Stephen H. Wright
Department of Physiology, College of Medicine, University of Arizona, Tucson, Arizona 85724
Address for reprint requests and other correspondence: S. H. Wright, Dept. of Physiology, College of Medicine, Univ. of Arizona, Tucson, AZ 85724 (E-mail: firstname.lastname@example.org )
This brief review is intended to serve as a refresher on theideas associated with teaching students the physiological basisof the resting membrane potential. The presentation is targetedtoward first-year medical students, first-year graduate students,or senior undergraduates. The emphasis is on general conceptsassociated with generation of the electrical potential differencethat exists across the plasma membrane of every animal cell.The intention is to provide students a general view of the quantitativerelationship that exists between 1) transmembrane gradientsfor K+ and Na+ and 2) the relative channel-mediated permeabilityof the membrane to these ions.
Key words: Nernst equation; Goldman equation; electrical potential difference
Source: ADV PHYSIOL EDUC 28:139-142, 2004.
IT WOULD BE DIFFICULT TO EXAGGERATE the physiological significanceof the transmembrane electrical potential difference (PD). Thisgradient of electrical energy that exists across the plasmamembrane of every cell in the body influences the transportof a vast array of nutrients into and out of cells, is a keydriving force in the movement of salt (and therefore water)across cell membranes and between organ-based compartments,is an essential element in the signaling processes associatedwith coordinated movements of cells and organisms, and is ultimatelythe basis of all cognitive processes. For those reasons (andmany more), it is critical that all students of physiology havea clear understanding of the basis of the resting membrane potential(so called to distinguish the steady-state electrical conditionof all cells from the electrical transients that are the “actionpotentials” of excitable cells: i.e., neurons and muscle cells).1
How, then, does this electrical gradient arise? It is the consequenceof the influence of two physiological parameters: 1) the presenceof large gradients for K+ and Na+ across the plasma membrane;and 2) the relative permeability of the membrane to those ions.The gradients for K+ and Na+ are the product of the activityof the Na+-K+-ATPase, a primary active ion pump that is ubiquitouslyexpressed in the plasma membrane of (for all intents and purposes)all animal cells. This process develops and then maintains thelarge outwardly directed K+ gradient, and the large inwardlydirected Na+ gradient, that are hallmarks of animal cells. Forthe purpose of this discussion, we will assume that the requisitegradients are in place (acknowledging that the mechanism ofion transport is beyond the scope of this presentation).
The second parameter, the relative permeability of the plasmamembrane to Na+ and K+, reflects the open versus closed statusof ion-selective membrane channels. Importantly, cell membranesdisplay different degrees of permeability to different ions(i.e., “permselectivity”), owing to the inherent selectivityof specific ion channels. The combination of 1) transmembraneion gradients, and 2) differential permeability to selectedions, is the basis for generation of transmembrane voltage differences.This idea can be developed by considering the hypothetical situationof two solutions separated by a membrane permselective to asingle ionic species. Side 1 (the “inside” of our hypotheticalcell) contains 100 mM KCl and 10 mM NaCl. Side 2 (the “outside”)contains 100 mM NaCl and 10 mM KCl. In other words, there isan “outwardly directed” K+ gradient, and inwardly directed Na+gradient, and no transmembrane gradient for Cl–. For thepurpose of this discussion, this ideal permselective membraneis permeable only to K+.
Let the experiment begin! The outwardly directed K+ gradientsupports a net diffusive flux from inside to outside. As K+diffuses from the cell it leaves behind “residual” negativecharge, in this case, its counter anion, Cl– (recall that,in our hypothetical scenario, neither Cl– nor Na+ cancross the membrane). The resulting local imbalance of negativecharge at the inside face of the membrane represents a forcethat can draw mobile positive charge at the external face ofthe membrane back into the cell; in this case, the only mobileion capable of crossing the membrane is K+. Significantly, thiselectrical force can draw K+ from the low chemical concentrationfound outside the cell into the higher [K+] inside the cell.As long as the chemical force of the outwardly directed K+ chemicalgradient is larger than the oppositely oriented electrical force,there will be a net efflux of K+ from the cell. However, asmore and more K+ leaves, the residual negative charge withinthe cell, i.e., the attractive electrical force that draws K+into the cell, gradually increases to a level that exactly balancesthe chemical force, resulting in an “equilibrium” condition.Whereas there may continue to be large unidirectional fluxesinto and out of the cell, the net flux under this equilibriumcondition is zero. The equation that describes this balanceof electrical and chemical forces is called the Nernst equation:
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.