Nondegenerate multistationarity in small reaction networks

Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this problem for certain types of small networks, and our techniques point the way forward for larger networks.

[1]  Badal Joshi,et al.  Atoms of multistationarity in chemical reaction networks , 2011, Journal of Mathematical Chemistry.

[2]  Ezra Miller,et al.  A Geometric Approach to the Global Attractor Conjecture , 2013, SIAM J. Appl. Dyn. Syst..

[3]  Bernd Sturmfels,et al.  Algebraic Systems Biology: A Case Study for the Wnt Pathway , 2015, Bulletin of Mathematical Biology.

[4]  David J. Grabiner.  Descartes' Rule of Signs: Another Construction , 1999 .

[5]  M. Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems , 1987 .

[6]  Anne Shiu,et al.  Analyzing multistationarity in chemical reaction networks using the determinant optimization method , 2015, Appl. Math. Comput..

[7]  F R Adler,et al.  How to make a biological switch. , 2000, Journal of theoretical biology.

[8]  Kenneth L. Ho,et al.  Bistability in Apoptosis by Receptor Clustering , 2009, PLoS Comput. Biol..

[9]  Timo de Wolff,et al.  Norms of roots of trinomials , 2014, 1411.6552.

[10]  V. Vassiliev.  Complements of Discriminants of Smooth Maps: Topology and Applications , 1992 .

[11]  Carsten Conradi,et al.  Dynamics of Posttranslational Modification Systems: Recent Progress and Future Directions. , 2017, Biophysical journal.

[12]  Alicia Dickenstein,et al.  The Structure of MESSI Biological Systems , 2016, SIAM J. Appl. Dyn. Syst..

[13]  Birkett Huber,et al.  A Family of Sparse Polynomial Systems Arising in Chemical Reaction Systems , 2002, J. Symb. Comput..

[14]  Badal Joshi,et al.  A survey of methods for deciding whether a reaction network is multistationary , 2014, 1412.5257.

[15]  Badal Joshi,et al.  Which Small Reaction Networks Are Multistationary? , 2016, SIAM J. Appl. Dyn. Syst..

[16]  Q. I. Rahman,et al.  Analytic theory of polynomials , 2002 .

[17]  Murad Banaji,et al.  The Inheritance of Nondegenerate Multistationarity in Chemical Reaction Networks , 2016, SIAM J. Appl. Math..

[18]  Elisenda Feliu,et al.  Identifying parameter regions for multistationarity , 2016, PLoS Comput. Biol..

[19]  Karin Gatermann,et al.  BIOCHEMICAL REACTION NETWORKS : AN INVITATION FOR ALGEBRAIC , 2015 .

[20]  Mark Sweeney.  Conditions for Solvability in Chemical Reaction Networks at Quasi-Steady-State , 2017, 1712.05533.

[21]  Jeremy Gunawardena,et al.  Invariants reveal multiple forms of robustness in bifunctional enzyme systems. , 2015, Integrative biology : quantitative biosciences from nano to macro.