Structural identifiability of equilibrium ligand-binding parameters

Understanding the interactions of proteins with their ligands requires knowledge of molecular properties, such as binding site affinities and the effects that binding at one site exerts on binding at other sites (cooperativity). These properties cannot be measured directly and are usually estimated by fitting binding data with models that contain these quantities as parameters. In this study, we present a general method for answering the critical question of whether these parameters are identifiable (i.e., whether their estimates are accurate and unique). In cases in which parameter estimates are not unique, our analysis provides insight into the fundamental causes of nonidentifiability. This approach can thus serve as a guide for the proper design and analysis of protein-ligand binding experiments. We show that the equilibrium total binding relation can be reduced to a conserved mathematical form for all models composed solely of bimolecular association reactions and to a related, conserved form for all models composed of arbitrary combinations of binding and conformational equilibria. This canonical mathematical structure implies a universal parameterization of the binding relation that is consistent with virtually any physically reasonable binding model, for proteins with any number of binding sites. Matrix algebraic methods are used to prove that these universal parameter sets are structurally identifiable (SI; i.e., identifiable under conditions of noiseless data). A general approach for assessing and understanding the factors governing practical identifiability (i.e., the identifiability under conditions of real, noisy data) of these SI parameter sets is presented in the companion paper by Middendorf and Aldrich (2017. J. Gen. Physiol. https://doi.org/10.1085/jgp.201611703).