[Paper Review] Observability and Structural Identifiability of Nonlinear Biological Systems

Alejandro F. Villaverde|arXiv (Cornell University)|Dec 11, 2018

Gene Regulatory Network Analysis62 references4 citations

TL;DR

This paper presents a comprehensive tutorial on observability and structural identifiability in nonlinear biological systems using differential geometry, unifying both concepts under a single mathematical framework. It demonstrates that structural identifiability is a special case of observability when parameters are treated as constant states, and highlights key challenges and open problems in applying these concepts to complex biological models.

ABSTRACT

Observability is a modelling property that describes the possibility of inferring the internal state of a system from observations of its output. A related property, structural identifiability, refers to the theoretical possibility of determining the parameter values from the output. In fact, structural identifiability becomes a particular case of observability if the parameters are considered as constant state variables. It is possible to simultaneously analyse the observability and structural identifiability of a model using the conceptual tools of differential geometry. Many complex biological processes can be described by systems of nonlinear ordinary differential equations, and can therefore be analysed with this approach. The purpose of this review article is threefold: (I) to serve as a tutorial on observability and structural identifiability of nonlinear systems, using the differential geometry approach for their analysis; (II) to review recent advances in the field; and (III) to identify open problems and suggest new avenues for research in this area.

Motivation & Objective

• To serve as a tutorial on observability and structural identifiability in nonlinear systems using differential geometry.
• To review recent advances in the analysis of observability and structural identifiability in biological models.
• To identify open problems and suggest new research directions in the field, particularly regarding computational scalability and input design.
• To clarify the distinction between structural and practical identifiability, especially in light of concepts like sloppiness and dynamical compensation.
• To address the limitations of current methods and advocate for hybrid approaches combining differential geometry with global methods like differential algebra.

Proposed method

• Uses differential geometry to analyze observability and structural identifiability, treating parameters as constant state variables.
• Applies the observability rank condition and the tangent linearization method to assess whether internal states or parameters can be inferred from outputs.
• Reviews symbolic methods such as power series and differential algebra as alternatives or complements to differential geometry.
• Considers the role of input signals—constant, time-varying, or piecewise constant—in enabling structural identifiability.
• Proposes that hybridization of differential geometry with global methods (e.g., differential algebra) could extend local results to global identifiability.
• Leverages high-performance computing and parallelization to reduce computational burden in large-scale biological models.

Experimental results

Research questions

• RQ1 How can observability and structural identifiability be systematically analyzed in nonlinear biological systems using differential geometry?
• RQ2 What are the limitations of local analysis methods like differential geometry, and how might they be extended to provide global results?
• RQ3 In what ways do time-varying or piecewise constant inputs improve structural identifiability compared to single constant input experiments?
• RQ4 How do concepts like sloppiness and dynamical compensation relate to structural identifiability, and are they equivalent to unidentifiability?
• RQ5 What is the relationship between observability/identifiability and the predictive accuracy of biological models?

Key findings

• Structural identifiability is a special case of observability when parameters are treated as constant state variables, unifying both concepts under a single framework.
• The differential geometry approach provides a powerful but inherently local analysis method, which may miss global identifiability properties that other methods like differential algebra can capture.
• Sloppiness is linked to practical identifiability, not structural identifiability, and does not imply unidentifiability—sloppy models can still be structurally identifiable.
• Dynamical compensation (DC) is equivalent to structural unidentifiability under the original definition, but a refined definition clarifies its role in preserving dynamic behavior despite parameter changes.
• Time-varying or multiple constant input experiments can render structurally unidentifiable models identifiable, but the conditions under which these input types are equivalent remain unresolved.
• Lack of observability or identifiability can compromise a model’s ability to yield biological insight, even if it fits experimental data, highlighting the need for early analysis in model development.