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The relationship between electrolyte concentrations and membrane potential is a cornerstone of any theory of electrical activity in biological cells. These relationships are given by the Nernst and Goldman-Hodgkin-Katz (GHK) equations.

The Nernst potential

The permeability of the cell membrane for specific ions is determined by the number of ion channels of various species that are open at the prevailing membrane potential. Ion channels can exhibit various degrees of ion selectivity: some are permeable to only a single species of ion, others can be permeable to a wide range. In a case where the plasma membrane is permeable to a single ion, two forces are exerted on this specific ion: the first is diffusion of ions down their concentration gradient (from high to low concentrations); the second is an electrical driving force resulting from a charge separation between the inside and the outside of the cell, caused by the active displacement of ions from one side of the membrane to the other, which is mediated by ion pumps such as Na/K ATPase. The electrical driving force depends on the prevailing potential difference across the channel. Thus there will be a value of the membrane potential such that the drift due to diffusion and due to the electromotive force exactly cancel. This value is called theequilibrium potential for that ion.

Glial cells provide an instructive elementary example, since in these cells, the membrane is permeable to almost solely potassium ions at the RMP. The cell has a higher concentration of potassium inside than outside. Net diffusion of potassium ions takes place from inside to outside the cell. As a result, excess potassium ions accumulate on the outside of the membrane (and negative charge on the inside of the membrane because of the deficit of potassium ions and the excess of anions inside the cell). A separation of charge resulting from the diffusion of potassium gives rise to a potential difference. The latter opposes further diffusion of potassium to the outside of the cell. When both forces exactly balance each other, the potential difference across the membrane is called the potassium equilibrium potential, EK.

In this cell, which is permeable to potassium only,EK determines the RMP.

The equilibrium condition depends only on the electric energy per particle of a given species on either side of the membrane. Hence the potential difference is

directly related to the concentration difference by the Boltzmann equation. Solving the latter for the potential difference, theNernst equationis obtained:

Ei= RT zF ln [ion+]o [ion+]i , (1.1)

whereR is the universal gas constant,F is the Faraday constant, T is the absolute temperature, z is the valency of the the ion, [ion+]o and [ion+]i are the extracellular

and the intracellular ion concentrations for that particular cell. Values of these constants, and their units are given in Table 4.33.

The Goldman-Hodgkin-Katz potential

The Goldman-Hodgkin-Katz (GHK) equation describes the steady-state membrane potential for a cell with a membrane permeable to various ionic species. This RMP is determined not only by the intracellular and extracellular ion concentrations but also by how readily the ions cross the membrane. A constant steady-state membrane potential occurs when the sum of fluxes add up to zero (zero net current) as follows:

Itotal =IK+INa+ICl= 0. (1.2)

When the net flux of permeable ions is not equal to zero, there will be fluxes of all the ions involved driven by their concentration gradients. To obtain a constant steady-state membrane potential in a membrane permeable to K+, Na+, and Cl−

(dV /dt= 0, where V =Vin−Vout), there must be zero net current.

The RMP lies between the individual equilibrium potentials of the different ions that are crossing the membrane. The assumption for the GHK model is that the potential drops linearly over the membrane; this means that the electric field (E = ∆V) is constant and proportional to the membrane potentialV as illustrated in Figure 1.8. The GHK constant-field equation is as follows:

Em= RT

2F ln

PK[K+]o+PNa[Na+]o+PCl[Cl−]i

PK[K+]i+PNa[Na+]i+PCl[Cl−]o

, (1.3)

where [K+]o, [Na+]o, and [Cl−]o are the extracellular ion concentrations and [K+]i,

[Na+]i, and [Cl−]i are intracellular ion concentrations for that particular cell. The

permeabilities of the membrane to K+, Na+, and Cl− are denoted byPK,PNa, and

PCl respectively. Goldman et al [52] give a derivation of the GHK equation. This

equation is valid only for permeant ions that have the same valence. When the ions differ in valence the equation is more complicated. Several expressions were derived

for combinations of different valences [53–56]. For instance, Fatt et al [53] derived a modified equation that includes currents carried by divalent ions, such as Ca2+, as well as monovalent ions. The equation by Fatt et al [53] of extracellular Ca2+ versus intracellular K+ is as follows:

Em= RT F ln s 4PCa[Ca2+]o PK[K+]i +1 4 − 1 2 . (1.4)

When the relative permeability of the membrane for a specific ion is significantly higher than it is for the remaining ions, the GHK equation reduces to the Nernst equation for that particular ion.

out in membrane [K+_] o [Cl-_] o [Na+_] o [K+_] i [Cl-_] i [Na+_] i y (n m) 0 l E Vout Vin V (y ) (mV) 0 _{l y (nm)}

Figure 1.8: A section of a membrane. The following symbols are indicated: E: electric

field; `: membrane thickness; Vout: extracellular potential; Vin: intracellular potential;

[ion+/−]o: extracellular ions concentration; [ion+/−]i: intracellular ions concentration.

Voltage clamp

The dynamics of the conductances of various ions can be both explicitly time- dependent and time-dependent through the dependence on the membrane poten- tial [57]. It was realised that the former of these two dependencies could be made more manageable to study if the latter could be eliminated by keeping the membrane potential constant. This is accomplished using the voltage clamp technique. The voltage across the membrane is clamped, by means of an electronic servo circuit, to a constant desired value, as shown in Figure 1.9. The explicit time-dependence of the conductances can then be studied at various constant ‘set’ values of the voltage, or, in more modern variations of the technique, under imposed voltage time series. A specialised version of voltage clamp was introduced later. This was the patch clamp, which measures the current across a specific patch allowing us to measure the current across single or multiple channels. In the pioneering studies, a giant axon

from a squid was used in view of the technical difficulties involved in introducing an electrode into the cell. Using specific blocking agents that block either sodium or potassium channels, the two corresponding currents could be further isolated. Thus, the combined application of voltage clamp and pharmacology essentially dissected out the dynamic behaviour of isolated ionic currents.

Extracellular electrode Intracellular electrode Cell V amplifier Signal generator Feedback amplifier Current monitor V_c V I_amp

Figure 1.9: Two-electrodes voltage clamp circuit.

Let Vc denote the command voltage or the desired membrane voltage. As

soon as the membrane potential V deviates ever so slightly from Vc, a difference

signal is generated instructing the amplifier to produce currentIapp that is injected

into the cell forcingV to be equal toVc. This current needed to keep V =Vc is the

recorded current which gives an insight into the opening and closing response of the ion channels (whose very existence was still at issue in 1952, Hodgkin and Huxley developed the theory).

When a depolarising step is applied, the sodium channels activate rapidly, leading to high Na+ permeability and an inward Na+ current is produced. Then the channel inactivates spontaneously because Na+ permeability decreases, the K+ permeability rises slowly causing an outward current. In the actual squid axon cell, both these currents shape the AP as a response to stimulus.