regimes or provide a statistical test showing that the majority of weakly epistatic landscapes align with the other models, weakening the claim that the empirical pattern is 'consistent with theoretical predictions for most landscape models'.

minor comments (2)

[theoretical derivations] Notation for fitness effects and sign changes could be standardized across the theoretical derivations and empirical tables to avoid ambiguity in comparing models.
[path diagrams] Figure legends for path diagrams should explicitly state the number of mutations and whether back-mutations are required for the detour.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have improved the clarity of our work. We address each major comment point by point below and indicate revisions made to the manuscript.

read point-by-point responses

Referee: Empirical frequency claim (results section on experimental landscapes): the assertion that simple sign epistasis occurs 'much more frequently' than reciprocal sign epistasis rests on a small set of resolved landscapes whose selection criteria, measurement-error thresholds for sign flips, and exclusion rules are not fully specified; without these, it is impossible to rule out that the reported difference is an artifact of detection biases rather than a general feature.

Authors: We thank the referee for highlighting this issue. The original manuscript referenced standard datasets from the literature but did not provide exhaustive methodological details. In the revised manuscript, we have added a dedicated subsection in the Methods describing: the exact sources and inclusion criteria for the experimental landscapes (publicly available microbial and viral datasets with sufficient mutational resolution and classified as weakly epistatic via low epistasis measures such as roughness-to-slope ratio); the sign-flip detection thresholds (fitness effect sign changes must exceed twice the reported measurement error); and exclusion rules (omission of high-noise or incomplete landscapes). These additions enable readers to evaluate potential biases directly. The frequency difference is observed consistently across the qualifying datasets. revision: yes
Referee: Theoretical consistency statement (discussion of model exceptions): while the block model and RNA stability landscapes are correctly identified as exceptions, the manuscript does not quantify the fraction of real data that fall into these regimes or provide a statistical test showing that the majority of weakly epistatic landscapes align with the other models, weakening the claim that the empirical pattern is 'consistent with theoretical predictions for most landscape models'.

Authors: We agree that a quantitative meta-analysis of the prevalence of block-model or pairwise-incompatibility-dominated regimes across all published fitness landscapes is not provided and would require substantial additional data collection beyond the scope of this study. We have revised the Discussion to explicitly acknowledge this limitation, to reiterate that the block model and RNA stability cases represent specific mechanistic regimes, and to note that the majority of the weakly epistatic empirical landscapes we examined (from microbial and viral systems) align with the standard models for which the prediction holds. The consistency statement is therefore framed as applying to the predominant classes of models and data rather than a universal claim; we view this as a fair characterization while highlighting the need for future surveys. revision: partial

Circularity Check

0 steps flagged

No circularity: proof and predictions derived independently from model definitions and external data

full rationale

The paper's core result is a mathematical proof that simple sign epistasis implies evolutionary detours in fitness-increasing paths, derived from standard definitions of sign epistasis and landscape structure. Theoretical predictions are obtained by direct analysis of common models (NK, block, RNA) without fitting parameters to the experimental frequency data. The empirical frequency comparison uses independently resolved experimental landscapes as input and reports consistency as an observation rather than a forced outcome. No self-citations are invoked as load-bearing uniqueness theorems, no fitted inputs are relabeled as predictions, and no ansatz or renaming reduces the derivation to its inputs by construction. The chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters or invented entities; the work relies on standard assumptions of fitness landscape models.

axioms (1)

domain assumption Fitness effects of mutations can be decomposed into additive and epistatic components
Implicit foundation for distinguishing sign epistasis types.

pith-pipeline@v0.9.0 · 5553 in / 1136 out tokens · 31043 ms · 2026-05-08T08:48:23.125353+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 4 canonical work pages · 1 internal anchor

[1]

https://doi.org/10.1016/j

[Ban22] Claudia Bank. “Epistasis and adaptation on fitness landscapes”. In:Annual review of ecology, evolution, and systematics53.1 (2022), pp. 457–479. [Bro+15] Sophie Brouillet et al. “MAGELLAN: a tool to visualize small fitness landscapes”. In:bioRxiv(2015). [CGB13] Kristina Crona, Devin Greene, and Miriam Barlow. “The peaks and geometry of fitness lan...

work page doi:10.1016/j 2022
[2]

Measuring epistasis in fitness landscapes: The correlation of fitness effects of mutations , Volume =

[Dia+23] Juan Diaz-Colunga et al. “Global epistasis on fitness landscapes”. In:Philosophical Transactions of the Royal Society B: Biological Sciences378.1877 (2023). [DK14] J Arjan Gm De Visser and Joachim Krug. “Empirical fitness landscapes and the predictability of evolution”. In:Nature Reviews Genetics15.7 (2014), pp. 480–490. 20 [Dra93] John W Drake. ...

work page doi:10.1016/j.jtbi.2016.01.037 2023
[3]

The expected time to cross extended fitness plateaus

[GW19] Mahan Ghafari and Daniel B Weissman. “The expected time to cross extended fitness plateaus”. In:Theoretical Population Biology129 (2019), pp. 54–67. [HPK17] Sungmin Hwang, Su-Chan Park, and Joachim Krug. “Genotypic complexity of Fisher’s geometric model”. In:Genetics206.2 (2017), pp. 1049–1079. [Hwa+18] Sungmin Hwang et al. “Universality classes of...

2019
[4]

Epistasis--the essential role of gene interactions in the structure and evolution of genetic systems , Volume =

[NSK14] Johannes Neidhart, Ivan G Szendro, and Joachim Krug. “Adaptation in tunably rugged fitness landscapes: the rough Mount Fuji model”. In:Genetics198.2 (2014), pp. 699–721. 21 [Pap+23] Andrei Papkou et al. “A rugged yet easily navigable fitness landscape”. In:Science 382.6673 (2023), eadh3860. [Phi08] Patrick C Phillips. “Epistasis–the essential role...

work page doi:10.1038/nrg2452 2014
[5]

Structure of the ribosome-bound cricket paralysis virus IRES RNA

[Sch+06] Martin Sch¨ uler et al. “Structure of the ribosome-bound cricket paralysis virus IRES RNA”. In:Nature structural & molecular biology13.12 (2006), pp. 1092–1096. [SH99] Peter F Stadler and Robert Happel. “Random field models for fitness landscapes”. In:Journal of Mathematical Biology38.5 (1999), pp. 435–478. [Sil+10] Jack da Silva et al. “Fitness ...

2006
[6]

Evolutionary Accessibility of Modular Fitness Land- scapes

[SK14] B Schmiegelt and J Krug. “Evolutionary Accessibility of Modular Fitness Land- scapes”. In:Journal of Statistical Physics154 (2014), pp. 334–355. 22 [SKK22] Raimundo Saona, Fyodor A Kondrashov, and Ksenia A Khudiakova. “Relation be- tween the number of peaks and the number of reciprocal sign epistatic interactions”. In:Bulletin of Mathematical Biolo...

2014
[7]

On the incongruence of genotype-phenotype and fitness landscapes

[SP22] Malvika Srivastava and Joshua L Payne. “On the incongruence of genotype-phenotype and fitness landscapes”. In:PLoS computational biology18.9 (2022), e1010524. [Sri+26] Malvika Srivastava et al.Evolution as fitness landscape navigation: Concepts, Mea- sures, and Emerging Questions

2022
[8]

Evolution as fitness landscape navigation: Concepts, Measures, and Emerging Questions

arXiv:2604.17036 [q-bio.PE].url:https: //arxiv.org/abs/2604.17036. [Ste92] DL Stein, ed.Spin Glasses and Biology. Singapore: World Scientific,

work page internal anchor Pith review Pith/arXiv arXiv
[9]

Quantitative analyses of empirical fitness landscapes

[Sze+13] Ivan G Szendro et al. “Quantitative analyses of empirical fitness landscapes”. In: Journal of Statistical Mechanics: Theory and Experiment(2013). [Tan+11] Longzhi Tan et al. “Hidden randomness between fitness landscapes limits reverse evolution”. In:Physical review letters106.19 (2011), p. 198102. [VHE97] J A de Visser, R F Hoekstra, and H van de...

2013
[10]

The highly rugged yet navigable regulatory landscape of the bacterial transcription factor TetR

[WGW24] Cau˜ a Antunes Westmann, Leander Goldbach, and Andreas Wagner. “The highly rugged yet navigable regulatory landscape of the bacterial transcription factor TetR”. In:Nature Communications15.1 (2024), p. 10745. [Wri32] S Wright. “The roles of mutation, inbreeding, crossbreeding and selection in evolu- tion”. In:Proc. 6th Int. Cong. Genet.1 (1932), p...

2024