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where V_{K} is the equilibrium electrical PD, which^{}exactly opposes the chemical energy of the chemical gradient,^{}the intracellulartoextracellular K^{+} concentration ratio ([K]_{in}/[K]_{out}).^{}R is the gas constant with units of 8.31 J/(Kmol), T is absolute^{}temperature in Kelvin (37°C = 310 K), F is Faraday’s^{}constant at 96,500 coulombs/mol, and z is the valance of the^{}ion question; +1 for K^{+}. It is instructive to insert the relevant^{}values for R, T, F, and z, and to convert from the natural log^{}to the common (base 10) log by multiplying by 2.303. The Nernst^{}equation then becomes (at 37°C)
^{}
It is convenient to simplify this equation to an adequate (and^{}useful) approximation
^{}
When we consider the K^{+} gradient of our example (100 mM inside,^{}10 mM outside) we find that this outwardly directed 10fold^{}gradient of a monovalent cation is balanced by a 60 mV electrical^{}PD (in this case, inside negative). This introduces an additional^{}valuable concept evident in another term synonymous with equilibrium^{}potential, i.e., the reversal potential. In our example, the^{}direction of net K^{+} flux was from inside to out until the equilibrium^{}PD of –60 mV was reached. If the potential were, by some^{}means, to become even more negative (say, –70 mV), then^{}the direction of net K^{+} flux would reverse, i.e., a net flux^{}from the low chemical concentration of K^{+} outside the cell into^{}the higher K^{+} concentration inside, with this "uphill" flux^{}driven by the imposed electrical force.^{}
At this point, it reasonable to ask, "If we started with 100^{}mM 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 is^{}sufficiently small that it cannot be measured chemically, despite^{}the 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 transmembrane^{}electrical potential of the type discussed here.^{}
So, in our hypothetical "cell," with the constraint that the^{}membrane is permeable only to K^{+}, the membrane potential is^{}precisely defined by the K^{+} chemical gradient. Although it was^{}emphasized above that intracellular ion concentrations generally^{}do not change as a consequence of the downhill fluxes associated^{}with transmembrane voltage changes, it is instructive to consider^{}what would happen if the K^{+} gradient were to change. In fact,^{}changes in the K^{+} gradient, typically the result of changes^{}in [K]_{out}, can be extremely important, both physiologically^{}and clinically; see Sidebar 1. So, go ahead, grab a calculator^{}and determine the new equilibrium potential that would arise^{}if [K^{+}]_{out} were suddenly increased to 20 mM or if the internal^{}[K^{+}] fell to 50 mM (be advised, these macroscopic changes in^{}K^{+} concentration would be associated with parallel changes in^{}the concentration of one or more anions). Here is the rule of^{}thumb: any manipulation that reduces the K^{+} gradient (i.e.,^{}either decreasing intracellular K^{+} or increasing extracellular^{}K^{+}), will decrease the equilibrium potential for K^{+} (i.e., a^{}voltage value closer to zero). In other words, if there is less^{}energy in the chemical gradient, it will take less energy in^{}an electrical gradient to "balance" it (go ahead, do the math...).^{}
You will not be surprised to learn that biological membranes^{}do not show "ideal" permselectivity. Real membranes have a finite^{}permeability to all the major inorganic ions in body fluids.^{}For most cells, the only ions that can exert any significant^{}influence on bioelectrical phenomena are the "big three" (in^{}terms of concentration): K^{+}, Na^{+}, and Cl^{–} (Ca^{2+} also contributes^{}to 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 realistic^{}case in which the membrane shows a finite permeability to these^{}three major players. The new equation is called the "GoldmanHodgkinKatz^{}Constant Field equation"; or, more typically, the "Goldman equation"
where V_{m} is the actual PD across the membrane, and^{}P_{i} is the membrane permeability (in cm/s) for the indicated^{}ion. Close inspection reveals that the Nernst equation is lurking^{}within the Goldman equation: if the membrane were to become^{}permeable only to K^{+}, i.e., if P_{Na} and P_{Cl} were zero, then the^{}equation simplifies to the Nernstian condition for K^{+}.^{2} Note^{}that to account for the differences in valence, the anionic^{}Cl^{–} concentrations are presented as "out over in," rather^{}than as the "in over out" convention used here for cations.^{}
It is worthwhile to consider the transmembrane ion gradients^{}and ionspecific membrane permeabilities of a "typical" neuron^{}(Table 1). The calculated Nernstian equilibrium potential for^{}K^{+}, Na^{+}, and Cl^{–} establish the "boundary conditions" for^{}the electrical PD across the cell membrane; i.e., our cell cannot^{}be more negative than –92 mV or more positive that +64^{}mV (Fig. 1) because there are no relevant chemical gradients^{}sufficiently 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 distinct^{}populations of K^{+} channels that share the general characteristic^{}of being constitutively active under normal resting conditions.^{}The relative contribution to the resting potential played by^{}these channels varies with cell type, but in neurons relevant^{}players include members of the family of inwardly rectifying^{}K^{+} channels (KIR) and the K(2P) family of K^{+} "leak" channels.^{}
Fig. 1. Graphical representation of the Nernstian equilibrium potentials (V_{i}) for Na^{+} (V_{Na}), K^{+} (V_{K}), and Cl^{–} (V_{Cl}); and the resting membrane potential (V_{m}) 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 product^{}of NaKATPase activity) and a high resting permeability to^{}K^{+} makes the interior of animal cells electrically negative^{}with respect to the external solution. However, the finite permeability^{}of the membrane to Na^{+} (and to Cl^{–}; see Sidebar 2) prevents^{}the membrane potential from ever actually reaching the Nernstian^{}K^{+} potential. The extent to which each ion gradient influences^{}the PD is defined by the permeability of the membrane to each^{}ion, as is evident from inspection of the Goldman equation.^{}Even very large concentrations exert little influence if the^{}associated P_{i} value is small. However, if the membrane were^{}suddenly to become permeable only to Na^{+}, the result would be^{}a Nernstian condition for Na^{+}, with a concomitant change in^{}membrane potential.^{}
Although under normal physiological conditions the concentration^{}terms of the Goldman equation remain relatively constant, the^{}permeability terms do not. Indeed, large, rapid changes in the^{}ratios of permeability for different ions represent the basis^{}for the control of bioelectric phenomena. On a molecular level,^{}membrane permeability to ions is defined by the activity of^{}membrane channels (the molecular basis of which, like so many^{}other things, is outside the scope of this review ). Indeed,^{}a large increase in P_{Na} (owing to activation of a population^{}of voltagegated Na^{+} channels) is the basis of the transient^{}depolarization of membrane potential that is associated with^{}the neuronal action potential.^{}
In summary, the combination of an outwardly directed K gradient^{}(the product of NaKATPase activity) and a high resting permeability^{}to K^{+} makes the interior of animal cells electrically negative^{}with respect to the external solution. Changes in either of^{}these controlling parameters, i.e., transmembrane ion gradients^{}or channelbased ion permeability, can have large, and immediate^{}consequences. Although the "stability" of ion gradients has^{}been emphasized here, in fact, changes in these gradients can^{}occur, generally with pathological consequences (see Sidebar^{}2). 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 liter^{}can have large effects on resting membrane potential (prove^{}this to yourself using the Goldman equation and the permeability^{}parameters listed in Table 1). Such changes can occur as a consequence^{}of, for example, crushing injuries that rapidly release into^{}the blood stream large absolute amounts of K^{+} (from the K^{+}rich^{}cytoplasm in cells of the damaged tissue). Alternatively, failure^{}of the NaKATPase during ischemia can result in local increases^{}in the [K^{+}]_{out}, a problem exacerbated by both the low starting^{}concentration of K^{+} and the low volume of fluid in the restricted^{}extracellular volume of "densely packed" tissues (e.g., in the^{}heart or brain). The "other side of the coin," i.e., alteration^{}in membrane permeability to ions, can arise as a consequence^{}of an extraordinary number of pathological defects in ion channel^{}proteins (or "channelopathies"). Of particular relevance to^{}the resting membrane potential are lesions in one or more subunits^{}of the KIR channels mentioned previously. Mutations in these^{}channels, and the consequent changes in resting membrane potential,^{}have been linked to persistent hyperinsulinemic hypoglycemia^{}of infancy, a disorder affecting the function of pancreatic^{}beta cells; Bartter’s syndrome, characterized by hypokalemic^{}alkalosis, hypercalciuria, increased serum aldosterone, and^{}plasma renin activity; and to several polygenic central nervous^{}system diseases, including white matter disease, epilepsy, and^{}Parkinson’s disease.^{}
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