title: Application of the one- and two-dimensional Ising models to studies of cooperativity between ion channels.
abstract: The Ising model of statistical physics provides a framework for studying systems of protomers in which nearest neighbors interact with each other. In this article, the Ising model is applied to the study of cooperative phenomena between ligand-gated ion channels. Expressions for the mean open channel probability, rho o, and the variance, sigma 2, are derived from the grand partition function. In the one-dimensional Ising model, interactions between neighboring open channels give rise to a sigmoidal rho o versus concentration curve and a nonquadratic relationship between sigma 2 and rho o. Positive cooperativity increases the slope at the midpoint of the rho o versus concentration curve, shifts the apparent binding affinity to lower concentrations, and increases the variance for a given rho o. Negative cooperativity has the opposite effects. Strong negative cooperativity results in a bimodal sigma 2 versus rho o curve. The slope of the rho o versus concentration curve increases linearly with the number of binding sites on a protomer, but the sigma 2 versus rho o relationship is independent of the number of ligand binding sites. Thus, the sigma 2 versus rho o curve provides unambiguous information about channel interactions. In the two-dimensional Ising model, rho o and sigma 2 are calculated numerically from a series expansion of the grand partition function appropriate for weak interactions. Virtually all of the features exhibited by the one-dimensional model are qualitatively present in the two-dimensional model. These models are also applicable to voltage-gated ion channels.
authors.affiliation: Department of Anesthesiology, State University of New York, Stony Brook 11794-8480.
authors.forename: Y
authors.initials: Y
authors.lastname: Liu
authors.forename: J P
authors.initials: JP
authors.lastname: Dilger
mesh_terms: D000818:Animals; D001665:Binding Sites; D055592:Biophysical Phenomena; D001703:Biophysics; D004563:Electrochemistry; D015640:Ion Channel Gating / Q000502:physiology; D007473:Ion Channels / Q000737:chemistry*; D008024:Ligands; D008956:Models, Chemical
chemical_list: D007473:Ion Channels; D008024:Ligands