{"id":"fd076d3b-9ffe-421d-b6dd-76325e8e38b6","arxiv_id":"2602.18028","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"Signaling fidelity is two-dimensional: mutual information scores state discrimination, while the inverse 2-Wasserstein distance scores distributional correspondence, and feedback motifs shift along this trade-off.","lead":"This paper argues that cellular signaling fidelity has two separate dimensions: how well a pathway distinguishes input states, measured by mutual information, and how faithfully the output distribution mirrors the input distribution, measured by the inverse 2-Wasserstein distance. It finds that negative feedback tends to trade state resolution for distributional stability, and reinterprets TNF signaling data along these two axes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Fig. 6c WT/A20−/− separation is likely an artifact of the condition-specific a.u.-to-ng/mL rescaling (SI S5, Figs. S3a–d): WT output is divided by a ~5–7× larger slope than A20−/−, mechanically lowering W for WT.","rationale":"The strongest claim is that feedback architectures sacrifice informational fidelity for geometric fidelity, and that geometric fidelity is a fundamental dimension of signaling fidelity. The theoretical part (Fig. 5) is internally coherent: Eqs. (8) and (10) follow from Gaussian/LNA assumptions, and the motif curves are clearly parameterized. The external anchor for the 'fundamental' claim is the TNF experiment, Fig. 6c. The reader's weakest assumption is exactly right: the a.u.-to-ng/mL mapping is condition-specific and directly changes W. Because the 2-WD is not invariant under monotone rescaling of one coordinate, and because the mapping factors differ by ~5–7× between WT and A20−/−, the observed separation is not robust until a common scaling is tested. I agree with the reader's assessment; no verdict change beyond the existing CONDITIONAL is needed, but the manuscript should add the common-scaling robustness check and, ideally, a unit-invariant comparison before the broadest claims are accepted.","tokens_in":31689,"tokens_out":6529,"duration_ms":62317,"concrete_test":"Recompute W from Eq. (S26) for all four experimental conditions using one common mapping factor — for example, a single Hill fit to the pooled WT + A20−/− dose-response data, with the reciprocal of its half-maximum slope used for both genotypes — and re-plot Fig. 6c. If the WT/A20−/− ordering reverses or collapses, the experimental trade-off claim is an artifact of the condition-specific rescaling; if the ordering persists under this and other reasonable common scalings, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The experimental support for the central claim — that negative feedback increases geometric fidelity at the expense of informational fidelity — reduces to Fig. 6c. But the geometric-fidelity axis in that figure is not measured in a unit-invariant way. Because X is in ng/mL and Z in arbitrary fluorescence units, Eq. (S26) requires putting both on the same scale. The authors do this with the reciprocal slope of a fitted dose–response curve, using a Hill fit at half-maximum for WT (slopes 215.0 and 156.9 a.u. per ng/mL for NF-κB and ATF-2) and a linear fit for A20−/− (slopes 29.2 and 28.1 a.u. per ng/mL; SI S5, Fig. S3). Dividing raw output quantiles by a ~5–7-fold smaller slope for A20−/− inflates the mapped z-quantiles and hence the 2-WD relative to WT, mechanically lowering the A20−/− geometric fidelity. The two genotypes are therefore compared under different mapping rules; the claimed WT/KO separation may be encoded in the unit conversion rather than in feedback biology. A second feature compounds the problem: MI is not independently measured but is forced to match Cheong et al.'s published values by optimizing P_X, so the only genuinely new quantity in Fig. 6c is W, and W is exactly the coordinate affected by the arbitrary rescaling.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a dual-fidelity framework for cell signaling, defining informational fidelity as mutual information I(X;Z) and geometric fidelity as the inverse 2-Wasserstein distance W(X;Z)^{-1}. A Lagrangian L = I - λW^2 is optimized over input noise for six gene-regulatory motifs under the LNA/Gaussian approximation, producing trade-off curves in the (I, W^{-1}) plane. The authors then apply the framework to TNF signaling data from Cheong et al. (2011), reporting that WT cells (with A20-mediated negative feedback) lie at higher geometric fidelity and lower informational fidelity than A20-deficient cells, and argue that this supports the theoretical prediction that negative feedback sacrifices information for distributional correspondence.","tokens_in":32209,"tokens_out":6630,"duration_ms":59002,"significance":"If the results were fully supported, the paper would offer a biologically meaningful second dimension of signaling fidelity and a practical computational framework. The analytic Gaussian formulas (Eqs. 8–10) are correct, and the LNA derivations in the SI are standard and clearly laid out. The conceptual distinction between state discrimination and distributional correspondence is valuable and likely of broad interest. The paper also includes explicit discussion of the limitations of the Gaussian approximation. However, the current experimental validation and the built-in character of the trade-off leave the central claims inadequately supported.","major_comments":[{"comment":"The unit rescaling used to compute the 2-WD is not unit-invariant and likely generates the WT/A20-/- separation in Fig. 6c. WT output is divided by Hill slopes at half-maximum of 215.0 and 156.9 a.u. per ng/mL, while A20-/- output is divided by linear slopes of 29.2 and 28.1 a.u. per ng/mL. This 5–7-fold difference in mapping factors compresses WT output quantiles toward the input scale and inflates A20-/- quantiles, mechanically increasing W for A20-/- and decreasing W for WT. The two genotypes are thus compared under different mapping rules, so Fig. 6c does not provide evidence for a feedback-dependent geometric-fidelity trade-off. A unit-invariant comparison (e.g., rank-transform both marginals to a common scale, or use a single calibration curve for both genotypes) is needed.","section":"SI S5, Eq. (S26), Figs. S3a–d"},{"comment":"The abstract states that 'RAS-MAPK data analysis further shows that jointly considering INF and GMF better characterizes intracellular signal relay than INF alone,' but no RAS-MAPK analysis appears anywhere in the main text, Methods, or SI. This is a claimed result with missing support. Either provide the missing analysis or revise the abstract so that it reflects the content of the paper.","section":"Abstract / Full text"},{"comment":"The trade-off between I and W^{-1} is largely built in by construction. Optimizing L = I - λW^2 over η_X^2 with increasing λ necessarily shifts the optimum toward smaller W (larger GMF) at the expense of I, for any pair of continuous functionals. Thus the qualitative statement that 'feedback architectures typically sacrifice informational fidelity to enhance geometric fidelity' is not an independent prediction of the model. The genuine content is in the motif-specific shape and location of the curves (e.g., which regimes are accessible at fixed λ, or the value of I at λ=0). The manuscript should be reframed accordingly, and the claims should be restricted to comparisons at fixed λ or to the predicted accessible regimes.","section":"Eq. (6) and Fig. 5"},{"comment":"The experimental MI is not independently measured: the input marginal P_X is optimized so that the computed I matches the values reported by Cheong et al. Therefore the I-coordinate in Fig. 6c is constrained rather than estimated. The only freely estimated quantity is W, which is exactly the coordinate affected by the rescaling in the first major comment. Thus Fig. 6c cannot provide independent confirmation of the dual-fidelity trade-off. The analysis should either estimate both coordinates from a pre-specified P_X (e.g., the empirical TNF dose distribution) or demonstrate that the conclusions are robust to the MI-matching procedure.","section":"SI S5, 'MI-constrained reconstruction'"}],"minor_comments":[{"comment":"The statement that 'the general form of the Lagrangian L can be considered the Sinkhorn distance' is inaccurate: the Sinkhorn distance refers to entropy-regularized optimal transport, not to an MI-minus-W^2 Lagrangian. Please correct or remove.","section":"Introduction, after Eq. (1)"},{"comment":"The theoretical and experimental curves are plotted on the same axes despite having different units (reciprocal copy number vs reciprocal ng/mL). The comparison is only qualitative; the figure should state this explicitly and ideally use normalized coordinates.","section":"Fig. 6e"},{"comment":"The output variable is described as being in 'concentration units,' but fluorescence is in arbitrary units (a.u.). The text should distinguish these consistently, as the unit conversion is a central issue.","section":"SI S5"},{"comment":"The regime thresholds (I ≥ 0.3 bits, W^{-1} ≥ 1) are arbitrary and unit-dependent. The paper should emphasize that they are illustrative and do not carry intrinsic biological meaning in experimental units.","section":"Results, 'Motif-specific patterns'"},{"comment":"The model uses Hill coefficient h=1, equal mean copy numbers, and fixed degradation timescales. A sensitivity analysis over these choices would strengthen the generality claims in the Discussion.","section":"Materials and methods / SI S3"},{"comment":"The bootstrap error bars appear to include resampling of the reconstructed distributions, but not uncertainty in the slope-based mapping factors. This limitation should be stated.","section":"Fig. 6c / SI S5"}],"recommendation":"major_revision","confidential_remarks":"The paper offers an interesting conceptual framework, and the theoretical machinery is competently presented. However, the central experimental support (Fig. 6c) appears to be strongly influenced—possibly dominated—by the condition-specific a.u.-to-ng/mL rescaling, and the theoretical trade-off is partly a consequence of the chosen objective. The missing RAS-MAPK analysis promised in the abstract is a significant gap. The revision therefore needs substantial work: a unit-invariant experimental comparison, a reframing of the theory claims, and either inclusion or removal of the RAS-MAPK result. I believe the work is salvageable, but the current version is not publishable as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's core proposal is genuinely new: treat signaling fidelity as two-dimensional, with MI capturing state discrimination and inverse 2-Wasserstein distance capturing distributional correspondence. That is a useful conceptual addition, and the theoretical machinery is competently assembled. The Gaussian formulas for MI and W2 are correct, the LNA derivations are standard, and the motif-by-motif maps — C1-FFL sitting in the precise regime, feedback motifs biased toward geometric fidelity — are clean and interpretable. The analytical expressions in Eqs. 8–10 check out, and the framework is clearly presented.\n\nThe soft spot is the experimental section, and it is load-bearing. The stress-test note holds up. In Fig. 6c, the WT/A20 separation is driven almost entirely by the geometric-fidelity coordinate, and that coordinate depends on rescaling output fluorescence (a.u.) to TNF concentration (ng/mL) using the reciprocal slope of a fitted dose–response curve. WT gets a Hill slope at half-max (215 or 157 a.u. per ng/mL); A20−/− gets a linear fit slope (29 or 28 a.u. per ng/mL). Dividing output quantiles by a 5–7× smaller slope inflates the A20−/− output units, inflates W, and mechanically lowers geometric fidelity. The two genotypes are compared under different mapping rules, and the claimed separation may be an artifact of that. The authors do flag in SI S5 that A20−/− lacks a sigmoidal profile and defend the linear fit, but they never address the magnitude mismatch.\n\nA second issue compounds this: MI is not independently measured. The input marginal is optimized to match Cheong et al.'s published MI values, so the only genuinely new experimental quantity is W, and W is exactly the coordinate affected by the arbitrary rescaling. Third, the theoretical trade-off is partly built in by construction: optimizing L = I − λW² with λ > 0 will produce a trade-off along the curve. The motif-level differences are still informative, but the \"fundamental trade-off\" language is stronger than the evidence supports. The \"previously unrecognized dimension\" framing is also overblown — distributional correspondence has been discussed in other forms, and the authors cite that literature.\n\nThe theoretical framework deserves a serious referee. The experimental validation does not, as written, support the central claim about negative feedback. If the authors can show the WT/A20 separation persists under unit-invariant comparisons — e.g., using a common slope, z-scoring within each genotype, or a sensitivity analysis across mapping choices — the claim would be credible. As it stands, the strongest experimental conclusion should be toned down or reanalyzed. I would send this to peer review with a request for major revision on the experimental section.","headline":"The dual-fidelity framework is a real idea worth engaging, but the TNF experiment doesn't establish the feedback claim: the unit rescaling looks like it manufactures the WT/A20 separation.","tokens_in":32607,"tokens_out":2169,"would_cite":true,"duration_ms":23338,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that signaling fidelity is two-dimensional—mutual information plus the inverse 2-Wasserstein distance—and that feedback circuits trade the first for the second.","keywords":["mutual information","2-Wasserstein distance","optimal transport","geometric fidelity","dual-fidelity framework","feedback regulation","TNF signaling","gene regulatory motifs"],"falsifier":"Compute the dual-fidelity coordinates using an independent calibration (e.g., a reporter whose output is measured in absolute molecule numbers, or single-molecule RNA counts) and see whether wild-type cells still show higher geometric fidelity than A20-knockout cells. Alternatively, recompute the 2-Wasserstein distance after rescaling output quantiles by the dose-response slope at several different TNF doses; if the ordering flips depending on the chosen dose, the gain-based mapping is driving the result.","tokens_in":31586,"feed_emoji":"🧬","tokens_out":6480,"duration_ms":64739,"temperature":0.7,"pith_summary":"The paper argues that measuring how many input states a cell can distinguish (mutual information, MI) is not enough to judge signaling fidelity; a cell also needs to preserve the statistical shape of the input distribution. It introduces the inverse 2-Wasserstein distance as 'geometric fidelity' and proposes that reliable signaling balances these two dimensions. In six canonical gene regulatory motifs, the theory predicts a topology-dependent trade-off: coherent feed-forward loops can score high on both, while feedback loops give up information to better track the input distribution. The authors find this predicted trade-off in single-cell TNF response data, where removing a negative-feedback regulator shifts cells toward higher MI but lower geometric fidelity. If right, the result rewrites how signaling performance should be measured and designed.","feed_headline":"Feedback loops trade information for geometric fidelity","feed_subtitle":"Inverse Wasserstein distance adds a second axis to cell-signaling fidelity and separates feedback from non-feedback circuits in TNF data.","key_machinery":"The load-bearing object is the 2-Wasserstein distance (2-WD), the minimal 'transport cost' needed to reshape the input distribution into the output distribution; its inverse becomes geometric fidelity, while mutual information becomes informational fidelity. The two are combined in a Lagrangian, L = MI − λ·(2-WD)², optimized over the input noise level, with λ setting whether the cell prioritizes state resolution or distributional correspondence. Analytical Gaussian formulas express both metrics through means, squared coefficients of variation, and input-output covariance, and binding-affinity parameters θX and θZ act as tunable biochemical levers that move a circuit across the dual-fidelity","core_discovery":"The central claim is that distributional correspondence—how faithfully the output distribution mirrors the input distribution—is a distinct, measurable dimension of signaling fidelity, alongside the usual informational dimension measured by mutual information. Using Gaussian-channel closed forms for MI and the 2-Wasserstein distance, the paper shows that six canonical regulatory motifs occupy different regions of the resulting dual-fidelity space as binding affinities and a trade-off parameter λ are varied. The motif-specific result is that feedback architectures (especially negative feedback) suppress noise and dynamic range, improving geometric fidelity while lowering informational fidelit","pith_inferences":["The paper leaves implicit that this two-dimensional view could reinterpret other evolution experiments where knockouts exceed wild-type in MI; those cases may be feedback-loss artifacts rather than improvements.","A direct extension is to test the predicted binding-affinity lever in synthetic circuits: vary θX and θZ experimentally and see whether the dual-fidelity movement follows the computed direction for each motif, especially I1-FFL's inversion.","With single-cell reporters calibrated to absolute molecule counts, the unit-rescaling step could be eliminated; such data would test whether the WT versus A20-knockout separation is a real biological trade-off or a product of the slope-based mapping.","The Lagrangian resembles a rate-distortion problem with a geometric distortion; if general bounds on the 2-Wasserstein distance exist, it might be possible to derive universal inequalities between information rate and distributional distortion in signaling networks, which the paper only hints at."],"forward_implications":["If geometric fidelity is real, MI-only comparisons mis-rank circuits: a mutant with higher information transmission (A20-knockout) can be worse at faithful signaling, not better.","Topology predetermines strategy: coherent type-1 feed-forward loops can reach the 'precise' regime (both fidelities high), while feedback motifs—especially negative feedback—are biased toward geometric fidelity and low information.","Binding affinities provide a practical control knob: weak binding generally boosts informational fidelity, strong binding boosts geometric fidelity, with characteristic inversions in incoherent and feedback motifs.","The framework yields a synthetic-design rule: tune promoter binding affinities (and, in principle, production/degradation rates) to place a circuit in the fidelity regime a task demands.","The trade-off supplies a resolution to a paradox: wild-type cells can look information-suboptimal relative to a knockout, yet be better optimized once distributional correspondence is included."],"fun_headline_variants":["Signal fidelity's second axis: geometry, not just info","Feedback loops trade info for geometric precision","Two metrics decode signaling: info and geometry","Geometric fidelity: what mutual information misses","Wasserstein adds shape to cell signaling analysis"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The experimental part assumes that arbitrary fluorescence units can be converted to TNF-concentration units using the reciprocal slope of a fitted dose-response curve, and that wild-type and A20-knockout cells can be compared under different mapping rules (sigmoidal versus linear); if that rescaling is invalid, the observed separation between feedback and no-feedback cells could be an artifact.","fun_headline_variants_meta":{"raw":{"variants":["Signal fidelity's second axis: geometry, not just info","Feedback loops trade info for geometric precision","Two metrics decode signaling: info and geometry","Geometric fidelity: what mutual information misses","Wasserstein adds shape to cell signaling analysis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000159,"raw_usage":{"total_tokens":1051,"prompt_tokens":717,"completion_tokens":334,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":265}},"tokens_in":461,"tokens_out":334,"duration_ms":4612,"temperature":1.0,"reasoning_tokens":265,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T22:02:03.841522+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the dual-fidelity coordinates using an independent calibration (e.g., a reporter whose output is measured in absolute molecule numbers, or single-molecule RNA counts) and see whether wild-type cells still show higher geometric fidelity than A20-knockout cells. Alternatively, recompute the 2-Wasserstein distance after rescaling output quantiles by the dose-response slope at several different TNF doses; if the ordering flips depending on the chosen dose, the gain-based mapping is driving the result.","supporting_citations":[],"review_version":1}