Pith. sign in

REVIEW 4 major objections 5 minor 22 references

Mixed-Integer Optimization for Loopless Flux Distributions in Metabolic Networks

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that a combinatorial Benders' decomposition solves most genome-scale loopless flux balance analysis instances, handling 93% of 107 BiGG models in about a minute, while direct big-M, indicator, and no-good-cut approaches…

desk verdict Useful, honest application of combinatorial Benders to loopless FBA on BiGG/yeast models, but the big-M baseline comparison needs a validity argument for M and the hull reformulation is incomplete as written. read the letter →

arxiv 2502.00807 v2 pith:IAK5GOWO submitted 2025-02-02 math.OC

Hannah Troppens , Mathieu Besançon , St. Elmo Wilken , Sebastian Pokutta This is my paper · ORCID
classification math.OC MSC 90C1190C9092C42
keywords fluxbalanceanalysislooplessFBAcombinatorialBendersdecompositionmixed-integerprogrammingdisjunctiveminimalinfeasiblesubsystemsgenome-scalemetabolicmodels
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

Loopless Flux Balance Analysis (ll-FBA) predicts metabolic fluxes without thermodynamically infeasible internal cycles, but it is an NP-hard disjunctive program that is hard to solve on genome-scale models. This paper compares direct mixed-integer reformulations, a convex hull formulation, and decomposition approaches on 107 BiGG models and on larger yeast models. Its central claim is that a combinatorial Benders' decomposition, which separates the flux/activation master problem from a thermodynamic feasibility subproblem, solves 93% of the BiGG instances with a geometric mean time of about 58 seconds, while direct big-M solves 33%, direct indicator solves 17%, and no-good cuts solve 8% within the same 1800-second limit. The paper also reports that the method fails on most large yeast models and that numerical instability remains a challenge.

What carries the argument

The combinatorial Benders' decomposition is the central object: the master problem is the relaxed ll-FBA with binary variables indicating forward/backward flux, and the subproblem asks whether there exists a vector $\mu$ satisfying $\Delta\mu = S_I^\top \mu$ with the sign pattern forced by the master's activation variables. Infeasibility of this subproblem is certified by a minimal infeasible subsystem (MIS), obtained by solving a dual LP whose support gives a small set of reactions responsible; each MIS generates a combinatorial Benders' cut that forbids that entire infeasible sign combination. The paper also uses the standard big-M and indicator reformulations of ll-FBA from the literature and a convex hull reformulation, against which the decomposition is compared.

What would settle it

Compare the indicator and big-M formulations on every BiGG instance: if any instance has a strictly larger optimal objective with the indicator formulation than with the big-M formulation, then the bound $M = \max(|l|,|u|)$ excludes valid loopless solutions, and part of the reported speed-up of combinatorial Benders' over the big-M baseline is an artifact of the baseline.

Watch

Extended reading notes

Core claim

The paper establishes that ll-FBA can be solved at genome scale by a combinatorial Benders' decomposition. The approach alternates between a master mixed-integer program that chooses fluxes and reaction directions, and a linear subproblem that checks whether a thermodynamic potential $\mu$ exists with $\Delta\mu = S_I^\top \mu$ having the required signs; when the subproblem is infeasible, minimal infeasible subsystems are computed from a dual LP and each yields a Benders' cut that blocks a whole internal cycle rather than a single binary assignment. With a big-M master problem and one cut per iteration, this solves 93 of 107 BiGG instances, with a geometric mean time of 58 seconds, outperforming direct solution of the big-M and indicator reformulations and a no-good-cut decomposition. The same method solves most instances when indicator constraints are used in the master, though more slowly, and it extends naturally to enzyme-constrained models. On the larger yeast models, however, neither the direct formulations nor most combinatorial Benders' setups solve any instance to optimality within the time limit, and some enzyme instances terminate with numerical errors.

Load-bearing premise

The headline performance comparison assumes that the big-M formulation with $M$ equal to the largest absolute flux bound is a correct and fair encoding of ll-FBA; because the energy variables $\Delta\mu$ have no fixed scale, this bound is not proven safe, and if it bites, the big-M baseline is solving a restricted problem.

Editorial extensions

If this is right

  • Genome-scale loopless flux predictions become practical on typical BiGG models, with most instances solved in about a minute.
  • The large gap between combinatorial Benders' cuts and no-good cuts shows that blocking minimal infeasible subsystems, not single binary assignments, is what makes the decomposition efficient.
  • The number of cuts added per iteration is a performance-critical parameter: a small fraction of reactions per iteration speeds up solving, while too many cuts slow it down and can even cause numerical errors.
  • Enzyme-constrained ll-FBA models solve much faster than unconstrained ones, suggesting that enzyme capacity constraints remove many loops from the relaxed solution space.
  • On the larger yeast models, neither direct formulations nor most combinatorial Benders' setups solve any instance, so scalable ll-FBA algorithms remain an open problem.

Reading between the lines

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

  • The energy variables in ll-FBA have no natural scale, so the big-M constant used in the direct baseline is not validated; a fair cross-formulation comparison would check that this bound never excludes a valid loopless solution, and any violation would put part of the observed speed-up on the baseline's weakness.
  • The same decomposition pattern — a flux/activation master and a linear thermodynamic subproblem — should transfer to other disjunctive metabolic models, such as those with gene regulation or resource allocation, whenever the subproblem is a linear feasibility problem.
  • An adaptive cut-addition schedule, starting with one cut and increasing only when the master stays infeasible, might avoid the slowdown that a fixed percentage of cuts causes on large instances.
  • The enzyme-constrained results hint that adding biologically motivated constraints may be a more promising route to tractable ll-FBA than purely algorithmic improvements.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies mixed-integer reformulations and solution algorithms for loopless Flux Balance Analysis (ll-FBA), a disjunctive program that excludes thermodynamically infeasible internal cycles from steady-state flux solutions. The authors present indicator and big-M reformulations, a convex hull reformulation, and a combinatorial Benders (CB) decomposition that separates the flux/sign master problem from a thermodynamic feasibility subproblem whose infeasible subsystems generate CB cuts. They benchmark these approaches on 107 BiGG metabolic models, 11 yeast models, and enzyme-constrained variants. The main reported result is that CB solves 93% of the BiGG instances within 1800 s (geometric mean time about 58 s), versus 33% for direct big-M, 17% for direct indicator, and 8% for no-good cuts. The paper is honest about limitations: yeast instances remain mostly unsolved, numerical instability occurs on several instances, and enzyme-data experiments use random parameters and make no biological claims.

Significance. If the reported results hold, the CB decomposition is a practically useful improvement for a genuinely hard problem in constraint-based metabolic modeling. The paper has clear strengths: the implementation is open-source, the benchmarks use public standard models, the algorithm follows standard and reproducible Benders/MIS machinery, and the manuscript explicitly reports failures and numerical difficulties rather than only successes. The significance is tempered, however, by two formulation issues that affect the fairness of the comparisons: the big-M baseline is not formally equivalent to the disjunctive ll-FBA model without an additional validity argument, and the convex hull reformulation appears incomplete as written. Because the exact direct indicator baseline also performs far worse than CB, the central conclusion that CB is the most promising tested approach is defensible, but the numerical claims need revision and additional validation.

major comments (4)
  1. [Section 3.1, ll-FBA (big-M)] The big-M reformulation is not established as equivalent to the disjunctive ll-FBA model. In the constraints -M ai + eps(1-ai) <= Δμ_i <= -eps ai + M(1-ai), the quantities Δμ_i = (S_I^T μ)_i are homogeneous in the free variable μ: if a sign pattern is realizable, so is the same pattern for tμ for any t>0. Fixing M = max(|l|,|u|) restricts Δμ to the box [-M,M], and a sign pattern is representable only if the feasible cone contains a point with max_i |Δμ_i| / min_i |Δμ_i| <= M/eps. No argument is given that every thermodynamically feasible sign pattern required for an optimal ll-FBA solution in the BiGG instances satisfies this ratio bound, so the direct big-M baseline (33% solved) may be solving a restricted problem rather than ll-FBA. Since the direct indicator formulation is exact and still performs worse (17%), the central conclusion that CB is competitive survives; however, the headline 93%-versus-33% comparison is quantitatively overstated. The authors should either prove the validity of this M for the instances, report a check that the optimal objectives of direct big-M and CB coincide on instances solved by both, or explicitly label the big-M results as an approximate baseline.
  2. [Section 3.3, Convex Hull Reformulation] The hull reformulation is incomplete as written. The disjunctions in (5) require Δμ_i <= -eps in the first disjunct and Δμ_i >= eps in the second, but the constraints listed below (5) contain only Δμ_i1 <= -y_i and -Δμ_i2 <= -y_{i+|I|}; the eps factor is missing, and there are no bounds forcing the disaggregated variables to zero when the corresponding binary variable is zero. Without such bounds, a disaggregated variable can take nonzero values of the wrong sign or magnitude, so the formulation does not represent the convex hull of the disjunctive feasible region. Consequently the poor performance reported for 'DP (convex-hull)' in Figure 3 is not a meaningful comparison until the formulation is corrected.
  3. [Section 4.2, Tables 1-2 and Appendix B.2] The recommended cut budget k (for example MIS 0.5% or 0.1%) and the cut selection strategies are selected on the same 107 BiGG instances used for the reported performance numbers. This is in-sample model selection, so the superiority of the tuned settings (for example 95% solved in about 28 s) is not an unbiased estimate of out-of-sample performance. The yeast experiments provide partial external validation, but almost all settings fail there, and the single success (CB big-M MIS 0.5%) is not enough to validate the specific budget. The authors should either use a separate validation set or explicitly state that the recommended parameters are tuned on the benchmark and may not transfer.
  4. [Section 4.2, objective consistency] The manuscript does not report whether direct big-M and CB produce the same optimal objective values on the instances where both methods succeed. This check would directly reveal whether the big-M bound is restrictive in practice. It would strengthen the comparison to report the distribution of objective differences or at least state that all matched instances agree to solver tolerance.
minor comments (5)
  1. [Section 3.2, subproblem and LP (4)] The notation for the dual-based MIS computation is unclear: the vector \tilde b is written as [-eps|C|], which suggests a scalar rather than a vector of length |C| with entries -eps; please clarify the dimensions. It would also help to state explicitly why a vertex of the feasible region of (4) corresponds to a minimal infeasible subsystem, rather than only referring the reader to [5].
  2. [Section 4.1 and Tables 1-2] The tables report 'time (s)' as a geometric mean, but it is not stated whether the mean is taken over solved instances only or whether time-limit instances are included as 1800 s; this should be specified in the table captions.
  3. [Section 3.3] The sentence 'the number of additional constraints grows exponentially with the number of disjunctions' is misleading for the disaggregated hull formulation presented here: for two-term disjunctions, the extended formulation adds a linear number of variables and constraints per disjunction. The exponential statement may be true in the original variable space, but it should be qualified.
  4. [Appendix B.2] There is a typo in the first sentence: 'te combinatorial Benders approach' should be 'the combinatorial Benders approach'.
  5. [Section 4.2] The discussion of instances iAM_Pc455, iAM_Pf480, iCN718, and iAM_Pv561 mentions that tighter tolerances resolve some numerical failures; it would be useful to state the default tolerance used and the tighter tolerance applied, so the experiments are reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the performance comparison is empirical and self-contained against public benchmarks; self-citations are tooling only.

full rationale

The paper's central claim is an empirical performance comparison of MIP reformulations and a combinatorial Benders' decomposition on public BiGG and yeast models, not a derivation of a biological quantity from a fitted constant. The big-M, indicator, no-good-cut, and CB formulations are standard constructions from external references [5,21] and from the paper's own equations in Section 3; no parameter is fitted to a subset of the data and then reported as a prediction of a closely related quantity. The only self-citations are to COBREXA [10] and SCIP [3,4] as software tools used to load models and solve LPs/MIPs; these are not load-bearing arguments. The choice M = max(|l|,|u|) in Section 3.1 may be a heuristic restriction because Delta_mu = S_I^T mu is homogeneous in mu, but this is a baseline-validity caveat, not a circular step: the paper does not derive the CB advantage from that M. Likewise, tuning the MIS cut budget on the same BiGG benchmark is in-sample model selection, not circular derivation. The convex hull formulation's stated inequalities omit the epsilon threshold, which is a correctness issue rather than circularity. Accordingly there is no step where a claimed result is equivalent by construction to its input.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new biological entities. Its central claim depends on standard Benders theory, on the choice of epsilon and M in the MIP reformulations, and on the correctness of the convex hull representation. The epsilon and M choices are not fitted to biological data, but M is a free modeling constant whose validity is not demonstrated.

free parameters (3)
  • epsilon threshold in ll-FBA = 1
    Set to 1 to match Schellenberger et al. (2011). This threshold replaces strict inequalities in the disjunction and affects numerical behavior of all formulations.
  • big-M constant M = max absolute flux bound from l and u
    Used in the big-M reformulation and in the big-M master problem. No proof is given that this M is large enough for the unbounded Delta_mu variables.
  • MIS cut budget per iteration k = 0.1% to 10% of reactions depending on configuration
    The number of combinatorial Benders cuts added per iteration is a tunable parameter. The reported best values are selected on the same BiGG benchmark instances used for evaluation.
assumptions (4)
  • standard math The dual LP in Equation (4) yields valid minimal infeasible subsystems for the sign-only subproblem.
    The paper cites Codato and Fischetti (2006) for this derivation and does not re-prove it in the ll-FBA setting.
  • ad hoc to paper The big-M reformulation with M equal to the maximal flux bound is equivalent to ll-FBA.
    The paper states this choice without proof. Since Delta_mu is homogeneous and unbounded, a finite M can exclude feasible thermodynamic sign patterns unless a scale argument is supplied.
  • ad hoc to paper The convex hull reformulation in Section 3.3 correctly models the disjunctive ll-FBA feasible region.
    As written, the hull constraints do not tie the disaggregated variables to the binaries via bounds, so the disjunction on v and Delta_mu is not enforced. This assumption is asserted, not derived.
  • domain assumption Randomly generated enzyme parameters are sufficient to test algorithmic flexibility of the framework.
    The paper explicitly disclaims biological realism for these models and uses them only to show that enzyme constraints can be added to the decomposition, which is a reasonable testing assumption for an algorithmic claim.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Mixed-Integer Optimization for Loopless Flux Distributions in Metabolic Networks." pith.science (2026). https://pith.science/paper/IAK5GOWO

@misc{pith2026250200807,
  author       = {Pith},
  title        = {Pith review of: Mixed-Integer Optimization for Loopless Flux Distributions in Metabolic Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IAK5GOWO}},
  note         = {Machine review of arXiv:2502.00807}
}
read the original abstract

Constraint-based metabolic models can be used to investigate the intracellular physiology of microorganisms. These models couple genes to reactions, and typically seek to predict metabolite fluxes that optimize some biologically important metric. Classical techniques, like Flux Balance Analysis (FBA), formulate the metabolism of a microbe as an optimization problem where growth rate is maximized. While FBA has found widespread use, it often leads to thermodynamically infeasible solutions that contain internal cycles (loops). To address this shortcoming, Loopless-Flux Balance Analysis (ll-FBA) seeks to predict flux distributions that do not contain these loops. ll-FBA is a disjunctive program, usually reformulated as a mixed-integer program, and is challenging to solve for biological models that often contain thousands of reactions and metabolites. In this paper, we compare various reformulations of ll-FBA and different solution approaches. Overall, the combinatorial Benders' decomposition is the most promising of the tested approaches with which we could solve most instances. However, the model size and numerical instability pose a challenge to the combinatorial Benders' method.

Figures

Figures reproduced from arXiv: 2502.00807 by the authors.

Figure 6
Figure 6. The results indicate that both variants of the combinatorial Benders’ approach [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 13 canonical work pages

  1. [1]

    Disjunctive programming

    Egon Balas. Disjunctive programming. In P.L. Hammer, E.L. Johnson, and B.H. Korte, editors, Discrete Optimization II , volume 5 of Annals of Discrete Mathematics , pages 3--51. Elsevier, 1979. https://doi.org/10.1016/S0167-5060(08)70342-X doi:10.1016/S0167-5060(08)70342-X

    doi:10.1016/s0167-5060(08)70342-x 1979

  2. [2]

    Beard, Eric Babson, Edward Curtis, and Hong Qian

    Daniel A. Beard, Eric Babson, Edward Curtis, and Hong Qian. Thermodynamic constraints for biochemical networks . Journal of Theoretical Biology , 228(3):327--333, 2004. https://doi.org/10.1016/j.jtbi.2004.01.008 doi:10.1016/j.jtbi.2004.01.008

    work page doi:10.1016/j.jtbi.2004.01.008 2004

  3. [3]

    u bbecke, Stephen J. Maher, Frederic Matter, Erik M\

    Ksenia Bestuzheva, Mathieu Besan c on, Wei-Kun Chen, Antonia Chmiela, Tim Donkiewicz, Jasper van Doornmalen, Leon Eifler, Oliver Gaul, Gerald Gamrath, Ambros Gleixner, Leona Gottwald, Christoph Graczyk, Katrin Halbig, Alexander Hoen, Christopher Hojny, Rolf van der Hulst, Thorsten Koch, Marco L\" u bbecke, Stephen J. Maher, Frederic Matter, Erik M\" u hme...

    2023

  4. [4]

    The SCIP Optimization Suite 9.0 , 2024

    Suresh Bolusani, Mathieu Besançon, Ksenia Bestuzheva, Antonia Chmiela, João Dionísio, Tim Donkiewicz, Jasper van Doornmalen, Leon Eifler, Mohammed Ghannam, Ambros Gleixner, et al. The SCIP Optimization Suite 9.0 , 2024. http://arxiv.org/abs/2402.17702 arXiv:2402.17702

    arXiv 2024

  5. [5]

    Combinatorial Benders' Cuts for Mixed-Integer Linear Programming

    Gianni Codato and Matteo Fischetti. Combinatorial Benders' Cuts for Mixed-Integer Linear Programming . Operations Research , 54(4):756--766, 2006. Publisher: INFORMS . https://doi.org/10.1287/opre.1060.0286 doi:10.1287/opre.1060.0286

    work page arXiv 2006

  6. [6]

    Abdelmoneim Amer Desouki, Florian Jarre, Gabriel Gelius-Dietrich, and Martin J. Lercher. CycleFreeFlux: Efficient Removal of Thermodynamically Infeasible Loops from Flux Distributions . Bioinformatics , 31(13):2159--2165, 2015. https://doi.org/10.1093/bioinformatics/btv096 doi:10.1093/bioinformatics/btv096

    work page doi:10.1093/bioinformatics/btv096 2015

  7. [7]

    Henry, Linda J

    Christopher S. Henry, Linda J. Broadbelt, and Vassily Hatzimanikatis. Thermodynamics-Based Metabolic Flux Analysis . Biophysical Journal , 92(5):1792--1805, 2007. https://doi.org/10.1529/biophysj.106.093138 doi:10.1529/biophysj.106.093138

    doi:10.1529/biophysj.106.093138 2007

  8. [8]

    Huangfu and J

    Q. Huangfu and J. A. J. Hall. Parallelizing the dual revised simplex method . Mathematical Programming Computation , 10(2):119--142, 2018. https://doi.org/10.1007/s12532-017-0130-5 doi:10.1007/s12532-017-0130-5

    doi:10.1007/s12532-017-0130-5 2018

Show all 22 references
  1. [9]

    King, Justin Lu, Andreas Dräger, Philip Miller, Stephen Federowicz, Joshua A

    Zachary A. King, Justin Lu, Andreas Dräger, Philip Miller, Stephen Federowicz, Joshua A. Lerman, Ali Ebrahim, Bernhard O. Palsson, and Nathan E. Lewis. BiGG Models: A platform for integrating, standardizing and sharing genome-scale models . Nucleic Acids Research , 44(D1):D515...

    2015 doi

  2. [10]

    COBREXA.jl: constraint-based reconstruction and exascale analysis

    Miroslav Kratochvíl, Laurent Heirendt, St Elmo Wilken, Taneli Pusa, Sylvain Arreckx, Alberto Noronha, Marvin van Aalst, Venkata P Satagopam, Oliver Ebenhöh, Reinhard Schneider, Christophe Trefois, and Wei Gu. COBREXA.jl: constraint-based reconstruction and exascale analysis . ...

    2021 doi

  3. [11]

    MathOptInterface : a data structure for mathematical optimization problems

    Beno \^ t Legat, Oscar Dowson, Joaquim Dias Garcia, and Miles Lubin. MathOptInterface : a data structure for mathematical optimization problems . INFORMS Journal on Computing , 34(2):672--689, 2021. https://doi.org/10.1287/ijoc.2021.1067 doi:10.1287/ijoc.2021.1067

    2021

  4. [12]

    Lewis, Kim K

    Nathan E. Lewis, Kim K. Hixson, Tom M. Conrad, Joshua A. Lerman, Pep Charusanti, Ashoka D. Polpitiya, Joshua N. Adkins, Gunnar Schramm, Samuel O. Purvine, Daniel Lopez-Ferrer, Karl K. Weitz, Roland Eils, Rainer König, Richard D. Smith, and Bernhard Ø. Palsson. Omic Data from E...

    2010 doi

  5. [13]

    Yeast metabolic innovations emerged via expanded metabolic network and gene positive selection

    Hongzhong Lu, Feiran Li, Le Yuan, Iv\' a n Domenzain, Rosemary Yu, Hao Wang, Gang Li, Yu Chen, Boyang Ji, Eduard J Kerkhoven, and Jens Nielsen. Yeast metabolic innovations emerged via expanded metabolic network and gene positive selection . Molecular Systems Biology , 17(10):e...

    2021 doi

  6. [14]

    JuMP 1.0: Recent Improvements to a Modeling Language for Mathematical Optimization

    Miles Lubin, Oscar Dowson, Joaquim Dias Garcia, Joey Huchette, Benoît Legat, and Juan Pablo Vielma. JuMP 1.0: Recent Improvements to a Modeling Language for Mathematical Optimization . Mathematical Programming Computation , 2023. https://doi.org/10.1007/s12532-023-00239-3 doi:...

    2023 doi

  7. [15]

    Lewis, and Ron Milo

    Elad Noor, Nathan E. Lewis, and Ron Milo. A proof for loop-law constraints in stoichiometric metabolic networks . BMC Systems Biology , 6(1):140, 2012. https://doi.org/10.1186/1752-0509-6-140 doi:10.1186/1752-0509-6-140

    2012 doi

  8. [16]

    Orth, Ines Thiele, and Bernhard Ø

    Jeffrey D. Orth, Ines Thiele, and Bernhard Ø. Palsson. What is flux balance analysis? Nature Biotechnology , 28(3):245--248, 2010. https://doi.org/10.1038/nbt.1614 doi:10.1038/nbt.1614

    2010 doi

  9. [17]

    Bernhard Ø. Palsson. Systems Biology: Constraint-based Reconstruction and Analysis . Cambridge University Press, 2015. https://doi.org/10.1017/CBO9781139854610 doi:10.1017/CBO9781139854610

    2015 doi

  10. [18]

    An Introduction to Computational Systems Biology: Systems-Level Modelling of Cellular Networks

    Karthik Raman. An Introduction to Computational Systems Biology: Systems-Level Modelling of Cellular Networks . Chapman and Hall/ CRC , 2021. https://doi.org/10.1201/9780429486951 doi:10.1201/9780429486951

    2021 doi

  11. [19]

    Arne C. Reimers. Metabolic Networks, Thermodynamic Constraints, and Matroid Theory . Phd dissertation, Freie Universität Berlin, 2014. URL: https://refubium.fu-berlin.de/handle/fub188/11116

    2014

  12. [20]

    Improving the phenotype predictions of a yeast genome‐scale metabolic model by incorporating enzymatic constraints

    Benjam\' n J S\' a nchez, Cheng Zhang, Avlant Nilsson, Petri‐Jaan Lahtvee, Eduard J Kerkhoven, and Jens Nielsen. Improving the phenotype predictions of a yeast genome‐scale metabolic model by incorporating enzymatic constraints . Molecular Systems Biology , 13(8):935, 2017. ht...

    2017 doi

  13. [21]

    Lewis, and Bernhard Ø

    Jan Schellenberger, Nathan E. Lewis, and Bernhard Ø. Palsson. Elimination of Thermodynamically Infeasible Loops in Steady-State Metabolic Models . Biophysical Journal , 100(3):544--553, 2011. https://doi.org/10.1016/j.bpj.2010.12.3707 doi:10.1016/j.bpj.2010.12.3707

    2011 doi

  14. [22]

    Gunderson, Eduard M

    Matthew Scott, Carl W. Gunderson, Eduard M. Mateescu, Zhongge Zhang, and Terence Hwa. Interdependence of Cell Growth and Gene Expression: Origins and Consequences . Science , 330(6007):1099--1102, 2010. https://doi.org/10.1126/science.1192588 doi:10.1126/science.1192588

    2010 doi

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.