Sensitivity Analysis and Robust Experimental Design of a Signal Transduction Pathway System

Experimental design for cellular networks based on sensitivity analysis is studied in this work. Both optimal and robust experimental design strategies are developed for the IκBNF-κB signal transduction model. Based on local sensitivity analysis, the initial IKK intensity is calculated using an optimal experimental design process, and several scalarization measures of the Fisher information matrix are compared. Global sensitivity analysis and robust experimental design techniques are then developed to consider parametric uncertainties in the model. The modified Morris method is employed in global sensitivity analysis, and a semidefinite programming method is exploited to implement the robust experimental design for the problem of measurement set selection. The parametric impacts on the oscillatory behavior of NF-κB in the nucleus are also discussed. C © 2008 Wiley Periodicals, Inc. Int J Chem Kinet 40: 730–741, 2008 Correspondence to: Hong Yue; e-mail: [email protected]. ac.uk. Contract grant sponsor: National Natural Science Foundation of China. Contract grant number: 30770560. Contract grant sponsor: UK Biotechnology and Biological Sciences Research Council. Contract grant number: BB/C007158/1. Contract grant sponsor: Hong Kong Research Grants Council (to Fei He). Contract grant numbers: CityU SRG 7001821 and CityU 122305. c © 2008 Wiley Periodicals, Inc. INTRODUCTION Sensitivity analysis is used to understand how a model’s output depends on variations in parameter values or initial conditions, and is perhaps best known in metabolic systems biology via metabolic control analysis [1–4]. It is particularly useful for complex biological networks that involve a large number of variables and parameters in which it is crucial to identify either the most important or the least relevant parameters. ANALYSIS AND DESIGN OF SIGNAL TRANSDUCTION PATHWAY SYSTEM 731 Based on the nominal parameter values, local sensitivity analysis (LSA) measures the effects that small changes in the parameters have on the output. It is widely used in modeling and analysis of biological systems, in which the nominal parameter values are estimated using experimental data or computation [4–7]. For continuous dynamic systems, the local sensitivities are defined as the first-order partial derivatives of the system output with respect to the input parameters. Such information reveals the gradient of a mathematical model’s output in parameter space at a given set of parameter values, and therefore plays a central role in many system identification problems. LSA has a wide spectrum of applications in systems biology. However, for a complex and/or uncertain model in which some parameter estimates are most likely far from the true values, or for a significantly nonlinear and interactive system, it is more relevant to study global sensitivities. Global sensitivity analysis (GSA) examines the effects of simultaneous “arbitrary” variations of multiple parameters on the dependent variables under conditions in which the variations are not local [8–10]. There are different ways to perform GSA, such as screening techniques, variancebased methods, Monte Carlo filtering approaches and regression methods, and so on. In principle, GSA is valid in a bounded region around the nominal value for each parameter, and the effect of each parameter is either aggregated [11] or some worst case measures are taken for evaluation. It is not simply the result from weighted local sensitivities, but a multidimensional averaging over the whole parameter space, since when one parameter is evaluated over its interval, all the other parameters are also varying instead of keeping their nominal values. It, therefore, reveals interactions between parameters from simultaneous parameter variations. GSA has been applied in modeling, analysis, and experimental design for a range of biological systems [12–16]. Optimal experimental design (OED) is one of the techniques developed from local sensitivities, whose purpose is to devise the necessary dynamic experiments in such a way that the parameters are estimated from the resulting experimental data with the best possible statistical quality. The tasks of experimental design include input signal design, sampling rate optimization, measurement set selection, and so on. Under the assumption of uncorrelated measurement noise with zero-mean Gaussian distribution, the information content of measurements can be quantified by the Fisher information matrix (FIM) [17,18]. In general, the smaller the joint confidence interval is for the estimated parameters, the more information is contained in the measurements. In many recent works on modeling of biochemical networks, the FIM was used to design the experiments to optimize the quality of parameter estimation in a certain statistical sense [18–23]. Several strategies for solving OED problems in the context of parameter estimation for biochemical models are discussed in [24]. The quality of optimal experimental design is dependent on the accuracy of the mathematical models. The true model parameters are in most cases rarely known, and nominal or estimated parameter values are used instead. These nominal parameters may be obtained from preliminary experiments, the literature, or from previous parameter estimation. When the quality of nominal parameters is poor, the experimental design results may be overoptimistic or even misleading. In inverse modeling of complex biochemical networks, the normal way to surmount this problem is to go through an iterative/sequential process for parameter estimation and experimental design. In each iteration, the OED is implemented to provide “rich” information for a better parameter estimation in the subsequent iteration [19,25]. Using this approach, the costs associated with experiments for several iterations are nontrivial, especially for the expensive and timeconsuming data collection in cellular experiments. An alternative method is the minimax experimental design, which makes the OED at the worst case in a region around the nominal parameter values. However, for a nonlinear model with a large amount of parameters, identifying the worst case for optimization of the FIM can be computationally challenging and impractical. In this work, the problem of experimental design based on local sensitivities is addressed for biochemical systems with particular interest in models with parametric uncertainties. An optimal design on the input activation intensity is studied first using the FIM, so as to illustrate the principle of optimal experimental design. The endeavors are then focused on the robust experimental design and global sensitivity analysis, so as to take into account model uncertainties. With the results given by local and global sensitivity analyses, the influence of some important parameters on the oscillatory behavior of NF-κB in the nucleus is investigated. LOCAL SENSITIVITIES AND OPTIMAL EXPERIMENTAL DESIGN System Model: An Example of 1κB-NF-κB Signal Pathway For a biochemical model with n reaction species and m parameters, denoteX = [x1 x2 · · · xn] as the state vector, θ = [k1 k2 · · · km] as the vector of International Journal of Chemical Kinetics DOI 10.1002/kin