Goldman constant field equation

We next explored the fit of the data to a more detailed relationship than that forming the basis of Nernst equation. We wrote the Goldman constant field equation (3) in the form used by Hodgkin and Katz (4) in the exponential form shown here as equation 3. 

An equation (also referred to as the constant field equation, the Goldman-Hodgkin-Katz equation, and the GHK equation) which relates the membrane potential (Ai/r) to the individual permeabilities of the ions (and their concentrations) on both sides of the membrane. Thus,... 

Q.23.3 (a) What are the assumptions in the Goldman-Hodgkin-Katz constant field equation (b) Under what circumstances does the equation tend to fail (c) Choose one assumption and hypothesize a specific situation that it might not be applicable. 

In this expression, there are no jumps in values of electrical potential and ionic concentration at the phase boundary. Although Eqs. (122) and (123) have a quite complex form, this equation can be reduced easily to the Nernst diffusion equation and the constant field equation (Goldman ), by assuming proper physical conditions. The former, the Nernst diffusion equation [Eq. (119)], can be obtained from Eqs. (122) and (123) for the case of solutions containing only a single salt (such as NaCl) and the latter, the constant field... 

Tli e assiunp tion of a constant elec trie field in th e membrane is a ctuall y no t essential for ob taining Equation 3.20 we could invoke Gauss s law and perform a more difficult integration. See Goldman (1943) for a consideration of the constant field situation in a general case. 

Three assumptions are usually made in calculating ion current density through a membrane (a) an ion does not interact with any other ion while traversing the membrane (independence principle) (b) the membrane is homogeneous and (c) the electric field in the membrane is constant in space. The linear transport equation then leads to the Goldman-Hodgkin-Katz equation - for the I -V relation ... 

Equation 14.1 contains derivatives of both the concentration and electrical potential, and requires both quantities to be known as a function of position in the membrane so as to predict the flux. To simplify matters, Goldman proposed the approximation that the electric field across the membrane be considered constant (i.e., the electric field is not affected by the presence of ions in the membrane). Under these conditions, the electrical potential gradient reduces to E=—d

applied potential and h is the membrane thickness. This approximation is appropriate for membranes that are relatively thin compared to the Debye length. This is not the case for the skin, even when heat-separated ( 100 pm) or dermatomed ( 0.5 mm). This approximation is also reasonable when the total ion concentrations on both sides of the membrane are equal. However, this is rarely the case in iontophoresis for which the applied ionic concentration is typically much less than that subdermally. 

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