A theory for the membrane potential of living cells

Abstract

We give an explicit formula for the membrane potential of cells in terms of the intracellular and extracellular ionic concentrations, and derive equations for the ionic currents that flow through channels, exchangers and electrogenic pumps. We demonstrate that the work done by the pumps equals the change in potential energy of the cell, plus the energy lost in downhill ionic fluxes through the channels and exchangers. The theory is illustrated in a simple model of spontaneously active cells in the cardiac pacemaker. The model predicts the experimentally observed intracellular ionic concentration of potassium, calcium and sodium. Likewise, the shapes of the simulated action potential and five membrane currents are in good agreement with experiment. We do not see any drift in the values of the concentrations in a long time simulation, and we obtain the same asymptotic values when starting from the full equilibrium situation with equal intracellular and extracellular ionic concentrations.

Computational biology in the study of cardiac ion channels and cell electrophysiology

“…In the LRd model, drift is only observed when ions (usually K + ) that carry the stimulus current are not accounted for. If the stimulus is properly implemented and the ions that it carries are included in computing concentrations, no drift is observed even at fast pacing for long intervals When dynamic intracellular ion concentrations are accounted for, V m can also be computed directly from the concentrations by integrating the differential equation for voltage (see Varghese & Sell, 1997;Endresen et al 2000;Dokos & Lovell, 2001;Hund et al 2001 for details) to formulate the 'algebraic' method: 3. Ion-channel-based formulation of the action potential…”

“…The primary pump is the Na + /K + ATPase (NaK) that converts energy produced by the metabolic system into potential energy (in the form of transmembrane ion concentration gradients) that is used to generate the AP. (Faber & Rudy, 2000).Recently, there have been reports that the values of intracellular ion concentrations in secondgeneration AP models that account for dynamic concentration changes do not reach a steady state when paced over a long period of time, but drift until their values leave the physiological range (Guan et al 1997;Yehia et al 1999;Endresen et al 2000;Krogh-Madsen et al 2005). In the LRd model, drift is only observed when ions (usually K + ) that carry the stimulus current are not accounted for.…”

A Differential Model of Controlled Cardiac Pacemaker Cell

“…is much larger than that for N a + (Endresen et al, 2000). This implies that a significant background influx of Ca 2+ is possible during diastole, and that this current might be responsible for pacemaking activity in sinoatrial node cells (Boyett et al, 2001).…”

“…The same model structure has been used in (Djabella and Sorine, 2005) to represent excitation-contraction coupling in a ventricular cell. It consists of two parts: 1) a cell membrane with capacitance, voltage-dependent ion channels, electrogenic pump and exchanger, the ionic currents model being derived using conservation laws as in (Endresen et al, 2000), and 2) a lumped compartmental model that accounts for intracellular changes in concentrations of N a Hund et al, 2001) and the main processes that regulate intracellular calcium concentration: release and uptake by the sarcoplasmic reticulum (SR), buffering in the SR (Tusscher et al, 2004) and in the bulk cytosol (Shannon et al, 2004).…”

A Reduced Differential Model of the Electrical Activity of Cardiac Purkinje Fibres

“…It has only eight state variables. Three variables are used to describe the membrane (membrane potential and two gate variables of ionic channels), taking into consideration the dynamics of the main ionic currents (inward sodium, Ltype calcium and outward potassium), Na + /Ca 2+ exchangers and Na + /K + pumps, derived using physical principles and conservation laws [6]. The remaining five variables are associated with the fluid compartment model accounting for intracellular changes in concentrations of Na + , K + [7] and the main processes that regulate intracellular calcium concentrations: release and uptake by the sarcoplasmic reticulum (SR), buffering in the SR [8] and in the bulk cytosol [9].…”