Multi-equilibrium property of metabolic networks: SSI module

BackgroundRevealing the multi-equilibrium property of a metabolic network is a fundamental and important topic in systems biology. Due to the complexity of the metabolic network, it is generally a difficult task to study the problem as a whole from both analytical and numerical viewpoint. On the other hand, the structure-oriented modularization idea is a good choice to overcome such a difficulty, i.e. decomposing the network into several basic building blocks and then studying the whole network through investigating the dynamical characteristics of the basic building blocks and their interactions. Single substrate and single product with inhibition (SSI) metabolic module is one type of the basic building blocks of metabolic networks, and its multi-equilibrium property has important influence on that of the whole metabolic networks.ResultsIn this paper, we describe what the SSI metabolic module is, characterize the rates of the metabolic reactions by Hill kinetics and give a unified model for SSI modules by using a set of nonlinear ordinary differential equations with multi-variables. Specifically, a sufficient and necessary condition is first given to describe the injectivity of a class of nonlinear systems, and then, the sufficient condition is used to study the multi-equilibrium property of SSI modules. As a main theoretical result, for the SSI modules in which each reaction has no more than one inhibitor, a sufficient condition is derived to rule out multiple equilibria, i.e. the Jacobian matrix of its rate function is nonsingular everywhere.ConclusionsIn summary, we describe SSI modules and give a general modeling framework based on Hill kinetics, and provide a sufficient condition for ruling out multiple equilibria of a key type of SSI module.

[1]  C. Frenzen,et al.  L'Hospital's Rule , 2002 .

[2]  A. Hill,et al.  The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves , 1910 .

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

[4]  J. Demongeot,et al.  Positive and negative feedback: striking a balance between necessary antagonists. , 2002, Journal of theoretical biology.

[5]  M. Feinberg,et al.  Understanding bistability in complex enzyme-driven reaction networks. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[6]  陈彭年,et al.  A proof of the Jacobian conjecture on global asymptotic stability , 1996 .

[7]  El Houssine Snoussi Necessary Conditions for Multistationarity and Stable Periodicity , 1998 .

[8]  C. Raines The Calvin cycle revisited , 2004, Photosynthesis Research.

[9]  G. Briggs,et al.  A Note on the Kinetics of Enzyme Action. , 1925, The Biochemical journal.

[10]  H. Gutfreund,et al.  Enzyme kinetics , 1975, Nature.

[11]  C. Soulé Graphic Requirements for Multistationarity , 2004, Complexus.

[12]  Thomas Mestl,et al.  FEEDBACK LOOPS, STABILITY AND MULTISTATIONARITY IN DYNAMICAL SYSTEMS , 1995 .

[13]  G. Pettersson,et al.  A mathematical model of the Calvin photosynthesis cycle. , 1988, European journal of biochemistry.

[14]  Xin Wang,et al.  Analysis on steady states of photosynthetic carbon metabolic system , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[15]  Rutherford Aris,et al.  Elementary Chemical Reactor Analysis , 1969 .

[16]  Kazuyuki Aihara,et al.  Modeling Biomolecular Networks in Cells: Structures and Dynamics , 2010 .

[17]  Kazuyuki Aihara,et al.  Modeling Biomolecular Networks in Cells , 2010 .

[18]  Jifeng Zhang,et al.  Multiple equilibria in SSN metabolic module , 2010, Proceedings of the 29th Chinese Control Conference.

[19]  J. Gouzé Positive and Negative Circuits in Dynamical Systems , 1998 .

[20]  René Thomas On the Relation Between the Logical Structure of Systems and Their Ability to Generate Multiple Steady States or Sustained Oscillations , 1981 .

[21]  Luonan Chen,et al.  Biomolecular Networks: Methods and Applications in Systems Biology , 2009 .

[22]  Maciej Ogorzalek,et al.  Stability analysis of SSN biochemical networks , 2011, 2011 IEEE International Symposium of Circuits and Systems (ISCAS).

[23]  M. Chamberland,et al.  A Mountain Pass to the Jacobian Conjecture , 1998, Canadian Mathematical Bulletin.

[24]  Hong-Bo Lei,et al.  Property of Multiple Equilibria for SSI Metabolic Module , 2010 .

[25]  Miroslav Krstic,et al.  Modular approach to adaptive nonlinear stabilization , 1996, Autom..

[26]  N. Wilhelmová,et al.  Cornish-Bowden, A.:Fundamentals of Enzyme Kinetics , 1996, Biologia Plantarum.

[27]  J. Demongeot,et al.  Numerical methods in the study of critical phenomena : proceedings of a colloquium, Carry-le-Rouet, France, June 2-4, 1980 , 1981 .

[28]  Eric de Sturler,et al.  A simple model of the Calvin cycle has only one physiologically feasible steady state under the same external conditions , 2009 .

[29]  Jifeng Zhang,et al.  Multi‐equilibrium property of metabolic networks: Exclusion of multi‐stability for SSN metabolic modules , 2011 .

[30]  Xin Wang,et al.  A parameter condition for ruling out multiple equilibria of the photosynthetic carbon metabolism , 2011 .

[31]  Peng Nian Chen,et al.  A Proof of the Jacobian Conjecture on Global Asymptotic Stability , 1996 .

[32]  Martin Feinberg,et al.  Multiple Equilibria in Complex Chemical Reaction Networks: Ii. the Species-reactions Graph , 2022 .

[33]  R. Richards Selectable traits to increase crop photosynthesis and yield of grain crops. , 2000, Journal of experimental botany.

[34]  S. Benson,et al.  Chemical Kinetics , 2021, Fuel Effects on Operability of Aircraft Gas Turbine Combustors.

[35]  Edward J. Davison,et al.  Modular model reduction for interconnected systems, , 1990, Autom..

[36]  S. Long,et al.  Can improvement in photosynthesis increase crop yields? , 2006, Plant, cell & environment.

[37]  D. Fell,et al.  Computer modelling and experimental evidence for two steady states in the photosynthetic Calvin cycle. , 2001, European journal of biochemistry.

[38]  R. Thomas,et al.  A new necessary condition on interaction graphs for multistationarity. , 2007, Journal of Theoretical Biology.