Cooperativity, Absolute Interaction, and Algebraic Optimization

Johannes W. R. Martini; Mohab Safey El Din; Nidhi Kaihnsa; Yue Ren

arxiv: 1906.10006 · v1 · pith:J4NOI4GNnew · submitted 2019-06-24 · 🧬 q-bio.BM · math.OC

Cooperativity, Absolute Interaction, and Algebraic Optimization

Nidhi Kaihnsa , Yue Ren , Mohab Safey El Din , Johannes W. R. Martini This is my paper

Pith reviewed 2026-05-25 16:54 UTC · model grok-4.3

classification 🧬 q-bio.BM math.OC

keywords cooperativityabsolute interactionalgebraic optimizationbinding polynomialshemoglobintitration behaviorHill slope

0 comments

The pith

Minimal absolute interaction provides a cooperativity measure for hemoglobin that ranks molecules differently from the maximal Hill slope.

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

The paper defines cooperativity through the smallest absolute interaction strength that can still produce a given set of titration curves in a binding polynomial. This quantity is recovered by solving an algebraic optimization problem with existing nonlinear algebra software. When the method is applied to oxygen-binding data from several hemoglobins and their chemically modified forms, the resulting values match the direction of change expected from the modifications. At the same time the ordering of the molecules by this new measure differs from the ordering produced by the largest Hill slope. A reader would care because the approach supplies an alternative numerical handle on cooperativity that is directly tied to the parameters of the binding model rather than to a derived slope statistic.

Core claim

The minimal absolute interaction required to generate an observed titration behavior is proposed as a measure of cooperativity; the corresponding algebraic optimization problem can be solved with SCIP; and numerical values computed for hemoglobin binding polynomials are consistent with the effects of chemical modifications yet produce a different ranking of cooperativity than the maximal Hill slope.

What carries the argument

Minimal absolute interaction: the smallest value of the absolute interaction parameters in a binding polynomial that is still sufficient to reproduce the observed titration curve, recovered by algebraic optimization.

If this is right

The same optimization procedure applies to any binding polynomial that describes titration data.
Computed values remain consistent with the direction of change produced by known chemical modifications.
The ranking of cooperativity obtained this way differs from the ranking obtained from maximal Hill slopes.
Existing nonlinear algebra solvers suffice to obtain the numerical values for typical hemoglobin data sets.

Where Pith is reading between the lines

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

The same optimization could be run on binding polynomials for other oligomeric proteins to compare cooperativity across different molecular systems.
If the minimal-interaction values correlate with structural distances between binding sites, they could supply a quantitative link between sequence changes and cooperative strength.
Repeated application to families of mutants might allow systematic testing of whether the measure isolates interaction terms more cleanly than slope-based statistics.

Load-bearing premise

That the minimal absolute interaction required to generate an observed titration behavior constitutes a biologically meaningful and preferable measure of cooperativity.

What would settle it

A set of binding curves for a chemically modified hemoglobin in which the computed minimal absolute interaction is larger than for the unmodified protein, yet experimental measures show the modification actually increases cooperativity.

Figures

Figures reproduced from arXiv: 1906.10006 by Johannes W. R. Martini, Mohab Safey El Din, Nidhi Kaihnsa, Yue Ren.

**Figure 1.** Figure 1: A molecule with 4 binding sites. Definition 2.2 (Binding Polynomial) Given a molecule W = (wI )I⊆[n] with n sites, its binding polynomial Φ(W) is a univariate polynomial of degree n, Φ(W) := anΛ n + · · · + a1Λ + a0, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Minimal molecules for binding polynomial P1, P2, P3. 3. The Algebraic Optimization Problem In this section, we consider the computation of the minimal absolute interaction as an optimization problem with the absolute interaction as objective function and the set of all molecules sharing the same binding polynomial as feasible set: minimize W kWk subject to Φ(W) = P (1) This problem seems simple and its con… view at source ↗

**Figure 3.** Figure 3: Computing the minimal absolute interaction for a molecule with 3 sites using SCIP. To simplify our problem computationally, we compute an upper bound b+ and a lower bound b− for the minimal absolute interaction of a given binding polynomial. If both bounds are identical, the minimum is found. To compute the upper bound we minimize the minimal absolute interaction in a single region. For the experiments in… view at source ↗

**Figure 4.** Figure 4: Conditions under which binding polynomials 1 to 8 were derived and the relation of their degree of cooperativity according to the maximal slope of the Hill plot nmax (≻ compares the hill slope nmax, DPG = 2,3-diphosphoglycerate). Clam HbII at 10◦ at 15◦ at 20◦ at 25◦ at 30◦ P9 P10 P11 P12 P13 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Temperature in degree Celsius at which binding polynomials 9 to 13 were determined. there remains a nontrivial gap between both bounds for P2, P3, P4. Let us assume that the correct value for P2 is close to the upper bound which we determined. Then, nearly all relations of the degree of cooperativity of polynomials P1 to P8 described by the maximal slope of the Hill plot nmax ( [PITH_FULL_IMAGE:figures/f… view at source ↗

**Figure 6.** Figure 6: Binding polynomials, bounds on the minimal absolute interaction and nmax as reported by [Ike+83; Ima73]. Coefficients a0 and a4 are equal to 1 for all polynomials. The polynomials P9 to P13 were derived from the same protein at different temperatures [Ike+83]. We see that the reported nmax is not behaving monotonously with increasing temperature, but that it varies between 2.08 and 2.12. Since one might e… view at source ↗

**Figure 7.** Figure 7: Molecules of polynomials P1 to P13 realizing the upper bound b+. The wI s are rounded to four digits. The incidences of a wI being smaller than 1 are numerical impressions when resolving from the sI coordinates. All interactions larger than 1 are highlighted in blue. 5. Discussion and Outlook 5.1. Cooperativity and minimal absolute interaction. A commonly used macroscopic conceptualization of cooperativit… view at source ↗

read the original abstract

We consider a measure of cooperativity based on the minimal absolute interaction required to generate an observed titration behavior. We describe the corresponding algebraic optimization problem and show how it can be solved using the nonlinear algebra tool \texttt{SCIP}. Moreover, we compute the minimal absolute interactions for various binding polynomials that describe the oxygen binding of various hemoglobins under different conditions. While calculated minimal absolute interactions are consistent with the expected outcome of the chemical modifications, it ranks the cooperativity of the molecules differently than the maximal Hill slope.

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

The paper defines cooperativity as the smallest absolute interaction term needed to match an observed binding curve, solves that via algebraic optimization with SCIP, and shows it ranks modified hemoglobins differently from the max Hill slope while staying consistent with chemical changes.

read the letter

The core contribution is a reframing of cooperativity as an algebraic minimization problem: find the minimal absolute interaction strength that can still reproduce a given titration polynomial. They implement this with the SCIP solver and run it on oxygen-binding data for several hemoglobins under different conditions. The results line up with what the chemical modifications should do, yet the ordering of cooperativity differs from the usual maximal Hill slope values. That difference is the main empirical point they report. The setup itself is cleanly stated as an optimization task over the binding polynomial coefficients, and the application to real hemoglobin variants gives a concrete test case rather than pure theory. The math appears reproducible in principle once the exact objective and constraints are written down. The main limitation is that the preference for this new measure over Hill slope is presented as an observation rather than a demonstrated advantage. There is no systematic comparison against other established models such as MWC, no synthetic-data recovery tests, and no discussion of how sensitive the minimal-interaction values are to noise in the titration curves. Without those checks it is hard to judge whether the new ranking reflects a biologically sharper quantity or simply a different mathematical emphasis. The work sits squarely in mathematical biology and binding thermodynamics. Readers who already work with algebraic methods on binding polynomials or who want an alternative scalar for cooperativity will find the framing useful. It is narrow enough that it will not change how most experimentalists measure cooperativity tomorrow, but the optimization angle is distinct enough to merit referee attention. I would send it for review.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a new measure of cooperativity given by the minimal absolute interaction required to reproduce an observed titration curve. It formulates the corresponding algebraic optimization problem, solves it with the SCIP solver, and applies the procedure to binding polynomials for oxygen binding by various hemoglobins under different conditions. The resulting minimal-interaction values are reported to be consistent with the expected effects of chemical modifications while producing a different ordering of cooperativity than the maximal Hill slope.

Significance. If the optimization is correctly posed and the measure is shown to be robust, the work supplies a concrete algebraic framework that links cooperativity directly to interaction parameters rather than to phenomenological descriptors such as the Hill coefficient. The explicit use of a global nonlinear solver (SCIP) on binding polynomials constitutes a reproducible computational contribution that could be extended to other multi-site systems.

major comments (2)

[Optimization problem formulation] The central definition equates the cooperativity measure with the objective value of the algebraic optimization itself. The manuscript should clarify, in the section that introduces the optimization problem, whether this construction yields an independent diagnostic or is tautological with the fitting procedure used to obtain the binding polynomial.
[Results on hemoglobin data] The claim that the new ranking is preferable rests on consistency with known chemical effects and divergence from the Hill slope. The results section should supply a quantitative metric (e.g., correlation with independent structural data or predictive accuracy on held-out titrations) rather than leaving preference as an empirical observation.

minor comments (2)

The abstract would benefit from stating the number of distinct hemoglobins and experimental conditions examined so that the scope of the empirical comparison is immediately clear.
All decision variables and constraints in the SCIP formulation should be explicitly numbered and cross-referenced in the text to facilitate independent verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the recommendation for minor revision. We respond to each major comment below.

read point-by-point responses

Referee: [Optimization problem formulation] The central definition equates the cooperativity measure with the objective value of the algebraic optimization itself. The manuscript should clarify, in the section that introduces the optimization problem, whether this construction yields an independent diagnostic or is tautological with the fitting procedure used to obtain the binding polynomial.

Authors: The binding polynomial is obtained by fitting titration data to a phenomenological model containing no explicit interaction terms. The subsequent algebraic optimization identifies the minimal absolute interaction parameters in a microscopic model whose binding polynomial exactly matches the observed coefficients. This step is independent of the initial fitting and supplies a diagnostic of required interaction strength. We will add a clarifying paragraph in the optimization-problem section to make this distinction explicit. revision: yes
Referee: [Results on hemoglobin data] The claim that the new ranking is preferable rests on consistency with known chemical effects and divergence from the Hill slope. The results section should supply a quantitative metric (e.g., correlation with independent structural data or predictive accuracy on held-out titrations) rather than leaving preference as an empirical observation.

Authors: The manuscript reports consistency with known chemical modifications and a different ordering from the maximal Hill slope as empirical observations; it does not assert that the new ranking is preferable. Because the study uses published binding polynomials and does not include independent structural data or held-out titrations, a quantitative metric of the requested type cannot be computed from the present material. We will insert a sentence in the results section noting that the comparison remains observational and that such metrics lie outside the current scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines a new cooperativity measure directly as the solution to an algebraic optimization problem (minimal absolute interaction reproducing a given titration curve) and applies it to existing binding polynomials for hemoglobins. Results are reported as consistent with chemical modifications while differing in ranking from the Hill slope. No derivation reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing claim rests on self-citation chains or imported uniqueness theorems. The framework is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract alone; therefore the ledger is necessarily incomplete. The central claim rests on the domain assumption that titration curves are faithfully captured by binding polynomials and on the new invented entity of absolute interaction.

axioms (1)

domain assumption Titration behavior is described by binding polynomials
The computations are performed on binding polynomials that model oxygen binding.

invented entities (1)

Absolute interaction no independent evidence
purpose: Quantifies the minimal interaction strength needed to produce observed titration behavior
New scalar introduced to serve as the cooperativity measure

pith-pipeline@v0.9.0 · 5617 in / 1135 out tokens · 28053 ms · 2026-05-25T16:54:06.342811+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel matches

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matches
MATCHES: this paper passage directly uses, restates, or depends on the cited Recognition theorem or module.

absolute interaction of a molecule W ... ∥W∥ := ∏_{|I|>1} max(w_I, w_I^{-1}) ... minimal absolute interaction ∥P∥ := min {∥W∥ | Φ(W)=P}
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

minimize ∏ max(w_I, w_I^{-1}) subject to Φ(W)=P ... lifted to linear objective with monomial constraints after s_I = ∏_{I'⊆I} w_{I'} change of variables

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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An empirical extremum principle fo r the hill coeﬃcient in ligand-protein interactions showing negative cooperativ ity

[Abe05] H. Abeliovich. “An empirical extremum principle fo r the hill coeﬃcient in ligand-protein interactions showing negative cooperativ ity”. In: Biophysical Journal 89.1 (2005), pp. 76–79. [Abe16] H. Abeliovich. “On Hill coeﬃcients and subunit inte raction energies”. In: Journal of Mathematical Biology 73.6-7 (2016), pp. 1399–1411. [Bar13] J. Barcroft...

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[2]

Ueber einen i n biologischer Beziehung wichtigen Einﬂuss, den die Kohlens¨ aurespannung des Blutes auf dessen Sauer- stoﬀbindung ¨ ubt 1

[BHK04] C. Bohr, K. Hasselbalch, and A. Krogh. “Ueber einen i n biologischer Beziehung wichtigen Einﬂuss, den die Kohlens¨ aurespannung des Blutes auf dessen Sauer- stoﬀbindung ¨ ubt 1”. In:Skandinavisches Archiv f¨ ur Physiologie16.2 (1904), pp. 402–412. [Bri83] W. Briggs. “A new measure of cooperativity in protei n-ligand binding”. In: Biophysical Chemi...

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Probabilistic algorit hm for polynomial optimization over a real algebraic set

url: http://www.optimization-online.or g/DB_HTML/2018/07/6692.html. [GS14] A. Greuet and M. Safey El Din. “Probabilistic algorit hm for polynomial optimization over a real algebraic set”. In: SIAM Journal on Optimization 24.3 (2014), pp. 1313–1343. [GYZ14] J. L. Gross, J. Yellen, and P. Zhang, eds. Handbook of graph theory . Second. Discrete Mathematics a...

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What is cooperativit y?

[HA09] C. A. Hunter and H. L. Anderson. “What is cooperativit y?” In: Angewandte Chemie International Edition 48.41 (2009), pp. 7488–7499. [Ike+83] M. Ikeda-Saito et al. “Thermodynamic properties o f oxygen equilibria of dimeric and tetrameric hemoglobins from Scapharca inaequi valvis”. In: Jour- nal of Molecular Biology 170.4 (1983), pp. 1009–1018. [Ima7...

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Global optimization with polynomial s and the problem of mo- ments

Graduate Studies in Mathematics. American Mathematical Society, Pr ovidence, RI, 2012, pp. xx+439. [Las01] J. Lasserre. “Global optimization with polynomial s and the problem of mo- ments”. In: SIAM Journal on Optimization 11.3 (2001), pp. 796–817. [LFSR09] T. Lenaerts, J. Ferkinghoﬀ-Borg, J. Schymkowitz, a nd F. Rousseau. “Infor- mation theoretical quant...

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A measure to quantify the degree of c ooperativity in overall titration curves

[Mar17a] J. W. Martini. “A measure to quantify the degree of c ooperativity in overall titration curves”. In: Journal of Theoretical Biology 432 (2017), pp. 33–37. 20 REFERENCES [Mar17b] J. W. Martini. “On the relation of diﬀerent deﬁnitio ns of cooperative binding for systems with two binding sites”. In: Match-Communications in Mathe- matical and in Comp...

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A novel vi ew of pH titration in biomolecules

[OCU01] A. Onufriev, D. A. Case, and G. M. Ullmann. “A novel vi ew of pH titration in biomolecules”. In: Biochemistry 40.12 (2001), pp. 3413–3419. [OU04] A. Onufriev and G. M. Ullmann. “Decomposing complex c ooperative lig- and binding into simple components: connections between mi croscopic and macroscopic models”. In: The Journal of Physical Chemistry B...

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Kinetic regulation of mult i-ligand binding proteins

REFERENCES 21 [Sal+16] D. V. Salakhieva et al. “Kinetic regulation of mult i-ligand binding proteins”. In: BMC Systems Biology 10.1 (2016), p

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Macromolecular binding

[Sch75] J. A. Schellman. “Macromolecular binding”. In: Biopolymers 14.5 (1975), pp. 999–1018. [Sch91] B. Schweizer. “Thirty years of copulas”. In: Advances in Probability Distri- butions with Given Marginals . Springer, 1991, pp. 13–50. [SW81] B. Schweizer and E. F. Wolﬀ. “On nonparametric measure s of dependence for random variables”. In: The Annals of S...

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[1] [1]

An empirical extremum principle fo r the hill coeﬃcient in ligand-protein interactions showing negative cooperativ ity

[Abe05] H. Abeliovich. “An empirical extremum principle fo r the hill coeﬃcient in ligand-protein interactions showing negative cooperativ ity”. In: Biophysical Journal 89.1 (2005), pp. 76–79. [Abe16] H. Abeliovich. “On Hill coeﬃcients and subunit inte raction energies”. In: Journal of Mathematical Biology 73.6-7 (2016), pp. 1399–1411. [Bar13] J. Barcroft...

work page 2005

[2] [2]

Ueber einen i n biologischer Beziehung wichtigen Einﬂuss, den die Kohlens¨ aurespannung des Blutes auf dessen Sauer- stoﬀbindung ¨ ubt 1

[BHK04] C. Bohr, K. Hasselbalch, and A. Krogh. “Ueber einen i n biologischer Beziehung wichtigen Einﬂuss, den die Kohlens¨ aurespannung des Blutes auf dessen Sauer- stoﬀbindung ¨ ubt 1”. In:Skandinavisches Archiv f¨ ur Physiologie16.2 (1904), pp. 402–412. [Bri83] W. Briggs. “A new measure of cooperativity in protei n-ligand binding”. In: Biophysical Chemi...

work page 1904

[3] [3]

Probabilistic algorit hm for polynomial optimization over a real algebraic set

url: http://www.optimization-online.or g/DB_HTML/2018/07/6692.html. [GS14] A. Greuet and M. Safey El Din. “Probabilistic algorit hm for polynomial optimization over a real algebraic set”. In: SIAM Journal on Optimization 24.3 (2014), pp. 1313–1343. [GYZ14] J. L. Gross, J. Yellen, and P. Zhang, eds. Handbook of graph theory . Second. Discrete Mathematics a...

work page 2018

[4] [4]

What is cooperativit y?

[HA09] C. A. Hunter and H. L. Anderson. “What is cooperativit y?” In: Angewandte Chemie International Edition 48.41 (2009), pp. 7488–7499. [Ike+83] M. Ikeda-Saito et al. “Thermodynamic properties o f oxygen equilibria of dimeric and tetrameric hemoglobins from Scapharca inaequi valvis”. In: Jour- nal of Molecular Biology 170.4 (1983), pp. 1009–1018. [Ima7...

work page 2009

[5] [5]

Global optimization with polynomial s and the problem of mo- ments

Graduate Studies in Mathematics. American Mathematical Society, Pr ovidence, RI, 2012, pp. xx+439. [Las01] J. Lasserre. “Global optimization with polynomial s and the problem of mo- ments”. In: SIAM Journal on Optimization 11.3 (2001), pp. 796–817. [LFSR09] T. Lenaerts, J. Ferkinghoﬀ-Borg, J. Schymkowitz, a nd F. Rousseau. “Infor- mation theoretical quant...

work page 2012

[6] [6]

A measure to quantify the degree of c ooperativity in overall titration curves

[Mar17a] J. W. Martini. “A measure to quantify the degree of c ooperativity in overall titration curves”. In: Journal of Theoretical Biology 432 (2017), pp. 33–37. 20 REFERENCES [Mar17b] J. W. Martini. “On the relation of diﬀerent deﬁnitio ns of cooperative binding for systems with two binding sites”. In: Match-Communications in Mathe- matical and in Comp...

work page 2017

[7] [7]

A novel vi ew of pH titration in biomolecules

[OCU01] A. Onufriev, D. A. Case, and G. M. Ullmann. “A novel vi ew of pH titration in biomolecules”. In: Biochemistry 40.12 (2001), pp. 3413–3419. [OU04] A. Onufriev and G. M. Ullmann. “Decomposing complex c ooperative lig- and binding into simple components: connections between mi croscopic and macroscopic models”. In: The Journal of Physical Chemistry B...

work page 2001

[8] [8]

Kinetic regulation of mult i-ligand binding proteins

REFERENCES 21 [Sal+16] D. V. Salakhieva et al. “Kinetic regulation of mult i-ligand binding proteins”. In: BMC Systems Biology 10.1 (2016), p

work page 2016

[9] [9]

Macromolecular binding

[Sch75] J. A. Schellman. “Macromolecular binding”. In: Biopolymers 14.5 (1975), pp. 999–1018. [Sch91] B. Schweizer. “Thirty years of copulas”. In: Advances in Probability Distri- butions with Given Marginals . Springer, 1991, pp. 13–50. [SW81] B. Schweizer and E. F. Wolﬀ. “On nonparametric measure s of dependence for random variables”. In: The Annals of S...

work page 1975