Quantitative techniques for steady-state calculation and dynamic integrated modelling of membrane potential and intracellular ion concentrations

The membrane potential (E_m) is a fundamental cellular parameter that is primarily determined by the transmembrane permeabilities and concentration gradients of various ions. However, ion gradients are themselves profoundly influenced by E_m due to its influence upon transmembrane ion fluxes and cell volume (V_c). These interrelationships between E_m, V_c and intracellular ion concentrations make computational modelling useful or necessary in order to guide experimentation and to achieve an integrated understanding of experimental data, particularly in complex, dynamic, multi-compartment systems such as skeletal and cardiac myocytes. A variety of quantitative techniques exist that may assist such understanding, from classical approaches such as the Goldman–Hodgkin–Katz equation and the Gibbs–Donnan equilibrium, to more recent “current-summing” models as exemplified by cardiac myocyte models including those of DiFrancesco & Noble, Luo & Rudy and Puglisi & Bers, or the “charge-difference” modelling technique of Fraser & Huang so far applied to skeletal muscle. In general, the classical approaches provide useful and important insights into the relationships between E_m, V_c and intracellular ion concentrations at steady state, providing their core assumptions are fully understood, while the more recent techniques permit the modelling of changing values of E_m, V_c and intracellular ion concentrations. The present work therefore reviews the various approaches that may be used to calculate E_m, V_c and intracellular ion concentrations with the aim of establishing the requirements for an integrated model that can both simulate dynamic systems and recapitulate the key findings of classical techniques regarding the cellular steady state. At a time when the number of cellular models is increasing at an unprecedented rate, it is hoped that this article will provide a useful and critical analysis of the mathematical techniques fundamental to each of them.