A Survey of Methods for Deciding Whether a Reaction Network is Multistationary

B. Joshi¹ ; A. Shiu²

¹ Dept. of Mathematics, CSU San Marcos, 333 S. Twin Oaks Valley, San Marcos CA 92096, USA
² Dept. of Mathematics, Mailstop 3368, Texas A&M University, College Station TX 77843–3368, USA

Mathematical modelling of natural phenomena, Tome 10 (2015) no. 5, pp. 47-67

Résumé

Which reaction networks, when taken with mass-action kinetics, have the capacity for multiple steady states? There is no complete answer to this question, but over the last 40 years various criteria have been developed that can answer this question in certain cases. This work surveys these developments, with an emphasis on recent results that connect the capacity for multistationarity of one network to that of another. In this latter setting, we consider a network N that is embedded in a larger network G, which means that N is obtained from G by removing some subsets of chemical species and reactions. This embedding relation is a significant generalization of the subnetwork relation. For arbitrary networks, it is not true that if N is embedded in G, then the steady states of N lift to G. Nonetheless, this does hold for certain classes of networks; one such class is that of fully open networks. This motivates the search for embedding-minimal multistationary networks: those networks which admit multiple steady states but no proper, embedded networks admit multiple steady states. We present results about such minimal networks, including several new constructions of infinite families of these networks.

Détail

DOI : 10.1051/mmnp/201510504

Bibliographie

[1] D. Angeli, P. De Leenheer, E. Sontag J. Math. Biol. 2010 581 616

[2] M. Banaji Dyn. Syst. 2009 1 30

[3] M. Banaji, G. Craciun Adv. Appl. Math. 2010 168 184

[4] M. Banaji, J. Mierczyński J. Differential Equations 2013 1359 1374

[5] M. Banaji, C. Pantea. Some results on injectivity and multistationarity in chemical reaction networks. preprint, http://arXiv.org/abs/1309.6771, (2013).

[6] C. Conradi, D. Flockerzi, J. Raisch, J. Stelling Proc. Natl. Acad. Sci. USA 2007 19175 19180

[7] C. Conradi, M. Mincheva. Graph-theoretic analysis of multistationarity using degree theory. preprint, http://arxiv.org/abs/1411.2896, (2014).

[8] G. Craciun, M. Feinberg SIAM J. Appl. Math. 2005 1526 1546

[9] G. Craciun, M. Feinberg IEE Proceedings-Systems Biology 2006 179 186

[10] G. Craciun, M. Feinberg SIAM J. Appl. Math. 2006 1321 1338

[11] G. Craciun, M. Feinberg SIAM J. Appl. Math. 2010 1859 1877

[12] G. Craciun, L. Garcia-Puente, F. Sottile. Some geometrical aspects of control points for toric patches. In M. Dæhlen, M. S. Floater, T. Lyche, J.-L. Merrien, K. Morken, L. L. Schumaker, eds., Mathematical Methods for Curves and Surfaces, vol. 5862 of Lecture Notes in Comput. Sci., pages 111–135. Springer, Heidelberg, 2010.

[13] G. Craciun, C. Pantea, E. Sontag. Graph-theoretic characterizations of multistability and monotonicity for biochemical reaction networks, pages 63–72. Springer, 2011.

[14] P. Donnell, M. Banaji SIAM J. Appl. Dyn. Syst. 2013 899 920

[15] P. Donnell, M. Banaji, A. Marginean, C. Pantea. CoNtRol: an open source framework for the analysis of chemical reaction networks. Bioinformatics, (2014), to appear.

[16] P. Ellison. The advanced deficiency algorithm and its applications to mechanism discrimination. Ph.D. thesis, University of Rochester, 1998.

[17] P. Ellison, M. Feinberg, H. Ji, D. Knight. Chemical Reaction Network Toolbox, 2011. Available at http://www.crnt.osu.edu/CRNTWin.

[18] M. Feinberg Arch. Rational Mech. Anal. 1972 187 194

[19] M. Feinberg Chem. Eng. Sci. 1987 2229 2268

[20] M. Feinberg Arch. Rational Mech. Anal. 1995 371 406

[21] E. Feliu. Injectivity, multiple zeros, and multistationarity in reaction networks. preprint, http://arXiv.org/abs/1407.2955, (2014).

[22] E. Feliu, C. Wiuf Appl. Math. Comput. 2012 1449 1467

[23] E. Feliu, C. Wiuf J. R. Soc. Interface 2013

[24] D. Fouchet, R. Regoes PLoS ONE 2008

[25] G. Gnacadja Linear Algebra Appl. 2012 612 622

[26] P. Grassberger Zeitschrift für Physik B Condensed Matter 1982 365 374

[27] J. W. Helton, I. Klep, R. Gomez SIAM J. Matrix Anal. Appl. 2009 732 754

[28] A. Ivanova Kinet. Katal. 1979 1019 1023

[29] H. Ji. Uniqueness of Equilibria for Complex Chemical Reaction Networks. Ph.D. thesis, Ohio State University, 2011.

[30] B. Joshi Applied Mathematics and Computation 2013 6931 6945

[31] B. Joshi, A. Shiu SIAM J. Appl. Math. 2012 857 876

[32] B. Joshi, A. Shiu Journal of Mathematical Chemistry 2013 153 178

[33] K. P. Lesch, D. Bengel CNS drugs 1995 302 322

[34] G. Marin, G. S. Yablonsky. Kinetics of Chemical Reactions. Wiley-VCH, Wienheim, Germany, 2011.

[35] M. Mincheva, G. Craciun Proceedings of the IEEE 2008 1281 1291

[36] S. Müller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu, A. Dickenstein. Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. to appear in Found. Comput. Math.

[37] M. Santillán, M. C. Mackey Proceedings of the National Academy of Sciences 2001 1364 1369

[38] F. Schlögl Zeitschrift für Physik 1972 147 161

[39] P. M. Schlosser, M. Feinberg Chemical Engineering Science 1994 1749 1767

[40] G. Shinar, M. Feinberg Math. Biosci. 2012 92 113

[41] G. Shinar, M. Feinberg Math. Biosci. 2013 1 23

[42] D. Siegal-Gaskins, M. K. Mejia-Guerra, G. D. Smith, E. Grotewold PLoS Comput. Biol. 2011

[43] C. Wiuf, E. Feliu SIAM J. Appl. Dyn. Syst. 2013 1685 1721

[44] G. Yablonskii, V. Bykov, A. Gorban, V. Elokhin. Kinetic Models of Catalytic Reactions. Comprehensive Chemical Kinetics, vol. 32, ed. by R.G. Compton, Elsevier, Amsterdam, 1991.