GHK Current Equation

Substitution of Equation 8.49 in Equation 8.45 yields the steady-state charge flux of the /-type ions as 

Assuming the partition coefficients Ki and K[ defined in Equation 8.12 to be the same, the above result is usually rewritten in terms of the ion concentrations in the inside and outside compartments as 

The GHK current equation reduces to the Nernst equation and the Ohm s law in the appropriate limits. In equilibrium, E is zero so that the term inside the square brackets of Equation 8.51 results in the Nernst equation (Equation 8.15). If the ion concentrations are uniform in the system, c,- (jc = 0) = c x=L) = a, Equation 8.50 gives the Ohm s law for the /-type ions 

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The results of the GHK current equation (Equation 8.51) are given in Figure 8.4 as a graphical representation, where li/iztePi) is plotted against zteVm/kBT for different ratios of c /c". The asymptotic limits of Equation 8.54 are evident in Figure 8.4. 

Figure 8.4 Plot of ///z eP/ against ZieV /ksT according to the GHK current equation. The asymptotic limits correspond to the Ohm s law. The curves are illustrated for c./c = 4,1, and 1 /4 (with the choice of cf =1/2).

Figure 8.6 (a) Potential barrier with a maximum value at x = i/2. (b) Plot of li/ZisPi gff with the barrier against ZieV /k T is steeper than the corresponding plot of /,/z eP, from GHK current equation (cj=2c° c = 0.5). 

Figure 7.6 Current-voltage relationship for passive channel models of Equations (7.27) and (7.28). Sodium concentrations typical for the squid giant axon are used [Na+ ] = 437 mM [Na J = 50 mM. The sodium equilibrium potential is VNa = 58.5 mV. Conductance g a is set to 0.01 mS-cm-2. The permeability for the GHK model of Equation (7.28) is set so that both models predict the same current density at AT = 0. Figure adapted from [108],...

In the steady state, when the drift contribution dominates, the ionic current obeys the Ohm s law. In the absence of either drift or barriers, the behavior of ions is according to the Pick s law. The GHK equations offer a convenient way to describe the crossover behavior between the diffusion-, drift-, and barrier-dominated regimes. We have also shown the utility of the numerically solved results from the PNP equations for the ionic currents through the GA channel and the aHL protein pore. The PNP calculations show that the steepest gradient in the electrical potential is only very near and across the pore. We have... 

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