On Quantitative Comparison of Chemical Reaction Network Models

Department of Mathematics); Ozan Kahramano\u{g}ullar{\i} (University of Trento

arxiv: 1907.04319 · v1 · pith:HSNMIVWQnew · submitted 2019-07-09 · 🧬 q-bio.MN · cs.DM· cs.LO

On Quantitative Comparison of Chemical Reaction Network Models

Ozan Kahramano\u{g}ullar{\i} (University of Trento , Department of Mathematics) This is my paper

Pith reviewed 2026-05-25 00:14 UTC · model grok-4.3

classification 🧬 q-bio.MN cs.DMcs.LO

keywords chemical reaction networksstochastic simulationsflux graphsmodel comparisonequivalencedistance metricsystems biology

0 comments

The pith

Flux graphs from stochastic simulations enable quantitative comparison and distance measurement between any two chemical reaction networks over any time interval.

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

The paper proposes using flux graphs generated by stochastic simulations as abstract representations to compare the dynamic behaviors of chemical reaction networks. This method supports direct comparison of any two networks for any chosen time interval and introduces a notion of equivalence between them. The equivalence overlaps with graph isomorphism at the lowest representation level, and similarity is quantified through a distance measure. Such comparisons could support model refinement or the substitution of larger models with smaller ones that produce matching behavior.

Core claim

Flux graphs delivered by stochastic simulations serve as abstract representations of dynamic behaviour. This allows comparison of the behaviour of any two chemical reaction networks for any time interval, definition of a notion of equivalence that overlaps with graph isomorphism at the lowest level of representation, and quantification of similarity in terms of their distance. The results support refinement of models or replacement of a larger model with a smaller one that produces the same behaviour.

What carries the argument

Flux graphs from stochastic simulations, used as abstract representations of dynamic behaviour to enable comparison and distance quantification.

If this is right

Any two chemical reaction networks become comparable for arbitrary time intervals via their flux graphs.
Equivalence between networks can be defined in a way that aligns with graph isomorphism at the base level.
Similarity receives a numerical distance value derived from the flux graph comparison.
Larger models can be replaced by smaller ones when the distance indicates matching behaviour.

Where Pith is reading between the lines

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

The distance could serve as a basis for clustering networks by behavioral type in large model collections.
If extended to deterministic cases, the same flux-graph approach might unify stochastic and deterministic model comparisons.
Model reduction workflows in systems biology could use the equivalence check as an automated validation step.

Load-bearing premise

Flux graphs from stochastic simulations capture enough of the underlying dynamic behaviour to support meaningful equivalence and distance between networks.

What would settle it

Finding two networks whose stochastic trajectories match closely but whose flux graphs differ substantially, or whose flux graphs match but trajectories differ, would show the distance does not track true behavioral similarity.

Figures

Figures reproduced from arXiv: 1907.04319 by Department of Mathematics), Ozan Kahramano\u{g}ullar{\i} (University of Trento.

**Figure 2.** Figure 2: The fSSA algorithm [11, 8] extends the stochastic simulation algorithm [6] by logging the de [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Flux graphs of the simulations in Figure 1. The graph on the left displays the fluxes between [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: A comparison of the CRN in Example 1 with different time intervals and [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: The mesh of the normalised fluxes resulting from 81 simulations in different time intervals, [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: The CRN in [16] that models the interactions of plasmin ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: The CRN that models the Rho GTP-binding proteins and its flux graph [3, 11]. In the time [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: The molecular machinery of the Gemcitabine metabolism, representative time series plots in [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: The CRN that implements the model depicted in Figure 8. [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

read the original abstract

Chemical reaction networks (CRNs) provide a convenient language for modelling a broad variety of biological systems. These models are commonly studied with respect to the time series they generate in deterministic or stochastic simulations. Their dynamic behaviours are then analysed, often by using deterministic methods based on differential equations with a focus on the steady states. Here, we propose a method for comparing CRNs with respect to their behaviour in stochastic simulations. Our method is based on using the flux graphs that are delivered by stochastic simulations as abstract representations of their dynamic behaviour. This allows us to compare the behaviour of any two CRNs for any time interval, and define a notion of equivalence on them that overlaps with graph isomorphism at the lowest level of representation. The similarity between the compared CRNs can be quantified in terms of their distance. The results can then be used to refine the models or to replace a larger model with a smaller one that produces the same behaviour or vice versa.

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 a flux-graph distance and equivalence for comparing stochastic CRN behaviors, but supplies no validation or argument that the graphs preserve the needed dynamics.

read the letter

The new piece is a concrete way to turn stochastic simulation output into flux graphs, then measure distance or check equivalence between any two CRNs over a chosen time window. The equivalence is said to line up with graph isomorphism at the lowest level. That gives a quantitative handle on model similarity that could support simplification or replacement of larger models with smaller ones that match on this metric. The framing moves away from steady-state analysis toward direct behavioral comparison, which is a useful shift for systems biology work that relies on stochastic trajectories. The soft spot is the unexamined assumption that flux graphs are faithful enough abstractions. The stress-test concern holds: these graphs aggregate firing counts and may drop variance, correlations between reactions, waiting-time distributions, or rare paths that matter for the underlying continuous-time Markov chain. If two networks produce close flux graphs but differ on those higher-order features, the distance will report similarity that is not really there. The abstract gives no derivation, no error bounds, and no worked examples, so the central claim rests on an untested step. This is aimed at people who already run stochastic CRN simulations and need a practical way to decide when two models are close enough to swap. A reader who wants a new comparison primitive might find the idea worth following up, but only if the full paper shows that the flux-graph map actually preserves the relevant probability measures. I would send it to peer review so the authors can supply the missing justification and tests; the idea is narrow enough that referees could check it quickly.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a method for quantitative comparison of chemical reaction networks (CRNs) by extracting flux graphs from stochastic simulations as abstract representations of dynamic behaviour. This is claimed to enable comparison of any two CRNs over arbitrary time intervals, to induce an equivalence relation that overlaps with graph isomorphism at the base level, and to yield a distance metric quantifying similarity, with potential uses in model refinement or reduction.

Significance. If the flux-graph abstraction can be shown to faithfully capture behavioural equivalence classes of the underlying CTMCs, the method would supply a practical tool for stochastic model comparison and simplification that extends beyond deterministic steady-state analysis.

major comments (2)

[Abstract] Abstract: the central claim that flux graphs 'delivered by stochastic simulations' constitute sufficient abstract representations for defining usable distance and equivalence rests on unshown implementation steps, with no derivation, error analysis, or validation examples supplied.
[Abstract, paragraph 3] Abstract, paragraph 3: the assertion that flux graphs serve as faithful abstractions is load-bearing for the equivalence and distance claims, yet no argument is given that the map from the continuous-time Markov chain to the flux graph is injective on the relevant equivalence classes (e.g., that it preserves variance, joint firing correlations, or waiting-time distributions).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their detailed review and constructive comments. Below we provide point-by-point responses to the major comments.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that flux graphs 'delivered by stochastic simulations' constitute sufficient abstract representations for defining usable distance and equivalence rests on unshown implementation steps, with no derivation, error analysis, or validation examples supplied.

Authors: The manuscript describes the extraction of flux graphs from stochastic simulations as the basis for the comparison. The implementation steps are outlined in the body of the paper, including how the graphs are constructed from simulation traces. We acknowledge the absence of explicit error analysis and validation examples in the current version and will incorporate these in the revision to better support the claims. revision: yes
Referee: [Abstract, paragraph 3] Abstract, paragraph 3: the assertion that flux graphs serve as faithful abstractions is load-bearing for the equivalence and distance claims, yet no argument is given that the map from the continuous-time Markov chain to the flux graph is injective on the relevant equivalence classes (e.g., that it preserves variance, joint firing correlations, or waiting-time distributions).

Authors: The flux graph is presented as an abstract representation chosen for its utility in comparing dynamic behavior over time intervals. The equivalence relation is defined on these representations and is shown to coincide with graph isomorphism when applied to isomorphic networks. The method does not assert that the abstraction is injective with respect to all properties of the underlying CTMC, such as variance or waiting times; instead, it offers a quantifiable similarity based on flux counts. We will add a clarifying statement in the abstract and a discussion of the abstraction's scope in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity: equivalence and distance are explicitly defined from flux graphs

full rationale

The paper introduces a comparison method by directly defining equivalence and distance on flux graphs obtained from stochastic simulations. No derivation chain reduces a claimed result to its own inputs by construction, no parameters are fitted then relabeled as predictions, and no load-bearing uniqueness theorems or ansatzes are imported via self-citation. The central construction is a new metric on the graphs themselves, which is self-contained and does not presuppose the behavioral equivalence it aims to quantify.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the method assumes standard graph theory and stochastic simulation outputs without introducing new free parameters, axioms beyond domain assumptions, or invented entities.

axioms (1)

domain assumption Flux graphs from stochastic simulations are adequate abstract representations of CRN dynamic behaviour
Invoked in abstract paragraph 3 to justify using graphs for comparison and equivalence.

pith-pipeline@v0.9.0 · 5702 in / 1179 out tokens · 18558 ms · 2026-05-25T00:14:33.653877+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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The similarity between the compared CRNs can be quantified in terms of their distance. ... δ(F1,F2) = sqrt(∑(w1−w2)² ... ) Proposition 5 The distance function δ is a metric.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

flux graphs that are delivered by stochastic simulations as abstract representations of their dynamic behaviour

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

16 extracted references · 16 canonical work pages

[1]

Bouajjani, C

A. Bouajjani, C. Enea & S. Lahiri (2017): Abstract Semantic Difﬁng of Evolving Concurrent Programs . In: International Static Analysis Symposium SAS 2017: Static Analysis , LNCS 10422, pp. 46–65, doi:10.1007/978-3-319-66706-5 3

work page doi:10.1007/978-3-319-66706-5 2017
[2]

Boutillier, J

P. Boutillier, J. Feret, J. Krivine & W. Fontana (2018): The Kappa Language and Kappa Tools, A User Man- ual and Guide,v4 . Available at https://kappalanguage.org/sites/kappalanguage.org/files/ inline-files/Kappa_Manual.pdf

work page 2018
[3]

Cardelli, E

L. Cardelli, E. Caron, P. Gardner, O. Kahramano ˘gulları & A. Phillips (2009): A Process Model of Rho GTP- binding Proteins. Theoretical Computer Science 410/33-34, pp. 3166–3185, doi:10.1016/j.tcs.2009.04.029

work page doi:10.1016/j.tcs.2009.04.029 2009
[4]

Cardelli, M

L. Cardelli, M. Kwiatkowska & L. Laurenti (2016): Stochastic Analysis of Chemical Reaction Networks Using Linear Noise Approximation. Biosystems 149, pp. 26–33, doi:10.1016/j.biosystems.2016.09.004

work page doi:10.1016/j.biosystems.2016.09.004 2016
[5]

Cardelli, M

L. Cardelli, M. Tribastone, M. Tschaikowski & A. Vandin (2017): Maximal aggregation of polynomial dy- namical systems. Proceedings of the National Academy of Sciences, doi:10.1073/pnas.1702697114

work page doi:10.1073/pnas.1702697114 2017
[6]

D. T. Gillespie (1977): Exact Stochastic Simulation of Coupled Chemical Reactions. The Journal of Physical Chemistry 81(25), pp. 2340–2361, doi:10.1021/j100540a008

work page doi:10.1021/j100540a008 1977
[7]

Kahramano ˘gulları (2009): On Linear Logic Planning and Concurrency

O. Kahramano ˘gulları (2009): On Linear Logic Planning and Concurrency . Information and Computation 207, pp. 1229–1258, doi:10.1016/j.ic.2009.02.008

work page doi:10.1016/j.ic.2009.02.008 2009
[8]

Kahramano ˘gulları (2017): Quantifying Information Flow in Chemical Reaction Networks

O. Kahramano ˘gulları (2017): Quantifying Information Flow in Chemical Reaction Networks. In: Proceedings of 4th International Conference on Algorithms for Computational Biology, AlCoB 2017 , LNCS 10252, pp. 155–166, doi:10.1007/978-3-319-58163-7 11

work page doi:10.1007/978-3-319-58163-7 2017
[9]

Kahramano ˘gulları & L

O. Kahramano ˘gulları & L. Cardelli (2015): Gener: a minimal programming module for chem- ical controllers based on DNA strand displacement . Bioinformatics 31 (17), pp. 2906–2908, doi:10.1093/bioinformatics/btv286

work page doi:10.1093/bioinformatics/btv286 2015
[10]

Kahramano ˘gulları, G

O. Kahramano ˘gulları, G. Fantaccini, P. Lecca, D. Morpurgo & C. Priami (2012): Algorithmic modeling quantiﬁes the complementary contribution of metabolic inhibitions to gemcitabine efﬁcacy. PLoS ONE 7(12), p. e50176, doi:10.1371/journal.pone.0050176

work page doi:10.1371/journal.pone.0050176 2012
[11]

Kahramano ˘gulları & J

O. Kahramano ˘gulları & J. Lynch (2013): Stochastic Flux Analysis of Chemical Reaction Networks . BMC Systems Biology 7(133), doi:10.1186/1752-0509-7-133

work page doi:10.1186/1752-0509-7-133 2013
[12]

M. R. Lakin, S. Youssef, F. Polo, S. Emmott & A. Phillips (2011):Visual DSD: a design and analysis tool for DNA strand displacement systems. Bioinformatics 27, pp. 3211–3213, doi:10.1093/bioinformatics/btr543

work page doi:10.1093/bioinformatics/btr543 2011
[13]

Nielsen, G

M. Nielsen, G. Plotkin & G. Winskel (1981): Petri Nets, Event Structures and Domains, part 1. Theoretical Computer Science 13(1), pp. 85–108, doi:10.1016/0304-3975(81)90112-2. 1The python scripts and tools of our method are available for download at ozan-k.com/distance.zip. O. Kahramano˘gulları 25

work page doi:10.1016/0304-3975(81)90112-2 1981
[14]

J. A. P. Sekar & J. R. Faeder (2012): Rule-Based Modeling of Signal Transduction: A Primer. Computational Modeling of Signaling Networks. Methods in Molecular Biology (Methods and Protocols)880, pp. 139–218, doi:10.1007/978-1-61779-833-7 9

work page doi:10.1007/978-1-61779-833-7 2012
[15]

Shannon, A

P. Shannon, A. Markiel, O. Ozier, N. S. Baliga, J. T. Wang, D. Ramage, N. Amin, B. Schwikowski & T. Ideker (2003): Cytoscape: a software environment for integrated models of biomolecular interaction networks. Genome Research 13(11), pp. 2498–2504, doi:10.1101/gr.1239303

work page doi:10.1101/gr.1239303 2003
[16]

Venkatraman, H

L. Venkatraman, H. Li, C. F. Jr. Dewey, J. K. White, S. S. Bhowmick, H. Yu & L. Tucker-Kellogg (2011): Steady states and dynamics of urokinase-mediated plasmin activation in silico and in vitro . Biophysical Journal 101 (8), pp. 1825–1834, doi:10.1016/j.bpj.2011.08.054. 26 On Quantitative Comparison of Chemical Reaction Network Models Figure 8: The molecu...

work page doi:10.1016/j.bpj.2011.08.054 2011

[1] [1]

Bouajjani, C

A. Bouajjani, C. Enea & S. Lahiri (2017): Abstract Semantic Difﬁng of Evolving Concurrent Programs . In: International Static Analysis Symposium SAS 2017: Static Analysis , LNCS 10422, pp. 46–65, doi:10.1007/978-3-319-66706-5 3

work page doi:10.1007/978-3-319-66706-5 2017

[2] [2]

Boutillier, J

P. Boutillier, J. Feret, J. Krivine & W. Fontana (2018): The Kappa Language and Kappa Tools, A User Man- ual and Guide,v4 . Available at https://kappalanguage.org/sites/kappalanguage.org/files/ inline-files/Kappa_Manual.pdf

work page 2018

[3] [3]

Cardelli, E

L. Cardelli, E. Caron, P. Gardner, O. Kahramano ˘gulları & A. Phillips (2009): A Process Model of Rho GTP- binding Proteins. Theoretical Computer Science 410/33-34, pp. 3166–3185, doi:10.1016/j.tcs.2009.04.029

work page doi:10.1016/j.tcs.2009.04.029 2009

[4] [4]

Cardelli, M

L. Cardelli, M. Kwiatkowska & L. Laurenti (2016): Stochastic Analysis of Chemical Reaction Networks Using Linear Noise Approximation. Biosystems 149, pp. 26–33, doi:10.1016/j.biosystems.2016.09.004

work page doi:10.1016/j.biosystems.2016.09.004 2016

[5] [5]

Cardelli, M

L. Cardelli, M. Tribastone, M. Tschaikowski & A. Vandin (2017): Maximal aggregation of polynomial dy- namical systems. Proceedings of the National Academy of Sciences, doi:10.1073/pnas.1702697114

work page doi:10.1073/pnas.1702697114 2017

[6] [6]

D. T. Gillespie (1977): Exact Stochastic Simulation of Coupled Chemical Reactions. The Journal of Physical Chemistry 81(25), pp. 2340–2361, doi:10.1021/j100540a008

work page doi:10.1021/j100540a008 1977

[7] [7]

Kahramano ˘gulları (2009): On Linear Logic Planning and Concurrency

O. Kahramano ˘gulları (2009): On Linear Logic Planning and Concurrency . Information and Computation 207, pp. 1229–1258, doi:10.1016/j.ic.2009.02.008

work page doi:10.1016/j.ic.2009.02.008 2009

[8] [8]

Kahramano ˘gulları (2017): Quantifying Information Flow in Chemical Reaction Networks

O. Kahramano ˘gulları (2017): Quantifying Information Flow in Chemical Reaction Networks. In: Proceedings of 4th International Conference on Algorithms for Computational Biology, AlCoB 2017 , LNCS 10252, pp. 155–166, doi:10.1007/978-3-319-58163-7 11

work page doi:10.1007/978-3-319-58163-7 2017

[9] [9]

Kahramano ˘gulları & L

O. Kahramano ˘gulları & L. Cardelli (2015): Gener: a minimal programming module for chem- ical controllers based on DNA strand displacement . Bioinformatics 31 (17), pp. 2906–2908, doi:10.1093/bioinformatics/btv286

work page doi:10.1093/bioinformatics/btv286 2015

[10] [10]

Kahramano ˘gulları, G

O. Kahramano ˘gulları, G. Fantaccini, P. Lecca, D. Morpurgo & C. Priami (2012): Algorithmic modeling quantiﬁes the complementary contribution of metabolic inhibitions to gemcitabine efﬁcacy. PLoS ONE 7(12), p. e50176, doi:10.1371/journal.pone.0050176

work page doi:10.1371/journal.pone.0050176 2012

[11] [11]

Kahramano ˘gulları & J

O. Kahramano ˘gulları & J. Lynch (2013): Stochastic Flux Analysis of Chemical Reaction Networks . BMC Systems Biology 7(133), doi:10.1186/1752-0509-7-133

work page doi:10.1186/1752-0509-7-133 2013

[12] [12]

M. R. Lakin, S. Youssef, F. Polo, S. Emmott & A. Phillips (2011):Visual DSD: a design and analysis tool for DNA strand displacement systems. Bioinformatics 27, pp. 3211–3213, doi:10.1093/bioinformatics/btr543

work page doi:10.1093/bioinformatics/btr543 2011

[13] [13]

Nielsen, G

M. Nielsen, G. Plotkin & G. Winskel (1981): Petri Nets, Event Structures and Domains, part 1. Theoretical Computer Science 13(1), pp. 85–108, doi:10.1016/0304-3975(81)90112-2. 1The python scripts and tools of our method are available for download at ozan-k.com/distance.zip. O. Kahramano˘gulları 25

work page doi:10.1016/0304-3975(81)90112-2 1981

[14] [14]

J. A. P. Sekar & J. R. Faeder (2012): Rule-Based Modeling of Signal Transduction: A Primer. Computational Modeling of Signaling Networks. Methods in Molecular Biology (Methods and Protocols)880, pp. 139–218, doi:10.1007/978-1-61779-833-7 9

work page doi:10.1007/978-1-61779-833-7 2012

[15] [15]

Shannon, A

P. Shannon, A. Markiel, O. Ozier, N. S. Baliga, J. T. Wang, D. Ramage, N. Amin, B. Schwikowski & T. Ideker (2003): Cytoscape: a software environment for integrated models of biomolecular interaction networks. Genome Research 13(11), pp. 2498–2504, doi:10.1101/gr.1239303

work page doi:10.1101/gr.1239303 2003

[16] [16]

Venkatraman, H

L. Venkatraman, H. Li, C. F. Jr. Dewey, J. K. White, S. S. Bhowmick, H. Yu & L. Tucker-Kellogg (2011): Steady states and dynamics of urokinase-mediated plasmin activation in silico and in vitro . Biophysical Journal 101 (8), pp. 1825–1834, doi:10.1016/j.bpj.2011.08.054. 26 On Quantitative Comparison of Chemical Reaction Network Models Figure 8: The molecu...

work page doi:10.1016/j.bpj.2011.08.054 2011