The zoo of the gene networks capable of pattern formation by extracellular signaling

Reviewer #1 (Public review):

Summary:

The authors tackle a long-standing question in developmental theory: given a gene-regulatory network that includes extracellular signaling, which topologies are even capable of transforming an initial spatial profile into a genuinely new pattern? Building on the classical reaction-diffusion framework in one dimension, but imposing biologically motivated constraints, they prove that every one-signal sub-network must be either Hierarchical (H), self-activating (L+), or self-inhibiting (L-). They further demonstrate that only three composite classes of full networks - pure H, a coupled L+ L- "Turing" pair, and an L- module fed by an intracellular positive loop ("noise-amplifying")-can create non-trivial spatial transformations. Analytical criteria and illustrative simulations are provided, together providing a closed taxonomy, which is supposed to be relevant for real systems.

Strengths:

- Useful classification framework. Reducing a vast number of possible gene circuits to three canonical pattern-forming motifs is a valuable organizing insight for both theorists and experimentalists.

- Practical interpretability. Given a reaction network diagram, one can now decide (assuming the model applies to real systems) whether spatial patterning is even possible, saving experimental effort on in silico screens that could never succeed.

Weaknesses:

- Theoretical limitations in the application of Linear Stability Analysis (LSA): I remain uncertain about the framework's reliance on LSA as a necessary condition for non-trivial pattern transformation, especially for large initial perturbations ("spikes"). The revised manuscript itself states that spike amplitudes must be sufficiently small for the linearization to hold. In the rebuttal, the authors argue that large spikes can nevertheless be treated because their influence is initially small outside the spike. However, linear stability of a homogeneous steady state only describes the response to infinitesimal perturbations around that state; it does not generally exclude finite-amplitude perturbations from entering a different nonlinear basin of attraction and producing a heterogeneous stationary state, e.g., as in subcritical Turing patterns. Thus, I do not think the rebuttal establishes the stronger claim that a linearly stable network cannot produce a non-trivial pattern regardless of nonlinear terms.

- Presentation: The manuscript remains difficult to follow. The argument is distributed across many named requirements and topology classes, long prose descriptions of network structures, and repeated cross-references to the Supplementary Information. Given that the main contribution is a conceptual classification, I think the logical hierarchy should be considerably easier to reconstruct.

Discussion:

The study offers a solid conceptual organization of pattern-forming networks. However, the theoretical bridge between infinitesimal linear stability and macroscopic, non-linear pattern emergence still presents some uncertainties. The way the current framework formally treats large initial perturbations leaves some questions open regarding its broad analytical applicability to real biological tissues.

Reviewer #1 (Public review):

Summary:

The authors tackle a long-standing question in developmental theory: given a gene-regulatory network that includes extracellular signaling, which topologies are even capable of transforming an initial spatial profile into a genuinely new pattern? Building on the classical reaction-diffusion framework in one dimension, but imposing biologically motivated constraints, they prove that every one-signal sub-network must be either Hierarchical (H), self-activating (L+), or self-inhibiting (L-). They further demonstrate that only three composite classes of full networks - pure H, a coupled L+ L- "Turing" pair, and an L- module fed by an intracellular positive loop ("noise-amplifying")-can create non-trivial spatial transformations. Analytical criteria and illustrative simulations are provided, together providing a closed taxonomy, which is supposed to be relevant for real systems.

Strengths:

(1) Useful classification framework. Reducing a vast number of possible gene circuits to three canonical pattern-forming motifs is a valuable organizing insight for both theorists and experimentalists.

(2) Practical interpretability. Given a reaction network diagram, one can now decide (assuming the model applies to real systems) whether spatial patterning is even possible, saving experimental effort on in silico screens that could never succeed.

Weaknesses:

(1) After the resubmission, I still have concerns regarding the formal definition of "non-trivial transformations" (P1/P2) and its application to noisy or multi-dimensional systems. The criteria rely on counting "new" critical points (maxima/minima). In their response, the authors argue that the diffusion operator instantly smooths discontinuous white noise, allowing critical points to be properly defined. However, this very smoothing process passively generates a landscape of new, smooth local extrema from the initial noise. Consequently, trivial diffusive regularization could inadvertently fulfil the criteria for a "non-trivial" transformation, leaving the definition conceptually problematic. Furthermore, when extending the framework to 2D/3D, the manuscript assumes that starting from a central "spike" will robustly preserve radial symmetry, yielding concentric rings or shells. This overlooks the fundamental nature of macroscopic mean-field models like reaction-diffusion equations. The realization of the final multidimensional pattern depends strictly on the stability of the solution against ubiquitous perturbations (including angular modes) rather than solely on the deterministic symmetry of the initial condition. It remains unclear how the current framework accounts for spontaneous symmetry breaking in cases where these angular modes become unstable, challenging the assumption that radial symmetry will strictly dictate the outcome. We note that the authors' use of noise as an initial condition does not resolve this fundamental issue. Reaction-diffusion equations inherently describe mean-field dynamics, meaning that microscopic fluctuations are continuously present in any real system, regardless of whether explicit stochastic terms are written into the equations. Ultimately, if a symmetric mean-field solution is structurally unstable to these inherent fluctuations, it simply cannot be realized in nature.

(2) Theoretical limitations in the application of Linear Stability Analysis (LSA): I remain uncertain about the framework's reliance on LSA to categorize macroscopic transformations, especially those arising from large initial perturbations (spikes). In their rebuttal letter, the authors justify this by assuming the perturbation remains small over a short time interval. However, because the study aims to describe stationary, asymptotic states, applying a linear approximation that relies on transient t->0 conditions to predict long-term global stability is not fully resolved.

(3) In the previous round of the review, I suggested that a biomolecular sink, such as A+B -> AB reaction, could break the approach. In their response letter, the authors defend their approach by arguing that such reactions can be accommodated by their abstract constraints (R1-R5) as long as the signs of the Jacobian elements remain invariant. However, the problem I see here is not the sign of the interactions, but the severe loss of spatial homogeneity.

When a macroscopic initial perturbation (a "spike" of morphogen) is introduced into a domain with a strong bimolecular sink, it will inevitably cause massive local depletion of the consumed substrate near the source. Consequently, the background state of the system will rapidly evolve into a profile with macroscopic spatial gradients long before any spontaneous pattern-forming instability takes over. Mathematically, this dictates that the system no longer possesses a homogeneous steady state, and the Jacobian matrix becomes explicitly space-dependent, which should break the classical LSA approach.

Discussion:

The study offers a solid conceptual organization of pattern-forming networks. However, the theoretical bridge between infinitesimal linear stability and macroscopic, non-linear pattern emergence still presents some uncertainties. The way the current framework formally treats noise, multi-dimensional symmetry breaking, and large initial perturbations leaves some questions open regarding its broad analytical applicability to real biological tissues.

Reviewer #3 (Public review):

Pattern formation is responsible for generating the spatial organization of cells, tissues, and organs during embryogenesis. It operates within a multifactorial system including initial conditions, gene regulatory networks, extracellular signals, mechanical forces, stochastic noise and environmental inputs, and finally ensures the functional anatomy of an organism.

This study focuses on the one central aspect in pattern formation: how spatial heterogeneity arises from an initial condition and evolves into a more complex or distinct spatial pattern (non-trivial pattern formation as they termed). The authors made efforts to explore and characterize all possible ways to achieve the pattern formation by discussing how extracellular signals spread, how individual cells respond to those signals, and how those responses, in turn, modulate signal propagation.

Finally, their comprehensive analysis summarizes that there are three classes of interactions between extracellular signal and intracellular responses, corresponding to previously known mechanisms that can generate spatial patterns: Difference in morphogen concentrations in space, noise-amplification, and Turing pattern.