Editing Resting potential (section)

== Resting potentials ==

The resting membrane potential is not an equilibrium potential as it relies on the constant expenditure of energy (for [[ion transporter|ionic pumps]] as mentioned above) for its maintenance.  It is a dynamic diffusion potential that takes this mechanism into account&mdash;wholly unlike the pillows equilibrium potential, which is true no matter the nature of the system under consideration.  The resting membrane potential is dominated by the ionic species in the system that has the greatest [[Electrical conductance|conductance]] across the membrane.  For most cells this is potassium.  As potassium is also the ion with the most negative equilibrium potential, usually the resting potential can be no more negative than the potassium equilibrium potential.  The resting potential can be calculated with the [[Goldman equation|Goldman-Hodgkin-Katz voltage equation]] using the concentrations of ions as for the equilibrium potential while also including the relative [[semipermeable membrane|permeabilities]] of each ionic species.  Under normal conditions, it is safe to assume that only potassium, [[sodium]] (Na<sup>+</sup>) and [[chloride]] (Cl<sup>−</sup>) ions play large roles for the resting potential:

:<math>E_{m} = \frac{RT}{F} \ln{ \left( \frac{ P_{Na^+}[Na^+]_{o} + P_{K^+}[K^+]_{o} + P_{Cl^-}[Cl^-]_{i} }{ P_{Na^+}[Na^+]_{i} + P_{K^+}[K^+]_{i} + P_{Cl^-}[Cl^-]_{o} } \right) }</math>

This equation resembles the Nernst equation, but has a term for each permeant ion. Also, ''z'' has been inserted into the equation, causing the intracellular and extracellular concentrations of Cl<sup>−</sup> to be reversed relative to K<sup>+</sup> and Na<sup>+</sup>, as chloride's negative charge is handled by inverting the fraction inside the logarithmic term. *''E''<sub>m</sub> is the membrane potential, measured in volts *''R'', ''T'', and ''F'' are as above *''P''<sub>s</sub> is the relative permeability of ion s *[s]<sub>Y</sub> is the concentration of ion s in compartment Y as above. Another way to view the membrane potential, considering instead the conductance of the ion channels rather than the permeability of the membrane, is using the Millman equation (also called the Chord Conductance Equation): 
:<math>E_{m} = \frac{g_{K^+}E_{eq,K^+} + g_{Na^+}E_{eq,Na^+} + g_{Cl^-}E_{eq,Cl^-}} {g_{K^+}+g_{Na^+}+g_{Cl^-}}</math>

or reformulated

:<math>E_{m} = \frac{g_{K^+}} {g_{tot}} E_{eq,K^+} + \frac{g_{Na^+}} {g_{tot}} E_{eq,Na^+} + \frac{g_{Cl^-}} {g_{tot}} E_{eq,Cl^-}</math>

where ''g''<sub>tot</sub> is the combined conductance of all ionic species, again in arbitrary units.  The latter equation portrays the resting membrane potential as a ''[[weighted mean|weighted average]]'' of the reversal potentials of the system, where the weights are the relative conductances of each ion species (''g''<sub>X</sub>/''g''<sub>tot</sub>).  During the action potential, these weights change.  If the conductances of Na<sup>+</sup> and Cl<sup>−</sup> are zero, the membrane potential reduces to the Nernst potential for K<sup>+</sup> (as ''g''<sub>K<sup>+</sup></sub> = ''g''<sub>tot</sub>).  Normally, under resting conditions ''g''<sub>Na+</sub> and ''g''<sub>Cl−</sub> are not zero, but they are much smaller than ''g''<sub>K+</sub>, which renders ''E''<sub>m</sub> close to ''E''<sub>eq,K+</sub>.  Medical conditions such as [[hyperkalemia]] in which [[blood]] [[blood plasma|serum]] potassium (which governs [K<sup>+</sup>]<sub>o</sub>) is changed are very dangerous since they offset ''E''<sub>eq,K+</sub>, thus affecting ''E''<sub>m</sub>.  This may cause [[Heart arrhythmia|arrhythmia]]s and [[cardiac arrest]].  The use of a [[Bolus (medicine)|bolus]] injection of potassium chloride in executions by [[Lethal injection#Potassium chloride|lethal injection]] stops the heart by shifting the resting potential to a more positive value, which depolarizes and contracts the cardiac cells permanently, not allowing the heart to [[repolarization|repolarize]] and thus enter [[diastole]] to be refilled with blood.

Although the GHK voltage equation and Millman's equation are related, they are not equivalent. The critical difference is that Millman's equation assumes the current-voltage relationship to be ohmic, whereas the GHK voltage equation takes into consideration the small, instantaneous rectifications predicted by the [[GHK flux equation]] caused by the concentration gradient of ions.  Thus, a more accurate estimate of membrane potential can be calculated using the GHK equation than with Millman's equation.<ref>Hille, Bertil (2001) Ion Channels of Excitable Membranes, 3 ed.</ref>