{"id":"f10b4e94-65da-4cfe-bb68-c8f05b793cfa","arxiv_id":"2607.17743","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under a linear-noise approximation, node-wise noise in two-node feedback motifs decomposes into intrinsic, extrinsic, and a loop-closure 'cyclic' term whose sign marks feedback amplification vs attenuation, and whose dynamics shape autocorrelation decay.","lead":"This paper presents an analytical framework that splits the noise in two-gene feedback circuits into three components, including a new 'cyclic' part that appears only when the regulatory loop is closed. Using a linear-noise approximation and stochastic simulations, it shows that reinforcing loops amplify and prolong fluctuations while opposing loops leave them nearly unchanged.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Temporal-signature claim uses isolated birth-death as 'no-feedback' baseline, conflating direct regulation with loop closure; proper open-loop control would test whether cyclic noise really controls persistence.","rationale":"The reader identified the LNA as the weakest assumption. That is a legitimate limitation, but it is acknowledged by the authors and does not undermine the internal consistency of the derivation. A more specific and less noted issue is the baseline for the temporal-signature claim. The paper's central pitch includes the new temporal readout: 'feedback-mediated noise circulation leaves a temporal signature in the decay of steady-state autocorrelation.' The evidence for this is the divergence of ρ_NN from ρ^(0)_NN. However, ρ^(0)_NN is not the no-feedback counterpart of a two-node motif; it is the autocorrelation of an unregulated birth-death process. Removing both regulatory edges is not the right counterfactual for isolating loop closure, because it removes the direct regulation along with the loop. The open-loop cascade keeps the direct edge and breaks the loop, so the difference between closed-loop and open-loop autocorrelation is the actual loop-closure effect. Without this control, the observed persistence in reinforcing motifs could be a generic property of positive input, not of circulation. This concern is directly testable and, if it lands, would require the temporal interpretation in Section III.C to be scaled back or reanalyzed. It does not affect the algebraic decomposition, but it affects a major advertised contribution. Hence, I recommend CONDITIONAL rather than UNCHANGED: the paper is largely sound, but the temporal-signature claim needs either a corrected baseline or a more careful interpretation.","tokens_in":16230,"tokens_out":19267,"duration_ms":197697,"concrete_test":"For each motif and each parameter set in Figs. 3–4, compute the LNA autocorrelation of node N in the open-loop system obtained by setting the reverse regulatory sensitivity (e.g., f'_YX = 0 for X) to zero, keeping all other parameters fixed. Compare ρ_NN(τ) in the full closed loop with this open-loop reference. If the closed-loop autocorrelation coincides with the open-loop reference for reinforcing motifs, the temporal signature does not arise from loop closure and the claim in §III.C should be revised; if a significant difference remains, the loop-closure interpretation is supported. This can be done analytically via Eq. (C6) or by Gillespie simulation.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In §III.C and Appendix C, the paper defines the 'no-feedback' reference as ρ^(0)_NN(τ)=1−β_Nτ (Eq. 10), the short-time autocorrelation of an isolated birth-death process. But in a two-node motif, the correct no-feedback control for isolating loop closure is the open-loop cascade obtained by deleting the reverse edge (e.g., delete X→Y to test X), so the node retains its direct regulatory input but the loop is broken. The autocorrelation of the downstream node in such an open loop already deviates from 1−βτ because the incoming edge propagates the upstream node's fluctuations into it. Consequently, the observation in Figures 3 and 4 that reinforcing motifs show ρ_NN above ρ^(0) cannot be attributed specifically to loop closure; it may simply reflect the presence of a positive regulatory input. The paper's conclusion that 'positive cyclic noise is associated with ... longer temporal persistence' is not supported by the chosen comparison, since ρ_NN(τ) includes both extrinsic noise propagation and cyclic noise (as acknowledged in Eq. 11). To substantiate the temporal signature of feedback circulation, the comparison must be made against the open-loop cascade with the same direct edge, not against an isolated node. This is a logical gap in the central argument for the temporal readout, even though the algebraic decomposition (Eqs. 3–6) is correct within the LNA.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript develops a linear-noise-approximation (LNA) framework for two-node gene-regulatory feedback motifs and proposes an additive decomposition of node-wise noise into intrinsic, extrinsic, and a new 'cyclic' component. The cyclic component, η_cyc,N = T_MN H/(1−H) η_int,N, is positive for reinforcing motifs and negative for opposing motifs, and the authors verify the analytic expressions with stochastic simulations. The paper further claims a temporal signature of feedback circulation based on a short-time expansion of the steady-state autocorrelation, arguing that reinforcing loop closure prolongs fluctuation persistence relative to a no-feedback reference. The steady-state decomposition is clean and analytically well grounded; the temporal interpretation is less well controlled.","tokens_in":16557,"tokens_out":4546,"duration_ms":54515,"significance":"If the steady-state decomposition stands, it gives a compact and interpretable topological readout: the sign of the feedback gain H determines whether loop closure amplifies or attenuates node-wise fluctuations, and the explicit formula for η_cyc is a useful analytic result. The derivation is self-contained, the definitions are clear, and the simulation agreement is a genuine strength. The temporal-signature claim, however, is not yet supported by the chosen comparison, so the present version overstates one of its advertised contributions. With the temporal claim either corrected or appropriately softened, the paper is a solid contribution to the noise-decomposition literature for feedback motifs.","major_comments":[{"comment":"The 'no-feedback reference' is taken as ρ_NN^(0)(τ)=1−β_N τ, the autocorrelation of an isolated birth–death process. This does not control for the direct regulatory edge in the two-node motif. Equation (11) shows that the feedback-mediated correction actually contains both extrinsic noise propagation and cyclic noise: ρ_NN(τ)=1−β_N τ + τ f'_NM... (η^2_MN/η^2_N). The second term is nonzero even in an open-loop cascade with the same direct edge and H=0, because upstream fluctuations propagate through that edge. Therefore, the deviations from ρ_NN^(0) seen in Figs. 3 and 4 cannot be attributed specifically to loop closure, and the conclusion that 'positive cyclic noise is associated with ... longer temporal persistence' is not supported by the chosen comparison. To isolate the temporal signature of cyclic circulation, the authors should compare the closed-loop motif against the open-loop ca","section":"§III.C, Eqs. (10)–(11), Figs. 3–4"}],"minor_comments":[{"comment":"The caption says 'panels (e)-(f)' for the Y autocorrelations; this should be '(e)-(h)'.","section":"Fig. 4 caption"},{"comment":"Typo: 'The quantitie ρ_NN^(0)' should read 'The quantities ρ_NN^(0)'.","section":"Fig. 3 caption"},{"comment":"The notation η^2_MN is used in the main text before it is explicitly defined. Define it as the normalized covariance σ^2_{MN}/⟨n_M⟩⟨n_N⟩ in the main text, not only in Appendix C.","section":"Eq. (11) and Appendix C"},{"comment":"The phrase 'no-feedback reference' is misleading because the reference is not the open-loop two-node system but an isolated birth–death process. Consider renaming it 'isolated-node reference' to avoid conflating direct regulation with absence of feedback, especially once the temporal comparison is revised.","section":"§III.C"}],"recommendation":"major_revision","confidential_remarks":"The steady-state cyclic-noise decomposition appears sound and is worth publishing. The main issue is the temporal-signature claim in §III.C, which compares against an inappropriate control; this is fixable either by performing the open-loop-cascade comparison or by removing the overstrong temporal language from the abstract and conclusion. I do not see a need for rejection, but the revision should address the control comparison directly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the algebraic part is sound—the cyclic noise decomposition is a clean and genuinely useful way to think about LNA noise in two-node motifs—but the temporal-signature section compares against the wrong baseline, so the claim that loop closure prolongs persistence is not actually supported.\n\nWhat's new: the explicit additive term η_cyc,N^2 = T_MN H/(1−H) η_int,N^2, with sign set by the feedback gain H, is a nice interpretive re-arrangement of known Lyapunov results. It gives a practical rule: reinforcing motifs amplify node-wise noise, opposing motifs attenuate it. The derivation in the appendices is transparent and the stochastic simulations match the formulas. The authors are honest about the LNA's small-noise limitation.\n\nThe soft spot is Section III.C. The no-feedback reference ρ^(0)_NN(τ)=1−β_N τ is an isolated birth-death process. In a two-node motif, the right control for isolating loop closure is the open-loop cascade with the reverse edge deleted. The direct incoming edge already causes the autocorrelation to deviate from 1−βτ, because the upstream node's fluctuations propagate downstream. The paper even acknowledges in Eq. (11) that ρ_NN contains both extrinsic propagation and cyclic noise. So when reinforcing motifs show slower decay than the birth-death baseline, that could be the direct edge, not the loop. To claim cyclic noise controls temporal persistence, you need to compare against the open-loop cascade, or at least explicitly separate the cyclic part of the autocorrelation. Without that, the temporal conclusion is overstated.\n\nEverything else is fine. The covariance decomposition, the motif-specific analysis, and the simulation checks all look solid. This is a moderate-novelty paper but a legitimate one. I'd send it to a serious referee, with a request that the temporal-signature analysis be reworked or carefully reworded.\n\nFor a reading group it would be worth discussing, mainly as an example of why controls matter in noise decomposition.","headline":"Solid LNA decomposition with a useful cyclic-noise term, but the temporal persistence claim is undercut by the wrong no-feedback baseline—worth reading, needs revision.","tokens_in":17022,"tokens_out":3612,"would_cite":true,"duration_ms":38247,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that in two-node gene regulatory motifs, total node-wise noise decomposes into intrinsic, extrinsic, and a new feedback-generated cyclic component whose sign—set by the product of the two regulatory sensitivities—determine","keywords":["cyclic noise","feedback gain","linear noise approximation","two-node feedback motifs","noise decomposition","gene regulatory circuits","autocorrelation","Lyapunov equation"],"falsifier":"Measure the total squared coefficient of variation of a node in a synthetic two-node motif with known sign of H, then disconnect the feedback loop (e.g., by removing the second regulator) while keeping the direct regulation and degradation rates fixed. If total noise does not increase when H>0 or decrease when H<0 relative to the open-loop reference, the cyclic-noise term as stated is wrong. More directly, compute H from independently measured regulatory sensitivities and check whether the extracted cyclic-noise sign from variance data matches the sign of H.","tokens_in":16159,"feed_emoji":"🧬","tokens_out":3452,"duration_ms":40792,"temperature":0.7,"pith_summary":"The paper tries to establish that feedback loops in the simplest two-node gene circuits do more than rescale existing noise: they inject a distinct, additive noise component caused purely by loop closure. This 'cyclic noise' is positive for reinforcing motifs (mutual activation or mutual repression), amplifying node-wise fluctuations, and negative for opposing motifs (one activation, one repression), attenuating them. The same mechanism leaves a temporal fingerprint: reinforcing loops slow the decay of autocorrelation, while opposing loops leave it close to the no-feedback baseline. The decomposition follows analytically from the linear noise approximation and is checked against stochastic simulation. If correct, it gives an experimental route to read feedback sign and strength from measured means, variances, covariances, and autocorrelations.","feed_headline":"Feedback loops add cyclic noise to gene circuits","feed_subtitle":"In two-node motifs, loop closure amplifies or dampens fluctuations by feedback sign, leaving a telltale autocorrelation decay.","key_machinery":"The load-bearing object is the Lyapunov equation JΣ + ΣJᵀ + 2D = 0 for the steady-state covariance matrix Σ, with the 2×2 Jacobian J containing the two regulatory sensitivities f′_XY and f′_YX. Solving this equation analytically yields the three-term noise decomposition. The new term, cyclic noise, is controlled by the dimensionless feedback gain H = f′_XY f′_YX/(β_X β_Y); its sign sets whether loop closure adds to or subtracts from node-wise noise, and H = 1 marks the stability boundary for reinforcing motifs. The temporal analysis relies on the short-time expansion C(τ) ≈ (I + Jτ)Σ, which separates the birth-death relaxation 1 − β_N τ from the feedback-coupled autocorrelation term.","core_discovery":"For each transcription factor N in a two-node feedback motif, the paper argues that the squared coefficient of variation obeys η_N² = η_int,N² + η_ext,N² + η_cyc,N², with cyclic noise η_cyc,N² = T_MN H/(1−H) η_int,N², where H = f′_XY f′_YX/(β_X β_Y) is the dimensionless feedback gain and T_MN = β_M/(β_N+β_M) is a time-averaging factor. H is positive for reinforcing motifs (++ and −−), making cyclic noise positive and divergent as H→1−; H is negative for opposing motifs (+− and −+), making cyclic noise negative. The paper further derives the short-lag autocorrelation ρ_NN(τ) = 1 − β_N τ + τ f′_NM (⟨n_M⟩/⟨n_N⟩) η²_MN/η²_N, where the feedback-dependent term is large enough in reinforcing motifs","pith_inferences":["Editorial extension: for networks with more than two nodes, the same Lyapunov-based decomposition should generalize to a sum over closed loops, with each loop contributing a term proportional to its own feedback gain; this would make cyclic noise a graph-theoretic quantity.","Editorial extension: the negative cyclic noise in opposing motifs resembles a noise-cancellation term; it may connect to control-theoretic limits on noise suppression, suggesting that opposing feedback achieves attenuation by redirecting fluctuations rather than by adding dissipation.","Editorial extension: a direct testable prediction beyond the paper is that in a synthetic (+−) motif, the total squared coefficient of variation should fall below the value obtained by breaking the loop (open-loop cascade) while keeping all other parameters fixed; this would confirm the negative cyclic contribution.","Editorial extension: the autocorrelation slowdown in reinforcing motifs implies that positive feedback lengthens the effective memory of a gene circuit, which could be exploited in synthetic memory devices and may be observable in cell-fate decision circuits before commitment."],"forward_implications":["If the decomposition is correct, measuring a node's mean copy number, variance, and covariance with its partner suffices to extract cyclic noise experimentally from single-cell time-series data.","Reinforcing feedback motifs (++ and −−) will show amplified node noise and longer-lived autocorrelation, providing a quantitative signature of positive-feedback-driven fluctuation persistence even below the bistability threshold.","Opposing motifs (+− and −+) will show reduced node noise and autocorrelation decay close to the no-feedback baseline, offering a way to identify negative feedback from noise statistics alone.","The divergence of cyclic noise as H→1− predicts that strong reinforcing feedback can produce large, slow fluctuations before any bifurcation, a precursor that could be observed in synthetic toggle circuits.","Because the formulas are expressed in measurable quantities, the framework can be applied directly to existing single-cell reporter data from synthetic gene circuits without additional fitting assumptions."],"fun_headline_variants":["Cyclic noise: how feedback loops reshape gene circuit fluctuations","Gene circuit feedback spawns cyclic noise that amplifies or dampens","Feedback loop sign sets cyclic noise, altering noise persistence in gene circuits","Cyclic noise from feedback loops leaves temporal fingerprint in gene circuits","Feedback loops inject cyclic noise, changing how gene fluctuations persist"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The linear noise approximation is the load-bearing premise: the covariance decomposition and the cyclic-noise sign rely on fluctuations staying small enough that the linearized Lyapunov equation accurately describes the steady state, so the results fail near bistability, bursting, or large-amplitude oscillations.","fun_headline_variants_meta":{"raw":{"variants":["Cyclic noise: how feedback loops reshape gene circuit fluctuations","Gene circuit feedback spawns cyclic noise that amplifies or dampens","Feedback loop sign sets cyclic noise, altering noise persistence in gene circuits","Cyclic noise from feedback loops leaves temporal fingerprint in gene circuits","Feedback loops inject cyclic noise, changing how gene fluctuations persist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000563,"raw_usage":{"total_tokens":2509,"prompt_tokens":749,"completion_tokens":1760,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":1674}},"tokens_in":493,"tokens_out":1760,"duration_ms":13922,"temperature":1.0,"reasoning_tokens":1674,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T17:03:55.109999+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the total squared coefficient of variation of a node in a synthetic two-node motif with known sign of H, then disconnect the feedback loop (e.g., by removing the second regulator) while keeping the direct regulation and degradation rates fixed. If total noise does not increase when H>0 or decrease when H<0 relative to the open-loop reference, the cyclic-noise term as stated is wrong. More directly, compute H from independently measured regulatory sensitivities and check whether the extracted cyclic-noise sign from variance data matches the sign of H.","supporting_citations":[],"review_version":1}