Clustering reveals limits of parameter identifiability in multi-parameter models of biochemical dynamics

Abstract

Background: Compared to engineering or physics problems, dynamical models in quantitative biology typically depend on a relatively large number of parameters. Progress in developing mathematics to manipulate such multi-parameter models and so enable their efficient interplay with experiments has been slow. Existing solutions are significantly limited by model size.

Results: In order to simplify analysis of multi-parameter models a method for clustering of model parameters is proposed. It is based on a derived statistically meaningful measure of similarity between groups of parameters. The measure quantifies to what extend changes in values of some parameters can be compensated by changes in values of other parameters. The proposed methodology provides a natural mathematical language to precisely communicate and visualise effects resulting from compensatory changes in values of parameters. As a results, a relevant insight into identifiability analysis and experimental planning can be obtained. Analysis of NF-κB and MAPK pathway models shows that highly compensative parameters constitute clusters consistent with the network topology. The method applied to examine an exceptionally rich set of published experiments on the NF-κB dynamics reveals that the experiments jointly ensure identifiability of only 60% of model parameters. The method indicates which further experiments should be performed in order to increase the number of identifiable parameters.

Conclusions: We currently lack methods that simplify broadly understood analysis of multi-parameter models. The introduced tools depict mutually compensative effects between parameters to provide insight regarding role of individual parameters, identifiability and experimental design. The method can also find applications in related methodological areas of model simplification and parameters estimation.

Fig. 1

Canonical correlations and identifiability. a Illustrative view of the sensitivity vectors S _i. b Conceptual illustration of the canonical correlations. Two subsets of sensitivity vectors represented as linear subspaces (planes Ω _A and Ω _B). Canonical vectors on the planes are found to yield maximum cosine. In a two-dimensional subspace case, the second canonical vectors u ₂,v ₂ are required to be perpendicular to the first ones. c The introduced δ-condition requires that each parameter θ _i is correlated less than δ with the remaining parameters θ _−i=(θ ₁,..,θ _i−1,θ _i+1,…,θ _l). It can be interpreted in terms of how variance of the estimates changes when a single parameter and all model parameters are estimated. Parameter θ ₀ denotes the linear combination of θ _−i maximally correlated with θ _i, i.e. θ ₀=lin^∗{θ _−i}. d Mutual information as a measure of similarity between two parameter sets θ _A,θ _B, which span linear subspaces Ω _A,Ω _B interpreted in terms of the asymptotic posterior $P\left(\widehat{\theta }\right|\theta )$

Fig. 2

a Agglomerative hierarchical clustering of model parameters. b Verification of the δ-condition. Recursively, at each level, a pair of most similar clusters is merged into a single cluster and δ-condition is verified. Linkages between clusters, at each stage of clustering, are plotted at high $\frac{1}{m}{\sum }_{i=1}^m(1-{\rho }^2)$, where m is the size of a new cluster, compared to a previous linkage. Identifiability results from violation of either of the δ-condition or ζ-condition therefore even parameters that have sensitivities above a threshold can be non-identifiable. Non-identifiable parameters are marked red

Fig. 3

The NF κB signalling pathway. a Schematic representation of the reactions involved. b Clustering of the model parameters reveals that parameters which characterise modules close in the network topology are generally more correlated than those far apart. Sensitivity vectors were calculated for a specific TNF- α stimulation to reflect physiological conditions (increase, plateau, decrease): see Figure 6B in Additional file 1

Fig. 4

Identifiability study of the NF- κB system. a Clustering results together with the identifiability analysis computed based on all the major published experiments. Non-identifiable parameters are marked in red. We used δ=0.95,ζ=1 to verify the identifiability condition. Sensitivity coefficients, i.e diagonal elements of the FIM, are shown below the dendrogram. b Clustering results as in (a) but for the published experiments together with the suggested experimental protocols