The genesis of the resting potential

When a membrane selectively permeable to a given ion separates two solutions containing different concentrations of that ion, an electrical potential difference is set up across it. In order to understand how this comes about, consider a compartment within which the ionic concentrations are [K]i and [Cl]i, and outside which they are [K]o and [Cl]o, bounded by a membrane that can discriminate perfectly between K+ and Cl— ions, allowing K+ to pass freely but being totally impermeable to Cl—. If [K]i is greater than [K]o there will initially be a net outward movement of potassium down the concentration gradient, but each K+ ion escaping from the compartment unaccompanied by a Cl— ion will tend to make the outside of the membrane electrically positive. The direction of the electric field set up by this separation of charge will be such as to assist the entry of K+ ions into the compartment and hinder their exit. A state of equilibrium will quickly be reached in which the opposed influences of the concentration and electrical gradients on the ionic movements will exactly balance one another, and although there will be a continuous flux of ions crossing the membrane in each direction, there will be no further net movement.

The argument may be placed on a quantitative basis by equating the chemical work involved in the transfer of potassium from one concentration to the other with the electrical work involved in the transfer against the potential gradient. In order to move 1 gram-equivalent of K+ from inside to outside, the

[K]o chemical work that has to be done is RT log . The corresponding electri-

cal work is —EF, where E is the membrane potential, inside relative to outside, and F is the charge carried by 1 gram-equivalent of ions. At equilibrium, no net work is done, and the sum of the two is zero, whence

This relationship was first derived by the German physical chemist Nernst in the nineteenth century, and the equilibrium potential EK for a membrane

Fig. 3.5. Variation in the resting potential of frog muscle fibres with the external potassium concentration [K]o. The measurements were made in a chloride-free sulphate-Ringer's solution containing 8 mM-CaSO4. Square symbols are potentials measured after equilibrating for 10 to 60 min; circles are potentials measured 20 to 60 s after a sudden change in concentration, filled ones after increase in [K]o, open ones after decrease in [K]o. For large [K]os the measured potentials agree well with

the Nernst equation, V = 58 log —o, taking [K] as 140 mM. The deviation at low [K]o

can partly be explained by taking PNa/PK = 0.01, so that V = 58 log ———-.

From Hodgkin and Horowicz (1959). 140

Fig. 3.5. Variation in the resting potential of frog muscle fibres with the external potassium concentration [K]o. The measurements were made in a chloride-free sulphate-Ringer's solution containing 8 mM-CaSO4. Square symbols are potentials measured after equilibrating for 10 to 60 min; circles are potentials measured 20 to 60 s after a sudden change in concentration, filled ones after increase in [K]o, open ones after decrease in [K]o. For large [K]os the measured potentials agree well with

the Nernst equation, V = 58 log —o, taking [K] as 140 mM. The deviation at low [K]o

can partly be explained by taking PNa/PK = 0.01, so that V = 58 log ———-.

From Hodgkin and Horowicz (1959). 140

permeable exclusively to K+ ions is known as the Nernst potential for potassium. The values of R and F are such that at room temperature the potential is given by

An e-fold change in concentration ratio therefore corresponds to a 25 mV change in potential, or a tenfold change to 58 mV.

On examining the applicability of the Nernst relation to the situation in nerve and muscle, it is found (Fig. 3.5) that it is well obeyed at high external potassium concentrations, but that for small values of [K]o the potential alters less steeply than eqn 3.2 predicts. It is evident that the membrane does not in

The Donnan equilibrium system in muscle 33

fact maintain a perfect selectivity for potassium over the whole concentration range, and that the effect of the other ions which are present must be considered. In order to derive a theoretical expression relating the membrane potential to the permeabilities and concentrations of all the ions in the system, whether positively or negatively charged, some assumption has to be made as to the manner in which the electric field varies within the membrane. Such an expression was first produced by Goldman, who showed that if the field was the same at all points across the membrane, the potential was given by

F ^ PKK + PNa[Na]i + Pa[Cl]o where the Ps are permeability coefficients for the various ions, and the suffixes o and i indicate external and internal concentrations respectively. Although the constant field equation has been shown in practice to fit rather well with experimental observation over a wide range of conditions, it does not follow that the field is indeed truly constant. Eqn 3.3 is, nevertheless, empirically very valuable for describing the behaviour of a membrane permeable to more than one species of ion. Thus in the experiment of Fig. 3.5, the deviation of the measured potential from a line with a slope of 58 mV can be accounted for nicely by taking PNa to be one hundred times smaller than PK.

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