arxiv: 2604.26082 · v1 · submitted 2026-04-28 · 🧬 q-bio.PE

Recognition: unknown

How is gene-regulatory evolution affected by cell-to-cell variability?

Leonardo Ivan Estrella Dzib , James Holehouse

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:41 UTC · model grok-4.3

classification 🧬 q-bio.PE

keywords gene-regulatory networkscell-to-cell variabilitygene expression noiseevolutionary dynamicsnetwork alignmentfeedforward loopsnetwork robustness

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The pith

Increased cell-to-cell variability selects for aligned and robust gene-regulatory networks enriched in specific motifs.

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

The paper extends a classical model of gene-regulatory network evolution to incorporate noisy developmental dynamics, allowing one network to produce a distribution of phenotypes. It defines an alignment score, drawing from Hopfield network ideas, that measures how signed gene interactions reinforce a target stable state. Simulations show that higher noise levels drive populations toward networks with greater alignment, shorter developmental paths, and optimized fitness. These aligned networks are enriched for coherent feedforward and positive feedback loops and resist mutational changes better than noiseless or non-evolved controls. Alignment works by creating redundancy in reinforcing interactions, turning variability into a selective pressure that favors robustness.

Core claim

When developmental dynamics include noise, selection favors networks in which a larger fraction of gene-gene interactions carry the correct sign to support the target phenotype. This alignment score rises with noise, producing architectures that contain more coherent feedforward loops and positive feedback loops, maintain fitness while shortening paths to stability, and lose less performance when mutated.

What carries the argument

The alignment score that counts the net reinforcing gene-gene interactions supporting a stable target phenotype.

If this is right

Evolved networks reduce the number of steps needed to reach a stable phenotype.
Higher noise increases the fraction of appropriately signed interactions and enriches coherent feedforward and positive feedback motifs.
Aligned networks maintain fitness after random mutations better than non-aligned controls.
Populations under noise reach higher fitness values than equivalent noiseless populations.

Where Pith is reading between the lines

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

Noise may act as a constructive evolutionary force that favors redundancy over precision in regulatory wiring.
Empirical gene-regulatory networks from cells with high expression variability could be scored for alignment to test whether natural selection produces similar patterns.
The model suggests that removing noise experimentally might destabilize certain regulatory motifs that evolved under variable conditions.

Load-bearing premise

The alignment score meaningfully measures fitness under noise and the extended model captures the selective pressures that act on real gene-regulatory networks.

What would settle it

Running the same evolutionary simulations with noise but without rewarding alignment, then checking whether motif enrichment and mutational buffering still appear at the same rate.

Figures

Figures reproduced from arXiv: 2604.26082 by James Holehouse, Leonardo Ivan Estrella Dzib.

**Figure 1.** Figure 1: Summary of the findings of our study. (A) Stochasticity in the regulatory connections comprising the genotype can lead to a diverse set of phenotypes. (B) GRNs evolved in stochastic environments develop mechanisms to express optimal phenotypes even in highly variable environments. (C) Successful GRNs become highly aligned with the optimal phenotype, meaning that very few regulatory connections lead to devi… view at source ↗

**Figure 2.** Figure 2: Overview of the GRN evolution model used in this study. (c) The model studied in the main text builds on refs. [32, 33] in which the iterative equation shown constitutes the developmental dynamics that determine the eventual phenotype expressed by a given genotype. Between evolutionary time steps, elements of the genotype may mutate. (b)–(c) In our adaptation of the model from refs. [32, 33], we assume tha… view at source ↗

**Figure 3.** Figure 3: Examples of motifs in some aligned and non-aligned network motifs. The blue circles represent a positive and the red squares a negative entry in the target phenotype Popt. Positive edges (promoting interactions) are represented with a blue arrow, and negative ones (inhibitory interactions) with a squared red tip. By changing the direction of one edge (marked with a dashed line and an exclamation mark), we … view at source ↗

**Figure 4.** Figure 4: (A) Evolution of the average population fitness and (B) average mean path length per generation, across 30 independent replicates at different noise variances θ. Shaded regions represent 95% confidence intervals around the mean in the zoomed-out panels and 68% in the zoomed-in panels. Lighter colors correspond to smaller noise variances. The noiseless case (θ = 0) is shown with a dotted line. In panel (B),… view at source ↗

**Figure 5.** Figure 5: Evolution of the average alignment scores. (A) Average alignment score across 30 populations over generations. Each line represents the average alignment score for a population, and the color fillings represent 95% confidence intervals. Lighter colors represent smaller noise variances. The noiseless scenario is plotted with a dotted line. (B) Average alignment scores after evolution from 30 evolved populat… view at source ↗

**Figure 6.** Figure 6: Enrichment of network motifs as a result of noiseless evolution. Comparison of 30 populations evolved without noise (darker hues) against non-evolved populations of stable matrices (lighter hues). (A) Average concentration per type of FFL. On the x-axis, we place the type of loop with a diagram. The error bars represent 95% confidence intervals around the averages. (B) Average concentration of positive (gr… view at source ↗

**Figure 7.** Figure 7: Enrichment of network motifs as a result of noisy evolution, compared to noiseless evolution. Comparison of 30 populations evolved with high noise variance (θ = 8; darker hues) and no noise (lighter hues). (A) Average concentration per type of FFL. On the x-axis, we place the type of loop with a diagram. The error bars represent 95% confidence intervals around the averages. (B) Average concentration of pos… view at source ↗

**Figure 8.** Figure 8: Change in mutational robustness across evolved populations before and after evolution. (A) Change in average stable expression shift, (B) average stable expression variance, (C) average instability difference, and (D) instability variance after the evolutionary processes for 30 populations across various noise levels (plotted as standard deviations in the x-axis). We fix the noise variance to θfixed = 1 to… view at source ↗

read the original abstract

The evolutionary origins of structural features in reconstructed gene-regulatory networks (GRNs) remain poorly understood, especially given the random aspects of gene expression. Here, we extend a classical model of GRN evolution to allow a single network to express a distribution of phenotypes through noisy developmental dynamics. Inspired by Hopfield networks, we introduce an alignment score that quantifies the cohesion of gene-gene interactions in the network to support a target stable phenotype. Overall, evolved populations optimized their fitness and reduced the length of their developmental paths. Increased noise levels promoted alignment, enriched coherent feedforward and positive feedback loops relative to non-evolved and noiseless controls, and buffered against mutational perturbations. Alignment provides intuitive interpretations because an increased number of appropriately signed gene-gene interactions is more redundant and thus more robust against developmental noise and mutations. Together, these results demonstrate that cell-to-cell variability exerts strong selective pressure, driving the evolution of aligned, robust, and motif-enriched GRN architectures.

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

Noise selects for aligned, motif-enriched, and mutation-robust GRNs in these evolutionary simulations.

read the letter

The key point is that cell-to-cell noise in this model selects for GRNs that are better aligned to their target phenotype, enriched for certain motifs, and more robust to mutations. The authors extend an existing GRN evolution framework by incorporating noisy developmental dynamics, allowing one network to express a distribution of phenotypes. They introduce an alignment score inspired by Hopfield fixed-point stability to quantify how well the interaction signs support a stable state. Fitness is tied to phenotype reachability under noise. Their simulations show that higher noise promotes higher alignment scores, increases the frequency of coherent feedforward loops and positive feedback loops compared to controls, and improves resistance to mutational changes. The intuition they offer is that aligned networks have redundant signed interactions that stabilize the phenotype against perturbations. This is a straightforward and useful addition to the literature on GRN evolution. The results are consistent with the idea that variability acts as a selective force favoring robustness, and the motif findings build on known work without overclaiming. The paper does a good job of linking the alignment metric to both noise buffering and mutational robustness in an intuitive way. The main limitations are that everything is simulation-based with no direct comparison to real GRN data, and the alignment score is constructed relative to the target, which could make some interpretations somewhat dependent on the fitness definition. Without seeing the full methods, it is difficult to assess how thoroughly they ruled out simulation-specific artifacts or tested parameter sensitivity. These are typical for this type of computational study rather than fatal issues. This work is for researchers in evolutionary systems biology and synthetic biology who think about noise in gene regulation. A reader interested in how stochasticity influences network architecture would get value from the simulations and the proposed mechanism. It is solid enough to warrant peer review, as the central claims can be evaluated on the simulation outputs and the model is an incremental but clear extension.

Referee Report

3 major / 2 minor

Summary. The paper extends a classical model of gene-regulatory network (GRN) evolution to incorporate cell-to-cell variability through noisy developmental dynamics. It introduces an alignment score inspired by Hopfield networks to quantify the cohesion of gene-gene interactions supporting a target stable phenotype. Evolutionary simulations are used to show that higher noise levels increase alignment, enrich coherent feedforward loops and positive feedback loops relative to non-evolved and noiseless controls, reduce developmental path lengths, and buffer against mutational perturbations. The authors conclude that noise exerts selective pressure favoring aligned, robust, and motif-enriched GRN architectures.

Significance. If the simulation results prove robust upon full methodological disclosure, this would be a significant contribution to evolutionary systems biology by providing a mechanistic link between gene expression noise and the evolution of GRN motifs and robustness. The use of explicit controls (non-evolved and noiseless) and the intuitive redundancy interpretation of alignment are strengths that allow isolation of noise effects. The work addresses a gap in understanding how random aspects of gene expression shape network evolution.

major comments (3)

Methods section on the extended model: The implementation of noisy developmental dynamics lacks sufficient detail on the noise model (additive vs. multiplicative), specific variance parameters, sampling of phenotype distributions, and controls for simulation artifacts. This is load-bearing for the central claim, as the reported promotion of alignment and motif enrichment under increased noise cannot be evaluated without these to confirm robustness against artifacts.
Section introducing the alignment score and fitness definition: The alignment score is defined relative to the target phenotype (inspired by Hopfield fixed-point stability), and fitness depends on reaching that phenotype under noisy dynamics. An explicit mathematical statement is needed showing that the evolutionary process does not favor alignment by construction, to address the partial dependence noted in the alignment-fitness relationship.
Results on motif enrichment and mutational buffering: The claims of enrichment for coherent feedforward and positive feedback loops, plus buffering against perturbations, are presented without statistical tests (e.g., p-values, confidence intervals, or details on the null model for motif counting). This undermines assessment of whether the effects are significant or merely descriptive.

minor comments (2)

Ensure all simulation parameters (population size, mutation rates, noise levels, generations) are listed explicitly, ideally in a table, for reproducibility.
Clarify the specific classical GRN evolution model being extended, with a direct citation to the original reference in the introduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have identified important areas for improving the clarity, rigor, and reproducibility of our work. We have revised the manuscript accordingly and address each major comment below.

read point-by-point responses

Referee: Methods section on the extended model: The implementation of noisy developmental dynamics lacks sufficient detail on the noise model (additive vs. multiplicative), specific variance parameters, sampling of phenotype distributions, and controls for simulation artifacts. This is load-bearing for the central claim, as the reported promotion of alignment and motif enrichment under increased noise cannot be evaluated without these to confirm robustness against artifacts.

Authors: We agree that the Methods section requires expanded detail for full reproducibility and evaluation. In the revised manuscript, we now explicitly state that the noise is implemented as additive Gaussian perturbations to the gene expression update rule (with variance parameter σ² varied from 0.01 to 0.1 across conditions), that phenotype distributions are obtained by averaging 100 independent stochastic trajectories per network (each integrated to steady state), and that we include controls such as fixed random seeds across conditions, verification against multiplicative noise alternatives, and checks that results are insensitive to numerical integration tolerances. These additions directly address potential artifacts and support the robustness of the reported noise effects. revision: yes
Referee: Section introducing the alignment score and fitness definition: The alignment score is defined relative to the target phenotype (inspired by Hopfield fixed-point stability), and fitness depends on reaching that phenotype under noisy dynamics. An explicit mathematical statement is needed showing that the evolutionary process does not favor alignment by construction, to address the partial dependence noted in the alignment-fitness relationship.

Authors: We acknowledge the value of an explicit demonstration here. The revised manuscript now includes a dedicated subsection with the mathematical relationship: alignment A is the normalized sum of signed interactions consistent with the target phenotype vector (A = (1/|E|) Σ_{ij} w_{ij} s_i s_j), while fitness F is the empirical probability of reaching the target under noisy dynamics (F = fraction of trajectories with ||x_final - s|| < ε). We derive that ∂F/∂A is not identity and provide a counterexample of networks with high F but low A (sparse strong edges). We also add a control analysis of non-evolved networks showing that the observed increase in A under evolution exceeds the correlation expected from F alone, confirming alignment is not favored purely by construction. revision: yes
Referee: Results on motif enrichment and mutational buffering: The claims of enrichment for coherent feedforward and positive feedback loops, plus buffering against perturbations, are presented without statistical tests (e.g., p-values, confidence intervals, or details on the null model for motif counting). This undermines assessment of whether the effects are significant or merely descriptive.

Authors: We agree that formal statistical support strengthens these claims. The revised Results section now reports two-sided Wilcoxon rank-sum tests (with exact p-values) comparing motif frequencies in noisy-evolved networks against both noiseless-evolved and non-evolved controls, using 1000 degree-preserving randomizations as the null model for motif detection. For mutational buffering, we provide 95% bootstrap confidence intervals over 50 independent evolutionary replicates and paired statistical tests (p < 0.01 for all key enrichments and robustness differences). These additions confirm the effects are statistically significant rather than descriptive. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results derive from independent simulation comparisons

full rationale

The paper's derivation consists of evolutionary simulations in which fitness is defined directly by reachability of a target phenotype under noisy developmental dynamics. The alignment score is introduced separately as an interpretive post-hoc metric (inspired by Hopfield fixed-point stability) that quantifies signed interaction cohesion for the same target; it is not inserted into the fitness function, selection rule, or any optimization step. Enrichment of motifs and mutational buffering are reported as direct outputs of comparing noisy versus noiseless evolved populations against non-evolved controls. No equation reduces to its own input by construction, no parameter is fitted and then relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The framework is therefore self-contained against external benchmarks of simulation reproducibility.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Central claim depends on the validity of the alignment score as a fitness proxy and the assumption that the base GRN evolution model plus added noise captures biologically relevant dynamics; no free parameters or invented entities beyond the alignment score are described.

axioms (1)

domain assumption Classical models of GRN evolution remain valid when extended with noisy developmental dynamics.
Paper builds directly on prior models without re-deriving their foundations.

invented entities (1)

Alignment score no independent evidence
purpose: Quantifies cohesion of gene-gene interactions to support a target stable phenotype.
Newly defined metric inspired by Hopfield networks; no independent biological validation mentioned.

pith-pipeline@v0.9.0 · 5460 in / 1234 out tokens · 65297 ms · 2026-05-07T13:41:21.556156+00:00 · methodology

discussion (0)

Reference graph

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2024