A strategy for determining the equilibrium constants for heteromeric ion channels in a complex model

Identifiability of equilibrium constants for receptors with two to five binding sites

Ligand-gated ion channels (LGICs) are regularly oligomers containing between two and five binding sites for ligands. Neither in homomeric nor heteromeric LGICs the activation process evoked by the ligand binding is fully understood. Here, we show on theoretical grounds that for LGICs with two to five binding sites, the cooperativity upon channel activation can be determined in considerable detail. The main requirements for our strategy are a defined number of binding sites in a channel, which can be achieved by concatenation, a systematic mutation of all binding sites and a global fit of all concentration-activation relationships (CARs) with corresponding intimately coupled Markovian state models. We take advantage of translating these state models to cubes with dimensions 2, 3, 4, and 5. We show that the maximum possible number of CARs for these LGICs specify all 7, 13, 23, and 41 independent model parameters, respectively, which directly provide all equilibrium constants within the respective schemes. Moreover, a fit that uses stochastically varied scaled unitary start vectors enables the determination of all parameters, without any bias imposed by specific start vectors. A comparison of the outcome of the analyses for the models with 2 to 5 binding sites showed that the identifiability of the parameters is best for a case with 5 binding sites and 41 parameters. Our strategy can be used to analyze experimental data of other LGICs and may be applicable to voltage-gated ion channels and metabotropic receptors.

Modeling the kinetics of heteromeric potassium channels

Mechanistic mathematical modeling has long been used as a tool for answering questions in cellular physiology. To mathematically describe cellular processes such as cell excitability, volume regulation, neurotransmitter release, and hormone secretion requires accurate descriptions of ion channel kinetics. One class of ion channels currently lacking a physiological model framework is the class of channels built with multiple different potassium protein subunits called heteromeric voltage gated potassium channels. Here we present a novel mathematical model for heteromeric potassium channels that captures both the number and type of protein subunits present in each channel. Key model assumptions are validated by showing our model is the reduction of a Markov model and through observations about voltage clamp data. We then show our model's success in replicating kinetic properties of concatemeric channels with different numbers of Kv1.1 and Kv1.2 subunits. Finally, through comparisons with multiple expression experiments across multiple voltage gated potassium families, we use the model to make predictions about the importance and effect of genetic mutations in heteromeric channel formation.