In the HH model of voltage-gated Na and K channels, there exist gating particles that respond to changes in voltage and thereby switch between the permissive and nonpermissive state (these are the m- and h-gates in the Na-channel and n-gates in the K-channel). The voltage-dependent changes in these molecular gates are thought to result from the interaction of Vm with charged amino acids in the channel, interactions that cause these miniature voltage sensors to move, or translocate, within the cell membrane. Since channels open during depolarization, we would expect positive gating charges to move in the extracellular direction when they enter the permissive state (equivalently, negative gating charges should move toward the inside of the cell).
Movement of a charge across the membrane is a current, the gating current, that should be observable (p. 57):
Hodgkin and Huxley pointed out that the necessary movement of charged gating particles within the membrane should also be detectable in a voltage clamp as a small electric current that would precede the ionic currents. At first the term "carrier current" was used for the proposed charge movement, but since we no longer think of channels as carriers, the term gating current is now universally used.The empirical measurement of gating currents wasn't actually performed until the early 70s.
Hille provides a brief summary of how to use general energetic considerations to calculate the number of charges that the voltage sensor contains, the gating charge. First, recall from chemistry that a protein can exist in many conformational states, and the energy associated with each conformation forms a "conformation energy landscape", a bumpy terrain usually with lots of local minima separated by hills that represent the energy needed to make the transition between the conformations. So we can assume that even when a channel is not subject to a voltage across the membrane, the conformational energy of a channel in the closed state is X, its conformational energy in the open state is X+w, and hence the difference in energy between the two conformations is w.
It is important to note that the conformational change in the channel takes place in a voltage field Vm. By hypothesis, the gating particle has a charge Q (Q=zgqe where qe is the elementary charge and zg is the number of charges) and the sensor must move against Vm. Hence, in addition to the purely conformational energy change described above, at a fixed Vm, the work done (i.e., energy) to move the charge from the intracellular to the extracellular edge of the membrane is -QVm (recall V is in J/Coulomb, so the units work out). This charge movement is exactly analogous to lifting a baseball above the ground and building up potential energy: similarly, a positive charge Q held just at the edge of the extracellular membrane will be pushed toward the intracellular side when the cell is at a hyperpolarized Vm. More explicitly, if the electrical potential in the closed state is Y, then the potential in the open state is Y-QVm, and the change in the electrical potential is -QVm. The sign is negative because we want a positive potential difference when the cell is hyperpolarized.
We can then examine both energies (conformational and electrical potential) simultaneously. The total energy associated with the closed state is Eclosed=X+Y, that associated with the open state is Eopen=(X+w)+(Y-QVm). Hence, the change in energy associated with the transition from the closed to open state is Eopen-Eclosed=w-QVm. Since we are describing a channel with only two states, and we know the energy difference between the states, we can apply the Boltzmann equation to describe the ratio of open to closed channels at thermodynamic equilibrium: it is a decaying exponential function of (w-QVm) [see Equations 2.21 and 2.22].
If you substitute zgqe for Q in the Boltzmann distribution, you can fit the observed proportion of open channels to the Boltzmann distribution using different values of zg, and thereby get an estimate of the number of charges in the voltage sensor (Figure 2.20). Technically, the best fit only provides a lower bound on the number of charges in the voltage sensor. This is because the above calculation assumes that zg charges move completely from the inside of the membrane to the outside. However, the same electrical potential -QVm could be generated by moving a greater number of charges partly across the membrane. Indeed, this is now thought to be exactly what happens.