Differential Algebra and System Modeling in Cellular Biology

Among all the modeling approaches dedicated to cellular biology, differential algebra is particularly related to the well-established one based on nonlinear differential equations. In this paper, it is shown that differential algebra makes one of the model reduction methods both simple and algorithmic: the quasi-steady state approximation theory, in the particular setting of generalized chemical reactions systems. This recent breakthrough may suggest some evolution of modeling techniques based on nonlinear differential equations, by incorporating the reduction hypotheses in the models. Potential improvements of parameters fitting methods are discussed too.

[1]  John J. Tyson,et al.  Modeling Molecular Interaction Networks with Nonlinear Ordinary Differential Equations , 2006 .

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

[3]  Hidde de Jong,et al.  Genetic Network Analyzer: qualitative simulation of genetic regulatory networks , 2003, Bioinform..

[4]  G. Odell,et al.  The segment polarity network is a robust developmental module , 2000, Nature.

[5]  Vipul Periwal,et al.  Qualitative Approaches to the Analysis of Genetic Regulatory Networks , 2006 .

[6]  A. Goldbeter,et al.  Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila. , 2000, BioEssays : news and reviews in molecular, cellular and developmental biology.

[7]  Céline Noiret.  Utilisation du calcul formel pour l'identifiabilité de modèles paramètriques et nouveaux algorithmes en estimation de paramètres , 2000 .

[8]  P. Daoutidis,et al.  Nonlinear model reduction of chemical reaction systems , 2001 .

[9]  Alexandre Sedoglavic.  A probabilistic algorithm to test local algebraic observability in polynomial time , 2001, ISSAC '01.

[10]  Eric Walter,et al.  Identifiability of State Space Models , 1982 .

[11]  M. Fliess,et al.  Nonlinear observability, identifiability, and persistent trajectories , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[12]  Y. Saka,et al.  A mechanism for the sharp transition of morphogen gradient interpretation in Xenopus , 2007, BMC Developmental Biology.

[13]  Dongming Wang.  Elimination Practice - Software Tools and Applications , 2004 .

[14]  F. Boulier,et al.  A computer scientist point of view on Hilbert's differential theorem of zeros , 2007 .

[15]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[16]  Hassan K. Khalil,et al.  Singular perturbation methods in control : analysis and design , 1986 .

[17]  Marc Moreno Maza,et al.  Computing differential characteristic sets by change of ordering , 2010, J. Symb. Comput..

[18]  Ghislaine Joly-Blanchard,et al.  System Identifiability (Symbolic Computation) and Parameter Estimation (Numerical Computation) , 2003, Numerical Algorithms.

[19]  François Lemaire,et al.  On Proving the Absence of Oscillations in Models of Genetic Circuits , 2007, AB.

[20]  Hidde de Jong,et al.  Modeling and Simulation of Genetic Regulatory Systems: A Literature Review , 2002, J. Comput. Biol..

[21]  François Lemaire,et al.  Applying a Rigorous Quasi-Steady State Approximation Method for Proving the Absence of Oscillations in Models of Genetic Circuits , 2008, AB.

[22]  François Boulier,et al.  Representation for the radical of a finitely generated differential ideal , 1995, ISSAC '95.

[23]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[24]  Lennart Ljung,et al.  On global identifiability for arbitrary model parametrizations , 1994, Autom..

[25]  Wei Niu,et al.  Algebraic Approaches to Stability Analysis of Biological Systems , 2008, Math. Comput. Sci..

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

[27]  G. Bastin,et al.  Reduced order dynamical modelling of reaction systems: A singular perturbation approach , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[28]  François Boulier,et al.  Differential Elimination and Biological Modelling , 2006 .

[29]  F. Ollivier.  Le probleme de l'identifiabilite structurelle globale : approche theorique, methodes effectives et bornes de complexite , 1990 .

[30]  M. Bennett,et al.  Transient dynamics of genetic regulatory networks. , 2007, Biophysical journal.