Quasi steady state and partial equilibrium approximations: their relation and their validity

The quasi steady state and partial equilibrium approximations are analysed in the context of a system of nonlinear differential equations exhibiting multiscale behaviour. Considering systems in the most general and dimensional form , it is shown that both approximations are limiting cases of leading-order asymptotics. Algorithmic conditions are established which guarantee that the accuracy and stability delivered by the two approximations are equivalent to those obtained with leading-order asymptotics. It is shown that the quasi steady state approximation is a limiting case of the partial equilibrium approximation. Algorithms are reported for the identification of the variables in quasi steady state and/or of the processes in partial equilibrium.

[1]  H. M. Tsuchiya,et al.  On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics☆ , 1967 .

[2]  Thomas Erneux,et al.  Rescue of the Quasi-Steady-State Approximation in a Model for Oscillations in an Enzymatic Cascade , 2006, SIAM J. Appl. Math..

[3]  Habib N. Najm,et al.  A CSP and tabulation-based adaptive chemistry model , 2007 .

[4]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[5]  Banghe Li,et al.  Quasi-steady-state laws in enzyme kinetics. , 2008, The journal of physical chemistry. A.

[6]  G. Stewart.  Introduction to matrix computations , 1973 .

[7]  L. A. Segel,et al.  The Quasi-Steady-State Assumption: A Case Study in Perturbation , 1989, SIAM Rev..

[8]  Marcos Chaos,et al.  Computational singular perturbation analysis of two-stage ignition of large hydrocarbons. , 2006, The journal of physical chemistry. A.

[9]  Habib N. Najm,et al.  Skeletal mechanism generation and analysis for n-heptane with CSP , 2007 .

[10]  B. V. Leer,et al.  A quasi-steady state solver for the stiff ordinary differential equations of reaction kinetics , 2000 .

[11]  E. M. Bulewicz.  Combustion , 1964, Nature.

[12]  C. Law,et al.  Complex CSP for chemistry reduction and analysis , 2001 .

[13]  U. Maas,et al.  Investigation of the Hierarchical Structure of Kinetic Models in Ignition Problems , 2009 .

[14]  S. Benson,et al.  The Induction Period in Chain Reactions , 1952 .

[15]  Dimitris A. Goussis,et al.  Physical understanding of complex multiscale biochemical models via algorithmic simplification: Glycolysis in Saccharomyces cerevisiae , 2010 .

[16]  Mauro Valorani,et al.  An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems , 2006, J. Comput. Phys..

[17]  C. Westbrook,et al.  A comprehensive detailed chemical kinetic reaction mechanism for combustion of n-alkane hydrocarbons from n-octane to n-hexadecane , 2009 .

[18]  Hans G. Kaper,et al.  Two perspectives on reduction of ordinary differential equations , 2005 .

[19]  Ioannis G. Kevrekidis,et al.  Constraint-Defined Manifolds: a Legacy Code Approach to Low-Dimensional Computation , 2005, J. Sci. Comput..

[20]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[21]  S. Lam,et al.  The CSP method for simplifying kinetics , 1994 .

[22]  Tamás Turányi,et al.  On the error of the quasi-steady-state approximation , 1993 .

[23]  Mauro Valorani,et al.  Explicit time-scale splitting algorithm for stiff problems: auto-ignition of gaseous mixtures behind a steady shock , 2001 .

[24]  Hans G. Kaper,et al.  Fast and Slow Dynamics for the Computational Singular Perturbation Method , 2004, Multiscale Model. Simul..

[25]  J. Bowen,et al.  Singular perturbation refinement to quasi-steady state approximation in chemical kinetics , 1963 .

[26]  F. Spellman.  Combustion Theory , 2020 .

[27]  Habib N. Najm,et al.  Higher order corrections in the approximation of low-dimensional manifolds and the construction of simplified problems with the CSP method , 2005 .

[28]  N. Levinson,et al.  Small Periodic Pertubations of an Autonomous System with a Stable Orbit , 1950 .

[29]  S. H. Lam,et al.  Understanding complex chemical kinetics with computational singular perturbation , 1989 .

[30]  M. Bodenstein,et al.  Eine Theorie der photochemischen Reaktionsgeschwindigkeiten , 1913 .

[31]  F. Verhulst.  Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics , 2010 .

[32]  J. Giddings,et al.  VALIDITY OF THE STEADY-STATE APPROXIMATION IN UNIMOLECULAR REACTIONS , 1961 .

[33]  S. H. Lam,et al.  Using CSP to Understand Complex Chemical Kinetics ∗ , 1992 .

[34]  Sebastian Walcher,et al.  Quasi-Steady State and Nearly Invariant Sets , 2009, SIAM J. Appl. Math..

[35]  Dimitris A. Goussis,et al.  Asymptotic Solution of Stiff PDEs with the CSP Method: The Reaction Diffusion Equation , 1998, SIAM J. Sci. Comput..

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

[37]  Iliya V. Karlin,et al.  Method of invariant manifold for chemical kinetics , 2003 .

[38]  S. H. Lam,et al.  A study of homogeneous methanol oxidation kinetics using CSP , 1992 .

[39]  Sebastião J. Formosinho,et al.  Chemical Kinetics: From Molecular Structure to Chemical Reactivity , 2019, Focus on Catalysts.

[40]  M. Rein.  The partial-equilibrium approximation in reacting flows , 1992 .

[41]  S. Walcher,et al.  Quasi-steady state in the Michaelis–Menten system , 2007 .

[42]  J. D. Ramshaw.  Partial chemical equilibrium in fluid dynamics , 1980 .

[43]  John W. Dingee,et al.  A new perturbation solution to the Michaelis‐Menten problem , 2008 .

[44]  Habib N. Najm,et al.  CSP analysis of a transient flame-vortex interaction: time scales and manifolds , 2003 .

[45]  D. Chapman,et al.  LV.—The interaction of chlorine and hydrogen. The influence of mass , 1913 .

[46]  Eric L Haseltine,et al.  Two classes of quasi-steady-state model reductions for stochastic kinetics. , 2007, The Journal of chemical physics.

[47]  D. Siegel,et al.  Properties of the Lindemann Mechanism in Phase Space , 2010, 1003.3692.

[48]  Kevin J. Hughes,et al.  The application of the QSSA via reaction lumping for the reduction of complex hydrocarbon oxidation mechanisms , 2009 .

[49]  Habib N. Najm,et al.  Skeletal mechanism generation with CSP and validation for premixed n-heptane flames , 2009 .

[50]  D. Lauffenburger,et al.  Physicochemical modelling of cell signalling pathways , 2006, Nature Cell Biology.

[51]  S. H. Lam,et al.  CONVENTIONAL ASYMPTOTICS AND COMPUTATIONAL SINGULAR PERTURBATION FOR SIMPLIFIED KINETICS MODELLING , 1999 .

[52]  Simon J. Fraser,et al.  The steady state and equilibrium approximations: A geometrical picture , 1988 .

[53]  S. Schnell,et al.  Use and abuse of the quasi-steady-state approximation. , 2006, Systems biology.

[54]  Prodromos Daoutidis,et al.  Model reduction and control of multi-scale reaction-convection processes , 2008 .

[55]  Michael J. Davis,et al.  Geometric investigation of low-dimensional manifolds in systems approaching equilibrium , 1999 .

[56]  Elazer R Edelman,et al.  On the validity of the quasi-steady state approximation of bimolecular reactions in solution. , 2005, Journal of theoretical biology.

[57]  W. Kyner.  Invariant Manifolds , 1961 .

[58]  Ulrich Maas,et al.  Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space , 1992 .

[59]  Alberto Maria Bersani,et al.  Quasi steady-state approximations in complex intracellular signal transduction networks – a word of caution , 2008 .

[60]  Yang Cao,et al.  Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems , 2005 .

[61]  M. Bodenstein,et al.  Photochemische Kinetik des Chlorknallgases , 1913 .

[62]  H. Najm,et al.  Reactive and reactive-diffusive time scales in stiff reaction-diffusion systems , 2005 .

[63]  Habib N. Najm,et al.  Analysis of methane–air edge flame structure , 2010 .

[64]  Robert E. O'Malley,et al.  Analyzing Multiscale Phenomena Using Singular Perturbation Methods , 1999 .

[65]  J. Goddard,et al.  Consequences of the partial-equilibrium approximation for chemical reaction and transport , 1990, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[66]  Marc R. Roussel,et al.  Geometry of the steady-state approximation: Perturbation and accelerated convergence methods , 1990 .

[67]  Mauro Valorani,et al.  The G-Scheme: A framework for multi-scale adaptive model reduction , 2009, J. Comput. Phys..

[68]  W. Klonowski.  Simplifying principles for chemical and enzyme reaction kinetics. , 1983, Biophysical chemistry.