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This brief review is intended to serve as a refresher on the …
where VK is the equilibrium electrical PD, whichexactly opposes the chemical energy of the chemical gradient,the intracellular-to-extracellular K+ concentration ratio ([K]in/[K]out).R is the gas constant with units of 8.31 J/(Kmol), T is absolutetemperature in Kelvin (37°C = 310 K), F is Faraday’sconstant at 96,500 coulombs/mol, and z is the valance of theion question; +1 for K+. It is instructive to insert the relevantvalues for R, T, F, and z, and to convert from the natural logto the common (base 10) log by multiplying by 2.303. The Nernstequation then becomes (at 37°C)
It is convenient to simplify this equation to an adequate (anduseful) approximation
When we consider the K+ gradient of our example (100 mM inside,10 mM outside) we find that this outwardly directed 10-foldgradient of a monovalent cation is balanced by a 60 mV electricalPD (in this case, inside negative). This introduces an additionalvaluable concept evident in another term synonymous with equilibriumpotential, i.e., the reversal potential. In our example, thedirection of net K+ flux was from inside to out until the equilibriumPD of –60 mV was reached. If the potential were, by somemeans, to become even more negative (say, –70 mV), thenthe direction of net K+ flux would reverse, i.e., a net fluxfrom the low chemical concentration of K+ outside the cell intothe higher K+ concentration inside, with this "uphill" fluxdriven by the imposed electrical force.
At this point, it reasonable to ask, "If we started with 100mM K+ inside and there was a net efflux of K+ from the cell,shouldn’t the intracellular K+ concentration now be lower?"This is an important issue. As it turns out, the amount of K+that leaves the cell to produce the equilibrium potential issufficiently small that it cannot be measured chemically, despitethe substantial electrical effect it has (see Sidebar 1). Therefore,to a first approximation you can assume that [K]in (and Na+)remain effectively constant during the shifts in transmembraneelectrical potential of the type discussed here.
So, in our hypothetical "cell," with the constraint that themembrane is permeable only to K+, the membrane potential isprecisely defined by the K+ chemical gradient. Although it wasemphasized above that intracellular ion concentrations generallydo not change as a consequence of the downhill fluxes associatedwith transmembrane voltage changes, it is instructive to considerwhat would happen if the K+ gradient were to change. In fact,changes in the K+ gradient, typically the result of changesin [K]out, can be extremely important, both physiologicallyand clinically; see Sidebar 1. So, go ahead, grab a calculatorand determine the new equilibrium potential that would ariseif [K+]out were suddenly increased to 20 mM or if the internal[K+] fell to 50 mM (be advised, these macroscopic changes inK+ concentration would be associated with parallel changes inthe concentration of one or more anions). Here is the rule ofthumb: any manipulation that reduces the K+ gradient (i.e.,either decreasing intracellular K+ or increasing extracellularK+), will decrease the equilibrium potential for K+ (i.e., avoltage value closer to zero). In other words, if there is lessenergy in the chemical gradient, it will take less energy inan electrical gradient to "balance" it (go ahead, do the math...).
You will not be surprised to learn that biological membranesdo not show "ideal" permselectivity. Real membranes have a finitepermeability to all the major inorganic ions in body fluids.For most cells, the only ions that can exert any significantinfluence on bioelectrical phenomena are the "big three" (interms of concentration): K+, Na+, and Cl– (Ca2+ also contributesto bioelectric issues in a few tissues, including the heart).The Nernst equation, which represents an idealized situation,can be modified to represent the more physiologically realisticcase in which the membrane shows a finite permeability to thesethree major players. The new equation is called the "Goldman-Hodgkin-KatzConstant Field equation"; or, more typically, the "Goldman equation"
where Vm is the actual PD across the membrane, andPi is the membrane permeability (in cm/s) for the indicatedion. Close inspection reveals that the Nernst equation is lurkingwithin the Goldman equation: if the membrane were to becomepermeable only to K+, i.e., if PNa and PCl were zero, then theequation simplifies to the Nernstian condition for K+.2 Notethat to account for the differences in valence, the anionicCl– concentrations are presented as "out over in," ratherthan as the "in over out" convention used here for cations.
It is worthwhile to consider the transmembrane ion gradientsand ion-specific membrane permeabilities of a "typical" neuron(Table 1). The calculated Nernstian equilibrium potential forK+, Na+, and Cl– establish the "boundary conditions" forthe electrical PD across the cell membrane; i.e., our cell cannotbe more negative than –92 mV or more positive that +64mV (Fig. 1) because there are no relevant chemical gradientssufficiently large to produce larger PDs. At rest, importantly,the membrane permeability of most cells, including neurons,is greatest for K,+ due to the activity of several distinctpopulations of K+ channels that share the general characteristicof being constitutively active under normal resting conditions.The relative contribution to the resting potential played bythese channels varies with cell type, but in neurons relevantplayers include members of the family of inwardly rectifyingK+ channels (KIR) and the K(2P) family of K+ "leak" channels.
Fig. 1. Graphical representation of the Nernstian equilibrium potentials (Vi) for Na+ (VNa), K+ (VK), and Cl– (VCl); and the resting membrane potential (Vm) calculated by using the Goldman equation. The relevant ion concentrations and permeabilities are listed in Table 1.
The combination of an outwardly directed K+ gradient (the productof Na-K-ATPase activity) and a high resting permeability toK+ makes the interior of animal cells electrically negativewith respect to the external solution. However, the finite permeabilityof the membrane to Na+ (and to Cl–; see Sidebar 2) preventsthe membrane potential from ever actually reaching the NernstianK+ potential. The extent to which each ion gradient influencesthe PD is defined by the permeability of the membrane to eachion, as is evident from inspection of the Goldman equation.Even very large concentrations exert little influence if theassociated Pi value is small. However, if the membrane weresuddenly to become permeable only to Na+, the result would bea Nernstian condition for Na+, with a concomitant change inmembrane potential.
Although under normal physiological conditions the concentrationterms of the Goldman equation remain relatively constant, thepermeability terms do not. Indeed, large, rapid changes in theratios of permeability for different ions represent the basisfor the control of bioelectric phenomena. On a molecular level,membrane permeability to ions is defined by the activity ofmembrane channels (the molecular basis of which, like so manyother things, is outside the scope of this review ). Indeed,a large increase in PNa (owing to activation of a populationof voltage-gated Na+ channels) is the basis of the transientdepolarization of membrane potential that is associated withthe neuronal action potential.
In summary, the combination of an outwardly directed K gradient(the product of Na-K-ATPase activity) and a high resting permeabilityto K+ makes the interior of animal cells electrically negativewith respect to the external solution. Changes in either ofthese controlling parameters, i.e., transmembrane ion gradientsor channel-based ion permeability, can have large, and immediateconsequences. Although the "stability" of ion gradients hasbeen emphasized here, in fact, changes in these gradients canoccur, generally with pathological consequences (see Sidebar2). The [K+]out is particularly susceptible to such changes.Because in absolute terms it is comparatively "small" (i.e.,4 mM), increases in [K+]out of only a few millimoles per litercan have large effects on resting membrane potential (provethis to yourself using the Goldman equation and the permeabilityparameters listed in Table 1). Such changes can occur as a consequenceof, for example, crushing injuries that rapidly release intothe blood stream large absolute amounts of K+ (from the K+-richcytoplasm in cells of the damaged tissue). Alternatively, failureof the Na-K-ATPase during ischemia can result in local increasesin the [K+]out, a problem exacerbated by both the low startingconcentration of K+ and the low volume of fluid in the restrictedextracellular volume of "densely packed" tissues (e.g., in theheart or brain). The "other side of the coin," i.e., alterationin membrane permeability to ions, can arise as a consequenceof an extraordinary number of pathological defects in ion channelproteins (or "channelopathies"). Of particular relevance tothe resting membrane potential are lesions in one or more subunitsof the KIR channels mentioned previously. Mutations in thesechannels, and the consequent changes in resting membrane potential,have been linked to persistent hyperinsulinemic hypoglycemiaof infancy, a disorder affecting the function of pancreaticbeta cells; Bartter’s syndrome, characterized by hypokalemicalkalosis, hypercalciuria, increased serum aldosterone, andplasma renin activity; and to several polygenic central nervoussystem diseases, including white matter disease, epilepsy, andParkinson’s disease.
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