Lecture 8 Notes, Tuesday 15 September 2005

  • Hille, Chap. 10, pp. 312-319 (electrodiffusion and the Nernst-Planck equation)
  • Hille, Chap. 14, pp. 445-452 (Goldman-Hodgkin-Katz equations)
  • Homework due: Thur 22 Sept 2005

    • Today we will revisit the Nernst-Planck equation and show that ionic conductances do not behave strictly according to Ohm's law (i.e., current as a linear function of driving force) under all conditions. In fact we will solve the Nernst-Planck equation to obtain the Goldman-Hodgkin-Katz (GHK) current equation, which shows how ionic currents rectify (i.e., deviate from linear Ohm's law behavior) as functions of voltage and concentration. This is called Goldman rectification and is generally applicable to any situation where current rectification can be attributed to concentration gradients. As we will see, rectification can arise due to other mechanisms, Goldman rectification applies to one particular mechanism.

    The GHK current equation will then be used to find a general expression for the resting potential of a cell that is permeable to more than one ion species. We touched on this idea in Lecture 6 in the form of the fractional conductance equation. We will derive the Goldman-Hodgkin-Katz (GHK) voltage equation. We will apply the GHK voltage equation to simple membrane model where K+ and Na+ are membrane permeable, and K+ permeable greatly exceeds Na+'s.

    Goldman-Hodgkin-Katz current equation

    To derive the GHK current equation we must make some key assumptions

  • The constant field assumption: membrane potential changes linearly with position in the membrane (-∂v/∂x = V/x)
  • Ions in the membrane behave like in aqueous environments and obey the Nernst-Planck equation
  • The concentration of ions at the interface surfaces of the membrane are proportional to their concentration in bulk solution (C0=βc0).
  • Ions move independently.

    • The GHK current equation is:

    I = Pz2F2V/(RT)[(Ci - Coe-zFV/(RT))/(1 - e-zFV/(RT))]

    Important principles illustrated by the GHK current equation:

  • I ~ Ci(V) at large positive membrane potentials
  • I ~ Co(V) at large negative membrane potentials

    • Goldman-Hodgkin-Katz voltage equation

    We obtain a more accurate expression for resting membrane potential because the sum of all ionic currents is zero at rest, so:

    ΣxIx = 0

    Using the GHK current equation for each ionic current Ix,

    ΣxPxz2F2V/(RT)[(Ci - Coe-zFV/(RT))/(1 - e-zFV/(RT))] = 0

    Cancel terms to get,

    ΣxPx[Ci - Coe-zFV/(RT)] = 0

    And finally, the expression for VM (at rest),

    VM = RT/(zF) LN[(ΣxPxCo)/(ΣxPxCi)] = 0

    We will apply this general expression to a membrane that is permeable to K+ and Na+, and the ratio of their permeabilities (PNa/PK) is 0.01. This simple example shows that the resting membrane potential is largely determined by the Nernst potential for K+, but deviates significantly from the Nernst equation's predictions at low external K+ levels.

    Lecture 8 Top - Cellular Biophysics and Modeling - Del Negro Homepage - Applied Science - The College of William and Mary