- The Goldman-Hodgkin-Katz Equation for computing resting membrane potentials due the exchange of K+, Na+, and Cl- consists of two known constants, R and F, and 12 experimental terms. For these ions, n is always set to 1.
- Given all but one of the experimental terms, this tool solves for the missing one.
- To use the tool, one of its form fields must be left empty.
- To empty a field, just double click it.
- If no field is left empty, the tool will randomly empty one and recompute its value based on the values of the other fields.
- If a membrane is permeable to only one ion, its resting potential can be computed with Nernst Equation. However, when more than one ion can cross a membrane, their relative permeabilities play an important role and thus the Goldman-Hodgkin-Katz Equation (aka GHK or Goldman equation) applies (Physiologyweb.com, 2005a, 2005b, 2005c, 2005d).
The resting membrane potential, Em, is a product of all of the open "leak" channels. Em varies from cell to cell since each cell has a different complement of ion channels (Fitzakerley, 2014). The contribution of each channel is expressed in terms of ion permeabilities; i.e. how easy is for a given ion to cross a membrane under the influence of an electrochemical gradient.
Since for neuron cells there are more K+ leak channel than other type of channel open at rest, permeabilities are reported relative to this ion. This is why its relative permeability is usually set to PK+ = 1. The following are typical permeability ratios used for neuron cells at rest:
PK+ : PNa+ : PCl- = 1.00 : 0.05 : 0.45
PK+ : PNa+ : PCl- = 1.00 : 0.01 : 0.10
PK+ : PNa+ : PCl- = 1.00 : 0.03 : 0.10
For these ions, Em is computed with the GHK equation given above where
R is the gas constant defined as 8.3144598(48) J·K-1·mol-1.
F is the Faraday constant defined as 96485.33289(59) C·mol-1.
Em is the resting membrane potential in Voltz (V), but frequently reported in milliVoltz (mV).
T is the temperature in the Kelvin scale.
n is the ion valence; i.e., 1 for monovalent ions, 2 for divalent ions, and so forth.
[K+]out, [Na+]out, and [Cl-]out = ion concentrations in the extracellular fluid.
[K+]in, [Na+]in, and [Cl-]in = ion concentrations in the intracellular fluid. The units used to express these concentration terms do not really matter as long as they are all of the same kind. However, these are frequently reported in moles per cubic meter to match other SI units.
Once Em is computed, the electrodriving force potential, Edf, experienced by each ion can be estimated as
Edf = Em - Eeqwhere Eeq is computed from the Nernst Equation. The sign of the driving force acting on an ion can be used to predict the direction of ion flow, outward or inward, across a membrane; i.e.
- for cations: outward (+) or inward (-)
- for anions: outward (-) or inward (+)
All of these facts are well known (Physiology.Arizona.edu, 2006; Physiologyweb.com, 2005a; 2005b; 2005c; 2005d;Science.UWaterloo.ca, 2007; Wright, 2004).
If T is in Celsius, Fahrenheit, Rankine, or other units, it must be converted to kelvins. We have developed a Temperature Converter tool that simplifies all these conversions.
When changing units, you may want to follow NIST 2006 guidelines for expressing results to a given number of significant digits:
- If the first significant digit of the converted value is greater than or equal to the first significant digit of the original value, round the converted value to the same number of significant digits as there are in the original value.
- If the first significant digit of the converted value is smaller than the first significant digit of the original value, round to one more significant digit.
- Lab techs as well as chemistry and physiology teachers and their students.
- Mapping Temperature-to-Potential Changes: To determine the effect of changing the temperature by 1 Kelvin on the resting membrane potential, Em, do as follows. Press the "Try This Example" button and determine the resting membrane potential for the combination of values provided. Next, clear the Em field, change the temperature field to 311, and submit form to determine the new resting potential. Proceeding in this way for several variable intervals, you should be able to map temperature changes to resting potential changes.
- Mapping Temperature-to-Concentration Changes: Repeat previous exercise, this time determining the effect of changing the temperature by 1 Kelvin on the concentration of any of the ions.
- Suggest other types of variable maps.
- In Physiology, a practical rule of thumb states that the resting membrane potential is close in value to the reversal potential for the ion that carries the majority of the resting current. In Chemistry the reversal potential is called the standard electrode potential, E°. If a membrane allows the exchange of only K+, Na+, or Cl-, which ion is responsible for most of its resting current? Why?
- In the previous exercise: Report the percentage of resting current carried out by each ion using only permeability information.
- The following problem was taken from the Web (Google Groups):
A typical neuron is dissected from an animal's nervous system and placed in a recording setup. The neuron is impaled with electrodes that allow the experimenter to set the membrane potential to any desired level. The normal resting potential is -70 mV, E (Na) = +50 mV, E (K)= -90 mV, and E (Cl)= -80 mV. At rest, the cell membrane is 50 times more permeable to K+ than to Na+ or Cl-. In her first experiment, the scientist hyperpolarizes the membrane potential to -100 mV and then immediately uses a drug that opens K+ channels. Which one of the following would be the largest current that would flow under these conditions.
- Inward current will flow, carried by K+ ions moving into the cell.
- Inward current will flow, carried by Na+ ions moving into the cell.
- Outward current will flow, carried by Cl- ions moving into the cell.
- All three ions (Na+, K+, and Cl-) will carry current across the membrane, but the direction each ion will move cannot be determined from the information given in the passage.
- Fitzakerley, J. (2014). Ion Channel Physiology.
- NIST (2006). The International System of Units (SI) - Conversion Factors for General Use. Nist Special Publication 1038.
- Physiology.Arizona.edu (2006). The Nernst/Goldman Equation Simulator.
- Physiologyweb.com (2005a). Nernst Potential Calculator.
- Physiologyweb.com (2005b). Electrochemical Driving Force Calculator.
- Physiologyweb.com (2005c). Goldman-Hodgkin-Katz Equation Calculator.
- Physiologyweb.com (2005d). Electrochemical Driving Force Acting on Ions.
- Science.UWaterloo.ca (2007). Nernst Equation.
- Wright, S. H. (2004). Generation of resting membrane potential.
Contact us for any suggestion or question regarding this tool.