Structural identifiability of systems biology models: a critical comparison of methods

Abstract

Analysing the properties of a biological system through in silico experimentation requires a satisfactory mathematical representation of the system including accurate values of the model parameters. Fortunately, modern experimental techniques allow obtaining time-series data of appropriate quality which may then be used to estimate unknown parameters. However, in many cases, a subset of those parameters may not be uniquely estimated, independently of the experimental data available or the numerical techniques used for estimation. This lack of identifiability is related to the structure of the model, i.e. the system dynamics plus the observation function. Despite the interest in knowing a priori whether there is any chance of uniquely estimating all model unknown parameters, the structural identifiability analysis for general non-linear dynamic models is still an open question. There is no method amenable to every model, thus at some point we have to face the selection of one of the possibilities. This work presents a critical comparison of the currently available techniques. To this end, we perform the structural identifiability analysis of a collection of biological models. The results reveal that the generating series approach, in combination with identifiability tableaus, offers the most advantageous compromise among range of applicability, computational complexity and information provided.

Figure 1. Goodwin oscillator: Identifiability tableaus.

(a) Identifiability tableau obtained by means of the power series methods for the case of full observation, (b) Identifiability tableau obtained by means of the power series methods for the case of pure polynomial form and full observation. and regard the different generating series coefficients, H is used for zero order coefficients whereas V correspond to the successive Lie derivatives of along , for example, . A black square in the coordinates indicates that the corresponding non-zero generating series coefficient depends on the parameter .

Figure 2. Pharmacokinetics model .

Identifiability tableau obtained by means of the Taylor/generating series method

Figure 3. Glycolysis metabolic pathway: Identifiability tableaus.

(a) Identifiability tableau obtained by means of the Taylor series method (, regards the component of the order coefficients of the Taylor series, (b) Identifiability tableau obtained by means of the generating series method.

Figure 4. High dimensional nonlinear model: Identifiability tableaus.

(a) Identifiability tableau obtained by means of the Taylor series method, (b) Identifiability tableau obtained by means of the generating series method.

Figure 5. Arabidopsis Thaliana model: Reduced identifiability tableaus.

Reduced identifiability tableau obtained by means of the (a) Taylor series and (b) generating series methods applied to the polynomial form of the model.

Figure 6. Arabidopsis Thaliana model: Full identifiability tableau.

Identifiability tableau obtained by means of the generating series method applied to the polynomial form of the model. Despite the large number of terms included in the tableau some parameters are not appearing. The analysis may be complemented with global sensitivity analysis.