Chapter 3 (1): Toxins to the Rescue

Chapter 2 outlined the HH theory of the action potential: when they wrote their papers, nobody yet knew the mechanism of current flow. From p 61:

A variety of mechanisms were considered possible. These included permeation in a homogeneous membrane, binding and migration along charged sites, passage on carriers, and flow through pores...The pathways might be formed from phospholipid or from protein, or even from nucleic acid. Each of these ideas was seriously advanced and rationalized in published articles.

Between the early 1960s and 1976 (in 1976 Neher and Sakmann published the first single channel patch recording), the mystery of the currents was essentially solved. First, in the 1960s toxins were found that could selectively block I_na (TTX and STX) or I_k (TEA) while the other current remained unaltered (Figure 3.2). This, in addition to the results of HH's ion substitution experiments, was taken as evidence that there were two separate ion pathways, or channels, one for K and one for Na.

The discovery of neurotoxins has been a boon for neuroscience. In addition to letting us turn individual currents on and off at will, they have helped us count individual channels (channel density is an important parameter in realistic neuronal models), and have suggested structural models of channels.

Counting Channels
The action of TEA, TTX, and STX is well-described by a simple model in which a single molecule of the toxin T binds reversibly to a single receptor site R in an individual channel. This model is represented in Equation 3.1. The strength of binding is given by the equilibrium dissociation constant K_d. K_d is the toxin concentration at which half of the available receptors are occupied by the toxin. The model allows for a relatively simple algebraic derivation of the proportion of bound receptors, denoted with the variable y (see equation 3.2 and Figure 3.3).

If the above is unfamiliar, I highly recommend reading this site for a good introduction to the kinetics of such ligand-receptor interactions, which includes a discussion of K_d. As usual, Hille is leaving out lots of details in his exposition, but his terse summaries are very accurate.

The simple model of toxin-receptor interactions described above allows us to directly estimate the number of receptor sites, B_max in a preparation. This is typically done with radioactively labeled toxins denoted radioligands. After marinating the preparation with radioligand, the preparation is thoroughly washed out to remove all the radioligand that is not bound to the receptor. Often it is washed out with a specific competitor such as a nonradioactive form of the toxin. If the radioligand only bound to the receptor, then estimating B_max, the number of receptors in the tissue, would be relatively simple. We could apply different concentrations of radioligand and fit the data to the equation B_maxy, where y is the proportion of bound receptors as a function of toxin concentration (Equation 3.2), and B_max is the number of radioligand molecules at saturation. Note that B_maxy is the first term in Equation 3.4.

Unfortunately, reality isn't quite so simple: we are never able to wash out all of the radioligand that isn't bound to receptor. It binds "nonspecifically" to other molecules in the preparation. Indeed, there is evidence that certain drugs bind to the plastic out of which test tubes are made! Luckily, this general, nonspecific binding typically has very different properties than the toxin-receptor binding: it is a linear nonsaturating function (i.e., the radioligand remaining after washout continues to increase linearly without bound as the concentration of radioligand is increased). Because of such nonspecific binding, a second linear term must be added to the saturating function, as represented by the right-hand term a[T] in Equation 3.4. [T] is the toxin concentration and a is a free parameter that represents the slope of the linear relationship.

Figure 3.4 shows the various curves we have discussed in the context of a clever experiment. The amount of radiolabeled STX was measured at different concentrations before and after wash with a high concentration of TTX, which has a higher affinity for the Na-channel receptor than STX. The remaining radiolableed STX was due to nonspecific binding, and exhibits the expected linear relationship with STX concentration. The difference between the linear curve and the overall saturating curve observed before TTX wash reveals the amount of radioligand remaining due to specific binding of STX to the Na-receptor. Such data has shown us that the Na-channel density on membrane is about 100-400 channels per square micron of unmyelenated axon.

Initial channel models
In addition to helping us count receptors, toxins have helped generate predictions about channel structure. For instance, a model of K channels was suggested by the effects of intracellular TEA on I_k. Namely, intracellular TEA blocks I_k only in open channels (a phenomenon known as open channel block (Figure 3.5). This suggested that the channel is a pore with a wide cytoplasmic opening (large enough for a big toxin like TEA to block it), an opening which is typically blocked by a gate that moves out of the way when the channel opens, thus allowing TEA to enter and close the channel. Because the toxin does not simply flow out of the extracellular pore, the model also assumes that the pore narrows toward the extracellular side. A cartoon of this model of the voltage-gated channel is shown in Figure 3.6. I

Chapter 2 (3): Gating charges and gating currents

Hille discusses gating currents at greater length in later chapters (e.g., Chapters 9, 12, 18, and 19), and he will explicitly draw on the material from Chapter 2. Hence, I thought the topic deserved a few bits of bandwidth.

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 V_m 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 V_m. By hypothesis, the gating particle has a charge Q (Q=z_gq_e where q_e is the elementary charge and z_g is the number of charges) and the sensor must move against V_m. Hence, in addition to the purely conformational energy change described above, at a fixed V_m, the work done (i.e., energy) to move the charge from the intracellular to the extracellular edge of the membrane is -QV_m (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 V_m. More explicitly, if the electrical potential in the closed state is Y, then the potential in the open state is Y-QV_m, and the change in the electrical potential is -QV_m. 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 E_closed=X+Y, that associated with the open state is E_open=(X+w)+(Y-QV_m). Hence, the change in energy associated with the transition from the closed to open state is E_open-E_closed=w-QV_m. 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-QV_m) [see Equations 2.21 and 2.22].

If you substitute z_gq_e for Q in the Boltzmann distribution, you can fit the observed proportion of open channels to the Boltzmann distribution using different values of z_g, 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 z_g charges move completely from the inside of the membrane to the outside. However, the same electrical potential -QV_m could be generated by moving a greater number of charges partly across the membrane. Indeed, this is now thought to be exactly what happens.