Changes in the occupancy of the open-channel state of the receptor as a function of time (pAR*(t)) in response to a perturbation of the receptor equilibrium can be used to obtain information about the rates of channel gating and the interaction of drugs with ion-channel receptors. The system is said to relax towards a new equilibrium. The time course of the relaxation is used to measure rates from the average behavior of many ion channels in a recording, while noise analysis uses the frequency of the moment-to-moment fluctuations in occupancy of the open-channel state at equilibrium to provide information about the rates in the receptor mechanism.
For k states, a relaxation (or noise spectrum) will contain k_j exponential (or Lorentzian) components. Thus, the mechanism in Eq. (6.25) above will have two states in the absence of blocker and so give rise to relaxations (or noise spectra) that can be fitted with single exponential (or Lorentzian) functions. Addition of the blocker creates an extra state (the blocked state), giving k = 3. For k = 3, the occupancy of the open state as a function of time will be described by two exponentials:
PA2R*(t) = PA2R*(œ) + W1 exp(-~] + W2 exp(-~] (6.27)
The reciprocals of the time constants, t1 and t2, are the rate constants and X2. The weights of the exponentials (wj and w2) are complicated functions of the transition rates in Eq. (6.25). However, the rate constants are eigenvalues found by solving the system of differential equations that describe the above mechanism. and X2 are the two solutions of the quadratic equation:
Notice that when P' is small (i.e., when the occupancy of A2R is very small, as it will be if the agonist concentration is low), then k1 + X 2 = a + [B]k+B + k_B
With the simplifying assumption of a small P', the sum and the product of the rate constants measured in an experiment can be used to calculate k_B and k+B if a is known from experiments in the absence of the blocker. This is simply done by plotting the sum or the product of the measured rate constants against blocker concentration. From Eq. (6.32) above, the product of the rate constants should be independent of blocker concentration with a value equal to ak_B, while the sum of the rate constants (Eq. (6.31)) will give a straight line with slope equal to k+B and intercept of a + k_B. If the experimental data is consistent with these predictions, then the data points plotted in this way should lie on a straight line, and this is good evidence that the mechanism of action of the drug is to block the open ion channel.
The assumption that P' is very small has been used when studying the effects of channel blockers on synaptic currents, as the transmitter concentration (and hencepA2R) is probably small during the decay phase of the current. During noise analysis experiments, a low agonist concentration is used so that, again, under these conditions P' should be small.
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