Many drugs act by blocking a specific ion channel. There are numerous specific channel blockers, such as sodium channel blockers, potassium channel blockers, calcium channel blockers, and so on. In fact, the discovery of site-specific and channel-specific blockers has been of tremendous benefit to the experimental study of ion channels. Examples of important channel blockers include verapamil (calcium-channel blocker), quinidine, sotolol, nicotine, DDT, various barbiturates (potassium-channel blockers), tetrodotoxin (TTX, the primary ingredient of puffer fish toxin), and scorpion toxins (sodium-channel blockers).
To include the effects of a drug or toxin like TTX in a model of a sodium channel is a relatively simple matter. We assume that a population P of sodium channels is available for ionic conduction and that a population B is blocked because they are bound by the toxin. Thus,
where D represents the concentration of the drug. Clearly, P + B = P0, so that dP dt and the original channel conductance must be modified by multiplying by the percentage of unbound channels, P/P0. In steady state, we have
The remarkable potency of TTX is reflected by its small equilibrium constant Kd, as Kd « 1-5 nM for sodium channels in nerve cells, and Kd « 1-10 ^.M for sodium channels in cardiac cells. By contrast, verapamil has Kd «140-940 ^M.
Other important drugs, such as lidocaine, flecainide, and encainide are so-called use-dependent sodium-channel blockers, in that they interfere with the sodium channel only when it is open. Thus, the more the channel is used, the more likely it will be blocked. Lidocaine is an important drug used in the treatment of cardiac arrhythmias. The folklore explanation of why it is useful is that because it is use-dependent, it helps prevent high-frequency firing of cardiac cells, which is commonly associated with cardiac arrhythmias. In fact, lidocaine, flecainide, and encainide are officially classified as antiarrhythmic drugs, even though it is now known that flecainide and k encainide are proarrhythmic in certain postinfarction (after a heart attack) patients. A full explanation of this behavior is not known.
To keep track of the effect of a use-dependent drug on a two-state channel, we suppose that there are four classes of channels, those that are closed but unbound by the drug (C), those that are open and unbound by the drug (O), those that are closed and bound by the drug (CB), and those that are open and bound by the drug (OB) (but unable to pass a current). For this four-state model a reasonable reaction mechanism is a a
Notice that we have assumed that the drug does not interfere with the process of opening and closing, only with the actual flow of ionic current, and that the drug can bind the channel only when it is open. It is now a straightforward matter to find the differential equations governing these four states, and we leave this as an exercise.
This is not the only way that drugs might interfere with a channel. For example, for a channel with multiple subunits, the drug may bind only when certain of the subunits are in specific states. Indeed, the binding of drugs with channels can occur in many ways, and there are numerous unresolved questions concerning this complicated process.
1. Derive the extended independence principle. Assume that there are more than one species of ion present, all with the same valence, and assume that the reversal potential is given by the GHK potential. Show that r = jlji - jj exp(-F)
where the sum over j is over all the ionic species. Subscripts i and e denote internal and external concentrations, respectively.
2. Show that the GHK equation (3.2) satisfies both the independence principle and the Ussing flux ratio, but that the linear I-V curve (3.1) satisfies neither.
3. In Section 3.3.1 we used the PNP equations to derive I-V curves when two ions with opposite valence are allowed to move through a channel. Extend this analysis by assuming that two types of ions with positive valence and one type of ion with negative valence are allowed to move through the channel. Show that in the high concentration limit, although the negative ion still obeys a linear I-V curve, the two positive ions do not. Details can be found in Chen, Barcilon, and Eisenberg (1992), equations (43)-(45).
4. (a) Show that (3.64) satisfies the independence principle and the Ussing flux ratio.
(b) Show that (3.64) can be made approximately linear by choosing g such that ng = ln ^—J . (3.145)
Although we know that a linear I-V curve does not satisfy the independence principle, why does this result not contradict part (a)?
5. Show that (3.86) does not satisfy the independence principle, but does obey the Ussing flux ratio.
6. Derive (3.86) by solving the steady-state equations (3.82) and (3.83). First show that
Then show that k0C0X = kn—1Cn—10n—1 — xk—nCn^n—2, (3.147)
kn—1Cn—1/t . \ k—ncnx t . . \ , kjCj =--(0n—1 — 1)---(0n—2 — 1), (3.148)
nj nj for j = 1,... ,n — 1. Substitute these expressions into the conservation equation and solve for x.
7. Numerically plot some I-V curves of the Hille Na+ channel model for a selection of values for [Na+]e and [Na+]i and for a range of parameter values, not only those in Table 3.1. Compare to the linear and GHK I-V curves.
8. By making a guess at the shape of the curve for [Na+] = 14.5 mM in Fig. 3.9A, repeat the calculations to obtain the smooth curves in A and B of that figure. In other words, take an arbitrary curve of approximately the same shape as the [Na+] = 14.5 mM curve and calculate the other smooth curves, first by using the independence principle, and second, by using the Hille model. (This is best done numerically.)
9. Write down state diagrams showing the channel states and the allowed transitions for a multi-ion model with two binding sites when the membrane is bathed with a solution containing:
(a) Only ion S on the left and only ion S' on the right.
(b) Ion S on both sides and ion S' only on the right.
In each case write down the corresponding system of linear equations that determine the steady-state ionic concentrations at the channel binding sites.
10. By using an arbitrary symmetric energy profile with two binding sites, show numerically that the Ussing flux ratio is not obeyed by a multi-ion model with two binding sites. (Note that since unidirectional fluxes must be calculated, it is necessary to treat the ions on each side of the membrane differently. Thus, an 8-state channel diagram must be used.) Hodgkin and Keynes predicted that the actual flux ratio is the Ussing ratio raised to the (n + 1)st power (cf. (3.19)). How does n depend on the ionic concentrations on either side of the membrane, and on the energy profile?
11. Choose an arbitrary symmetric energy profile with two binding sites, and compare the I-V curves of the one-ion and multi-ion models. Assume that the same ionic species is present on both sides of the membrane, so that only a 4-state multi-ion model is needed.
12. Suppose the sodium Nemst potential of a cell is 56 mV, its resting potential is -70 mV, and the extracellular calcium concentration is 1 mM. At what intracellular calcium concentration is the flux of a three-for-one sodium-calcium exchanger zero? (Use that RT/F = 25.8 mV at 27°C.)
13. Modify the pump-leak model of Chapter 2 to include a calcium current and the 3-for-1 sodium-calcium exchanger. What effect does this modification have on the relationship between pump rate and membrane potential?
14. Because there is a net current, the sodium-potassium pump current must be voltage dependent. Determine this dependence by including voltage dependence in the rates of conformational change in expression (2.53). How does voltage dependence affect the pump-leak model of Chapter 2?
15. Intestinal epithelial cells have a glucose-sodium symport that transports one sodium ion and one glucose molecule from the intestine into the cell. Model this transport process. Is the transport of glucose aided or hindered by the cell's negative membrane potential?
16. Suppose that a channel consists of k identical, independent subunits, each of which can be open or closed, and that a current can pass through the channel only if all units are open.
(a) Let Sj denote the state in which j subunits are open. Show that the conversions between states are governed by the reaction scheme ka a
(b) Derive the differential equation for Xj, the proportion of channels in state j.
(c) Show that Xj = | | nj(1 — n)k—j, where | | = wj is the binomial coefficient, is a
stable invariant manifold for the system of differential equations, provided that dn
17. Consider the model of the Na+ channel shown in Fig. 3.14. Show that if a and f are large compared to y and S, then x21 is given (approximately) by
dt while conversely, if y and S are large compared to a and f, then (approximately)
18. Show that (3.128) has two negative real roots. Show that when f = 0 and a < —1 , then (3.129)-(3.131) have two possible solutions, one with a + S = — X1, y = —2, the other with a + S = — X2, y = —X1. In the first solution inactivation is faster than activation, while the reverse is true for the second solution.
19. Write a computer program to simulate the response of a stochastic three-state Na+ channel (Fig. 3.15A) to a voltage step. Take the ensemble average of many runs to reproduce the macroscopic behavior of Fig. 3.13. Using the data from simulations, reconstruct the open-time distribution, the latency distribution, and the distribution of N, the number of times the channel opens. From these distributions calculate the rate constants of the simulation.
20. Find the differential equations describing the interaction of a two-state channel with a use-dependent blocker.
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