Structural Properties of Dynamic Systems Biology Models: Identifiability, Reachability, and Initial Conditions

Abstract

Dynamic modelling is a powerful tool for studying biological networks. Reachability (controllability), observability, and structural identifiability are classical system-theoretic properties of dynamical models. A model is structurally identifiable if the values of its parameters can in principle be determined from observations of its outputs. If model parameters are considered as constant state variables, structural identifiability can be studied as a generalization of observability. Thus, it is possible to assess the identifiability of a nonlinear model by checking the rank of its augmented observability matrix. When such rank test is performed symbolically, the result is of general validity for almost all numerical values of the variables. However, for special cases, such as specific values of the initial conditions, the result of such test can be misleading-that is, a structurally unidentifiable model may be classified as identifiable. An augmented observability rank test that specializes the symbolic states to particular numerical values can give hints of the existence of this problem. Sometimes it is possible to find such problematic values analytically, or via optimization. This manuscript proposes procedures for performing these tasks and discusses the relation between loss of identifiability and loss of reachability, using several case studies of biochemical networks.

Input-Dependent Structural Identifiability of Nonlinear Systems

“…We seek to determine, a priori, what mathematical form the inputs must have in order to guarantee structural identifiability, that is, the question (a) in the above list. In regard to the other points, question (b) has been studied in [1] and [2], and question (c) in [8], [21], and [26]. As an example of the problem we want to address, imagine a model for which a constant input is not sufficiently exciting, and requires a ramp input to make its output different for different parameters.…”

“…To a certain extent, this situation resembles the relationship between structural identifiability and initial conditions (item 'c'). Generally, methods that analyse structural identifiability yield results that are valid for almost all initial conditions; however, a model classified as structurally identifiable may lose structural identifiability when started from particular initial conditions [8], [21], [26]. Likewise, for the case of inputs, structural identifiability analysis methods can determine whether a model is structurally identifiable provided that sufficiently exciting inputs are applied -but it is not straightforward to characterize the necessary inputs with the existing implementations of these methods [4], [15].…”

Special Issue on “Biological Networks”

“…The lack of structural identifiability implies that any attempt to determine model parameters from measurement data is futile. The paper by Villaverde and Banga [1] focuses on assessing structural identifiability by using the concept of observability. More specifically, for certain initial conditions, structurally unidentifiable models can mistakenly be ascertained to be identifiable.…”

A Discontinuous Unscented Kalman Filter for Non-Smooth Dynamic Problems

“…The examples reveal a consistently superior performance of the D-modification further highlighting the effects of the special observability properties of non-smooth problems. The proposed alternative opens up the way for robust online tracking and control of a variety of engineered systems including rocking Smyth, 2012a,b, 2013;Greenbaum et al, 2015), energy (Alavi et al, 2015), and biological systems (Villaverde et al, 2016;Villaverde and Banga, 2017).…”