This is known as the Hill-Langmuir equation. Its derivation assumes that the concentration of A does not change as drug receptor complexes are formed. In effect, the drug is considered to be present in such excess that the number of drug molecules in solution is many times greater than the number of receptor molecules available for the drug to bind to. It can be rearranged to
'ar _ [A] l — 'ar ka and taking logs gives log [ A]-log Ka log [ A]-log Ka
Hence a plot of log (pAR/(1 — pAR)) against log[A] gives a straight line of unit slope. This is known as a Hill plot.
In practice, it is not often possible to directly measure pAR except in radioligand binding experiments. In many experiments it is the relationship between agonist concentration [A] and percentage maximum response (y) which is measured (a dose-response curve) and the Hill plot is made by plotting
The slope of this log-log plot is known as the Hill coefficient (nH) or Hill slope. If the slope is 1 this may imply there is only one agonist binding site on the receptor, while a slope approaching 2 implies two binding sites. In practice, the slope of the line may be greater or less than unity and is rarely an integer. Factors which can affect the Hill slope are particularly the presence of more than one population of receptors with different affinities for the agonist contributing to the response (nH < 1), occurrence of receptor desensitisation (nH < 1), the presence of more than one agonist binding site on the receptor (as occurs with the ligand-gated ion channel receptors) where more than one site needs to be occupied for efficient activation of the receptor (nH > 1), and the presence of spare receptors in the tissue (nH > 1).
Concentration-response curves are often fitted empirically by the expression where nH is the Hill coefficient and ymax is the maximum response. [A]50 is the concentration of A at which y is half maximal. Equation (A3.5) is known as the Hill equation. [A]50 is sometimes denoted by K. However, the constant K obtained by fitting the Hill equation does not correspond to an equilibrium constant as defined above when deriving the Hill-Langmuir equation.
2 KINETICS AND MECHANISMS OF AGONIST ACTION (a) Ion channel receptors
Agonist responses at ligand-gated ion channels and drug effects at ion channels are often more amenable to mechanistic investigation because the response (ionic current through open ion channels when measured with voltage or patch-clamp techniques) is directly proportional to receptor activation. This is a great advantage and has allowed electrophysiological techniques to be used to study ion channel activation and drug block of ion channels in great detail.
The first physically plausible mechanism for receptor activation was proposed by del Castillo and Katz (1957). They made the important distinction that agonist binding and channel opening of the AChR must occur as two separate steps:
In the del Castillo and Katz model it is important to notice that the fraction of receptors occupied is the sum of both active (AR*) and occupied, but inactive (AR) receptors:
This can be rewritten as
where Keff is the effective dissociation equilibrium constant. Thus, most macroscopic estimates of the equilibrium constant for an agonist (radioligand binding, ECso from the occupancy versus agonist concentration-response curve) the estimated equilibrium constant will depend on both affinity for the receptor and subsequent activation steps on the receptor.
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