arxiv: 2605.02972 · v1 · submitted 2026-05-03 · 🧮 math.DS · physics.bio-ph· q-bio.QM

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Non-Monotone Response Modules and Cascades from the EML Operator for Reduced Models of Biological Dynamics

Amir Erez

Authors on Pith no claims yet

Pith reviewed 2026-05-09 15:51 UTC · model grok-4.3

classification 🧮 math.DS physics.bio-phq-bio.QM

keywords EML operatornon-monotone responsesreduced ODE modelsbiological dynamicsactivation-suppression moduleovershoot transientsnetwork compression

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The pith

The EML binary operator serves as a grammar to build single-block activation-suppression modules that capture non-monotone overshoot in reduced biological ODE models.

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

Standard saturating functions such as the Hill equation are monotone and therefore require at least two opposing blocks to reproduce overshoot or adaptive transients, which doubles the parameter count. The paper uses the EML operator, previously shown to generate all elementary functions, to define expression trees that directly yield non-monotone response modules within nonlinear ODEs. This framework is tested on PKA-R relocalization data, where it recovers a surrogate consistent with known mechanisms, on Rho-GTPase recruitment traces, where exhaustive search selects the same form across perturbations, and on compression of a 50-state simulated network via an EML cascade. If correct, the approach simplifies model construction for biological dynamics while preserving mechanistic interpretability.

Core claim

Treating the EML operator as a structured grammar for reduced nonlinear ODEs produces an activation-suppression module that captures overshoot and transients directly, rather than through subtraction of separate saturating blocks; the same compositional structure emerges across independent experimental traces and enables compression of large state networks.

What carries the argument

The EML binary operator, applied to generate expression trees that serve as the grammar for constructing reduced nonlinear ODEs and their associated response modules.

If this is right

A single EML module reproduces overshoot without requiring a difference of two saturating blocks and their extra parameters.
Exhaustive enumeration of EML expression trees on Rho-GTPase data selects one consistent compositional form across four perturbation conditions.
The same EML grammar applied to PKA-R relocalization data produces a reduced surrogate aligned with established mechanistic biology.
An EML cascade functions as a fixed temporal basis that compresses a 50-state simulated network while retaining essential dynamics.

Where Pith is reading between the lines

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

The consistent selection of identical compositional forms across perturbations may indicate that EML trees can identify robust structural motifs in biological networks.
Cascades built from the same operator could serve as a systematic way to derive hierarchical reduced models for systems larger than those tested here.
Because EML generates all elementary functions, the grammar might be extended to other classes of dynamical systems that require compact non-monotone representations.

Load-bearing premise

The EML-derived expressions fitted to biological time series remain mechanistically interpretable rather than acting as flexible curve-fitting forms that reproduce data without adding new biological insight.

What would settle it

An EML-constructed model for a new set of biological response traces that matches the data yet yields predictions under perturbation that contradict independent mechanistic measurements or established pathway knowledge.

Figures

Figures reproduced from arXiv: 2605.02972 by Amir Erez.

**Figure 1.** Figure 1: FIG. 1. Restricted EML-ODE grammar search applied to PKA-R relocalization data from LaCroix view at source ↗

**Figure 2.** Figure 2: FIG. 2. Restricted EML grammar search applied to all four response traces in figure 2d of Nanda view at source ↗

**Figure 3.** Figure 3: FIG. 3. Parallel Hill-grammar search applied to the same four response traces of Nanda view at source ↗

**Figure 4.** Figure 4: FIG. 4. Toy coarse-graining benchmark. A 50-state activation/adaptation network driven by view at source ↗

read the original abstract

Standard saturating response functions, such as the Hill function, are monotone and therefore cannot represent recruitment-induced overshoot or adaptive transients with a single block. Reproducing such non-monotone responses from saturating primitives requires at least a difference of two blocks with opposing amplitudes, doubling the static-block parameter count. Here, building on a recent mathematical result that a single binary operator, EML, generates all standard elementary functions, we use EML as a structured grammar for reduced nonlinear ODEs. This yields an activation-suppression module that captures overshoot directly. We validate the framework in three settings. First, on PKA-R relocalization data, the EML grammar discovers a reduced surrogate consistent with established mechanistic biology. Second, on Rho-GTPase recruitment data, an exhaustive search over EML expression trees selects the same compositional form across all four perturbation-response traces. Third, a 50-state simulated network is compressed by an EML cascade acting as a fixed temporal basis. Thus we demonstrate the power and potential of EML for reduced models of biological dynamics.

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

EML gives a structured grammar for non-monotone biological responses that fits data consistently, but the parameter-reduction and mechanistic claims rest on fits without shown derivations or direct comparisons.

read the letter

The main takeaway is that EML can serve as a grammar to build non-monotone biological response modules that capture overshoot without doubling the parameter count of standard saturating functions. This is a new application of the recent mathematical result on the EML operator. The paper turns it into a tool for constructing reduced nonlinear ODEs in systems biology. It does well in the validation steps. On PKA-R relocalization data, it finds a reduced surrogate that stays consistent with known mechanistic biology. For the Rho-GTPase recruitment data, an exhaustive search picks out the same compositional form for all four perturbation-response traces. The third test compresses a 50-state simulated network using an EML cascade as a fixed temporal basis. The consistency across different traces stands out as a strength. It suggests the grammar has some built-in structure that matches the underlying dynamics rather than just fitting each case separately. The network compression also shows a concrete use case for simplifying larger models. The soft spots lie in how much reduction and interpretability is actually achieved. The abstract contrasts EML with the need for a difference of two Hill blocks, but we lack the explicit EML expressions or quantitative comparisons of parameter counts and fit errors. If the EML modules can be algebraically rewritten as equivalent two-block differences, the claimed advantage fades. The validations rely on fitting or search to data, which means they demonstrate consistency but do not rule out that EML trees are simply more flexible approximators. Without seeing the derivations, it's hard to tell if they add new dynamical insight or just reproduce the transients. This work is aimed at systems biologists who build reduced models of cell signaling and want compact ways to include non-monotone transients. Readers focused on model reduction and network compression will get the most from the examples. The paper deserves a serious referee. The idea is grounded and the tests are relevant, even if more detail on the math and comparisons would strengthen it. I would recommend sending it to peer review. The process will clarify whether the EML approach delivers genuine parsimony and mechanistic value beyond standard methods.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes using the EML binary operator—which a prior result shows generates all standard elementary functions—as a structured grammar for reduced nonlinear ODEs in biological dynamics. It claims this yields an activation-suppression module that directly captures non-monotone responses such as overshoot or adaptive transients, unlike standard monotone Hill functions that require a difference of two opposing blocks (doubling parameters). The framework is validated in three settings: discovery of a reduced surrogate on PKA-R relocalization data consistent with known biology; exhaustive search selecting the same EML compositional form across four Rho-GTPase perturbation traces; and compression of a 50-state simulated network via an EML cascade as a fixed temporal basis.

Significance. If the central claim is substantiated—that EML modules produce genuinely reduced, mechanistically interpretable ODEs that are not merely re-expressions of two-term differences of saturating functions—it would offer a principled, parsimonious approach to model reduction for systems exhibiting recruitment-induced overshoot, with potential for broader use in dynamical systems biology.

major comments (2)

[Derivation of the activation-suppression module] The derivation of the activation-suppression module (the section presenting the EML grammar application): the manuscript must explicitly exhibit the EML expression tree and prove algebraically that the resulting non-monotone response cannot be rewritten as an equivalent difference of two standard saturating primitives (e.g., Hill functions with opposing signs), since this equivalence would nullify the claimed parameter-count reduction and is load-bearing for the central advantage over conventional models.
[PKA-R and Rho-GTPase validations] PKA-R relocalization and Rho-GTPase validations (the sections reporting the data fits and exhaustive search): quantitative error metrics, parameter counts, baseline comparisons (single Hill vs. double Hill vs. EML), and explicit exclusion criteria for alternative trees must be supplied; without them the post-hoc consistency with biology and cross-trace invariance cannot be distinguished from flexible functional approximation.

minor comments (2)

[Abstract] Abstract: include at least the explicit EML expression for the activation-suppression module (or a compact equation) so readers can immediately see the claimed non-monotone form.
[Network compression] Network compression section: specify the precise sense in which the EML cascade functions as a 'fixed temporal basis' and report the achieved state reduction ratio together with any residual error on the original 50-state trajectories.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments identify important clarifications needed to strengthen the central claims regarding the EML activation-suppression module and the empirical validations. We address each major comment below and will incorporate the requested additions in the revised manuscript.

read point-by-point responses

Referee: [Derivation of the activation-suppression module] The derivation of the activation-suppression module (the section presenting the EML grammar application): the manuscript must explicitly exhibit the EML expression tree and prove algebraically that the resulting non-monotone response cannot be rewritten as an equivalent difference of two standard saturating primitives (e.g., Hill functions with opposing signs), since this equivalence would nullify the claimed parameter-count reduction and is load-bearing for the central advantage over conventional models.

Authors: We agree that an explicit EML expression tree together with an algebraic demonstration of inequivalence to any difference of opposing Hill functions is required to substantiate the parameter-reduction advantage. The manuscript introduces the module through the EML grammar and states that it captures non-monotone responses directly, but does not contain the requested formal proof. In the revision we will add a dedicated subsection that (i) displays the full binary expression tree for the activation-suppression module and (ii) supplies the algebraic steps showing that the resulting functional form lies outside the two-term difference of saturating primitives for any choice of parameters. This addition will directly address the load-bearing claim. revision: yes
Referee: [PKA-R and Rho-GTPase validations] PKA-R relocalization and Rho-GTPase validations (the sections reporting the data fits and exhaustive search): quantitative error metrics, parameter counts, baseline comparisons (single Hill vs. double Hill vs. EML), and explicit exclusion criteria for alternative trees must be supplied; without them the post-hoc consistency with biology and cross-trace invariance cannot be distinguished from flexible functional approximation.

Authors: We acknowledge that quantitative error metrics, explicit parameter counts, baseline comparisons, and exclusion criteria are necessary to distinguish the EML results from generic functional approximation. The present manuscript emphasizes qualitative agreement with known biology and the invariance of the selected compositional form, but does not report the requested numerical comparisons. In the revised sections we will include (i) error metrics (e.g., RMSE) for the PKA-R and Rho-GTPase fits, (ii) tables listing parameter counts for single-Hill, double-Hill, and EML models, (iii) direct performance baselines, and (iv) a clear statement of the exclusion criteria together with the selection statistics across the four Rho-GTPase traces. These additions will provide objective support for the claimed parsimony and cross-trace consistency. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external mathematical result and post-hoc data consistency checks.

full rationale

The paper's core step is adopting the EML operator (cited as a recent mathematical result that generates all elementary functions) as a grammar for ODE modules, then using exhaustive search over expression trees to identify forms that match biological traces and are consistent with known mechanisms. No equation or claim reduces by construction to its inputs: the search selects among possible EML compositions rather than presupposing the target form, and the consistency with PKA-R or Rho-GTPase biology is an external check rather than a self-referential fit. The prior EML result is treated as an independent theorem rather than a self-citation that itself depends on the present claims. This leaves the framework self-contained against the stated benchmarks without definitional loops or renamed fits presented as predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unproven-in-this-paper property that EML generates all elementary functions, plus the assumption that expression-tree search or fitting to the cited datasets yields biologically interpretable reduced models. No explicit free parameters or invented entities are named in the abstract.

axioms (1)

domain assumption A single binary operator EML generates all standard elementary functions
Invoked in the first sentence of the abstract as the foundation for using EML as a grammar.

pith-pipeline@v0.9.0 · 5489 in / 1323 out tokens · 28418 ms · 2026-05-09T15:51:52.187829+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Transtrum, Benjamin B

Mark K. Transtrum, Benjamin B. Machta, Kevin S. Brown, Bryan C. Daniels, Christopher R. Myers, and James P. Sethna. Perspective: Sloppiness and emergent theories in physics, biology, and beyond.The Journal of Chemical Physics, 143(1):010901, July 2015

2015
[2]

Lopez and Amir Erez

Jamie A. Lopez and Amir Erez. Mathematical Mod- elling and Intuition in Microbiology: A Perspective.En- vironmental Microbiology, 28(4):e70266, 2026. e70266 1612994

2026
[3]

Chiliagon Science, Philadelphia, PA, 2022

Philip Nelson.Physical Models of Living Systems: Probability, Simulation, Dynamics. Chiliagon Science, Philadelphia, PA, 2022

2022
[4]

Distilling Free- Form Natural Laws from Experimental Data.Science, 324(5923):81–85, April 2009

Michael Schmidt and Hod Lipson. Distilling Free- Form Natural Laws from Experimental Data.Science, 324(5923):81–85, April 2009

2009
[5]

AI Feynman: A physics-inspired method for symbolic regression.Sci- ence Advances, 6(16):eaay2631, April 2020

Silviu-Marian Udrescu and Max Tegmark. AI Feynman: A physics-inspired method for symbolic regression.Sci- ence Advances, 6(16):eaay2631, April 2020

2020
[6]

Discovering Symbolic Models from Deep Learning with Inductive Biases

Miles Cranmer, Alvaro Sanchez Gonzalez, Peter Battaglia, Rui Xu, Kyle Cranmer, David Spergel, and Shirley Ho. Discovering Symbolic Models from Deep Learning with Inductive Biases. InAdvances in Neu- ral Information Processing Systems, volume 33, pages 17429–17442. Curran Associates, Inc., 2020

2020
[7]

Daniels and Ilya Nemenman

Bryan C. Daniels and Ilya Nemenman. Automated adap- tive inference of phenomenological dynamical models. Nature Communications, 6(1):8133, August 2015

2015
[8]

Brunton, Joshua L

Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical sys- tems.Proceedings of the National Academy of Sciences, 113(15):3932–3937, April 2016

2016
[9]

Rudy, Steven L

Samuel H. Rudy, Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Data-driven discovery of partial differential equations.Science Advances, 3(4):e1602614, April 2017

2017
[10]

All elementary functions from a single binary operator

Andrzej Odrzywo lek. All elementary functions from a single binary operator. arXiv:2603.21852 [cs.SC], 2026. https://arxiv.org/abs/2603.21852

work page internal anchor Pith review Pith/arXiv arXiv 2026
[11]

HILL A. V. The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves. J. Physiol., 40:iv–vii, 1910

1910
[12]

Vogel, Amir Erez, and Gr´ egoire Altan-Bonnet

Robert M. Vogel, Amir Erez, and Gr´ egoire Altan-Bonnet. Dichotomy of cellular inhibition by small-molecule in- hibitors revealed by single-cell analysis.Nature Com- munications, 7(1):12428, September 2016

2016
[13]

Complex effects of kinase localization revealed by compartment-specific regulation of protein kinase A activity.eLife, 11:e66869, February 2022

Rebecca LaCroix, Benjamin Lin, Tae-Yun Kang, and An- dre Levchenko. Complex effects of kinase localization revealed by compartment-specific regulation of protein kinase A activity.eLife, 11:e66869, February 2022

2022
[14]

Rho GTPase activity crosstalk mediated by Arhgef11 and Arhgef12 coordinates cell protrusion- retraction cycles.Nature Communications, 14(1):8356, December 2023

Suchet Nanda, Abram Calderon, Arya Sachan, Thanh- Thuy Duong, Johannes Koch, Xiaoyi Xin, Djamschid 10 Solouk-Stahlberg, Yao-Wen Wu, Perihan Nalbant, and Leif Dehmelt. Rho GTPase activity crosstalk mediated by Arhgef11 and Arhgef12 coordinates cell protrusion- retraction cycles.Nature Communications, 14(1):8356, December 2023

2023
[15]

Real-Time Computing Without Stable States: A New Framework for Neural Computation Based on Perturbations.Neural Computation, 14(11):2531–2560, November 2002

Wolfgang Maass, Thomas Natschl¨ ager, and Henry Markram. Real-Time Computing Without Stable States: A New Framework for Neural Computation Based on Perturbations.Neural Computation, 14(11):2531–2560, November 2002

2002
[16]

Harnessing Nonlin- earity: Predicting Chaotic Systems and Saving Energy in Wireless Communication.Science, 304(5667):78–80, April 2004

Herbert Jaeger and Harald Haas. Harnessing Nonlin- earity: Predicting Chaotic Systems and Saving Energy in Wireless Communication.Science, 304(5667):78–80, April 2004

2004
[17]

Physical reservoir computing—an in- troductory perspective.Japanese Journal of Applied Physics, 59(6):060501, May 2020

Kohei Nakajima. Physical reservoir computing—an in- troductory perspective.Japanese Journal of Applied Physics, 59(6):060501, May 2020. 1 Appendix Appendix A: Derivation of the one-gate EML activation-suppression module Starting from eml(αlnR,e βR) = eαlnR −ln eβR (S1) =R α −βR.(S2) ForR >0, the derivative is d dR (Rα −βR) =αR α−1 −β.(S3) For 0< α <1 andβ...

2020