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Goldman-Hodgkin-Katz Equation Calculator

In living cells, the resting membrane potential (Vm) is seldom governed by only one ion such as K+, Na+, Cl-, etc. If this were the case, the membrane potential could be predicted by the equilibrium potential (VEq.) for that ion, and could be easily calculated by using the Nernst equation. Instead, the membrane potential is generally established as a result of the relative contributions of several ions. In many cells, K+, Na+, and Cl- are the main contributors to the membrane potential. For example, in a typical mammalian neuron, K+, Na+, and Cl- contribute to a resting membrane potential of around -70 mV, a membrane potential value that is not at the equilibrium potential for K+, Na+, or Cl-. This is because in neurons at rest, there are K+-, Na+- and Cl--selective channels in the plasma membrane. These selective ion channels allow K+, Na+, and Cl- to each move down its own electrochemical gradient. The movement of any ion down its own electrochemical gradient will tend to move the membrane potential toward the equilibrium potential for that ion. Therefore, the transmembrane movements of all three ions (K+, Na+, and Cl-) collectively contribute to the membrane potential. When more than one ion channel is present (and open) in the plasma membrane, the membrane potential can be calculated by using the Goldman-Hodgkin-Katz equation (GHK equation). Moreover, the GHK equation can predict the reversal potential (Vrev) of the current-voltage (I-V) relationship obtained from a cell in which the predominant ion channels in the plasma membrane are K+, Na+, and Cl- channels.

In examining the GHK equation (see below), it is clear that the relative contribution of any given ion is determined not only by its concentration gradient across the plasma membrane, but also by its relative membrane permeability. 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 or if no K+ channels exist in the membrane, pK will be zero. 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. In contrast, approximate relative permeability values at the peak of a typical neuronal action potential are pK : pNa : pCl = 1 : 12 : 0.45.

When two or more ions contribute to the membrane potential, it is likely that the membrane potential would not be at the equilibrium potential for any of the contributing ions. Thus, no ion would be at its equilibrium (i.e., Veq. ≠ Vm). When an ion is not at its equilibrium, an electrochemical driving force (VDF) acts on the ion, causing the net movement of the ion across the membrane down its electrochemical gradient. The driving force is quantified by the difference between the membrane potential and the ion equilibrium potential (VDF = Vm − Veq.). The sign (i.e., positive or negative) of the driving force acting on an ion along with the knowledge of the valence of the ion (i.e., cation or anion) can be used to predict the direction of ion flow across the plasma membrane (i.e., into or out of the cell). For example, for cations (positively charged ions such as Na+, K+, H+, and Ca2+), a positive driving force (i.e., VDF > 0) predicts ion movement out of the cell (efflux) down its electrochemical gradient, and a negative driving force (i.e., VDF < 0) predicts ion movement into the cell (influx). The situation is reversed for anions (negatively charged ions such as Cl− and HCO3−), where a positive driving force predicts ion movement into the cell (influx), and a negative driving force predicts ion movement out of the cell (efflux). If the membrane potential (Vm) is exactly at the equilibrium potential (Veq.) for an ion, the driving force acting on the ion would be zero. If Vm = Veq., it can be seen that VDF = Vm − Veq. = 0. In this case, there would be no net movement of the ion across the plasma membrane into or out of the cell (i.e., no net flux of ion). See the Electrochemical Driving Force Calculator, and the lecture notes on the Resting Membrane Potential for additional details.

The Goldman-Hodgkin-Katz equation

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.

R is the universal gas constant (8.314 J.K-1.mol-1).

T is the temperature in Kelvin (K = °C + 273.15).

F is the Faraday's constant (96485 C.mol-1).

pK is the 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-.

[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-]o is the concentration of Cl- in the extracellular fluid. Note that the concentration units for all the ions must match.

[Cl-]i is the concentration of Cl- in the intracellular fluid. Note that the concentration units for all the ions must match.

Each calculator cell shown below corresponds to a term in the formula presented above. Enter appropriate values in all cells except the one you wish to calculate. Therefore, at least ten cells must have values, and no more than one cell may be blank. The value of the blank cell will be calculated based on the other values entered. After a calculation is performed, the calculated cell will be highlighted and subsequent calculations will calculate the value of the highlighted cell (with no requirement to have a blank cell). However, a blank cell has priority over a highlighted cell.

Please note that the unit of temperature used in the Goldman-Hodgkin-Katz equation is the Kelvin. It is also important to note that although this worksheet allows you to select different concentration units, during the calculation, the numerator and denominator concentration units for K+, Na+, and Cl- are converted so that they match. Moreover, the calculator ensures that there is consistency in the concentration units used for K+, Na+, and Cl-. Also note that based on the constants used (R = 8.314 J.K-1.mol-1 and F = 96485 C.mol-1), the unit of Vm will be in Volts. Keeping this fact in mind, this tool simplifies the calculation by allowing you to calculate directly to or from mV.

The calculated equilibrium potentials for K+ (VK), Na+ (VNa), and Cl- (VCl), as well as the calculated electrochemical driving forces acting on K+ (VDF, K), Na+ (VDF, Na), and Cl- (VDF, Cl), are read-only values. The sign of the electrochemical driving force (VDF = Vm − Veq.) acting on any given ion allows us to determine the direction of ion flow (i.e., into or out of the cell). This information is also provided after every calculation.

As mentioned above and as can be seen from the GHK equation shown above, the value of the membrane potential is determined by the concentration gradients and the relative permeability values of ions for which there are open channels in the plasma membrane. The physiological concentration gradients are homeostatically maintained within a very narrow range. The magnitude of the permeability (i.e., how many open channels in the plasma membrane) for any given ion can, in fact, be regulated physiologically, and determines the relative contribution of that ion to Vm. It is important to remember that the movement of any ion down its own electrochemical gradient will tend to move the membrane potential toward the equilibrium potential for that ion. The larger the permeability of a given ion, the larger the contribution of that ion will be in setting the membrane potential. For example, by examining the GHK equation, it can be seen that if pK is much larger than pNa and pCl, Vm will be closer to the equilibrium potential for K+ (VK) than it will be to the equilibrium potential for Na+ (VNa) or Cl- (VCl), although it will never be exactly at VK unless both pNa = 0 and pCl = 0. As another example, if pNa is very large compared to pK and pCl, Vm will be closer to VNa, although it will never be exactly at VNa unless both pK = 0 and pCl = 0. Notice that if the channels for a certain ion are all closed (i.e., the permeability for that ion is zero), the GHK equation is reduced and simplified to include only the terms regarding the other two ions.