## Donnan Equilibrium

The example of Figure 4-4b shows how we could construct a model cell that is simultaneously at osmotic and ionic equilibrium. However, the situation in Figure 4-4b is not very much like that in real animal cells. A major difference is that the principal internal cation in real cells is K+, not Na+. Also, there is some potassium in the ECF, and the cell membrane is permeable to K+ as well as Cl-.

In this situation, there are two ions that can cross the membrane: K+ and Cl-. If equilibrium is to be reached, the electrical potential across the cell membrane must simultaneously balance the concentration gradients for both K+ and Cl-. Because the membrane potential can have only one value, this equilibrium condition will be satisfied only when the equilibrium potentials for Cl- and K+ are equal. In equation form, this condition can be written as:

Here, the minus sign on the far right arises from the fact that the valence of chloride is -1. Canceling 58 mV from the above relation leaves log

The minus sign on the right side can be moved inside the parentheses of the logarithm to yield log([Cl-]/[Cl-]o). Thus, equilibrium will be reached when

Jo y

This equilibrium condition is called the Donnan or Gibbs-Donnan equilibrium, and it specifies the conditions that must be met in order for two ions that can cross a cell membrane to be simultaneously at equilibrium. Equation (4-4) is usually written in a slightly rearranged form as the product of concentrations:

In words, for a Donnan equilibrium to hold, the product of the concentrations of the permeant ions outside the cell must be equal to the product of the concentrations of those two ions inside the cell.

To see how the Donnan equilibrium might apply in an animal cell, consider the example shown in Figure 4-5a. Here a model cell containing K+, Cl-, and P is placed in ECF containing Na+, K+, and Cl-. As an exercise, we will calculate the values of all concentrations at equilibrium assuming that [Na+]o is 120 mM and [K+]o is 5 mM. From the principal of electrical neutrality, [Cl-]o must be 125 mM. Also, because P is assumed for the present to be uncharged, the principle of electrical neutrality requires that [K+] must equal [Cl-]. Because two ions K+ and Cl- can cross the membrane, the defining relation for a Donnan equilibrium shown in Equation (3-5) must be obeyed. Thus, if the model cell of Figure 4-5a is to be at equilibrium, [K+]i[Cl-]i must equal [K+]o[Cl-]o, which is 5 x 125, or 625 mM2. Because [K+] = [cl-], the Donnan condition reduces to [K+]2 = 625 mM2; thus, [K+] and [Cl-] must be 25 mM

INSIDE

Cell membrane rh

OUTSIDE

INSIDE

200 mM P Total osmolarity = 250 mOsm

Cell membrane i4-i

Cl- 125 mM

Total osmolarity = 250 mOsm Em = Ek = Eel « -40.5 mV

Figure 4-5 An example of a model cell at Donnan equilibrium. The cell membrane is permeable to both potassium and chloride. (a) The starting conditions, with initial values of some parameters provided. (b) The values of all parameters required for the cell to be at equilibrium.

at equilibrium. For osmotic balance, the internal osmolarity must equal the external osmolarity, which is 250 mOsm. This requires that [P]i must be 200 mM for the model cell to be at equilibrium. The results of this example are summarized in Figure 4-5b, which represents a model cell at equilibrium. What would be the membrane potential of this equilibrated model cell? The Nernst equation Equation (4-2) tells us that the membrane potential for a cell at equilibrium with [K+]o = 5 mM and [K+] = 25 mM is about -40.5 mV, inside negative. You should satisfy yourself that the Nernst equation for chloride yields the same value for membrane potential.

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