Identifiability analysis of empirical models used for quantifying “biased” ligands

Background: Model identifiability is an important attribute that a model must satisfy in order to derive the meaningful interpretation of estimated parameters.  In general, there are two types of identifiability: structural and deterministic.  Structural identifiability is rooted in the underlying mathematical structure of model.  Deterministic identifiability is concerned with the study design and its execution.

Recently, interest has arisen in the ability of ligands to differentially regulate multiple signalling pathways when coupled to a single receptor, termed functional selectivity.  Analysis of these data has been based on the concept of biased ligands using a metric, logR, to compare differences from a reference over different pathways (i.e. to determine “bias” of a ligand to a pathway). This metric is based on an application of the operational model proposed by Black and Leff [1].  An understanding of which parameters can be estimated from the data and which should be fixed is not clearly evident in the literature.

Aim: To formally assess the identifiability of this application of the operational model.

Methods & Results: Structural identifiability of the operational model was fully explored using the criterion developed previously [2].  It was demonstrated that the original operational model was not structurally identifiable.  The parameters R(total receptor density) and K(equilibrium transduction constant) are reduced into a single quantity τ (termed the transducer ratio, Rt/K) to eliminate this issue.  The result also showed that with the increment of τ value, especially for full agonists, the operational model was not deterministically identifiable under normal study designs.  This explains the reason why in most cases KA is assigned to 1 for full agonist in current practice.  Only the ratio between τ and KA (equilibrium dissociation constant) could be precisely estimated.  Therefore, a new quantity designated  R (transduction coefficient, τ/KA) was introduced.  Furthermore, since all the estimated parameters were conditional upon Em (system maximal response) and Em is rarely known for sure, a “normalized” transduction coefficient, namely logR, was constructed to cancel out the possible estimation bias caused by any misspecification of Em, though it was originally intended to eliminate the system bias.  Finally, relative activity ratios of each ligand across different pathways (i.e., ∆∆logR) could be determined as the metric for functional selectivity.

Discussion: The results from identifiability analysis provided additional pieces of rationale for the usage of current functional selectivity metrics from operational model.  The insights from this study may be valuable for the future work on the quantification of functional selectivity.


  1. 1. Black J W et al. (1983). Proceedings of the Royal Society of London B: Biological Sciences, 220(1219), 141-162.
  2. 2. Shivva V et al. (2013). CPT: pharmacometrics & systems pharmacology, 2(6), 1-9.