4-7 MEMBRANE POTENTIALS AND DONNAN EQUILIBRIUM We have already seen that

a) ion diffusion plays a central role in the life process of most cells and

b) the phenomena are extremely complex as many ion species are involved, with different concentrations and permeabilities.

In this section we discuss some of the basic physics determining the evolution and equilibrium of non-living systems. Although we are discussing these matters qualitatively, the principles are widely used by biophysicists in setting up the mathematical relationships which enable them to understand the electrodiffusive properties of living cells.

Suppose we have two compartments containing solutions of various ions in various concentrations, separated by a membrane which is permeable in differing degree to the various ions. To be quite general, there need be no membrane at all, just a liquid-liquid junction (such as that mentioned between the blood sample and the saturated potassium chloride (KCl) solution in the description of the blood pH apparatus). For the moment we assume no external electric connection between the compartments, and no active biological mechanism in operation. What then can physics tell us about this system?

Two basic principles apply (based on considerations discussed in chapters E1, E2 and E3). They are as follows.

1. Bulk charge neutrality

The bulk solutions on each side of the boundary must both be essentially electrically neutral, that is each solution must contain equal amounts of positive and negative ionic charges. There may of course be a surface charge - a charge double layer - but that exists only on the surfaces of the membrane. In symbols

ni i

S

E4: Ion Diffusion 53

2. Zero membrane current density

As ions can diffuse through the membrane, there must be zero mean current density through the membrane if bulk charge neutrality is to be maintained. There is no completed circuit for the current, and a continuing separation of charge would violate bulk charge neutrality. Of course there is a very small, transient current as any charge double layer is established or changed. In symbols

i ni

Â

where v _i is the component of mean velocity, along an axis perpendicular to the membrane, of ions of species i.

In the KCl example discussed in the lecture, K+ was the only ion which could pass through the membrane, so zero net current also meant zero net ion flow, and an equilibrium was quickly established. In the general case, when there is more than one kind of permeant ion, there is no such equilibrium. Ions can (and do) diffuse both ways through the boundary, subject only to the constraint of zero mean current density. Roughly this means that for every positive ion that diffuses to the left, either a negative ion also diffuses to the left, or another positive ion diffuses to the right.

If the different ions have different concentration ratios it is likely that the value of the Nernst potential will be different for each kind of ion. It is therefore impossible for all the ions to be in Nernst equilibrium, because there can be only one value of the potential difference across the membrane. One species of ion at most could be in equilibrium, but it is more likely that none are. Since there is no equilibrium the system must keep changing.

The rates at which the various ions diffuse are determined by the permeability of the membrane (and/or the mobility of the ions in the solutions). A junction potential (alias liquid junction potential, diffusion potential, membrane potential, boundary potential etc.) is usually set up by this ion diffusion. The sign and magnitude of the potential are determined by the concentrations and permeabilities of all the ion species. The most permeant ions are, understandably, dominant in determining the potential. (This point is of vital importance in understanding the propagated action potential of nerve cells). The junction potential influences the flow of the various ions, but unlike our single-ion example, it does not stop the flow of any particular ion. The potential builds up rapidly to just that value required to ensure that the mean current density carried by all ions is zero.

But that situation is not an equilibrium. As time goes on, the concentrations of the ions and the junction potential evolve gradually towards a final equilibrium state. The time scale for this evolution may be days or months for laboratory-scale apparatus but for typical living cells would be minutes or less - if it were allowed to happen.

The course of this evolution is controlled by the concentrations, the permeabilities and the zero current principle. The end point of the evolution, the ultimate equilibrium state, is governed by the principle of bulk charge neutrality.

To see how this works, suppose that among all the various ions, there is just one species to which the membrane is totally impermeant. To be definite, suppose that the species is negatively charged and exists only in the RH compartment of the system in figure 4.4. This system will evolve as discussed above. It is clear that in the final state there must be an excess of permeant positive ions (over permeant negative ions) in the RH compartment, if bulk neutrality is to be maintained. As a consequence there must be a potential across the membrane, with the left hand side positive, to stop these ions diffusing through the membrane.

E4: Ion Diffusion 54 Membrane Solution of many permeant ions This compartment contains totally impermeant ions X- Many permeant ions + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - No impermeant ions

Figure 4.4 Donnan equilibrium.

A vessel is divided into two compartments by a semipermeable membrane. Many ions species are present, but only the right hand compartment contains the totally impermeant ions X-. The system

evolves to a Donnan equilibrium. The totally impermeant X- ions in the right hand compartment maintain a junction potential, with the left hand side positive. All other ions have reached Nernst

equilibrium.

This end-point equilibrium must eventually be reached by all liquid-liquid interfaces unless some other processes intervene. It is called a Donnan equilibrium, and the potential difference is called the Donnan potential.

A Donnan equilibrium has two characteristic features by which it can be recognised.

1. All ion species which are permeant (i.e. able to pass through the boundary) are in Nernst equilibrium at the final junction potential, i.e. their concentration ratios have adjusted to satisfy the Nernst equation, where the potential in the Nernst equation is the Donnan potential.

2. The only ions not in Nernst equilibrium are totally impermeant; they can never pass through the boundary.

If the original solutions contain no totally impermeant ions, then the concentrations and the junction potential evolve until all ions are equally distributed between the two compartments, and the junction potential is zero. This is a special (degenerate) case of Donnan equilibrium: all the ions are in Nernst equilibrium at a junction potential of zero. For each ion the concentration ratio is 1:1.

Thus, physical arguments show that two liquids separated by a membrane permeable to some of the ion species present will evolve to an equilibrium sate - a Donnan equilibrium - in which all the permeant ions are in Nernst equilibrium.