Equation for calculating the potential difference across a membrane that will balance the osmotic gradient of an ion.

An osmotic gradient of ions is an interesting situation. Neutral molecules at two different concentrations on either side of a dividing membrane will, through diffusion, want to be of equal concentrations on both sides. This is the principle behind physical processes such as osmosis and biochemical techniques such as dialysis. However, if the molecules are charged ions, then a difference in concentration accross a membrane also causes an electrostatic potential to be formed across it. This potential also exerts a force on the ions and must be taken into account when characterizing the equilibrium.

    neutral molecules

              o o |  oooo
             o  o | o o ooo
               o  | oooooo

               <------     direction of osmotic potential


              + + |  +++
               +  |  ++
               +  | + ++

                <------  osmotic potential
                <------  membrane potential

Taking the second scenario above, if one uses a tool such as a voltage clamp to force the membrane potential in the other direction, although the ions will want to diffuse to the right through osmosis, they will also have a counterbalancing force pushing them to right due to the interaction with the electric field.

Often one knows the concentration of ions on two sides of the membrane, but wants to calculate the membrane potential. This is calculated using the Nernst Equation:

             R T       X1
     E2-1 = _____ log ____
             F z       X2

Where E2-1 is the membrane potential from side two minus side 1 in volts, R is the ideal gas constant, T is temperature, F is the Faraday constant (96,500 C/g), z is the valence (charge per ion), and X is the concentration of ions on side one or two of the membrane.

This equation has made it possible to realistically model nerve action potentials which involve a complex interplay between ion diffusion across a membrane and the membrane potential. Rather sophisticated models of ion currents in the heart are long aggregations of differential equations which describe the different ions and their temporal and spatial location as the heart beats.

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