Decoding cell signaling via optimal transport and information theory

Mintu Nandi; Sosuke Ito

REVIEW 4 major objections 6 minor 17 references

Decoding cell signaling via optimal transport and information theory

T0 review · 4 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2602.18028 v3 pith:T7FIKQS6 submitted 2026-02-20 physics.bio-ph cond-mat.stat-mech

Mintu Nandi , Sosuke Ito This is my paper · ORCID

classification physics.bio-phcond-mat.stat-mech

keywords mutualinformation2-Wassersteindistanceoptimaltransportgeometricfidelitydual-fidelityframeworkfeedbackregulationTNFsignalinggeneregulatorymotifs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

A 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.

The 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.

Referee Report

4 major / 6 minor

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.

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 (4)

[SI S5, Eq. (S26), Figs. S3a–d] 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.
[Abstract / Full text] 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.
[Eq. (6) and Fig. 5] 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.
[SI S5, 'MI-constrained reconstruction'] 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.

minor comments (6)

[Introduction, after Eq. (1)] 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.
[Fig. 6e] 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.
[SI S5] 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.
[Results, 'Motif-specific patterns'] 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.
[Materials and methods / SI S3] 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.
[Fig. 6c / SI S5] 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.

Circularity Check

1 steps flagged · score 6.0 of 10

The experimental WT/A20−/− geometric-fidelity separation in Fig. 6c is generated by condition-specific a.u.-to-ng/mL rescaling (different fitted slopes for WT vs A20−/−), so the main experimental support for the paper's central claim is partly constructed rather than independently measured.

fitted input called prediction [SI S5, Eq. (S26) and Figs. S3a–d; Results 'Experimental signature of dual-fidelity trade-offs' / Fig. 6c]
"The reciprocal of this slope, with units of ng/mL per a.u., is therefore used as the mapping factor to convert output quantiles into the same units as the input. ... Slope at half-max = 215.016 a.u./(ng/mL) ... Slope = 29.208 a.u./(ng/mL)"

Eq. (S26) defines W as the RMS difference between input and output quantile functions in ng/mL. Output quantiles are originally in a.u., so each condition's output quantiles are divided by a fitted condition-specific slope. WT is divided by 215.0/156.9 a.u. per ng/mL (Hill half-max slopes), while A20−/− is divided by 29.2/28.1 a.u. per ng/mL (linear slopes). The ~5–7× larger WT divisor compresses WT output quantiles toward the input scale, mechanically reducing W (raising geometric fidelity), while the small A20−/− divisor inflates W. Thus the WT/KO separation in Fig. 6c is not a unit-invariant property of the signaling distributions; it is imposed by applying different fitted mapping rules before computing the metric that is then presented as the new fidelity axis.

full rationale

The theoretical motif analysis is not substantially circular: the LNA supplies closed forms for output noise η_Z^2 and covariance ζ_XZ, and MI and 2-WD are evaluated from those expressions (Eqs. 8 and 10). Optimizing L = I − λ W² over input noise with a sweep of λ does parameterize a trade-off by construction, but the motif-specific ordering (e.g., C1-FFL high in both, NFL biased toward geometric fidelity) is a nontrivial output of the model, not written into the definition of L. Self-citations to Ito 2024 and Nandi et al. 2024 are methodological (OT bounds and LNA formulas) and are not load-bearing for the dual-fidelity dichotomy. The serious circularity is in the experimental validation. The Materials and methods state that the input marginal is optimized until MI matches Cheong et al.'s reported values, so the informational-fidelity coordinate is imported, not predicted. The genuinely new coordinate, geometric fidelity, is then computed from quantile functions after rescaling output a.u. quantiles by the reciprocal of a fitted, condition-specific dose-response slope. Because WT and A20−/− are rescaled with very different fitted slopes (Hill half-max for WT, linear for A20−/−), the resulting W values, and hence the Fig. 6c WT/KO separation, are largely determined by that arbitrary unit-conversion choice rather than by feedback biology. The paper's caveat that only trends and separations are compared does not neutralize this, since the separation itself is not invariant to the chosen rescaling. This makes the main experimental support for 'negative feedback increases geometric fidelity at the expense of informational fidelity' partially circular: the measured outcome is encoded in the condition-specific fitted mapping used to define it.

Assumptions & free parameters 9 free parameters · 8 assumptions · 1 invented entities

The framework's predictions depend on many modeling choices: the Gaussian/LNA approximation, the Lagrangian form, BAP sampling, and, in the experimental part, an MI-constrained reconstruction plus slope-based unit conversion. These are the main sources of freedom that the central claims sit on.

free parameters (9)

Lagrange multiplier lambda = 10^-4 to 1
Balances MI and W^2 in Eq. (1); range chosen so both terms enter comparably. Not measured.
Binding affinity parameters theta_X, theta_Z = {0.5, 1, 2}
Represent strong, half-maximal, and weak binding; chosen by hand, not fitted to data.
Input noise eta_X^2 = optimized via Eq. (6)
Variational parameter representing upstream variability; not measured in the theoretical part.
Mean copy numbers mu_X, mu_Y, mu_Z = 100
Chosen to justify LNA and keep noise comparable across species.
Degradation rates tau_Y^-1, tau_Z^-1 = 1.0 and 10.0
Chosen to impose a separation of timescales; arbitrary.
Hill coefficient h = 1
Chosen for monostability and analytical tractability.
Experimental input marginal P_X(x) = optimized to match Cheong et al. MI
SI S5: input distribution is fitted so that computed MI equals reported MI; this reconstruction feeds directly into the experimental 2-WD.
Experimental unit mapping factors = WT: 215.016, 156.863 a.u./(ng/mL); A20-/-: 29.208, 28.129 a.u./(ng/mL)
Slopes fitted to dose-response data and used to convert output a.u. to TNF ng/mL before computing W.
Regime thresholds for informative/precise/geometric/poor = 0.3 bits and W^-1 = 1 (copy numbers)^-1
Chosen for interpretability; affect qualitative regime labels and motif comparisons.

assumptions (8)

domain assumption Linear noise approximation gives accurate steady-state noise for copy numbers ~100.
Used throughout theoretical derivations; SI S1; requires large copy numbers and small fluctuations.
domain assumption Input X is a static extrinsic Gaussian variable not regulated by Y or Z.
SI S1; allows closed-form sigma and zeta; excludes downstream regulation of the input.
domain assumption Regulation uses Hill functions with Hill coefficient h=1 and first-order degradation.
SI Table S1; chosen for tractability and monostability; results may differ for h>1.
standard math For one-dimensional Gaussian X and Z, W2^2 = (mu_X - mu_Z)^2 + (sigma_X - sigma_Z)^2.
Gelbrich 1990; used in Eq. (9).
standard math For Gaussian joint distributions, MI is given by Eq. (8).
Cover and Thomas; used throughout.
domain assumption Reported mean and standard deviation from Cheong et al., with Gaussian conditionals, represent the true single-cell output distributions.
SI S5; empirical single-cell distributions were not published, so Gaussian reconstruction is assumed.
ad hoc to paper Gain-based rescaling via fitted dose-response slope is a valid common scale for computing 2-WD between input (ng/mL) and output (a.u.).
SI S5 and Figs. S3a-d; no independent calibration; different fitting forms are used for WT vs A20-/-.
ad hoc to paper Cellular priorities are captured by maximizing L = I - lambda W^2 over input noise eta_X^2.
Eqs. (1) and (6); the trade-off is encoded by construction, and lambda is not derived from biology.

invented entities (1)

Geometric fidelity (GMF), the inverse 2-Wasserstein distance, as a second dimension of signaling fidelity
purpose: Serves as the geometric counterpart to mutual information in the dual-fidelity landscape.
It is a defined metric, not an independently observed entity; its biological role is evidenced only through the model and reconstructed experimental distributions in this paper.

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Pith. "Pith review of Decoding cell signaling via optimal transport and information theory." pith.science (2026). https://pith.science/paper/T7FIKQS6

@misc{pith2026260218028,
  author       = {Pith},
  title        = {Pith review of: Decoding cell signaling via optimal transport and information theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7FIKQS6}},
  note         = {Machine review of arXiv:2602.18028}
}

read the original abstract

Cellular signal processing performs reliably despite molecular noise. Mutual information (MI) is widely used to quantify signaling fidelity, capturing how well outputs discriminate input states. However, it fails to capture whether the output preserves the statistical structure of the input, a property crucial in morphogen patterning and dose-dependent signaling. To address this gap, we introduce the 2-Wasserstein (2-WD) distance, which provides a geometric basis for comparing input and output distributions. We define MI as informational fidelity (INF) and the inverse of the 2-WD as geometric fidelity (GMF). Applying this dual-fidelity framework to canonical regulatory motifs under Gaussian channel approximation reveals topology-dependent trade-offs: coherent feed-forward loops can perform well in both dimensions, whereas feedback architectures reduce INF to enhance GMF. Experimental analysis of tumor necrosis factor signaling reveals dual-fidelity behavior qualitatively consistent with feedback regulation. RAS-MAPK data analysis further shows that jointly considering INF and GMF better characterizes intracellular signal relay than INF alone. Our results thus indicate that these signaling behaviors are not fully characterized by MI alone; instead, distributional correspondence provides a complementary dimension of signaling fidelity. Our study provides a practical framework for analyzing natural networks and guiding the design of task-specific synthetic circuits.

Figures

Figures reproduced from arXiv: 2602.18028 by the authors.

**Figure 1.** Schematics of signaling systems and the fidelity representations. Illustrative examples of signaling pathways: a morphogen signaling, b neuronal signaling, c intracellular signaling, d cell-to-cell communication, e bet-hedging, and f homeostatic control. g A general scheme of signal transduction. Here, the input X acts as an internal representation of the external signals. The input distribution PX(x) is processed i… view at source ↗

**Figure 2.** Schematic illustrations of informational and geometric fidelity. a Informational fidelity characterizes state-wise mapping between input and output distributions. This mapping resolves states accurately but sacrifices distributional correspondence. b Geometric fidelity accounts for shape, scale (variability) and mean-wise mapping of input distribution to output. While this geometric fidelity cannot measure the preci… view at source ↗

**Figure 3.** Schematics of gene regulatory motifs and governing dynamical equation. a Simple cascade: the input X activates Y , which subsequently activates Z. b Coherent type-1 feed-forward loop: X activates both Y and Z, and Y activates Z. c Incoherent type-1 feed-forward loop: X activates both Y and Z, but Y represses Z. d Positive feedback loop: X activates Y , which again activates Z, and Z activates Y forming the feedback.… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Binding affinity parameters θX and θZ . a Schematic illustration of θX and θZ across different network motifs. b Definition of the binding affinity parameters. θX and θZ represent the effective binding affinities of X and Z, respectively, to the promoter of gene Y . θX…

**Figure 5.** Figure 5: Trade-off between informational and geometric fidelities across network motifs. The relationships between the two fidelities obtained from the dual-fidelity optimization framework for six canonical motifs: a Simple cascade, b Coherent type-1 feed-forward loop, c Incohe…

**Figure 6.** Figure 6: Theory-experiment correspondence of dual-fidelity trade-offs. a Schematic of TNF signaling in wild-type (WT) cells and A20-deficient (A20−/−) mutant cells. b Summary of measured quantities: informational fidelity I(X;Z) and geometric fidelity W(X, Z) −1 for NF-κB and A…

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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