Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks

Xiong M (2008) Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks. PLoS ONE 3(11): e3758. doi:10.1371/journal.pone.0003758 Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks Xiaodian Sun 0 Li Jin 0 Momiao Xiong 0 Gustavo Stolovitzky, IBM Thomas J. Watson Research Center, United States of America 0 1 Laboratory of Theoretical Systems Biology and Center for Evolutionary Biology, School of Life Science and Institute for Biomedical Sciences, Fudan University , Shanghai , China , 2 CAS-MPG Partner Institute of Computational Biology, SIBS, CAS , Shanghai, China, 3 Human Genetics Center , University of Texas Health Science Center at Houston , Houston, Texas , United States of America It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks. - Funding: M. M. Xiong is supported by Grant from National Institutes of Health NIAMS P01 AR052915-01A1, NIAMS P50 AR054144-01 CORT, HL74735, and ES09912 and Shanghai Commission of Science and Technology Grant (04dz14003). X. D. Sun and L. Jin are supported by Grant from Shanghai Commission of Science and Technology (04dz14003). Competing Interests: The authors have declared that no competing interests exist. Cells are complex interconnected web of dynamic systems. They involve metabolites, genes and proteins which are organized into different biochemical reaction networks: metabolic, signal transduction and gene regulation networks, and protein interaction networks which form complex biological systems [1].These biochemical reaction networks control cell proliferation, differentiation, and survival [2]. To unravel the rules that govern behavior of biological systems is the focus of molecular biology researches. To gain a deep understanding about the biological systems requires modeling of biochemical reaction networks. Simple empirical description of biochemical reaction networks is insufficient for discovery of the general principles underlying biological process and prediction of dynamic response of biological networks to drug interventions or environmental perturbation [3]. The inherent properties of complex biochemical reaction networks are hard to elucidate by intuition [4]. Mathematical and computational modeling of biochemical reaction networks can comprehensively integrate experimental knowledge into forming and testing hypotheses and help to gain into system level understanding of biochemical networks, which will not been seen if the components of biochemical networks are separately studied. Therefore, developing mathematical models of biological systems holds a key to understanding and predicting the dynamic behaviors of the biological systems under perturbation of external stimuli and hence a major task of systems biology and is the keystones of systems biology [5]. Two basic types of approaches: bottom-up approach and topdown approach have been widely used in mathematical modeling of biochemical reaction networks [6]. Bottom-up approach usually assumes the mechanistic kinetic models. A full understanding of biochemical reaction networks requires quantitative information about the structure of the networks, kinetic laws and the concentrations of metabolites, enzymes and proteins [7]. The kinetic models allow us to test hypotheses, investigate the fundamental design principles of cell functions, and predict the dynamic changes of concentration of metabolites and proteins [8]. The kinetic models explicitly incorporate prior knowledge about biochemical mechanism underlying biological processes into the model and hence can serve as the basis for studying the effects of direct intervention for improving desired properties of biological systems. Top-down approach assumes black-boxes models about the molecular organization of biochemical networks and quantifies the input and output relations in biochemical networks. The kinetic models are undoubtedly a major tool for investigation of biochemical networks [9]. A great challenge in kinetic modeling of biochemical networks is to identify the structure of the networks and estimate kinetic parameters in the model [10]. Since most kinetic models of biochemical networks are nonlinear it is extremely difficult to identify the structure of the networks by computational methods. They are often determined by experiments. We are mainly concerned with estimation of kinetic parameters in this report. It is increasingly recognized that it is dynamics of the systems that determines the function of cells, tissues and organisms. Successful modeling which can unravel inherent dynamic properties of biochemical networks requires time-course quantitative measurements of metabolites, enzymes and proteins, although these measurements are still difficult to obtain [11]. A general framework for parameter estimation is to estimate the parameters in the mathematical model of the biochemical network, given time-course experimental data [12]. Parameter estimation in nonlinear dynamic systems is extremely important, but also extremely difficult. Most current methods for parameter estimation, in principle, are to formulate the parameter estimation problem as a nonlinear optimization problem with differentialalgebraic constraints that describe dynamics of biochemical networks. The objective function of the optimization is the discrepancy between model prediction, which are obtained from simulations using assumed model with estimated parameters, and the experimental data [13].Various deterministic and stochastic optimization methods have been used to solve the formulated nonlinear dynamic optimization problems [1418]. However, nonlinear dynamic optimizat (...truncated)