Reduction of chemical systems by delayed quasi-steady state assumptions

Mathematical analysis of mass action models of large complex chemical systems is typically only possible if the models are reduced. The most common reduction technique is based on quasi-steady state assumptions. To increase the accuracy of this technique we propose delayed quasi-steady state assumptions (D-QSSA) which yield systems of delay differential equations. We define the approximation based on D-QSSA, prove the corresponding error estimate, and show how it approximates the invariant manifold. Then we define a class of well mixed chemical systems and formulate assumptions enabling the application of D-QSSA. We also apply the D-QSSA to a model of Hes1 expression and to a cell-cycle model to illustrate the improved accuracy of the D-QSSA with respect to the standard quasi-steady state assumptions.

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

[2]  S. Leibler,et al.  Mechanisms of noise-resistance in genetic oscillators , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[3]  M. Mavrovouniotis,et al.  Simplification of Mathematical Models of Chemical Reaction Systems. , 1998, Chemical reviews.

[4]  H. Hirata,et al.  Oscillatory Expression of the bHLH Factor Hes1 Regulated by a Negative Feedback Loop , 2002, Science.

[5]  D. Volfson,et al.  Delay-induced stochastic oscillations in gene regulation. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Rui Zhu,et al.  Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression , 2006, Physical biology.

[7]  Simon J. Fraser,et al.  Geometrical picture of reaction in enzyme kinetics , 1989 .

[8]  Marc R. Roussel,et al.  On the geometry of transient relaxation , 1991 .

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

[10]  M. Roussel.  The Use of Delay Differential Equations in Chemical Kinetics , 1996 .

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

[12]  A. N. Gorban,et al.  Constructive methods of invariant manifolds for kinetic problems , 2003 .

[13]  S. Bhat,et al.  Modeling and analysis of mass-action kinetics , 2009, IEEE Control Systems.

[14]  Hans G Othmer,et al.  A multi-time-scale analysis of chemical reaction networks: II. Stochastic systems , 2015, Journal of Mathematical Biology.

[15]  Hans G. Kaper,et al.  Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics , 2004, J. Nonlinear Sci..

[16]  Hans G Othmer,et al.  A multi-time-scale analysis of chemical reaction networks: I. Deterministic systems , 2010, Journal of mathematical biology.

[17]  M. Mackey,et al.  Modelling transcriptional feedback loops: the role of Gro/TLE1 in Hes1 oscillations , 2006, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  Zsolt Tuza,et al.  Structure simplification of dynamic process models , 2002 .

[19]  N. Monk.  Oscillatory Expression of Hes1, p53, and NF-κB Driven by Transcriptional Time Delays , 2003, Current Biology.

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

[21]  Alexander N. Gorban,et al.  Robust simplifications of multiscale biochemical networks , 2008, BMC Systems Biology.

[22]  R. Jackson,et al.  General mass action kinetics , 1972 .

[23]  S. H. Lam,et al.  Using CSP to Understand Complex Chemical Kinetics , 1993 .

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

[25]  Ioannis G. Kevrekidis,et al.  Constraint-defined manifolds: A legacy code approach to low-dimensional computation , 2005 .

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

[27]  Rui Zhu,et al.  Stochastic kinetics description of a simple transcription model , 2006, Bulletin of mathematical biology.

[28]  J. Ferrell,et al.  Modeling the Cell Cycle: Why Do Certain Circuits Oscillate? , 2011, Cell.

[29]  S. Verblunsky,et al.  On Positive Polynomials , 1945 .

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

[31]  Extending the quasi-steady state approximation by changing variables , 1996 .

[32]  Ioannis G Kevrekidis,et al.  A constrained approach to multiscale stochastic simulation of chemically reacting systems. , 2011, The Journal of chemical physics.

[33]  D Lebiedz,et al.  Computing minimal entropy production trajectories: an approach to model reduction in chemical kinetics. , 2004, The Journal of chemical physics.

[34]  T. Vejchodský.  Accurate reduction of a model of circadian rhythms by delayed quasi-steady state assumptions , 2013, 1312.2825.