Variable elimination in post-translational modification reaction networks with mass-action kinetics

We define a subclass of chemical reaction networks called post-translational modification systems. Important biological examples of such systems include MAPK cascades and two-component systems which are well-studied experimentally as well as theoretically. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop a mathematical framework based on the notion of a cut (a particular subset of species in the system), which provides a linear elimination procedure to reduce the number of variables in the system to a set of core variables. The steady states are parameterized algebraically by the core variables, and graphical conditions for when steady states with positive core variables imply positivity of all variables are given. Further, minimal cuts are the connected components of the species graph and provide conservation laws. A criterion for when a (maximal) set of independent conservation laws can be derived from cuts is given.

[1]  С.,et al.  The Cell , 1997, Nature Medicine.

[2]  J. Gross,et al.  Graph Theory and Its Applications , 1998 .

[3]  Jeremy Gunawardena,et al.  The rational parameterization theorem for multisite post-translational modification systems. , 2009, Journal of theoretical biology.

[4]  Martin Feinberg,et al.  Multiple Equilibria in Complex Chemical Reaction Networks: Semiopen Mass Action Systems * , 2022 .

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

[6]  W. T. Tutte.  The dissection of equilateral triangles into equilateral triangles , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  Elisenda Feliu,et al.  An Algebraic Approach to Signaling Cascades with n Layers , 2010, Bulletin of Mathematical Biology.

[8]  Alejandra C. Ventura,et al.  A Hidden Feedback in Signaling Cascades Is Revealed , 2008, PLoS Comput. Biol..

[9]  Elisenda Feliu,et al.  Enzyme-sharing as a cause of multi-stationarity in signalling systems , 2011, Journal of The Royal Society Interface.

[10]  M. Feinberg,et al.  Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces , 1977 .

[11]  A. Cornish-Bowden.  Fundamentals of Enzyme Kinetics , 1979 .

[12]  Chi-Ying F. Huang,et al.  Ultrasensitivity in the mitogen-activated protein kinase cascade. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Boris N Kholodenko,et al.  Four‐dimensional dynamics of MAPK information‐processing systems , 2009, Wiley interdisciplinary reviews. Systems biology and medicine.

[14]  Elisenda Feliu,et al.  Variable Elimination in Chemical Reaction Networks with Mass-Action Kinetics , 2011, SIAM J. Appl. Math..

[15]  E. L. King,et al.  A Schematic Method of Deriving the Rate Laws for Enzyme-Catalyzed Reactions , 1956 .

[16]  María-Eugenia Guazzaroni,et al.  Bacterial sensor kinases: diversity in the recognition of environmental signals. , 2010, Annual review of microbiology.

[17]  J. Gunawardena,et al.  Unlimited multistability in multisite phosphorylation systems , 2009, Nature.

[18]  A. Ostman,et al.  Regulation of receptor tyrosine kinase signaling by protein tyrosine phosphatases. , 2001, Trends in cell biology.

[19]  Martin Feinberg,et al.  Multiple Equilibria in Complex Chemical Reaction Networks: I. the Injectivity Property * , 2006 .

[20]  David Angeli,et al.  Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates , 2010, Journal of mathematical biology.

[21]  Reinhart Heinrich,et al.  Mathematical models of protein kinase signal transduction. , 2002, Molecular cell.

[22]  B. Kholodenko,et al.  Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades , 2004, The Journal of cell biology.