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In Real Cells, Multiple Ions Contribute to the Membrane Potential
In real cells, the resting potential is seldom at the equilibrium potential for any one ion. For a typical mammalian neuron, the membrane potential is generally around −70 mV (not at the equilibrium potential for any ion). This is because in most cells at rest, there are both K+ and Na+ selective channels in the plasma membrane (and sometimes there are channels for other ions as well). In fact, it turns out that in many cells, K+, Na+, and Cl make the largest contribution to the resting membrane potential. Therefore, the contribution of all three ions (Na+, K+, and Cl) to Vm must be taken into account. When more than one ion channel is present in the membrane, the membrane potential can be calculated by using the Goldman-Hodgkin-Katz equation (GHK equation): Eq. 1
• Vm is the membrane potential. This equation is used to determine the resting membrane potential in real cells, in which K+, Na+, and Cl- are the major contributors to the membrane potential. Note that the unit of Vm is the Volt. However, the membrane potential is typically reported in millivolts (mV). If the channels for a given ion (Na+, K+, or Cl-) are closed, then the corresponding relative permeability values can be set to zero. For example, if all Na+ channels are closed, pNa = 0. Our online calculator can be used to perform calculations involving the GHK equation.
• R is the universal gas constant and is equal to 8.314 J.K−1.mol−1 (Joules per Kelvin per mole).
• T is the temperature in Kelvin (K = °C + 273.15).
• F is the Faraday's constant and is equal to 96485 C.mol−1 (Coulombs per mole).
• pK is the relative membrane permeability for K+. Normally, permeability values are reported as relative permeabilities with pK having the reference value of one (because in most cells at rest pK is larger than pNa and pCl). For a typical neuron at rest, pK : pNa : pCl = 1 : 0.05 : 0.45. Note that because relative permeability values are reported, permeability values are unitless.
• pNa is the relative membrane permeability for Na+.
• pCl is the relative membrane permeability for Cl−1.
• [K+]o is the concentration of K+ in the extracellular fluid. Note that the concentration units for all the ions must match.
• [K+]i is the concentration of K+ in the intracellular fluid. Note that the concentration units for all the ions must match.
• [Na+]o is the concentration of Na+ in the extracellular fluid. Note that the concentration units for all the ions must match.
• [Na+]i is the concentration of Na+ in the intracellular fluid. Note that the concentration units for all the ions must match.
• [Cl−1]o is the concentration of Cl−1 in the extracellular fluid. Note that the concentration units for all the ions must match.
• [Cl−1]i is the concentration of Cl−1 in the intracellular fluid. Note that the concentration units for all the ions must match.

Our online Goldman-Hodgkin-Katz (GHK) calculator can be used to calculate the membrane potential (Vm) under a variety of conditions. The Goldman-Hodgkin-Katz equation can be derived based on simple thermodynamic principles in the same manner that the Nernst equation can be derived based on simple thermodynamic principles (see Derivation of the Nernst Equation). Notice that whereas the extracellular concentrations of K+ and Na+ appear in the numerator, the intracellular concentration of Cl is in the numerator. Moreover, whereas the intracellular concentrations of K+ and Na+ are in the denominator, the extracellular concentration of Cl is in the denominator.
In the equation above, pK, pNa, and pCl are the relative membrane permeabilities for K+, Na+, and Cl, respectively. Permeability refers to the ease with which ions cross the membrane, and is directly proportional to the total number of open channels for a given ion in the membrane. Therefore, if many K+ channels are open, pK will be high. If only a few K+ channels are open, pK will be small. If all K+ channels are closed, pK will be zero.
As can be seen from the equation above, two major factors contribute to the membrane potential: concentration gradients and the relative permeabilities of the ions. Under most physiological conditions, the intracellular and extracellular concentrations of ions remain the same. Similarly, temperature remains the same in most cases. Therefore, the relative permeability values become the most important factors that contribute to the value of the membrane potential. The magnitude of the permeability (i.e., how many open channels) for each ion determines the relative contribution of that ion to Vm. The larger the permeability of a certain ion, the larger the contribution of that ion will be in setting the membrane potential. Notice that if the channels for a certain ion are all closed (i.e., the permeability for that ion is zero), the above expression is reduced and simplified to include only the terms involving the other two ions. It is important to see that the movement of each ion down its own electrochemical gradient will tend to move the membrane potential towards the equilibrium potential for that ion.
Normally, permeabilities are reported as relative permeabilities with pK having the reference value of one (because in most cells at rest pK is larger than pNa and pCl). For a typical neuron at rest, pK : pNa : pCl = 1 : 0.05 : 0.45. Given these values and typical mammalian intracellular and extracellular concentrations (see here), you should be able to calculate the membrane potential for a neuron (or any other cell) at rest. Using the GHK equation, and the ionic concentrations and relative permeabilities given above, it can be calculated that Vm ≈ −68 mV in a typical neuron. In most cells, the membrane potential (Vm) is relatively stable with little or no significant deviation from the resting value. Therefore, in most cells, Vm = Vrest. This is because in these cells, the relative ionic permeability values do not change appreciably over time. In excitable cells (such as neurons, muscle cells and some endocrine cells), on the other hand, there are large transient changes in the relative permeability values for ions and, therefore, the membrane potential transiently deviates from the "normal" resting membrane potential.
Figure 1 demonstrates how changes in the relative permeability values for K+, Na+, and Cl influence the value of the membrane potential. Changes in the relative permeability of ions can come about as a result of channel opening or closing caused by physiological stimuli (e.g., opening of voltage-gated channels) or by the application of agonistic or antagonistic drugs. If the values of pNa and pCl are set to zero (i.e., if all Na+ and Cl channels are closed), it can be see that K+ will be the only contributor to the membrane potential. In this case, the value of the membrane potential will be exactly the same as the K+ equilibrium potential (Vm = VK) (2 in Fig. 1). On the other hand, if the values of pK and pCl are set to zero (i.e., if all K+ and Cl channels are closed), it can be see that Na+ will be the only contributor to the membrane potential. In this case, the value of the membrane potential will be exactly the same as the Na+ equilibrium potential (Vm = VNa) (3 in Fig. 1). Similarly, if all K+ and Na+ channels are closed (i.e., pK = 0 and pNa = 0), then Vm = VCl (4 in Fig. 1). Figure 1. Changes in the membrane potential caused by changes in the relative permeability of ions.
This figure demonstrates a graphical representation of the membrane potential as calculated by the Goldman-Hodgkin-Katz (GHK) equation. The figure shows how the membrane potentials changes as the relative permeability values are changed for potassium, sodium, and chloride. See text for additional details.
We should also consider more realistic situation where pK, pNa, and pCl have non-zero values. For example, it can also be seen that if pK is much larger than pNa and pCl, then Vm will be closer to VK than it will be to VNa or VCl because K+ is now the major contributor to the membrane potential (5 in Fig. 1). Thus, if pK is much larger than pNa and pCl, then Vm will be closer to VK, although it will never be exactly at VK (unless pNa = 0 and pCl = 0). Similarly, if pNa is much larger than pK and pCl, then Vm will be closer to VNa (6 in Fig. 1). Finally, if pCl is much larger than pK and pNa, then Vm will be closer to VCl (7 in Fig. 1).
In summary, based on the information presented here and the equilibrium potential values for common ions found in physiological solutions (see here), it can be seen that under physiological ionic concentrations, the equilibrium potential for K+ (VK) sets the negative limit for the membrane potential (∼ &minus90 mV), whereas the equilibrium potential for Ca2+ (VCa) sets the positive limit for the membrane potential (∼ +137 mV). As discussed above, for these extreme values to be reached, K+ or Ca2+ has to be the only ion contributing to the membrane potential. Thus, under the physiological conditions of mammalian cells, membrane potentials outside of this range are not possible. Of course, experimentally, the membrane potential may be changed to almost any value (within limits of about ±200 mV where the membrane begins to break down) by changing the ionic concentrations or by injecting positive or negative current into the cell (see lecture on the Neuronal Action Potential).
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Posted: Saturday, February 15, 2014