1.5 Inhibitory Actions at Receptors: I. Surmountable Antagonism
1.5.2 Reversible Competitive Antagonism
We start by examining how a reversible competitive antagonist (for example, atropine) alters the concentration–response relationship for the action of an agonist (for example, acetylcholine). It is found experimentally that the presence of such an antagonist causes the log concentration–response curve for the agonist to be shifted to the right, often without a change in slope or maximal response.
The antagonism is surmountable, commonly over a wide range of antagonist concentrations, as illustrated in Figure 1.16.
The extent of the shift is best expressed as a concentration ratio,* which is defined as the factor by which the agonist concentration must be increased to restore a given response in the presence of the antagonist. The calculation of the concentration ratio is done as follows. First, a certain magnitude of response is selected. This is often 50% of the maximum attainable, but in principle any value would do;** 40% has been taken in the illustration. In the absence of antagonist, this response is elicited by a concentration of agonist, [A]. When the antagonist is present, the agonist concentration has to be increased by a factor r (i.e., to r[A]). Thus, for antagonist concentration [B]3 in Figure 1.16, the concentration ratio is r3 (= r3[A]/[A]).
The negative logarithm of the concentration of antagonist that causes a concentration ratio of x is commonly denoted by pAx. This term was introduced by H. O. Schild as an empirical measure of the activity of an antagonist. The value most often quoted is pA2, where
FIGURE 1.16 The predicted effect of three concentrations of a reversible competitive antagonist, B, on the log concentration–response relationship for an agonist. The calculation of the concentration ratio (r3) for the highest concentration of antagonist, [B]3, is illustrated.
* Or dose ratio — both terms are used.
** Clearly it is sensible to avoid the extreme ends of the range. The concentration ratio can also be estimated using a least-squares minimization procedure to fit the Hill equation (see Sections 1.2.2 and 1.2.4.3), or some other suitable function, to each of the concentration–response curves. This also allows the parallelism of the curves to be assessed. A further possibility is to fit all the curves (i.e., with and without antagonist) simultaneously by assuming that the Gaddum equation holds (see next page) and by making use of the Hill equation, or another function, to relate receptor activation to the measured tissue response.
pA2 = –log[B]r=2
To illustrate this notation we consider the ability of atropine to block the muscarinic receptors for acetylcholine. The presence of atropine at a concentration of only 1 nM makes it necessary to double the acetylcholine concentration required to elicit a given submaximal response of a tissue.
Hence, pA2 = 9 for this action of atropine (–log(10–9) = 9).
We next look at why a parallel shift in the curves occurs, and at the same time we will derive a simple but most important relationship between the amount of the shift, as expressed by the concentration ratio, and the concentration of the antagonist. We will assume for simplicity that when the tissue is exposed to the agonist and the antagonist at the same time, the two drugs come into equilibrium with the binding sites on the receptor. At a given moment, an individual site may be occupied by either an agonist or an antagonist molecule, or it may be vacant. The relative proportions of the total population of binding sites occupied by agonist and antagonist are governed, just as Langley had surmised (see Introduction (Section 1.1)), by the concentrations of agonist and antagonist and by the affinities of the sites for each. Because the agonist and the antagonist bind reversibly, raising the agonist concentration will increase the proportion of sites occupied by the agonist, at the expense of antagonist occupancy. Hence, the response will become greater.
The law of mass action was first applied to competitive antagonism by Clark, Gaddum, and Schild at a time before the importance of receptor activation by isomerization was established. It was assumed, therefore, that the equilibrium among agonist, antagonist, and their common binding site could be represented quite simply by the reactions:
As shown in Section 1.5.5, application of the law of mass action to these simultaneous equilibria leads to the following expression for the proportion of the binding sites occupied by agonist:
(1.48)
Here, KA and KB are the dissociation equilibrium constants for the binding of agonist and antagonist, respectively. This is the Gaddum equation, named after J. H. Gaddum, who was the first to derive it in the context of competitive antagonism. Note that if [B] is set to zero, we have the Hill–Langmuir equation (Section 1.2.1).
If, instead, we take as our starting point the del Castillo–Katz mechanism for receptor activation (see Eq. (1.7)), three equilibria should be considered:
Applying the law of mass action (see Section 1.5.5), we obtain the following expression for the proportion of receptors in the active state:
(1.49)
A R AR
B R BR
+ +
p
K K
AR A
B
A
B A
=
⎛ +
⎝⎜ ⎞
⎠⎟ + [ ]
[ ] [ ] 1
A R AR AR
B R BR
A
B
+ +
*
K E
K
p E
K K E
AR*
A B
A
B A]
=
⎛ +
⎝⎜ ⎞
⎠⎟ + + [ ] [ ] ( )[
1 1
Here, KA and E are as defined in Section 1.2.3, and KB, as before, is the dissociation equilibrium constant for the combination of the antagonist with the binding site. If [B] is set to zero, we have Eq. (1.32).
Equations (1.48) and (1.49) embody the law that Langley had concluded must relate the amounts of the “compounds” he postulated to the concentrations of the agonist and antagonist (see Section 1.1). However, in order to apply this law to the practical problem of understanding how a competitive antagonist will affect the response to the agonist, we need to make some assumption about the relationship between the response and the proportion of active receptors. Gaddum and Schild recog-nized that the best way to proceed was to assume that the same response (say, 30% of the maximum attainable) corresponded to the same receptor activation by agonist whether the agonist was acting alone or at a higher concentration in the presence of the competitive antagonist. This assumption makes it unnecessary to know the exact form of the relationship between receptor activation and response. This was a most important advance, however obvious it might seem on looking back.
We can now consider an experiment in which a certain response (e.g., 30% of the maximum) is elicited first by a concentration of agonist, [A], acting alone and then by a greater concentration (r[A]), when A is applied in the presence of the antagonist. Here, r is the concentration ratio, as already defined. Because pAR* is assumed to be the same in the two situations, we can then write, from Eq. (1.49):*
Here, the left-hand side gives the fraction of receptors in the active state when A is applied on its own. This fraction is assumed to be the same when an identical response is elicited by applying the agonist at an increased concentration (r[A]) in the presence of the antagonist at concentration [B] (right-hand side of the equation).
Dividing each term on the right-hand side by r, we have:
If the expressions on the left and right are to take the same value, the following equality must hold:
Hence,
(1.50)
This is the Schild equation, which was first stated and applied to the study of competitive antagonism by H. O. Schild in 1949. It is probably the most important single quantitative relationship in
* We assume here that the del Castillo–Katz model applies. Using the Gaddum equation, based on the simpler scheme explored by Hill and by Clark, leads to exactly the same conclusion, as the reader can easily show by following the same steps but starting with Eq. (1.48).
E
K E
Er
K K E r
[ ] ( )[ ]
[ ] [ ] ( ) [ A
A
A
B A]
A A
B
+ + =
⎛ +
⎝⎜ ⎞
⎠⎟ + +
1 1 1
E
K E
E
K K
r E
[ ] ( )[ ]
[ ] [ ]
( )[
A A
A B
A]
A
A B
+ + =
⎛ +
⎝
⎜⎜
⎞
⎠
⎟⎟ + +
1 1
1
1
1 +
= [ ]B
KB
r
r− =1 [ ]KB
B
pharmacology and has been shown to apply to the action of many competitive antagonists over a wide range of concentrations. Though originally derived on the basis of the simple scheme for receptor activation described in Sections 1.2.1 and 1.2.2, it holds equally for the del Castillo–Katz scheme, as we have just shown, as well as for more complex models in which the receptor is constitutively active.
One of the predictions of the Schild equation is that a reversible competitive antagonist should cause a parallel shift in the log agonist concentration–response curve (as illustrated in Figure 1.16;
see also Figure 1.18). This is because if the equation holds, the concentration ratio, r, is determined only by the values of [B] and of KB, regardless of the concentration and even the identity of the agonist (provided that it acts through the same receptors as the antagonist). With a logarithmic scale, a constant value of r corresponds to a constant separation of the concentration–response curves, i.e., parallelism, because log (r[A]) – log [A] = log r + log [A] – log [A] = log r, whatever the value of [A].
Probably the most important application of the Schild equation is that it provides a way of estimating the dissociation equilibrium constant for the combination of an antagonist with its binding site. A series of agonist concentration–response curves is established, first without and then with increasing concentrations of antagonist present, and is tested for parallelism. If this condition is met, the value of (r – 1) is plotted against the antagonist concentration, [B]. This should give a straight line of slope equal to the reciprocal of KB.
More usually, both (r – 1) and [B] are plotted on logarithmic scales (the Schild plot). The outcome should be a straight line with a slope of unity, and the intercept on the x-axis provides an estimate of log KB. The basis for these statements can be seen by expressing the Schild equation in logarithmic form:
log(r – 1) = log[B] – log KB (1.51)
A Schild plot (based on the results of a student class experiment on the effect of atropine on the contractile response of guinea-pig ileum to acetylcholine) is shown in Figure 1.17. Note that the line is straight, and its slope is close to unity, as Eq. (1.51) predicts.
FIGURE 1.17 Schild plot for the action of atropine in antagonizing the action of acetylcholine on guinea-pig ileum. Each point gives the mean ± the standard error of the mean of the number of observations shown.
How might the value of pA2 be interpreted in these terms? If the Schild equation is obeyed, pA2 then gives an estimate of –log KB, because, from Eq. (1.51):
The term pKB is often used to denote –log KB.*
To summarize to this point, reversible competitive antagonism has the following characteristics:
1. The action of the antagonist can be overcome by a sufficient increase in the concentration of agonist (i.e., the antagonism is surmountable).
2. In the presence of the antagonist, the curve relating the log of the agonist concentration to the size of the response is shifted to the right in a parallel fashion.
3. The relationship between the magnitude of the shift (as expressed by the concentration ratio) and the antagonist concentration obeys the Schild equation.
1.5.3 PRACTICAL APPLICATIONSOF THE STUDY OF REVERSIBLE COMPETITIVE