The voltage clamp measurements in Figure 4.1 show typical whole-cell Ca2+ currents for L-type channels from a neuron in the sea hare Aplysia. The control curve in Figure 4.1A shows that these channels rapidly activate and then slowly inactivate when the cell is depolarized to 20 mV. The Ca2+ dependence of the inactivation step is illustrated by the slowing of inactivation when the mobile Ca2+ chelator EGTA is injected into the cell. The Ca2+ dependence of inactivation is confirmed by the experiment in Figure 4.1B, in which Ba2+ replaces Ca2+ outside the cell.
A cartoon for the mechanism underlying Ca2+ inactivation of L-type channels is given in Figure 4.2A. The cartoon illustrates the formation of a domain of elevated Ca2+ at the cytoplasmic face of an open Ca2+ channel (i.e., a small localized region in the vicinity of the channel in which Ca2+ concentration can be quite high). Domains like this have been predicted to form within a few microseconds of the opening of a channel due to the combined effects of high Ca2+ concentrations outside the cell (ca. 2 mM) and low basal concentrations in the cytosol (ca. 0.1 ^M). When this is combined with slow diffusion of Ca2+ within the cell, calculations predict Ca2+ domains with peak values approaching 200—500 ^M within nanometers of the channel. The slow rate of diffusion of Ca2+ is caused by tight binding of Ca2+ to numerous buffering sites in the cytoplasm, which greatly retards its ability to diffuse. The high concentration of Ca2+ in a domain suggests that an open channel may be subject to direct block of the open state by binding of a Ca2+ ion at the cytoplasmic face of the channel. A simple mechanism that accounts for this is given in Figure 4.2B.
Ca external medium high
O fast activation
Figure 4.2 (A) Cartoon of domain calcium. (B) State diagram for Ca2+ channel.
The three states C, O, and I represent closed, open, and inactivated states of the channel. Step 1 is the activation step, whereas step 2 represents the binding of domain Ca2+, written as Ca+2. The mechanism postulates a low affinity site for Ca2+ binding, which means that the inactivated state can be reached only when the channel is open and the Ca2+ domain has formed. Since simulations show that the peak concentration in a domain falls rapidly when a Ca2+ channel closes, it is possible to associate a unique value of domain Ca2+ with the open state, whose value depends only on the current through the open channel.
This mechanism is easily translated into a mathematical model. All the steps are unimolecular, except for the bimolecular binding of domain Ca2+. If we represent the fractions of channels in the three states by xC, xO, and xj, then the kinetic equations for the model can be written dxC/dt = —V, (4.1)
the rates of steps 1 and 2. Because the concentration of Ca2+ in the domain, [Ca2+]d, depends only on the current, it is a function of the electrical driving force and the singlechannel conductance. (In general, the value of [Ca2+]d depends on the external Ca2+ concentration and the membrane potential and is proportional to the single channel conductance; cf. Exercise 6. Specific values for the rate constants are given in Exercise 3. Figure 4.3 shows a simulation with the model that depicts the Ca2+ current for a cell that is depolarized at t = 10 ms to a voltage where the channel is open. Simulations like this have been used to duplicate the time course of voltage clamp measurements for L-type Ca2+ currents in pancreatic beta cells. Key evidence that supports the domain model has come from recent experiments with genetically engineered L-type channels,
full model full model
Figure 4.3 Computed solution and approximate solution for open fraction of L-type calcium channels.
and it now seems certain that the essential ideas of the model are correct.
In both the experiments and the simulations, activation of the channel is fast compared to inactivation. In the model this is due to the fact that both the rate constants for step 1 are much larger than those for step 2. For example, the forward rate for V is about 47 times faster than that for V2. Because of this, step 1 rapidly "equilibrates" the states C and O.
To see how this equilibration takes place, it is a good approximation to ignore V2, at least at first. Assuming that xC(0) = 1 (all channels are initially closed), it follows that xj = 0 and xC + xO = 1. We use this to calculate that with K = and ract = 1/(k+ + k-). The number Tact is the time constant for activation. Because this process is fast, within a few milliseconds V ~ 0. This condition continues to hold even as the fractional occupancies xC, xO, and xj change.
The rapid equilibrium approximation is a method to exploit this observation that some kinetic steps are "fast." By "fast" we mean "faster than the time scales of physiological interest," i.e., faster than the slowest times scales in the process. Here the fast process is the process V, and for most times after the short initial phase, V ^ 0. According to (4.3), the condition V =0 implies that which is the equilibrium condition for step 1 in the mechanism.
Now, it might be tempting to set V = 0 in (4.2), but this is the wrong thing to do. Instead, we recognize that since xC and xO are in equilibrium, the quantity of interest is the total number of channels in the states C and O. Notice that by adding (4.1) and (4.2) together, V is eliminated, and we find the rate of change of the combined state
Using the equilibrium condition (4.6), we find that
This ODE can be rearranged into the familiar form dy/dt=^-nca^ ■ (4.9)
Written this way, (4.9) has the same form as (4.7) for a voltage gated channel, except that now [Ca2+]d replaces the voltage.
The only tricky part remaining about the solution to (4.9), which is an exponential, is finding the correct initial condition. Since we have assumed that step 1 is fast, the initial condition for (4.9) must take into account the rapid initial equilibration of C and O. As in Figure 4.1 the initial condition typically is at a hyperpolarized potential where all the channels are closed and xC (0) = 1. After the initial equilibration, some of the channels will have moved to state O, so that y = xC + xO = 1. This gives the equilibrated initial condition for (4.9) as y(0) = 1. (4.12)
Using this initial condition, the solution to the rapid equilibrium approximation for the model is plotted as the dashed line in Figure 4.3. Two things are notable in comparing the approximation to the complete solution. First, by neglecting the rapid activation of the channel, the approximation slightly overestimates the peak current, which is given analytically using the initial condition in (4.12) as /peak = g(1/(1 + K,))(V — VCa) (see Exercise 6). Second, the exponential decline in current predicted by (4.9) does a good job of approximating the rate of inactivation of the current. The approximation works well because the time scale for the fast process (ca. 1 ms) is much faster than that for inactivation (ca. 45 ms). As long as the time constant for the fast process is at least an order of magnitude faster than the remaining processes, the rapid equilibrium approximation provides a reasonable approximation to the complete model.
The rapid equilibrium approximation is a useful way to reduce the complexity of models. For the domain Ca2+ inactivation model, the simplification is not really necessary, because the full model involves only two linear differential equations that can be analyzed by the matrix methods in Appendix A. However, the fact that the equation for the simplified model resembles that for voltage—gated channels provides a conceptual bridge to the properties of ligand—gating of channels.
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