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Data-driven, variational model reduction of high-dimensional reaction networks. (English) Zbl 1453.62637

Summary: In this work we present new scalable, information theory-based variational methods for the efficient model reduction of high-dimensional deterministic and stochastic reaction networks. The proposed methodology combines, (a) information theoretic tools for sensitivity analysis that allow us to identify the proper coarse variables of the reaction network, with (b) variational approximate inference methods for training a best-fit reduced model. This approach takes advantage of both physicochemical modeling and data-based approaches and allows to construct optimal parameterized reduced dynamics in the number of variables, reactions and parameters, while controlling the information loss due to the reduction. We demonstrate the effectiveness of our model reduction method on several complex, high-dimensional chemical reaction networks arising in biochemistry.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
37M10 Time series analysis of dynamical systems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92E20 Classical flows, reactions, etc. in chemistry
68T05 Learning and adaptive systems in artificial intelligence

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PRMLT; TChem; CHEMKIN
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[1] Sutton, J. E.; Vlachos, D. G., Building large microkinetic models with first-principles accuracy at reduced computational cost, 2013 Danckwerts Special Issue on Molecular Modelling in Chemical Engineering. 2013 Danckwerts Special Issue on Molecular Modelling in Chemical Engineering, Chem. Eng. Sci., 121, 190-199 (2015)
[2] DiStefano, J., Dynamic Systems Biology Modeling and Simulation (2015), Academic Press
[3] Briggs, G. E.; Haldane, J. B.S., A note on the kinetics of enzyme action, Biochem. J., 19, 2, 338 (1925)
[4] Roussel, M. R.; Fraser, S. J., Invariant manifold methods for metabolic model reduction, Chaos, Interdiscip. J. Nonlinear Sci., 11, 1, 196-206 (2001) · Zbl 0992.92025
[5] Kooi, B.; Poggiale, J.; Auger, P.; Kooijman, S., Aggregation methods in food chains with nutrient recycling, Ecol. Model., 157, 69-86 (2002)
[6] Radulescu, O.; Gorban, A. N.; Zinovyev, A.; Lilienbaum, A., Robust simplifications of multiscale biochemical networks, BMC Syst. Biol., 2, 1, 86 (2008)
[7] Petrov, V.; Nikolova, E.; Wolkenhauer, O., Reduction of nonlinear dynamic systems with an application to signal transduction pathways, IET Syst. Biol., 1, 1, 2-9 (2007)
[8] Schneider, K. R.; Wilhelm, T., Model reduction by extended quasi-steady-state approximation, J. Math. Biol., 40, 5, 443-450 (2000) · Zbl 0970.92028
[9] Maas, U.; Pope, S. B., Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, Combust. Flame, 88, 3-4, 239-264 (1992)
[10] Vallabhajosyula, R. R.; Chickarmane, V.; Sauro, H. M., Conservation analysis of large biochemical networks, Bioinformatics, 22, 3, 346-353 (2005)
[11] Zobeley, J.; Lebiedz, D.; Kammerer, J.; Ishmurzin, A.; Kummer, U., A new time-dependent complexity reduction method for biochemical systems, (Transactions on Computational Systems Biology I (2005), Springer), 90-110 · Zbl 1117.92309
[12] Surovtsova, I.; Simus, N.; Lorenz, T.; König, A.; Sahle, S.; Kummer, U., Accessible methods for the dynamic time-scale decomposition of biochemical systems, Bioinformatics, 25, 21, 2816-2823 (2009)
[13] Liu, G.; Swihart, M. T.; Neelamegham, S., Sensitivity, principal component and flux analysis applied to signal transduction: the case of epidermal growth factor mediated signaling, Bioinformatics, 21, 7, 1194-1202 (2004)
[14] Degenring, D.; Froemel, C.; Dikta, G.; Takors, R., Sensitivity analysis for the reduction of complex metabolism models, Dynamics, Monitoring, Control and Optimization of Biological Systems. Dynamics, Monitoring, Control and Optimization of Biological Systems, J. Process Control, 14, 7, 729-745 (2004)
[15] Apri, M.; de Gee, M.; Molenaar, J., Complexity reduction preserving dynamical behavior of biochemical networks, J. Theor. Biol., 304, 16-26 (2012) · Zbl 1397.92248
[16] Turányi, T., Sensitivity analysis of complex kinetic systems. Tools and applications, J. Math. Chem., 5, 3, 203-248 (1990)
[17] Tomlin, A. S.; Pilling, M. J.; Merkin, J. H.; Brindley, J.; Burgess, N.; Gough, A., Reduced mechanisms for propane pyrolysis, Ind. Eng. Chem. Res., 34, 11, 3749-3760 (1995)
[18] Maurya, M.; Bornheimer, S.; Venkatasubramanian, V.; Subramaniam, S., Reduced-order modelling of biochemical networks: application to the GTPase-cycle signalling module, IEE Proc., Syst. Biol., 152, 4, 229-242 (2005)
[19] Jayachandran, D.; Rundell, A. E.; Hannemann, R. E.; Vik, T. A.; Ramkrishna, D., Optimal chemotherapy for leukemia: a model-based strategy for individualized treatment, PLoS ONE, 9, 10, Article e109623 pp. (2014)
[20] Maurya, M.; Bornheimer, S.; Venkatasubramanian, V.; Subramaniam, S., Mixed-integer nonlinear optimisation approach to coarse-graining biochemical networks, IET Syst. Biol., 3, 1, 24-39 (2009)
[21] Hangos, K. M.; Gábor, A.; Szederkényi, G., Model reduction in bio-chemical reaction networks with Michaelis-Menten kinetics, (2013 European Control Conference. 2013 European Control Conference, ECC (2013), IEEE), 4478-4483
[22] Locke, J. C.; Southern, M. M.; Kozma-Bognár, L.; Hibberd, V.; Brown, P. E.; Turner, M. S.; Millar, A. J., Extension of a genetic network model by iterative experimentation and mathematical analysis, Mol. Syst. Biol., 1, 1 (2005)
[23] Anderson, J.; Chang, Y.-C.; Papachristodoulou, A., Model decomposition and reduction tools for large-scale networks in systems biology, Automatica, 47, 6, 1165-1174 (2011) · Zbl 1235.93018
[24] Prescott, T. P.; Papachristodoulou, A., Guaranteed error bounds for structured complexity reduction of biochemical networks, J. Theor. Biol., 304, 172-182 (2012) · Zbl 1397.92261
[25] Danø, S.; Madsen, M. F.; Schmidt, H.; Cedersund, G., Reduction of a biochemical model with preservation of its basic dynamic properties, FEBS J., 273, 21, 4862-4877 (2006)
[26] Dokoumetzidis, A.; Aarons, L., Proper lumping in systems biology models, IET Syst. Biol., 3, 1, 40-51 (2009)
[27] Koschorreck, M.; Conzelmann, H.; Ebert, S.; Ederer, M.; Gilles, E. D., Reduced modeling of signal transduction-a modular approach, BMC Bioinform., 8, 1, 336 (2007)
[28] Sunnåker, M.; Cedersund, G.; Jirstrand, M., A method for zooming of nonlinear models of biochemical systems, BMC Syst. Biol., 5, 1, 140 (2011)
[29] Majda, A.; Abramov, R. V.; Grote, M. J., Information Theory and Stochastics for Multiscale Nonlinear Systems, vol. 25 (2005), American Mathematical Soc. · Zbl 1082.60002
[30] Constantino, P. H.; Kaznessis, Y. N., Maximum entropy prediction of non-equilibrium stationary distributions for stochastic reaction networks with oscillatory dynamics, Chem. Eng. Sci., 171, 139-148 (2017)
[31] Lee, C. H.; Kim, K.-H.; Kim, P., A moment closure method for stochastic reaction networks, J. Chem. Phys., 130, 13, Article 134107 pp. (2009)
[32] Gillespie, C. S., Moment-closure approximations for mass-action models, IET Syst. Biol., 3, 1, 52-58 (2009)
[33] Grima, R., A study of the accuracy of moment-closure approximations for stochastic chemical kinetics, J. Chem. Phys., 136, 15, Article 04B616 pp. (2012)
[34] Érdi, P.; Tóth, J., Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models (Nonlinear Science) (1989), Princeton University Press · Zbl 0696.92027
[35] Wilkinson, D. J., Stochastic Modelling for Systems Biology (2012), Chapman & Hall · Zbl 1300.92004
[36] Cover, T.; Thomas, J., Elements of Information Theory (1991), John Wiley & Sons · Zbl 0762.94001
[37] Kipnis, C.; Landim, C., Scaling Limits of Interacting Particle Systems (1999), Springer-Verlag · Zbl 0927.60002
[38] MacKay, D. J.C., Information Theory, Inference & Learning Algorithms (2003), Cambridge University Press · Zbl 1055.94001
[39] Bishop, C. M., Pattern Recognition and Machine Learning (Information Science and Statistics) (2006), Springer-Verlag: Springer-Verlag New York, Inc., Secaucus, NJ, USA · Zbl 1107.68072
[40] Pinski, F. J.; Simpson, G.; Stuart, A. M.; Weber, H., Kullback-Leibler approximation for probability measures on infinite dimensional spaces, SIAM J. Math. Anal., 47, 6, 4091-4122 (2015) · Zbl 1342.60049
[41] Wainwright, M. J.; Jordan, M. I., Graphical models, exponential families, and variational inference, Found. Trends Mach. Learn., 1, 1-2, 1-305 (2008) · Zbl 1193.62107
[42] Hoffman, M. D.; Blei, D. M.; Wang, C.; Paisley, J., Stochastic variational inference, J. Mach. Learn. Res., 14, 1, 1303-1347 (2013) · Zbl 1317.68163
[43] Shell, M. S., The relative entropy is fundamental to multiscale and inverse thermodynamic problems, J. Chem. Phys., 129, 14 (2008)
[44] Chaimovich, A.; Shell, M. S., Relative entropy as a universal metric for multiscale errors, Phys. Rev. E, 81, 6, Article 060104 pp. (2010)
[45] Rudzinski, J. F.; Noid, W. G., Coarse-graining, entropy, forces and structures, J. Chem. Phys., 135, 21 (2011)
[46] Bilionis, I.; Koutsourelakis, P., Free energy computations by minimization of Kullback-Leibler divergence: an efficient adaptive biasing potential method for sparse representations, J. Comput. Phys., 231, 9, 3849-3870 (2012) · Zbl 1252.82051
[47] Bilionis, I.; Zabaras, N., A stochastic optimization approach to coarse-graining using a relative-entropy framework, J. Chem. Phys., 138, 4, Article 044313 pp. (2013)
[48] Foley, T. T.; Shell, M. S.; Noid, W. G., The impact of resolution upon entropy and information in coarse-grained models, J. Chem. Phys., 143, 24 (2015)
[49] Katsoulakis, M. A.; Plecháč, P., Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems, J. Chem. Phys., 139, 7, Article 074115 pp. (2013)
[50] Harmandaris, V.; Kalligiannaki, E.; Katsoulakis, M.; Plecháč, P., Path-space variational inference for non-equilibrium coarse-grained systems, J. Comput. Phys., 314, 355-383 (2016) · Zbl 1349.65030
[51] Kalligiannaki, E.; Chazirakis, A.; Tsourtis, A.; Katsoulakis, M.; Plecháč, P.; Harmandaris, V., Parametrizing coarse grained models for molecular systems at equilibrium, Eur. Phys. J. Spec. Top., 225, 8, 1347-1372 (2016)
[52] Pantazis, Y.; Katsoulakis, M., A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics, J. Chem. Phys., 138, 5, Article 054115 pp. (2013)
[53] Pantazis, Y.; Katsoulakis, M.; Vlachos, D., Parametric sensitivity analysis for biochemical reaction networks based on pathwise information theory, BMC Bioinform., 14, 1, 311 (2013)
[54] Dupuis, P.; Katsoulakis, M. A.; Pantazis, Y.; Plecháč, P., Path-space information bounds for uncertainty quantification and sensitivity analysis of stochastic dynamics, SIAM/ASA J. Uncertain. Quantificat., 4, 1, 80-111 (2016) · Zbl 1371.65004
[55] Proctor, C. J.; Lorimer, I. A.J., Modelling the role of the Hsp70/Hsp90 system in the maintenance of protein homeostasis, PLoS ONE, 6, 7, 1-17 (2011)
[56] Kholodenko, B. N.; Demin, O. V.; Moehren, G.; Hoek, J. B., Quantification of short term signaling by the epidermal growth factor receptor, J. Biol. Chem., 274, 42, 30169-30181 (1999)
[57] Leloup, J.-C.; Goldbeter, A., Toward a detailed computational model for the mammalian circadian clock, Proc. Natl. Acad. Sci., 100, 12, 7051-7056 (2003)
[58] Vilanova, P., Multilevel Approximations of Markovian Jump Processes with Applications in Communication Networks (2015), King Abdullah University of Science and Technology, PhD dissertation
[59] Ethier, S. N.; Kurtz, T. G., Markov Processes: Characterization and Convergence, Wiley Series in Probability and Statistics (2005), Wiley-Interscience · Zbl 1089.60005
[60] Gillespie, D. T., Approximated accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115, 4, 1716-1733 (2001)
[61] Cao, Y.; Gillespie, D. T.; Petzold, L. R., Efficient step size selection for the tau-leaping simulation method, J. Chem. Phys., 124, Article 044109 pp. (2006)
[62] Tian, T.; Burrage, K., Binomial leap methods for simulating stochastic chemical kinetics, J. Chem. Phys., 121, Article 10356 pp. (2004)
[63] Chatterjee, A.; Vlachos, D. G.; Katsoulakis, M. A., Binomial distribution based tau-leap accelerated stochastic simulation, J. Chem. Phys., 122, Article 024112 pp. (2005)
[64] Moraes, A.; Tempone, R.; Vilanova, P., Multilevel hybrid Chernoff tau-leap, BIT Numer. Math., 1-51 (2015)
[65] Moraes, A.; Tempone, R.; Vilanova, P., Hybrid Chernoff tau-leap, Multiscale Model. Simul., 12, 2, 581-615 (2014) · Zbl 1338.65013
[66] Gardiner, C., Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences (1985), Springer · Zbl 1143.60001
[67] van Kampen, N. G., Stochastic Processes in Physics and Chemistry (2006), North Holland · Zbl 0974.60020
[68] Gillespie, D. T., The chemical Langevin equation, J. Chem. Phys., 113, 297-306 (2000)
[69] Kurtz, T. G., The relationship between stochastic and deterministic models for chemical reactions, J. Chem. Phys., 57, 2976 (1972)
[70] Kee, R. J.; Miller, J. A.; Jefferson, T. H., Chemkin: a General-Purpose, Problem-Independent, Transportable, Fortran Chemical Kinetics Code Package (1980), Tech. rep, Sandia Labs.
[71] Safta, C.; Najm, H. N.; Knio, O., Tchem-A Software Toolkit for the Analysis of Complex Kinetic Models (2011), Sandia Report, SAND2011-3282
[72] Arampatzis, G.; Katsoulakis, M. A.; Pantazis, Y., Accelerated sensitivity analysis in high-dimensional stochastic reaction networks, PLoS ONE, 10, 7, 1-24 (2015)
[73] Chu, Y.; Hahn, J., Parameter set selection via clustering of parameters into pairwise indistinguishable groups of parameters, Ind. Eng. Chem. Res., 48, 13, 6000-6009 (2008)
[74] Cintrón-Arias, A.; Banks, H.; Capaldi, A.; Lloyd, A. L., A sensitivity matrix based methodology for inverse problem formulation, J. Inverse Ill-Posed Probl., 17, 6, 545-564 (2009) · Zbl 1179.34008
[75] Yue, H.; Brown, M.; Knowles, J.; Wang, H.; Broomhead, D. S.; Kell, D. B., Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-κB signalling pathway, Mol. BioSyst., 2, 12, 640-649 (2006)
[76] Kurtz, T. G., Approximation of Population Processes (1981), Society for Industrial and Applied Mathematics (SIAM) · Zbl 0465.60078
[77] Jordan, M. I.; Ghahramani, Z.; Jaakkola, T. S.; Saul, L. K., An introduction to variational methods for graphical models, Mach. Learn., 37, 2, 183-233 (1999) · Zbl 0945.68164
[78] Wainwright, M. J.; Jordan, M. I., Graphical models, exponential families, and variational inference, Found. Trends® Mach. Learn., 1, 1-2, 1-305 (2008) · Zbl 1193.62107
[79] Blei, D. M.; Kucukelbir, A.; McAuliffe, J. D., Variational inference: a review for statisticians, J. Am. Stat. Assoc., 112, 518, 859-877 (2017)
[80] Efron, B., Computer Age Statistical Inference: Algorithms, Evidence, and Data Science (2016), Cambridge University Press: Cambridge University Press New York, NY · Zbl 1377.62004
[81] Aster, R., Parameter Estimation and Inverse Problems (2005), Elsevier Academic Press: Elsevier Academic Press Amsterdam Boston · Zbl 1088.35081
[82] Kaipio, J., Statistical and Computational Inverse Problems (2005), Springer: Springer New York · Zbl 1068.65022
[83] Saunders, M. G.; Voth, G. A., Coarse-graining methods for computational biology, Annu. Rev. Biophys., 42, 1, 73-93 (2013), pMID: 23451897
[84] Izvekov, S.; Voth, G. A., A multiscale coarse-graining method for biomolecular systems, J. Phys. Chem. B, 109, 7, 2469-2473 (2005)
[85] Brunton, S. L.; Proctor, J. L.; Kutz, J. N., Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci., 113, 15, 3932-3937 (2016) · Zbl 1355.94013
[86] Schmidt, M.; Lipson, H., Distilling free-form natural laws from experimental data, Science, 324, 5923, 81-85 (2009)
[87] Komorowski, M.; Costa, M. J.; Rand, D. A.; Stumpf, M. P.H., Sensitivity, robustness, and identifiability in stochastic chemical kinetics models, Proc. Natl. Acad. Sci. USA, 108, 8645-8650 (2011)
[88] EMBL-EBI, BioModels Database [link]. URL
[89] Li, C.; Donizelli, M.; Rodriguez, N.; Dharuri, H.; Endler, L.; Chelliah, V.; Li, L.; He, E.; Henry, A.; Stefan, M. I.; Snoep, J. L.; Hucka, M.; Le Novère, N.; Laibe, C., BioModels database: an enhanced, curated and annotated resource for published quantitative kinetic models, BMC Syst. Biol., 4, 92 (2010)
[90] Kurtz, T. G., Strong approximation theorems for density dependent Markov chains, Stoch. Process. Appl., 6, 3, 223-240 (1978) · Zbl 0373.60085
[91] Kang, H.-W.; Kurtz, T. G., Separation of time-scales and model reduction for stochastic reaction networks, Ann. Appl. Probab., 23, 2, 529-583 (2013) · Zbl 1377.60076
[92] Chassagnole, C.; Noisommit-Rizzi, N.; Schmid, J. W.; Mauch, K.; Reuss, M., Dynamic modeling of the central carbon metabolism of escherichia coli, Biotechnol. Bioeng., 79, 1, 53-73 (2002)
[93] Turanyi, T.; Berces, T.; Vajda, S., Reaction rate analysis of complex kinetic systems, Int. J. Chem. Kinet., 21, 2, 83-99 (1989)
[94] Degenring, D.; Froemel, C.; Dikta, G.; Takors, R., Sensitivity analysis for the reduction of complex metabolism models, J. Process Control, 14, 7, 729-745 (2004)
[95] Smets, I.; Bernaerts, K.; Sun, J.; Marchal, K.; Vanderleyden, J.; Van Impe, J., Sensitivity function-based model reduction: a bacterial gene expression case study, Biotechnol. Bioeng., 80, 2, 195-200 (2002)
[96] Tikhonov, A. N., Systems of differential equations containing small parameters in the derivatives, Mat. Sb., 73, 3, 575-586 (1952) · Zbl 0048.07101
[97] Choi, J.; Yang, K.-w.; Lee, T.-y.; Lee, S. Y., New time-scale criteria for model simplification of bio-reaction systems, BMC Bioinform., 9, 1, 338 (2008)
[98] West, S.; Bridge, L. J.; White, M. R.; Paszek, P.; Biktashev, V. N., A method of ‘speed coefficients’ for biochemical model reduction applied to the NF-κ B system, J. Math. Biol., 70, 3, 591-620 (2015) · Zbl 1306.92022
[99] Menten, L.; Michaelis, M., Die kinetik der invertinwirkung, Biochem. Z., 49, 333-369 (1913)
[100] Gerdtzen, Z. P.; Daoutidis, P.; Hu, W.-S., Non-linear reduction for kinetic models of metabolic reaction networks, Metab. Eng., 6, 2, 140-154 (2004)
[101] Noel, V.; Grigoriev, D.; Vakulenko, S.; Radulescu, O., Tropicalization and tropical equilibration of chemical reactions, (Tropical and Idempotent Mathematics and Applications, vol. 616 (2014)), 261-277 · Zbl 1320.92091
[102] Lam, S., Singular perturbation for stiff equations using numerical methods, (Recent Advances in the Aerospace Sciences (1985), Springer), 3-19
[103] Lam, S.; Coussis, D., Conventional asymptotics and computational singular perturbation for simplified kinetics modelling, (Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames (1991), Springer), 227-242
[104] Lam, S.; Goussis, D., The CSP method for simplifying kinetics, Int. J. Chem. Kinet., 26, 4, 461-486 (1994)
[105] Zagaris, A.; Kaper, H. G.; Kaper, T. J., Analysis of the computational singular perturbation reduction method for chemical kinetics, J. Nonlinear Sci., 14, 1, 59-91 (2004) · Zbl 1053.92051
[106] Surovtsova, I.; Simus, N.; Hübner, K.; Sahle, S.; Kummer, U., Simplification of biochemical models: a general approach based on the analysis of the impact of individual species and reactions on the systems dynamics, BMC Syst. Biol., 6, 1, 14 (2012)
[107] Kourdis, P. D.; Steuer, R.; Goussis, D. A., Physical understanding of complex multiscale biochemical models via algorithmic simplification: glycolysis in Saccharomyces cerevisiae, Phys. D: Nonlinear Phenom., 239, 18, 1798-1817 (2010) · Zbl 1228.92027
[108] Wei, J.; Kuo, J. C., Lumping analysis in monomolecular reaction systems. Analysis of the exactly lumpable system, Ind. Eng. Chem. Fundam., 8, 1, 114-123 (1969)
[109] Kuo, J. C.; Wei, J., Lumping analysis in monomolecular reaction systems. Analysis of approximately lumpable system, Ind. Eng. Chem. Fundam., 8, 1, 124-133 (1969)
[110] Hartwell, L. H.; Hopfield, J. J.; Leibler, S.; Murray, A. W., From molecular to modular cell biology, Nature, 402, 6761 supp, C47 (1999)
[111] Saez-Rodriguez, J.; Kremling, A.; Gilles, E. D., Dissecting the puzzle of life: modularization of signal transduction networks, Comput. Chem. Eng., 29, 3, 619-629 (2005)
[112] Saez-Rodriguez, J.; Kremling, A.; Conzelmann, H.; Bettenbrock, K.; Gilles, E. D., Modular analysis of signal transduction networks, IEEE Control Syst., 24, 4, 35-52 (2004) · Zbl 1395.93108
[113] Conzelmann, H.; Saez-Rodriguez, J.; Sauter, T.; Bullinger, E.; Allgöwer, F.; Gilles, E. D., Reduction of mathematical models of signal transduction networks: simulation-based approach applied to EGF receptor signalling, Syst. Biol., 1, 1, 159-169 (2004)
[114] Wolf, D. M.; Arkin, A. P., Motifs, modules and games in bacteria, Curr. Opin. Microbiol., 6, 2, 125-134 (2003)
[115] Craciun, G.; Pantea, C., Identifiability of chemical reaction networks, J. Math. Chem., 44, 1, 244-259 (2008) · Zbl 1145.92040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.