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Numerical Matrices Method for Nonlinear System Identification and Description of Dynamics of Biochemical Reaction Networks

Alexey V. Karnaukhov ^*, Elena V. Karnaukhova ^* and James R. Williamson 

^* Institute of Cell Biophysics Russian Academy of Sciences, Pushchino, Russia; and Departments of Molecular Biology and Chemistry, The Skaggs Institute for Chemical Biology, The Scripps Research Institute, La Jolla, California USA

Correspondence: Address reprint requests to James R. Williamson, E-mail: jrwill{at}scripps.edu.

A flexible Numerical Matrices Method (NMM) for nonlinear system identification has been developed based on a description of the dynamics of the system in terms of kinetic complexes. A set of related methods are presented that include increasing amounts of prior information about the reaction network structure, resulting in increased accuracy of the reconstructed rate constants. The NMM is based on an analytical least squares solution for a set of linear equations to determine the rate parameters. In the absence of prior information, all possible unimolecular and bimolecular reactions among the species in the system are considered, and the elements of a general kinetic matrix are determined. Inclusion of prior information is facilitated by formulation of the kinetic matrix in terms of a stoichiometry matrix or a more general set of representation matrices. A method for determination of the stoichiometry matrix beginning only with time-dependent concentration data is presented. In addition, we demonstrate that singularities that arise from linear dependencies among the species can be avoided by inclusion of data collected from a number of different initial states. The NMM provides a flexible set of tools for analysis of complex kinetic data, in particular for analysis of chemical and biochemical reaction networks.