sign in sign up
<<
Home , Nernst equation

670 PANEL 11–2: The Derivation of the Nernst Equation

THE NERNST EQUATION AND FLOW

The flow of any ion through a membrane channel protein At the point where the concentration and voltage is driven by the electrochemical gradient for that ion. This gradients just balance, gradient represents the combination of two influences: DG + DG = 0 the voltage gradient and the concentration gradient of conc the ion across the membrane. When these two influences and the ion distribution is at equlibrium across the just balance each other the electrochemical gradient for membrane. the ion is zero and there is no net flow of the ion through the channel. The voltage gradient () Thus, C at which this equilibrium is reached is called the zFV – RT In o = 0 equilibrium potential for the ion. It can be calculated Ci from an equation that will be derived below, called the and, therefore, Nernst equation. C V = RT In o zF Ci

or, using the constant that converts natural to The Nernst equation is base 10, RT Co V = 2.3 log10 zF Ci RT C For a univalent ion, V = In o zF Ci RT 2.3 = 58 mV at 20oC and 61.5 mV at 37oC F Thus, for such an ion at 37oC,

where V = + 61.5 mV for Co / Ci = 10, whereas V = the equilibrium potential in (internal V = 0 for Co / Ci = 1. potential minus external potential) The K+ equilibrium potential (V ), for example, is Co and Ci = outside and inside concentrations of the K + + ion, respectively 61.5 log10([K ]o / [K ]i) millivolts –1 –1 + R = the (2 cal mol K ) (–89 mV for a typical cell where [K ]o = 5 mM + T = the absolute temperature (K) and [K ]i = 140 mM). F = Faraday’s constant (2.3 ¥ 104 cal V–1 mol–1) + z = the valence (charge) of the ion At VK, there is no net flow of K across the membrane. In = to the base e Similarly, when the membrane potential has a value of + + 61.5 log10([Na ]o /[Na ]i), the Na+ equilibrium potential (V ), The Nernst equation is derived as follows: Na there is no net flow of Na+. A molecule in solution (a solute) tends to move from a region of high concentration to a region of low For any particular membrane potential, VM, the net concentration simply due to the random movement of force tending to drive a particular type of ion out of the molecules, which results in their equilibrium. cell, is proportional to the difference between VM and the Consequently, movement down a concentration gradient equilibrium potential for the ion: hence, is accompanied by a favorable free-energy change for K+ it is V – V (DG < 0), whereas movement up a concentration gradient M K + is accompanied by an unfavorable free-energy change and for Na it is VM – VNa. (DG > 0). (Free energy is introduced in Chapter 2, and The number of that go to form the layer of charge discussed in the context of reactions in adjacent to the membrane is minute compared with the Panel 14–1, p. 830.) total number inside the cell. For example, the movement of 6000 Na+ ions across 1 mm2 of membrane will carry The free-energy change per of solute moved across sufficient charge to shift the membrane potential by D the plasma membrane ( Gconc) is equal to –RT In Co / Ci. about 100 mV.

If the solute is an ion, moving it into a cell across a Because there are about 3 ¥ 107 Na+ ions in a typical cell membrane whose inside is at a voltage V relative to the (1 mm3 of bulk cytoplasm), such a movement of charge outside will cause an additional free-energy change (per will generally have a negligible effect on the ion D mole of solute moved) of Gvolt = zFV. concentration gradients across the membrane.