Geodesic
Model Reduction of Biochemical Reactions Networks by Tropical Analysis Methods
O. Radulescu1 ; S. Vakulenko2 ; D. Grigoriev3
1 University of Montpellier 2, France
2 University of Technology and Design, Saint-Petersburg, Russia
3 CNRS, Mathématiques, Université de Lille, Villeneuve d’Ascq, France
Mathematical modelling of natural phenomena, Tome 10 (2015) no. 3, pp. 124-138.

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We discuss a method of approximate model reduction for networks of biochemical reactions. This method can be applied to networks with polynomial or rational reaction rates and whose parameters are given by their orders of magnitude. In order to obtain reduced models we solve the problem of tropical equilibration that is a system of equations in max-plus algebra. In the case of networks with fast nonlinear cycles we have to compute the tropical equilibrations at least twice, once for the initial system and a second time for an extended system obtained by adding to the full system the differential equations satisfied by the conservation laws of the fast subsystem. Our method can be used for formal model reduction in computational systems biology.
DOI : 10.1051/mmnp/201510310

O. Radulescu 1 ; S. Vakulenko 2 ; D. Grigoriev 3

1 University of Montpellier 2, France
2 University of Technology and Design, Saint-Petersburg, Russia
3 CNRS, Mathématiques, Université de Lille, Villeneuve d’Ascq, France 
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O. Radulescu; S. Vakulenko; D. Grigoriev. Model Reduction of Biochemical Reactions Networks by Tropical Analysis Methods. Mathematical modelling of natural phenomena, Tome 10 (2015) no. 3, pp. 124-138. doi : 10.1051/mmnp/201510310. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201510310/

[1] T. Bogart, A.N. Jensen, D. Speyer, B. Sturmfels, R.R. Thomas Journal of Symbolic Computation 2007 54 73

[2] B.L. Clarke J. Phys. Chem. 1992 4066 4071

[3] E.M. Clarke, R. Enders, T. Filkorn, S. Jha Formal Methods in System Design 1996 77 104

[4] M. Einsiedler, M. Kapranov, D. Lind. Non-archimedean amoebas and tropical varieties. Journal für die reine und angewandte Mathematik (Crelles Journal), (601) (2006), 139–157.

[5] D. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag, 1995.

[6] J. Feret, V. Danos, J. Krivine, R. Harmer, W. Fontana Proceedings of the National Academy of Sciences 2009 6453 6458

[7] A.N. Gorban, I.V. Karlin. Invariant manifolds for physical and chemical kinetics, Lect. Notes Phys. 660. Springer, Berlin Heidelberg, 2005.

[8] A.N. Gorban, O. Radulescu. Dynamic and static limitation in reaction networks, revisited. In: D.W. Guy, B. Marin, G.S. Yablonsky, editors. Advances in Chemical Engineering – Mathematics in Chemical Kinetics and Engineering. vol. 34 of Advances in Chemical Engineering. Elsevier, 2008, 103–173.

[9] A.N. Gorban, O. Radulescu, A. Zinovyev Chemical Engineering Science 2010 2310 2324

[10] D. Grigoriev, A. Weber. Complexity of solving systems with few independent monomials and applications to mass-action kinetics. In: V. P. Gerdt, W. Koepf, E.W. Mayr, E.V. Vorozhtsov, editors. Computer Algebra in Scientific Computing, vol. 7442 of Lecture Notes in Computer Science. Springer, Berlin Heidelberg, 2012, 143–154.

[11] A. Katok, B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1996.

[12] E.L. King, C.A. Altman J. Phys. Chem. 1956 1375 1378

[13] S.H. Lam, D.A. Goussis International Journal of Chemical Kinetics 1994 461 486

[14] U. Maas, S.B. Pope Combustion and Flame 1992 239 264

[15] D. Maclagan, B. Sturmfels. Introduction to tropical geometry. Graduate Studies in Mathematics, vol. 161, 2009.

[16] M.P. Millán, A. Dickenstein, A. Shiu, C. Conradi Bulletin of mathematical biology 2012 1027 1065

[17] V. Noel, D. Grigoriev, S. Vakulenko, O. Radulescu. Tropical geometries and dynamics of biochemical networks application to hybrid cell cycle models. In: Jérôme Feret and Andre Levchenko, editors, Proceedings of the 2nd International Workshop on Static Analysis and Systems Biology (SASB 2011), vol. 284 of Electronic Notes in Theoretical Computer Science. Elsevier, 2012, 75–91.

[18] V. Noel, D. Grigoriev, S. Vakulenko, O. Radulescu. Tropicalization and tropical equilibration of chemical reactions. In: G. Litvinov and S. Sergeev, editors, Tropical and Idempotent Mathematics and Applications, vol. 616 of Contemporary Mathematics. American Mathematical Soc., 2014, 261–277.

[19] O. Radulescu, A.N. Gorban, A. Zinovyev, A. Lilienbaum BMC systems biology 2008 86

[20] O. Radulescu, A.N. Gorban, A. Zinovyev, V. Noel Frontiers in Genetics 2012 131

[21] S. Rao, A. Van Der Schaft, B. Jayawardhana Journal of Mathematical Chemistry 2013 2401 2422

[22] C. Robinson. Dynamical systems: stability, symbolic dynamics and chaos. CRC Press, 1999.

[23] S.S. Samal, O. Radulescu, D. Grigoriev, H. Fröhlich, A. Weber. A Tropical Method based on Newton Polygon Approach for Algebraic Analysis of Biochemical Reaction Networks. In: Proceedings of the 9th European Conference on Mathematical and Theoretical Biology, 2014.

[24] M.A. Savageau, E.O. Voit Mathematical biosciences 1987 83 115

[25] S. Soliman Algorithms for Molecular Biology 2012 15

[26] S. Soliman, F. Fages, O. Radulescu Algorithms for Molecular Biology 2014 24

[27] D. Speyer, B. Sturmfels Advances in Geometry 2004 389 411

[28] M.I. Temkin Dokl. Akad. Nauk SSSR 1965 615 618

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