محدوده تجزیه و تحلیل حساسیت مبتنی بر واریانس برای تست غیرقابل شناسایی در مدل پویا با ابعاد بالا

In systems biology, a common approach to model biological processes is to use large systems of nonlinear differential equations. The associated parameter estimation problem then requires a prior handling of the global identifiability question in a realistic experimental framework. The lack of a method able to solve this issue has indirectly encouraged the use of global sensitivity analysis to select the subset of parameters to estimate. Nevertheless, the links between these two global analyses are not yet fully explored. The present work reveals new bridges between sensitivity analyses and global non-identifiability, through the use of functions derived from the Sobol’ high dimensional representation of the model output. We particularly specify limits of variance-based sensitivity tools to completely conclude on global non-identifiability of parameters in a given experimental context.

In systems biology, the inference of biological networks from quantitative properties of their elementary constituents is a major area of research (Cassman et al., 2007 and Kitano, 2001). This raises particular challenges such as the identification of high-dimensional nonlinear dynamical systems and more precisely the analysis of their parameter identifiability (Mirsky et al., 2009, Raue et al., 2009 and Stelling and Gilles, 2001). Different classifications of parameter identifiability definitions exist. We refer herein to three classes (see Fig. 1): a priori identifiability, a posteriori identifiability and practical identifiability. The first class, known also as the theoretical or structural identifiability of model parameters, examines the question of existence and uniqueness of a solution to the parameter estimation problem ( Walter & Pronzato, 1997), in an idealized framework where (1) the system and the model have an identical structure (no characterization error); (2) the data are noise-free, and (3) the input signals and measurement times can be chosen at will. However, this is only a necessary condition which cannot guarantee successful parameter estimation from real data. The second class, namely the a posteriori identifiability only considers the first two working assumptions and is a particular case of the output distinguishability ( Grewal & Glover, 1976) for a finite collection of noise-free observations and a given input signal. The last class, practical identifiability, only relies on the first hypothesis and accounts for the noise factor but is generally established for a given estimation criterion ( Dochain et al., 1995 and Vanrolleghem et al., 1995). For that reason, this class of identifiability is often linked to the theory of optimization in mathematics. Full-size image (31 K) Fig. 1. Classification of identifiability definitions, where MM denotes the model structure, View the MathML sourceu the input signals, tktk the time measurements and vkvk the output noise. Figure options Sensitivity analysis of the model output with respect to changes in model parameters is another technique widely used in system modeling to discriminate influential and non influential parameters (Saltelli et al., 2008 and Streif et al., 2009). Dynamic sensitivity analysis has already been applied to biological networks for various purposes such as experimental design (Schlosser, 1994), parameter estimation (Miller & Frenklach, 2004) or the analysis of oscillatory systems (Rand, 2008 and Zak et al., 2005). Several investigations on the connections between dynamic sensitivity and parameter identifiability analyses have been carried out (Brun et al., 2002, Stelling and Gilles, 2001 and Yue et al., 2006), but the latter were only focused on local analysis. In genomics, proteomics or metabolomics, biological parameters may vary widely within different ranges. As a consequence, global a posteriori identifiability needs to be addressed. Unfortunately, there is no technique able to assess the global identifiability condition in a given experimental context, i.e. when the input signal and the sampling conditions are imposed by the experimental context. Consequently, authors generally prefer to apply global sensitivity analysis techniques without solid justifications related to identifiability. Indeed, while the relationship between local sensitivity and identifiability analysis, through the Fisher information matrix, is clearly established ( Brun et al., 2001, Yao et al., 2003 and Zak et al., 2003), the link between global studies is less obvious. As a matter of fact, only insensitive parameters are generally considered as being non-identifiable. This is not surprising since global sensitivity measures usually serve as model reduction principles (before parameter estimation) or in tandem with uncertainty analysis for model robustness analysis ( Saltelli et al., 2008). However, sensitive parameters could also be non-identifiable. The objective of this paper is to present new results on the connections between global a posteriori identifiability and global dynamic sensitivity analysis. This study is structured around the Sobol’ decomposition 2 method ( Sobol’, 2001). Specific functions, entitled ΨΨ and ΩΩ-functions, derived from the Sobol’ high dimensional model representation, are introduced. Their linear time-dependence and injectivity are examined, and their consequences on the non-identifiability of parameters are discussed. We show that variance-based sensitivity analysis can be used to test only one out of three causes of non-identifiability in a given experimental context, where the input signal and the measurement times are imposed. We also point out that the conclusions on parameter non-identifiability, in the case of colinear sensitivity measures, must be treated with caution. This paper is structured as follows: a priori and a posteriori identifiabilities are first defined. The global sensitivity analysis based on the Sobol’ high dimensional model representation is then briefly introduced in Section 3. Finally, the main contributions of this study are presented in Section 4 in which the links between sensitivity and global identifiability analyses are decomposed, in a theoretical framework. For simplicity reasons, the nomenclature used throughout this paper is detailed in Table 1. Table 1. Nomenclature. Symbol Definition tktk The kkth time instant, with k=0,…,N−1k=0,…,N−1 View the MathML sourcep∈P⊂Rn The parameters of the model View the MathML sourcep∼i The vector composed by all the parameters except pipi View the MathML sourceu∈Rnu Input signals View the MathML sourcex∈Rnx State variables View the MathML sourcey∈Rny Output variables E[⋅]E[⋅] The expectation operator V[⋅]V[⋅] The variance operator Vi1,…,ir(tk)Vi1,…,ir(tk) The rrth-order variance of the output yy with respect to collective effect of the parameters pi1,…,pirpi1,…,pir Si1,…,ir(tk)Si1,…,ir(tk) The rrth-order sensitivity of the output yy with respect to collective effect of the parameters pi1,…,pirpi1,…,pir View the MathML sourceIn=[0,1]n nn-dimensional unit hypercube View the MathML sourceΨi(tk,p) The total effect on the model output yy of the parameter pipi View the MathML sourceΩi(tk,p∼j) Represents the influence on the output yy of the parameter pipi independently of pjpj

The lack of available methods to test the global a posteriori identifiability of parameters in high dimensional dynamic models probably explains the success of global sensitivity techniques, particularly in systems biology and more precisely in the identification of metabolic pathways. Nevertheless, the links between these two analyses are not yet fully explored. This present work provides new insights into the relationships between them. We show that the lack of identifiability may be due to three causes: insensitivity, colinearity, or the non-injectivity of the functions involved in the global sensitivity analysis. While the first is in truth a general acknowledged association, between insensitive parameters and non-identifiable parameters, the other two are less straightforward. Indeed, these two points correspond to two limits which prevent a sure conclusion on global non-identifiability. The first limit is due to the impossibility of testing the non-injectivity from the variance functions. The second shows that, conversely to what is admitted in local identifiability, colinear analysis of sensitivity functions is no longer valid. Thus, the conclusions about parameter non-identifiability drawn from the analysis of global sensitivity functions must be carefully analyzed. This study also brings out the central role of some functions, entitled ΩΩ and ΨΨ functions, in both sensitivity and identifiability analyses. These functions are derived from the Sobol’ high dimensional representation of the model output. They could be regarded as a promising perspective to solve the global identifiability issue in practice. If the explicit expression of the output variable is known, those functions can be determined by computer algebra. Otherwise, their determination in a given experimental framework is another challenge in perspective.