The Nernst/Goldman Equation Simulator

It would be difficult to exaggerate the physiological significance of the transmembrane electrical potential difference, or ‘PD.’ This gradient of electrical energy that exists across the plasma membrane of every cell in the body influences nutrient transport into and out of cells, is a key driving 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 associated with coordinated movements of cells and organisms, and is ultimately the basis of all cognitive processes. The Nernst equation (named after its originator, the German Chemist and Nobel laureate, Walther Nernst), provides a quantitative measure of the equality that exists between chemical and electrical gradients and is the starting point for understanding the basis of the “membrane potential.” The Goldman-Hodgkin-Katz equation (named in honor of American David Goldman and the British Nobel laureates Sir Alan Hodgkin and Sir Bernard Katz; frequently simply referred to as “the Goldman equation”) calculates an estimated membrane potential that reflects the relative contributions of the chemical concentration gradients and relative membrane permeability for K+, Na+ and Cl-.

Understanding of the concepts associated with the simulator requires familiarity with the parameters involved in Nernst/Goldman calculations. Following is a brief discussion of these parameters as they pertain to the simulator’s use.

Ion Concentrations

For each ion (i.e., K+, Na+ and Cl-) a ‘slider’ is provided to adjust one of three parameters. Two of these sliders control the intracellular and extracellular concentrations of the ion in question, and these can be varied between values of 1 mM and 600 mM. All slider values can be adjusted by double clicking on the value reported in the slider box and typing in the desired value. The default values are: for K+, 10 mM out and 100 mM in; for Na+, 100 mM out and 10 mM in; and for Cl-, 100 mM out and 10 mM. In fact, these absolute values are not expected for any tissue and were selected for didactic reasons (factors of 10 to simply the math associated with logarithms). However, physiologically relevant values for ion concentrations, particularly intracellular values, are difficult to define with precision, in part because of the technical difficulty associated with measurement of intracellular ion concentrations (let alone “activities”), but, more importantly, every cell type typically maintains intracellular ion concentrations within a range of values (albeit, a narrow range typically), and this range can vary under different physiological condition. Providing “a value” for the intracellular concentration of K+ of 155 mM in muscle tissue (for example) runs the risk of planting in a student’s mind the notion that this value “and this one alone” is always found in muscle tissue. It’s not (though it’s probably close).

Nevertheless, it is useful to have some “starting values” for intra- and extracellular ion concentrations with which to begin working with the simulator. Therefore, at the bottom of the simulator page there are four buttons that will load the simulator with “representative” intra- and extracellular concentrations for K+, Na+, and Cl- for four cell different cell types: (1) a “generic” cell; (2) skeletal muscle; (3) squid giant axon; and (4), the red cell. In addition, the simulator loads representative values for the relative membrane permeability (P) for these ions in these cell types (see below). Please, take all these values with considerable “salt...”

Average Concentration Levels
ion
intra
extra
P
“Generic”1
K+
120
4.5
100
Na+
15
145
5
Cl-
20
116
10
Skeletal Muscle2
K+
150
4.5
100
Na+
12
145
1
Cl-
4.2
116
1000
Squid Axon3
K+
400
20
100
Na+
50
440
1
Cl-
40
560
10
Red Cell4
K+
140
4.5
100
Na+
11
145
54
Cl-
80
116
21

Sources:

1Values for intracellular ion concentrations and permeabilities for a “generic” cell are rife with caveats; finally, every cell type is different, and the concentrations and permeabilites for any given cell can differ rather markedly depending upon the physiological “status” of the cell. Use these values for their intended purpose: as a rough starting point.

2The values for intracellular ion concentrations in skeletal muscle are from W. Boron and E. Boulpaep’s excellent text, Medical Physiology (Saunders, Philadelphia, 2003). They, in turn are modified from similar values in previous texts and reviews, all of which stem, principally, from a review by E.J. Conway (Physiol. Rev., 37:84-132. 1957) that included data on ionic composition of frog and rat skeletal muscle. The values in Boron and Boulpaep, and that we use here, are “similar but different” from those reported by Conway, but reflect a consensus view of “typical” values and (importantly) reflect the fact that these numbers are not “absolutes.”

3There are many textbook values for the ionic composition of squid axon and blood (with sea water values often replacing those for squid plasma). Generally, such values are derived from values reported and/or used by Sir Alan Hodgkin, and the values presented here are taken from the 1958 Croonian Lecture to the Royal Society of London by Hodgkin (Proc. Royal Soc London B, 148(930):1-37, 1958).

4Red cell cytoplasmic ion concentrations are from J. Hoffman’s review, “Active transport of Na+ and K+ by red blood cells” (Physiology of Membrane Disorders, 2nded., Ed. Andreoli, T.E., et al., Plenum, New York, 1986, pp.221-234). Relative permeabilities of the red cell membrane are from London, et al., Am. J. Physiol., 257: F985-F993, 1989.

Membrane Permeability

A third slider controls the permeability of the membrane to the ion in question and can be varied between arbitrarily established values of 1 to 10,000. When running in ‘Nernst’ mode, only the concentration sliders are available for adjustment.

The permeability values used in the simulation warrant some discussion. The formal units of “membrane permeability” are cm/sec and, in the present case, reflect the net conductive flux of the ionic species in question. Although absolute values for ionic permeability have been determined for selected cell types (under ‘selected conditions’), permeability ratios, that is, the relative permeability of a membrane to selected ions, are far more pertinent to understanding the impact of ion gradients on the electrical potential across a membrane. It’s also worth noting that When a membrane is permeable to only one species, the Nernst equation applies. However, when more than one ionic species can cross the membrane in response to prevailing electrochemical gradients, the GHK equation applies (within the limits imposed by the assumptions behind its derivation; see B. Hille, Ion Channels of Excitable Membranes, 3rd Ed., 2001), and the ratio of permeabilities is the relevant parameter. In our simulation we limit the permeability parameters to arbitrary values of between 1 and 10,000. The default values for relative ion permeability for Na+ : Cl- : K+ are 1 : 10 : 100, values often used in discussions of the resting ion permeability of neuronal cells. The values associated with the four representative cell types should be used as very rough relative estimates.

Temperature

In addition to controls associated with K+, Na+ and Cl-, there is a slider for controlling the temperature. Although the slider reports values in degrees C, temperature values used for calculation of Nernstian or Goldman potentials are in degrees Kelvin. The tabs located at the upper right-hand aspect of the simulation page allow you to display Nernstian or Goldman potentials at temperatures controlled by this slider or, alternatively, to “force” the calculations to use 37°C (i.e., 310° K).

RT/F

“R” is the gas constant, and in this application has a value of 8.314 Joules · K-1 · mole-1.

“T” is the temperature in degrees Kelvin.

“F” is the Faraday constant (the amount of electric charge in one mole of electrons) and has a value of 96,485 coulomb · mole-1.

Background Reference

Wright SH. Generation of resting membrane potential. Adv Physiol Educ 28:139-142, 2004.

Appendix

“Expansion of the constant field equation to include both divalent and monovalent ions.” (Spangler, S.G., Ala J Med Sci, 9: 218-223, 1972)

Although K+, Na+ and Cl- exert the primary influence on resting membrane potential (their relative concentrations and permeabilities make each a player in the Goldman equation), students frequently ask “But what about calcium?”

Calcium does exert a large and physiologically critical influence on the bioelectric parameters of cardiac cells and selected smooth cells. Calcium has a comparatively low concentration (i.e., chemical activity) in the extracellular solution (about 1.6 mM), and a very low concentration/activity in the cytoplasm <<100 nM under “resting conditions” (and not much more than a few micromoles per liter in the contractile periods of muscle during which calcium channels tend to be open). Combine those low concentrations with the low permeability of cell membrane to calcium during rest (<<0.1% of K+ permeability), and it is apparent that calcium is not likely to play a significant role in defining membrane potential under “resting” conditions. However, an increase in calcium permeability (due to the opening of calcium channels and/or increase in activity of other conductive calcium pathways, including Na/Ca exchange) can increase the influence of calcium on membrane potential.

But calcium is a divalent cation; how does it “fit in” to a Goldman-type analysis? This issue has been addressed by several authors (see Hille’s text). One treatment, by S.G. Spangler (Ala J Med Sci, 9: 218-223, 1972), lent its name to this section. Spangler’s derivation has the general form:

EAB = (-RT/F) · (2.303) log10y

where

quadratic equation

and

a = 4∑Pi++CA,i++ + ∑Pi+CA,i+ + ∑Pi--CB,I-- + ∑P i-CB,i-

b = ∑Pi+CA,i+ - ∑Pi+CB,i+ + ∑Pi-CB,i- - ∑Pi-CA,i-

c = -(4∑Pi++CB,i++ + ∑Pi+CB,i+ + ∑Pi--CA,I-- + ∑Pi-CA,i-)

A and B refer to two compartments separated by a membrane (in this formulation, A can be considered the outside of the cell, and B the inside of the cell). C refers to concentration of ions, where i++, i+, i--, and i- refer to divalent and monovalent cations or anions, respectively. P refers to the permeability of the membrane to these ions. When expressed in terms of the relative influence of K+, Na+, Cl- and, now, Ca2+, a, b, and c can be rewritten as:

a = 4PCa[Ca]in + PK[K]in + PNa[Na]in + PCl[Cl]out

b = PK[K]in + PNa[Na]in PK[K]out + PNa[Na]out + PCl[Cl]out PCl[Cl]in

c = -(4PCa[Ca]out + PK[K]out + PNa[Na]out + PCl[Cl]in)

Inserting physiologically relevant values for the above listed parameters reveals that, at rest, the transmembrane gradient for Ca2+, despite its size (>10,000-fold!), has no substantive effect on membrane potential. For the Ca2+ gradient to exert its effect, PCa has to increase many thousand-fold (as occurs during activation in selected excitable cells!).