Model of transmembrane resting potential due to concentration differences in K, Na, and Ca with constant conductances for the ions, plus accounting for the NaKPump current using an implicit calculation of EQ.10.4 from Sperelakis (1979).
GoldmanNaKPump2 provides a solution to the implicit equation of Sperelakis Eq 10.4, page 258, 1979 for the steady-state resting potential, EGoldmanPump, due to three ion-selective channel conductances and the Na Pump. The model allows changing of concentrations and temperature in order to explore the Nernst and Goldman potential relationships. Plotpage is set up using log scaling on bottom two panels. Set rateo to a negative rate such as -1 mv/ms to show that Em and Egoldman changes as Ki decreases. Model accounts for two monovalent cations, K and Na, and one divalent ion, Ca, all driven by their electrochem gradients. Membrane separates 2 regions of fixed concentrations for Na and Ca and varying amounts of K. Solute activities are unity. Currents reach S.S. instantaneously. The NaKATPase, the sodium pump, creats a net efflux of positive charges, and therefore tends to depolarize the membrane. The pump current can produce mambrane potentials more negative than EK. Oubain, a blocker of this ATPase, causes depolarization of 4 to 8 mV. The NaKATPase current changes during an action potential. To generate a pump current there must be K outside and Na inside. The pump reversal potential is highly negative, perhaps below -200 mV. Nernst potentials are independent of conductance. Given a single channel conductance about 300 pSiemens; 10^6 chan -> 300 uS. What would be the the current flow at the starting concentrations Ko=5 and Ki=150 mM? Insert suitable concentrations for Ca, e.g. 1e-4 and 2 mM, and for Na, 12 and 144 mM and determine the slopes. See Notes within this .proj file for guidance.
The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.
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Goldman David E. Potential, impedance, and rectification in membranes. J Gen Physiol 27: 37-60, 1943. Hodgkin AL and Katz B. The effect of sodium ions on the electrical activity of the giant axon of the squid. J Physiol 108: 37-77, 1949. Sperelakis N. Origin of the cardiac resting potential. Appendix: Physical principles, derivations, and applications. In: Handbook of Physiology, Sec. 2, The Cardiovascular System. Vol. 1, edited by Berne RM, Sperelakis N, and Geiger SR. Baltimore: Waverly Press, Inc., 1979, pp 187-267. 9 Hille B. Ion Channels of Excitable Membranes, Third Edition. Sunderland, Massachusetts: Sinauer Associates, 2001, 814 pp. page 19, Figure 1.6
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