Recording and Analysing Concentration- Response Curves. should be slightly higher, or at least within the range of the dissociation constant K D

Transcription

1 Recording and Analysing Concentration- Response Curves Stefan Dhein Introduction In many cases it is the goal of a study to evaluate the effect of a physiological mediator or a drug on a given organ, such as the heart or vasculature, under physiological or pathophysiological conditions. This sounds simple, but unfortunately is not. Outlined in the following chapter are the principles of analysing a drug- (or mediator-) effect. The first thing to be proven is that the observed effect truly relates to the drug or mediator (subsequently refered to as drug). The first prerequisite, therefore, is to investigate several concentrations of the drug and to show that the drug s effect is related to the drug s concentration. Increasing concentrations should evoke an increased effect. This is unless there is a second (e.g. toxic) effect that counteracts the desired one at higher concentrations. However, in all cases, a concentration-response curve is needed for the analysis of the effect of a specific drug. To enable analysis, at least five concentration steps should be investigated. A concentration-dependent effect of a drug generally favours the hypothesis that it is related to the drug. However, the next question would be whether this drug effect is specific, i.e. is it related to the drug s action at the receptor or signal transduction system, which it is designed for, or is its effect unspecific (e.g. because of the drug s incorporation into the membrane bilayer)? The first hint of a specific effect can be obtained from analysing the concentrationresponse curve: the half-maximum concentration EC 50 should be slightly higher, or at least within the range of the dissociation constant K D of the drug, at its receptor. A difference, by several orders of magnitude, between EC 50 and K D is against the hypothesis that a drug effect is specifically related to a receptor. Further steps in testing the specificity of a drug effect would be to use a competitive antagonist of the drug and perform concentration-response curves in the absence, or presence, of various concentrations of the inhibitor. Assuming a specific drug effect at the receptor system, the competitive antagonist should induce a parallel rightward shift of the concentrationresponse curve. The magnitude of the shift depends on the affinity of the competitive antagonist, and can be analysed by Schild plot or pa 2 -analysis. The method is based on the search for an equally effective concentration of drug A, in the presence of drug B, and analysis of the dose ratio (see below). The pa 2 value should be in the range of the K D value of drug B in order to prove a specific action of the drug at the believed to be involved receptor.

2 Recording and Analysing Concentration-Response Curves Another point worth considering is the effect itself. It is often difficult to define an effect exactly, because initial values or basal activity might be present. Very often, the effect is expressed as a percentage of the maximum effect obtainable (with this or with any other drug; although it is necessary to define this exactly). Another possibility is to define the effect as a percentage of the maximum effect possible (regardless of whether this can be achieved with any drug). For example: it might be that a bradycardic effect is investigated in a given model. The maximum effect possible is cardiac arrest, i.e. heart rate = 0, regardless of whether this can be achieved. Thus, the effect is expressed in relation to the maximum effect possible. It is best to define a maximum and minimum effect, and to express all drug effects as a percentage of the difference between maximum and minimum. This is usually performed with vascular preparations: 1) A maximum vasoconstriction is produced with KCl; 2) The preparation is preconstricted with a vasoconstrictor such as noradrenaline, which is then 3) dilated with increasing concentrations of the vasodilatory drug that is being investigated. 4) At the end of the experiment, a maximum dilatation is induced by nifedipine or papaverine (this has to be done at the end of the experiment because it is difficult to wash out the effects of these drugs). The dilator effect of the drug (difference in tone between the noradrenaline-preconstriction and the drug-induced tone), is then expressed as the percentage of the maximal span between KCl and the papaverine-induced tone. Unfortunately, such a procedure is not possible with all preparations and is often impossible to establish in vivo. Sometimes (such as with hypertension) the drug effect can be expressed as a percentage normalization (defined as the normal tension) of the current value. Many preparations or experimental set-ups, show a certain basal activity or initial value. If a drug effect is only related to the initial value, e.g. is expressed as a percentage of the initial value, the results will be hard to compare with, and might vary greatly between, different experiments. It is therefore recommended to define a maximum or minimum effect to which to relate, or to prestretch or preconstrict the preparation, such that the initial values of all preparations are the same. Description of Methods and Practical Approach First of all, the classical concentration-response curve is described and analysed. In principle, two types of concentration-response curves can be distinguished: (a) the concentration-response curve of a group or (b) the concentration-response curve of an individual. In the first case, the concentration-response curve (or often dose-response curve) describes the percentage of individuals out of this group who exhibit the desired (or investigated) effect. The underlying mathematics are derived from the Gaussian normal distribution (see Tallarida and Jacob 1979; Tallarida and Murray 1987 for further reading). Most often in cardiovascular experimental research, concentration-response curves for an individual are actually investigated in a small number of individuals. Therefore, one individual is exposed to increasing concentrations of the drug or me-

3 6 786 Biochemical and Analytical Techniques diator that is to be investigated (cummulative concentration-response curve). In some cases it might be necessary to test not more than only one concentration at at time instead of a cumulative dose-response. The underlying mathematics are closely related to those already known from biochemistry, the Michaelis-Menten kinetics and the Lineweaver-Burk-diagram. To understand the concentration-response curve, one has to assume that everything is related to the interaction of a drug A with a receptor R, yielding the drug-receptor complex [RA], and depending on the drugs affinity constant K A, according to equation (1): [R] [A]/[RA] = K A (1) with [R total ] = [R] + [RA] (2) That means that the fraction of occupied receptors (RA/R total ) can be described as: [RA]/[R total ] = [RA]/[RA] + [R] (3) In-Vitro Techniques Because according to (1) one can rewrite equation (3) as [R]/[RA] = K A /[A] [RA]/[R total ] = 1/{1 + (K A /[A])} (4) This means that the number of occupied receptors depends on the concentration of the agonist. Because the effect of a given concentration of drug A, [A], is related to the effect (E) of this concentration, E A, and also because E A should be at its maximum (=E max ) when all receptors are occupied, equation (4) can be converted to: which is equivalent to: E A /E max = [RA]/[R total ] = 1/{1 + (K A /[A])} (5) E A = E max /{1 + (K A /[A])} (6) thus relating the effect of a drug to its concentration. Equation (6) is equivalent to the Michaelis-Menten equation known from biochemistry. To obtain the typical sigmoidal concentration-response curve, the effect E is plotted against log ([A]). Now, several values have to be calculated. Firstly, the EC 50 needs to be determined. This is the concentration of a drug at which its half-maximum effect is observed, or at which 50% of the population exhibit the desired effect. The EC 50 is related to the potency of the drug. However, it is important to note, that high potency does not mean a great effect, but a high affinity of the drug to its receptor. It is important not to confuse effect with potency. Second, E max needs to be determined by estimating the saturation effect. Moreover, a threshold and saturation concentration, need to be defined.

4 Recording and Analysing Concentration-Response Curves In some cases, it is not possible to define E max directly from the recorded concentration-response curve. Thus, if the positive inotropic effect of ouabain or other digitalis glycosides is investigated, the positive inotropic effect will be overrun by the toxic arrhythmogenic effect, before reaching saturation. In such cases, one can try to convert the concentration-response curve according to Lineweaver-Burk, in order to obtain a double-reciprocal linear concentration-response-relationship according to the equation: 1/[E] = (K A /E max ) (1/[A]) + 1/E max (7) which is of the form y = a x + b, and can be analysed using simple linear regression. Therefore, the data have to be converted and plotted as 1/E against 1/[A], and a linear regression can be performed on these data. The y-axis intercept of which E max can be calculated is 1/E max, whereas the slope is K A /E max, and the x-axis intercept is 1/K A. Using the E max value, the E 50 value can be calculated. Following this, 1/E 50 and 1/EC 50 can be read from the curve. Nowadays, data can usually be fitted to this curve by using non-linear regression algorithms for sigmoidal curves, which are commercially available in computer programs such as GraphPadPrism (GraphPad Inc., San Diego, USA) or SigmaPlot (Jandel Scientific, Erkrath, Germany). However, even when a computer program is used, the steepness of the concentration-response curve (slope), as described by the Hill coefficient, needs to be taken into account. Most computer programs have at least two options (a) Hill slope = 1 or (b) variable slope. This is important, because the slope can be very steep (most often when toxic effects are measured), rather flat, or and this is the normal case is ~1. If nothing is known, the slope should be calculated using a variable slope setting in the first analysis. This slope of the concentration-response curve means that in normal curves a doubling of the EC 50 causes an increase in the effect by 66%, whereas in steep concentration-response curves, this is 75% or even more. The slope is generally evaluated using a Hill plot (which is also derived from biochemistry) by plotting log (E/(E max -E)) against log ([A]), yielding the following linear function: log(e/(e max -E)) = h log([a]) log(k A ) (8) in which h is the slope of the function. A Hill slope of h = 1 is the normal slope, whereas a Hill slope of 1.6 defines a steep concentration-response curve. In biochemistry, this indicates possible cooperation, and is seen when an allosteric modulation is present. To enable analysis, at least five concentration steps should be investigated. Usually, the steps span several orders of magnitude, and because in most cases the logarithm of the concentration of the drug is plotted, steps are typically 1, 2, 10, 20, 100, 200, 1000, 2000 and so forth, to obtain a plot where intercept points are equal. A classic method to determine the affinity of a competitive antagonist is the Schild-plot or pa 2 -analysis. Here, in the first step, the concentration-response curve for agonist A is recorded. Next, the same concentration-response curve is repeated in the presence of the second antagonist, B, which is being investigated. This is repeated with several concentrations of the antagonist B. For analysis, the concentrations of

5 6 788 Biochemical and Analytical Techniques equal effectiveness are determined, i.e. for a given concentration [A], the concentration of equal effectiveness [A ], in the presence of the antagonist, is sought. The shift of the concentration-response curve for A by B depends on the concentration of B, and its affinity K B to the receptor. The dose ratio [A]/[A ] can be described as: [A]/[A ] = 1 + ([B]/K B ) (9) Next, the concentration of B is determined at which the dose-ratio is [A]/[A ] = 2. The equation can be written as: [A]/[A ] = 2 = 1 + ([B]/K B ) or [B] = K B (10) pa 2 = log([b]) = log([k B ]) (11) In-Vitro Techniques The higher the value for pa 2, the higher the affinity of B is to its receptor. For graphic analysis, one can plot log (DR 1) against log ([B]), yielding a linear relationship that can be analysed by linear regression. At y = log (DR 1) = 1 the x-value represents pa 2. The affinity of the antagonist (K i ) to inhibit the agonist-induced effect can be calculated according to the Cheng and Prusoff equation (Cheng and Prusoff 1973): which can also be written as: K i = IC 50 /([A]/EC 50 ) + 1 (12) K i = IC 50 /([A]/K D ) + 1 (13) IC 50 is the concentration of antagonist, yielding half-maximal inhibition of the agonist-induced effect; [A] is the concentration of the agonist in the assay; EC 50 is the concentration of agonist causing 50% of maximal effect. In many cases, a half-maximum effect is reached before binding to the receptor is half-maximum. This is because an amplifying signal transduction pathway exists between receptor and effector. Therefore, saturation of effect will be achieved before all receptors are occupied, i.e. EC 50 and BC 50 (=half maximum binding concentration) will be different. Thus, if for a full agonist BC 50 <<EC 50, this is the first hint that a receptor reserve (a number of spare receptors) is present. In some cases, the researcher works with partial agonists that have an intrinsic activity smaller than that of a full agonist. The effect of a partial agonist, however, may depend on the receptor reserve in the assay system. The receptor reserve is defined as the number of receptors that remain unused after the maximun effect is reached. The receptor reserve is usually larger in systems with a multicomponent-signal-transduction system. In systems without an amplifying signal transduction pathway, the receptor reserve is usually small or not present. When a partial agonist is used, all receptors have to be occupied to obtain the maximum effect. According to the above considerations it can be stated that, if a partial agonist is used in a system with a large receptor reserve, the partial agonist will

6 Recording and Analysing Concentration-Response Curves exhibit an intrinsic activity of 1, whereas in a system with a small receptor reserve, the intrinsic activity of the partial agonist will be considerably smaller than 1. If the number of receptors used are experimentally so critically reduced (by using an irreversible antagonist) that there is just enough to obtain the maximum effect (100%), the receptor reserve can be determined. The concentration-response curve of the agonist, and the agonist, in the presence of the critical concentration of the irreversible antagonist, can be compared. Examples Competitive Antagonism Firstly, we will consider competitive antagonism. A classic example in cardiovascular research for this type of antagonism is between the β 1 -adrenoceptor antagonist betaxolol and the β 1 -mediated positive inotropic effect of noradrenaline. Theoretically, we can expect in this case, a rightward shift of the concentration-response curve, which depends on the affinity of the β 1 -adrenoceptor antagonist betaxolol to the receptor. To analyse this interaction, a cumulative concentration-response curve is examined on an isolated papillary muscle, and the developing force is recorded. To characterise the interaction between both drugs, a cumulative concentrationresponse curve of the agonist is established in the absence or presence of increasing concentrations of the antagonist, that will shift the curve of the agonist to the right (Fig. 1a). Figure 1a shows concentration-response curves of an agonist in the presence of 0, 1, 10 or 100 nm of antagonist; the antagonist shifts the curves to the right in a concentration-dependent manner. Using these concentration-response curves, the dose ratios are determined by calculating the EC 50 values and defining A /A, i.e. agonist concentrations of equal effectiveness in absence and presence of the antagonist. EC 50 values are now determined yielding, in this example, in values of 10 nm (in absence of the antagonist), 50 nm (in presence of 1 nm antagonist), 300 nm (with 10 nm antagonist) and 2000 nm (with 100 nm antagonist), which means that the dose ratio (DR) is 5, 30 and 200 (and DR 1=4, 29, 199). Next, the log (DR 1) values are plotted against the negative logarithm of the antagonist s concentrations (Schild-plot) (Fig. 1b). These values should give a straight line, in the case of competetive antagonism, with a slope = 1. It is important that not only a narrow range of the antagonist concentration is investigated, because in some cases (e.g. allosteric modulators) a linear relationship might exist within a narrow range which, with an increasing antagonist concentration, can turn into a saturation curve. The Schild-plot data can be analysed using linear regression, which yields the X-intercept at (DR 1)=1 (when DR=2), indicating the pa 2 value. In the example the pa 2 value is A competitive antagonist should give a slope of 1.0 in the Schild-plot. In case of allosteric inhibition the slope of the Schild-plot might differ from 1. However, it is necessary to investigate a dose range that is large enough to evaluate linearity. Allosteric inhibitors might exhibit a linear initial part of the Schildplot.

7 6 790 Biochemical and Analytical Techniques In-Vitro Techniques Figure 1a,b a Typical example for a competitive antagonist. Increasing concentrations of the antagonist cause a parallel rightward shift of the concentration-response curve. b Schild-plot Analysis of the data of panel A, revealing a pa 2 value of 9.7 (corresponding to about M) for the antagonist Non-Competitive Antagonism Let us now consider a non-competitive antagonism such as phenoxybenzamine and noradrenaline. In the case of a non-competitive antagonism, it is not possible to displace the antagonist by higher concentrations of the agonist, possibly because of covalent binding of the agonist to the receptor or by different binding sites. As a consequence, in the presence of the antagonist, E max is lower than in its absence, even at high agonist concentrations (as shown in Fig. 2). Typically, the EC 50 is not altered. It is possible to calculate an inhibitor constant according to the methods used in biochemistry. For this purpose, data should be expressed and plotted as 1/[A] ([A] = agonist concentration) for x-values, and 1/E (E=effect) for y-values, and plotted as a Lineweaver-Burk diagram both in the presence and absence of inhibitor of various concentrations. The inhibitor constant K i can be calculated according to the equation: 1/E = [1 + ([I]/K i )] [(K D /E max ) (1/[A]) + (1/E max )] (14)

8 Recording and Analysing Concentration-Response Curves Figure 2 Example of a non-competitive antagonism With increasing inhibitor concentration [I] the gradient of the linear relationship 1/[A] versus 1/E increases, whereas the X-intercept (in biochemistry=k M, here =K D ) does not change. The Y-intercept 1/E max is typically enhanced, corresponding to a lower E max value. Functional Antagonism In many cases the interaction between two compounds is on a functional level, i.e. at two different receptors or signal transduction pathways that are functionally linked. To give an example, one can consider the functional antagonism between adrenaline and acetylcholine with regard to the heart rate. Adrenaline enhances the heart rate via a β 1 -adrenoceptor stimulation-dependent increase in adenylylcyclase activity, whereas acetylcholine decreases the heart rate by a M 2 -cholinoceptor-dependent inhibition of adenylylcyclase activity, and by inhibition of certain ionic channels, such as the pacemaker current I f, and by activation of the repolarizing current I K.ACh. In these cases, the typical concentration-response curve, in presence of the inhibitor, exhibits a reduced E max and a rightward shift of the curve with increased EC 50, as depicted in Fig. 3. Notice, that depending on the strength of the effect of the full agonist that is under investigation, the functional antagonist might induce a limited rightward shift of an agonist curve with high efficacy, and to a flattening of the concentration-response curve of an agonist with low efficacy. Type I Synergism This type of synergism is the opposite of a competitive antagonism. Drug A is a full agonist, and agonistic drug B is given in its presence. For example, if drug B inhibits the inactivation of drug A, this leads to a higher functional concentration of A at the

9 6 792 Biochemical and Analytical Techniques Figure 3 This example shows a typical functional antagonism with a limited rightward shift of the concentrationresponse curve and concomitant flattening In-Vitro Techniques receptor than in absence of B. As a consequence, the concentration-response curve shifts to the left. A classic example for this type of interaction is between noradrenaline, acting on adrenoceptors, and cocaine or desipramin, which inhibit the uptake of noradrenaline, thereby inhibiting the inactivation of noradrenaline. However, it is necessary to bear in mind that in this case of interaction, the only drug acting at the receptor itself, is drug A. Type II Synergism In contrast to type I synergism, type II synergism means that the efficacy of the agonistic drug A is enhanced by the presence of drug B. Typically, this is achieved by drug B acting on the post-receptor signal transduction pathway. The efficacy of isoprenaline, can be enhanced by phosphodiesterase inhibitors, such as 3-isobutyl-1-methylxanthine (IBMX): isoprenaline acts on β-adrenoceptors and enhances intracellular levels of the second messenger camp; IBMX inhibits the inactivation phosphodiesterase prolonging the effect of camp. This primarily affects the E max of the concentration-response curve of drug A, yielding a higher E max. Because neither the concentration of drug A at the receptor or its affinity are influenced, EC 50 is typically not altered. Additive Effects and Complex Interactions When interactions are investigated it is often necessary to consider basal effects. So might one of the two drugs have an effect and the other drug is tested on top of this (after drug B). For analysis of such effects, it is necessary to plot the total effect of drug A alone, and in combination with drug B, as well as the net effect of drug A in combination with drug B. In such complex interactions, drug B also has a separate effect

10 Recording and Analysing Concentration-Response Curves Figure 4 Functional antagonism of two drugs with basal activity of the investigated system. Drug A is given in either the absence or presence of a fixed concentration of drug B that is sufficient to inhibit the basal activity. The second part of the figure shows the net effect of drug A after the addition of drug B in the absence of drug A. It is therefore necessary to define exactly the maximum effect, if possible, as that which can maximally be achieved in this system with drug A. Let us consider the following situation (see Fig. 4): in a given system is a basal activity of 30% when drug A is present at low concentrations. This basal activity is completely antagonised by a specific concentration of drug B. When drug A is applied in the presence of drug B, the concentration-response curve of drug A is shifted to the right and flattened. To analyse the interaction in more detail, the net effect of drug A alone is compared with the net effect of drug A after the addition of drug B. In our case, the resulting curve is shifted to the right and the E max is lowered, indicating a type of functional antagonism with similar targets of drug action. Another type of complex interaction is the independent antagonism. In that case (Fig. 5), we also have a drug A with a basal effect of 30% that is completely antagonised by drug B. When drug A is tested after drug B, the curve is lowered but not shifted to the right. This becomes apparent when the net effect of drug A is analysed following the addition of drug B, yielding an identical curve to the one obtained by adding drug A only. This type of interaction is an independent antagonism and is often observed if two drugs are tested in combinations that have divergent targets of action.

11 6 794 Biochemical and Analytical Techniques In-Vitro Techniques Figure 5 Independent Antagonism of Two Drugs in a System with Basal Activity Similar to that Shown in Fig. 4. Again, drug B is given in a fixed concentration that is sufficient to antagonise the basal activity. However, as seen in the second part of the figure, the net effect of drug A is not altered by the presence of drug B, indicating independent actions In a similar way, complex synergism might occur (with or without a shift of the EC 50 ) when a second drug is present that exerts its effect in the same direction as drug A. When EC 50 of the net curve for drug A is changed in presence of drug B (i.e. shifted to the left), we have a functional synergism. An independent synergism is observed when the resulting net curve is identical to the one when drug B is absent. Thus, regarding independent interactions (synergistic or antagonistic), there is no effect of drug B on the net effect of drug A, hinting at different drug targets and different mechanisms of actions. In additive effects, the effect of a combination of two drugs might simply be the addition of both effects or at least more or less so (over- or under-additive). If B is equally effective as A, the combined effect of A+B should be 2 A. Meaning that, in the case of a normal concentration-response curve (Hill slope = 1) the effect of A+B should be 2 EC 50 of A. This is under the assumption that, the effect of 1A is the effect of the EC 50 of A, and would result in an E max value of 66 67%. In the case of a steep concentration-response curve, E max should be about 75%. Thus, the effect of the combined effect is larger when the concentration-response curve is steeper.

12 Recording and Analysing Concentration-Response Curves For evaluation purposes, it might be helpful to calculate an additive-concentration-response curve in theory, and compare this with the one being measured. In the case of an independent synergistic interaction, the theoretical curve of the combined effect (in %) can be calculated as: E A+B = E A + {E B [(100 E A )/100]} (15) In the case of a normal concentration-response curve (Hill slope = 1), the independent additive synergistic effect is usually larger than the additive effect with similar targets of action (approximately 66 67% of E max at 2 EC 50 (see above). In the case of a steep concentration-response curve, however, additive and independent effect might be equal or, in the case of a very steep curve, the additive effect might be larger than the independent additive effect. To construct an additive curve in theory in order to compare it with the registered curve (Fig. 6), the concentration of drug A must first be determined, whose effect is equal to the concentration of drug B. Next, the X-intercept of twice the concentration of A is sought along with the corresponding effect of 2A. In the subsequent step this (the found concentarion of A) is subtracted from the concentration-response curve by shifting it to the right by 1A. The same is carried out for 3A and 4A and so forth, yielding the theoretically-calculated additive curve for a combination of drugs with similar effects. The measured effect can now be classified as additive, over- or under-addi-tive or (see above) as independent. In a similar (but reverse) manner an antagonistic interaction can be classified. With regard to the interpretation of such effects, great care should be taken and over-interpretation should be avoided. However, in the case of a difference to the simple additive curve, this suggests that the targets of the drug action of drug A and B are not identical. Values of the measured curve that coincide with those of the theo- Figure 6 Construction of a theoretically-calculated additive concentration-response curve. First, the concentration of equal effectiveness is determined for drug A. Next, X-intercept of 2A and the corresponding point on the concentration-response curve are determined. From that point, 1A is subtracted and yields the first point of the theoretically-calculated additive curve. The same was done for 2A, 3A, 4A and so forth

13 6 796 Biochemical and Analytical Techniques retically-calculated additive curve, might indicate similar mechanisms of action for drugs with reversible effects or different mechanisms for drugs with irreversible effects. An over-additive effect normally contradicts the hypothesis of a similar or identical mechanism of action. Under-additive effects, for example, are observed when a full agonist is tested in the absence and presence of a partial agonist. If the registered effect corresponds to the theoretical independent effect, it might mean that either the effects of both drugs are mediated by different mechanisms and/or that the effects of a combination partner are independent from the effect of the previous dose (as is the case in exponential dose-response curves). However, this does not exclude an effect by a similar or identical mechanism. Interested readers are referred to the more detailed literature given in the references section. In-Vitro Techniques Troubleshooting Mathematical analysis of concentration-response curves is now done by a number of computer programs (e.g. GraphPadPrism, SigmaPlot, Origin 6.0 etc.). Generally, the data table has to be generated first, then the data are transformed [concentration to log (concentration)] and thereafter a non-linear fit for sigmoidal curves is performed. If nothing is known about the concentration-response relation, the slope of the nonlinear regression curve should be set as variable, otherwise it may be set to 1.0. The software will give the goodness of fit as r 2 which should be higher than The Hill slope, also given by the software, is the slope of the concentration-response curve. It is normally about 1.0; in case of toxicity curves, however, it is often steeper (around 1.6 or more). The accuracy of a concentration-response curve can be estimated from the goodness of fit (which should be higher than 0.85), the Hill slope (which should be around 1.0; very flat curves are suspect), EC 50 and E max as well as the standard deviation of E max and EC 50. A particular problem is incomplete concentration-response curves. They might be due to toxic effects of the investigated drug that do not allow for investigation into the effect of the drug in higher concentrations on the desired parameter. For example, if inotropy is investigated using certain phosphodiesterase inhibitors (such as amrinone or milrinone), these drugs might exert arrhythmogenic effects before reaching a saturation of the concentration-response curve in regard to the inotropic effect. In these cases, it can sometimes be helpful to convert the effect to 1/E and the concentration to 1/C, and to plot 1/E against 1/C according to Lineweaver and Burk. A linear regression yields the Y-intercept as 1/E max that can simply be converted to E max (for more details see above). Occasionally, the measured concentration-response curve does not look like a simple sigmoidal curve but more like a biphasic curve. If this is the case, the first step is to test whether it is truly biphasic or not. Therefore, two fits have to be performed, first the simple sigmoidal fit and second a biphasic fit. In the next step an F-test is performed to find out which fit is best. In the case of a biphasic curve, one has to con-

14 Recording and Analysing Concentration-Response Curves Figure 7 Effect of a partial antagonist with intrinsic activity on the concentration-response curve of a full antagonist. Notice the agonistic effect of the drug in absence or at very low concentrations of the full agonist sider that the drug might act on two different receptors, or might activate two different mechanisms. If known, one can try to repeat the concentration-response curve in the presence of an inhibitor of one of the receptors. If the inhibited receptor is involved in the biphasic curve, the curve should be converted to a monophasic curve in the presence of the inhibitor. Some drugs that are used as antagonists might exert intrinsic agonism. This means that the drug binds to the receptor and activates it but does not lead to its full activation. A classic example is the β-adrenoceptor antagonist pindolol. Intrinsic agonism might cause some trouble or confusion when testing it in combination. As outlined above and shown in Fig. 7, the addition of an inhibitor with intrinsic activity causes some agonistic effects that are either detected at very low concentrations or in the absence of a full agonist. At higher concentrations, the antagonist will inhibit the effects of the full agonist because this drug occupies a part of the receptor without activating it fully. This behaviour is similar to a partial antagonism. When antagonistic effects are investigated, the type of antagonism should be clearly described. With regard to drugs that act at the same receptor: it is possible that an inhibitor competes with the full agonist for the receptor-binding site without activating the receptor, this being a typically competitive antagonist without intrinsic activity. However, it should be kept in mind that this type of drug neither activates nor deactivates the receptor. Thus, in the absence of a full antagonist, this type of antagonist does not exhibit any effect. In the absence of a full agonist, a partial antagonist, a partial agonist or an antagonist with intrinsic activity causes some activation of the receptor but diminishes the effect of a full agonist. Degree of inhibition and shift of the curve depend on the affinity of the full agonist. Besides, another type of antagonist exists: the inverse agonist. An inverse agonist can bind to the receptor and deactivate it. It therefore exerts the opposite action of a full agonist. Thus, in contrast to a competitive antagonist, an inverse agonist will show inhibition of the receptor in absence of a full agonist. A classic example for this type of antagonisms or inverse agonisms comes from the GABA receptor. Benzodiazepines, such as diazepam, act as agonists by binding to the GABA receptor (at a binding site different to the GABA

15 6 798 Biochemical and Analytical Techniques In-Vitro Techniques binding site). They activate the receptor and enhance the effect of the endogenous transmitter GABA, thus allowing a chloride current. This action can be completely blocked with the competitive antagonist flumazenil without exerting intrinsic activity. Flumazenil is used in benzodiazepine intoxication and competes with benzodiazepines for their binding site. Bretazenile represents a partial agonist of the GABA receptor. Methyl-6,7-dimethyl-4-ethyl-β-carbolin-3-carboxylic acid (DMCM) acts as an inverse agonist at the GABA receptor, binding to the same binding site as benzodiazepines and reducing the effect of GABA. While the full agonist diazepam leads to sedation and reduces anxiety, the inverse agonist DMCM induces anxiety and agitation. It might be necessary to characterise a drug as either a receptor-blocker (competitive) or an effector-blocker (non-competitive). In such cases, a dose-ratio test might help. First, a known competitive blocker (A) is tested against a known agonist (C) by measuring the effect of a given concentration of A, compared with the effect of increasing concentrations of C. Subsequently, the unknown blocker B is tested against the agonist C in the same way. Finally, the combined effect of blockers A and B is tested against increasing concentrations of C. If the unknown blocker B is a competitive blocker, the dose ratio follows the equation: DR A+B = DR A + DR B 1 (16) If the unknown drug B is a non-competitive blocker, the dose ratio can be described as: DR A+B = DR A DR B (17) Sometimes, the duration of drug exposure might influence the effect. Normally, the interaction between drug and receptor is based on weak, reversible binding forces, so that an equilibrium is reached within a short period of time. However, toxic effects are often related to covalent binding or other effects, possibly related to the duration of exposure. In such cases, prolongation of exposure time will yield the true concentration-response curve. References Arunlakshana O and Schild HO (1959): Some Quantitative Uses of Drug Anatgonists. Brit J Pharmacol 14: Cheng Y-C and Prusoff WH (1973): Relationship Between the Inhibition Constant (K i ) and the Concentration of Inhibitor Which Causes 50 Percent Inhibition (IC 50 ) of an Enzymatic Reaction. Biochem Pharmacol 22: Goldstein A, Aronow L and Kalman SM (1987): Principles of Drug Action: The Basis of Pharmacology. 2 nd edition. Wiley, New York, 1974 Kenakin TP: Pharmacologic Analysis of Drug-Receptor Interaction. Raven Press, New York. Lazarena S and Birdsall NJM (1993): Estimation of Antagonist KB from Inhibition Curves in Functional Experiments: Alternatives to the Cheng-Prusoff Equation. Trends Pharmacol Sci 14: Pöch G and Juan H (1990): Wirkungen von Pharmaka. Thieme Verlag, Stuttgart. Tallarida RJ and Jacob LS (1979): The Dose Response Relation in Pharmacology. Springer-Verlag, Berlin. Tallarida RJ and Murray RB (1987): Manual of Pharmacologic Calculations with Computer Programs. Springer-Verlag, New York. Van den Brink FG (1977): General Theory of Drug-Receptor Interactions. Drug-receptor Interaction Models. Calculation of Drug Parameters. In: Van Rossum JM (Ed): Kinetics of Drug Action. Exp Pharmacol Volume 47, Springer- Verlag, Berlin, pp