Interpreting enzyme and receptor kinetics - CiteSeerX

Nuclear Medicine and Biology 30 (2003) 819 – 826

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Interpreting enzyme and receptor kinetics: keeping it simple, but not too simple1夞 Kenneth A. Krohn*, Jeanne M. Link Department of Radiology, University of Washington, Seattle, WA 98195-6004, USA

Abstract The hyperbolic parabola is commonly used to summarize kinetics for enzyme reactions and receptor binding kinetics. Depending on the experimental conditions, certain assumptions are valid but others might be violated so that the parameters used to fit this hyperbolic function, the maximum asymptote and the equilibrium constant, require different interpretations. The first topic of this review compares enzymeinduced transformations and receptor binding in terms of the appropriate assumptions. The second topic considers the complication of adding a competitive inhibitor as an enzyme substrate or a receptor ligand and the subtleties of inferring the equilibrium dissociation constant from the concentration of inhibitor (for example unlabeled drug) that leads to the midpoint, IC50, of an inhibition curve. Receptor binding may be measured directly by a concentration assay or as a pharmacodynamic response variable. © 2003 Elsevier Inc. All rights reserved. Keywords: Receptor kinetics; Enzyme kinetics; Kinetic modeling; Pharmacodynamic models; IC50; Cheng-Prusoff equation

1. The equilibrium model for enzyme kinetics involving only two reactants Radiopharmaceutical chemists borrow heavily from the rich literature of enzyme kinetics [1]. However, there are pitfalls in a direct translation to receptor studies that warrant a review of some experimental details and assumptions that influence both data analysis and mechanistic interpretation. k1

kP

E ⫹ S ^ ES 3 E ⫹ P k⫺1

E denotes an enzyme and S is a substrate and they form a reversible intermediate, ES, which then breaks up at a rate kp to form product P and the original enzyme. In the following analysis square brackets are used to indicate molar concentrations. The receptor-binding analog looks similar except that ligand L is substituted for S and receptor R is substituted for E. Characteristics of enzymes and their reactions: 1. Enzymes are catalysts; they are not consumed 2. E and S react rapidly to form ES complex 3. Stoichiometry is 1:1 and yields 1 product

夞 Contribution to the Receptors Meeting, San Diego, February 2003. * Corresponding author. E-mail address: [email protected] (K. A. Krohn). 0969-8051/03/$ – see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0969-8051(03)00132-X

4. E, S and ES are at equilibrium. ES returns to E⫹S faster than it goes to E⫹P 5. [S] ⬎⬎ [E] so that formation of ES does not alter [S] 6. The overall rate of catalysis is governed by breakup of ES to form E⫹P The rate of product formation is d[P]/dt, which is also -d[S]/dt and defines the velocity, v, of the enzyme reaction. We can define the equilibrium dissociation constant, k⫺1 [E][S] ⫽ , and invoke conservation, [E]t ⫽ KS ⫽ k1 [ES] d[P] [E]⫹[ES], so that v ⫽ ⫽ kp[ES] which, when dt v divided by the conservation equation, gives [E]t kp[ES] v . If we define Vmax⫽ kp[E]t, then ⫽ [E] ⫹ [ES] kp[E]t v [ES] . It is common to parameterize [ES] ⫽ ⫽ Vmax [E]⫹[ES] in terms of KS, which allows cancellation of [E] and can be rearranged to Eq. 1. v⫽

Vmax[S] KS ⫹ [S]

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This equation leads to the well-known hyperbolic parabola graph [2], as shown in Fig. 1. One approach to linearize the display of enzyme kinetic

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K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) 819 – 826

rather than an equilibrium constant. The function is the same but with a different interpretation that relates to the mechanistic details of ES. To understand the difference between Eqs. 1 and 2, we must consider the distinction between equilibrium and steady-state and understand what constitutes “irreversible”. d[ES] Steady-state only requires that ⫽ 0, but ES can dt decompose by two routes: kp

k⫺1

ES 3 P ⫹ E, or ES 3 S ⫹ E

Fig. 1. Simulated enzyme kinetic data plotted on a linear axis.

data has been suggested by Eadie and Scatchard [1]. StartVmax 1 v ⫽ ⫺ v , which ing with Eq. 1, one can derive [S] KS KS v leads to a linear plot of versus v. This plot is particularly [S] helpful to visually identify when two enzymes catalyze the same reaction, which is identified by a break in the slope of the line. Linearization processes were developed before computer programs were available to evaluate the binding data directly. These methods are still useful for illustration purposes but they are not appropriate for data analysis because the linear transformation introduces a non-uniform distribution of error that violates the assumption of a regression model. Nonlinear regression must be used for accurate analysis of the hyperbolic parabola graphs that characterize enzyme and receptor reaction kinetics [2].

2. The steady-state model for enzyme kinetics involving only two reactants Van Slyke has suggested an alternative analysis of enzyme kinetic experiments [1]. The enzyme reaction involves two irreversible steps and the rate equation describes the time required for the overall reaction, written as the sum of the times for each step. k1

kp

3. Enzyme kinetics. Advanced concepts

E ⫹ S 3 ES 3 E ⫹ P 1 1 ⫹ k1[S] kp kp ⫹ k1[S] 1 1 ⫽ ; the rate constant is and v ⫽ [E]tot. k1kp[S] t t Again, Vmax⫽kp[E]tot but now K⫽kp/k1. Algebraic arrangement yields The time required is expressed as t ⫽

v [S] ⫽ Vmax K ⫹ [S]

Steady-state is close to equilibrium whenever kp⬎⬎k⫺1, but when kp⬇k⫺1, then [ES]SS⬍[ES]eq. If [P]⯝0, any backd[ES] reaction can be neglected and ⫹ ⫽ k1[E][S]. The dt d[ES] rate of decomposition of ES is ⫺ ⫽ k⫺1[ES] dt d[ES] d[ES] ⫹ kp[ES] . At steady-state ⫹ ⫽ ⫺ , and so dt dt k1[E][S] ⫽ 共k⫺1 ⫹ kp)[ES] . By combining velocity and kp[ES] v and, taking [ES] from conservation, ⫽ [E]t [E] ⫹ [ES] k1[S] v k⫺1 ⫹ kp ⫽ . above, k1[S] Vmax 1⫹ k⫺1 ⫹ kp The cluster of rate constants, termed the “Michaelis” [E][S] k⫺1 ⫹ kp ⫽ Km ⫽ , so Km is Kss, not constant, is k1 [ES] v [S] Keq. Again the equation is hyperbolic, , ⫽ Vmax Km ⫹ [S] but now the assumptions are different. The Ks are also k⫺1 kp different; KS ⫽ ⯝ Km only when kp30 and Km ⯝ k1 k1 when kp⬎⬎k-1. It is important to appreciate that the experimentalist has the option of conducting enzyme reactions under either equilibrium or steady state conditions. Receptor binding experiments in vitro are invariably analyzed at equilibrium.

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but now the K is a ratio of two forward rate constants

From a complete kinetics perspective, we can write differential equations for the rate of formation and consumption of each of S, E, ES, and P plus a mass balance equation for Etot, thus avoiding any requirements about steady-state or equilibrium. However, we cannot write a general integral equation for [S]time or [P]time unless we invoke the condition d[ES] [S]0 that ⫽0, the steady-state requirement. If is high, dt [E]tot the time to reach steady-state is short and the equation is valid.

K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) 819 – 826

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Conditions where the kinetic equations become more complicated: ● If kp is a reversible process; k⫺p⫽0. ● As product builds up, we can no longer assume [P]⫽0. ● When a sequence of steps is involved: ES13ES23 3E ⫹P. For example, consider the rate equation that includes the reverse reaction of E ⫹ P to make ES. For parallel nomenclature, use k2 and k⫺2 rather than kp, so that vnet ⫽ k2[ES] ⫺ k⫺2[E] [P] and the conservation equation is [E]tot⫽[E]⫹[ES]⫹[EP]. The algebra follows the original [S] [P] Vmax f ⫺ Vmax r KmS KmP . scheme and results in vnet ⫽ [S] [P] 1⫹ ⫹ KmS KmP Note that the general hyperbolic parabola form of both Eq. 1 and 2 persists and we can still interpret a simple experiKm 1 ment. When [S] ⫽ Km, v ⫽ V ⫽ Vmax . For Km ⫹ Km max 2 an efficient enzyme system, Km⬵[S]intracellular. Under this condition, a molecule-induced change in Km provides effective “regulation” of the enzyme. Some enzymes are composed of sub-units with each one bearing a catalytic site. If these sites are the same and are independent, then the kinetics can be described by the same hyperbolic velocity curves; n molecules of one site are indistinguishable from one molecule of n sites. The algebra Vmax[S] . Alternatively, if S at one will still simplify to v ⫽ K ⫹ [S] site influences the binding of additional S to open sites, either by activation or inhibition, then the kinetic curves no longer follow hyperbolic parabolas. These are called allosteric enzymes and they yield sigmoidal velocity curves. One classic example involves the O2 saturation curve of hemoglobin [1].

Fig. 2. In this figure the data is depicted as a receptor binding experiment.

the receptor system also results in a hyperbolic parabola Bmax [L] . The reader may show that nonplot, [LR]⫽ KD⫹[L] specific binding, NSB, results in [L-NSB]⫽knsb[L]. Both of these functions are graphed hypothetically in Fig. 2. The same table of numbers was used here as for the previous enzyme graphs. This hyperbolic function is often transformed to a linear form for viewing: [LR]KD ⫹ [LR][L] ⫽ Bmax[L] leads to the classical Scatchard Eq. 3: [LR] Bmax 1 ⫽ ⫺ [LR] [L] KD KD which suggests a plot of

[LR] versus [LR], where the slope [L]

⫺1 . The simulated data of Fig. 2 is transformed to this KD equation in Fig. 3.

is

4. How does enzyme kinetics relate to receptor models? kon

The ligand-receptor reversible reaction L⫹ R ^ LR, koff looks much like our initial E ⫹ S reaction and both generally behave according to second-order rate laws that koff lead to a steady state. In the receptor case, KD ⫽ kon [L][R] at equilibrium. When [R] ⫽ [LR], then [L] ⫽ ⫽ [LR] KD. The fractional occupancy of receptor is described by [R][L] [L] KD [LR] ⫽ ⫽ . To conserve mass, [R][L] [L]⫹KD [R]⫹[LR] [R]⫹ KD [R]tot ⫽ [R]⫹[LR] and is often denoted as Bmax, so that

(3)

Fig. 3. A Scatchard plot of the simulated receptor binding data.

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K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) 819 – 826

v kp[ES] by the equilibrium model. ⫽ [E]tot [E] ⫹ [ES] ⫹ [EI] Substitute for [ES] and [EI] and recall Vmax⫽kp[E]tot, so [S] KS v ⫽ , which can be rearranged to that [S] [I] Vmax 1⫹ ⫹ KS Ki v Vmax

Fig. 4. Simulated receptor binding data for a radioligand binding to a receptor with a high affinity site as in Fig. 3 and an order of magnitude lower affinity site.

A radioligand that binds to two independent sites, both with the same Bmax and KD, can also be modeled with the hyperbolic function. [LR]tot ⫽ n

Bmax[L] KD⫹[L]

(4)

A radioligand that binds to two specific sites, with different associated Bmax and KD, can also be modeled with the hyperbolic equation but the two-site property is best appreciated when the data is viewed as a Scatchard plot [3]. [LR]tot ⫽

Bmax1[L] Bmax2[L] ⫹ KD1 ⫹ [L] KD2 ⫹ [L]

(5)

A simulated example is shown in Fig. 4, which is plotted as a Scatchard graph to visually appreciate the two-component nature of binding. This is the only example where we will consider a single ligand. The more common situation in radiopharmaceutical development involves two ligands competing for a single binding site. Note that radioactive contaminants can produce a Scatchard plot with a similar shape, as described in [4]. 5. Competitive inhibition experiments: enzymes In enzyme studies, one encounters situations where an inhibitor I competes for an enzyme’s catalytic site but EI complex does not result in any product. An equivalent situation exists in receptor binding studies. KS

Ki

E ⫹ I 7 EI [E][I] [E][S] and Ks ⫽ . As before, v⫽kp[ES] where Ki ⫽ [EI] [ES] but now [E]tot⫽[E]⫹[ES]⫹[EI].

[S] . [I] KS 1 ⫹ ⫹ [S] Ki

冉

冊

(6)

The steady-state model gives the same equation, but with Km rather than KS in the denominator. These equations have the general form of Eq. 1 except that KS is now modified by [I] the term 1 ⫹ . Vmax is unchanged but the apparent KS Ki is increased. In this model EI does not interact with S; the limited supply of enzyme is simply shared between two [I] , is substrates, I and S, and, as Segel [1] explains 1 ⫹ KI the distribution of enzyme available for binding S. Consider the fractional difference in velocity in the presence of inhibitor, vi/v0, where v0 is the reaction velocity Vmax[S] [I] KS 1 ⫹ ⫹ [S] Ki vi when [I]⫽0: ⫽ v0 Vmax[S] KS ⫹ [S] KS ⫹ [S] . It is convenient to convert from ⫽ [I] KS 1 ⫹ ⫹ [S] Ki v0 ⫺ vi relative velocities to fractional inhibition: ⫽ v0 [I] , which has a value of 0.5 when [I] has a [S] Ki 1 ⫹ ⫹ [I] KS [I]0.5 concentration of IC50. Thus, 0.5 ⫽ , [S] Ki 1 ⫹ ⫹ [I]0.5 KS [S] Ki 1 ⫹ KS and or 2 ⫽ 1 ⫹ [I]0.5

冉

冊

冉

冊

冉

冉

冊

冊

冊

冉

冉

冊

冉

[I]0.5 ⫽ Ki 1 ⫹

kP

E ⫹ S 7 ES 3 E ⫹ P

⫽

冊

[S] ⫽ IC50 . KS

冉

冊

(7)

Reciprocal plots are also used to interpret competitive 1 inhibition data. Eq. 6 can be inverted to give v KS [I] 1 1 ⫽ 1 ⫹ ⫹ . For each [I], a new Vmax Ki [S] Vmax 1 straight line can be drawn, but the y-intercept stays at . Vmax

冉

冊

K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) 819 – 826

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kept as low as possible while still maintaining acceptable counting statistics for accurately assaying separated [L*] and [L*R]. It is also important to remember that the analysis requires the ability to separate bound and free species and assay each without disturbing equilibrium. Clearly systematic errors can occur during this process. 7. Relationship between IC50 and KD

Fig. 5. Plot of simulated radioligand binding assay where a fixed amount of L* and R are competed with varying amounts of additional unlabeled drug L.

The slope changes, related to x-intercept.

冉

冊

KS [I] 1 ⫹ , as does the Vmax Ki

6. Competitive inhibition experiments: receptors In a competitive radioligand binding study, the experimenter selects an amount of [L*] and [R] and then makes measurements of total radioligand binding, [L*R], in the presence of varying amounts of unlabeled ligand. The algebra for this analysis follows the principles developed for competitive enzyme inhibition by two substrates. In this case L* and L may have the same rate parameters if the labeling procedure did not change the forward or reverse binding rate constants. Fig. 5 shows simulated data for a single receptor type. Half way between the non-specific binding NSB asymptote and the maximum binding asymptote (L* with no competing L) is the 50% inhibition concentration, IC50. If the experiment involves a single class of binding sites, it will follow the law of mass action Bmax[L] with a sigmoidal dose-response. [LR]tot ⫽ , KD ⫹ [L] where [L]⫽[L*]⫹[L]cold and total binding includes NSB. If [L*R] is the binding of total ligand times the fraction of ligand that is labeled, then Bmax([L*] ⫹ [L]cold) [L*] [L*R] ⫽ ⫻ KD ⫹ [L*] ⫹ [L]cold [L*] ⫹ [L]cold Bmax[L*] ⫽ . In receptor binding assays where KD ⫹ [L*] ⫹ [L]cold L and L* are the same drug with a single KD, and [LR]⫽0.5[LR]tot, Bmax[L*] Bmax[L*] 0.5 ⫽ KD ⫹ [L*] KD ⫹ [L*] ⫹ [L]cold

(8)

Experiments measure [L]cold for half-maximum inhibition: IC50 ⫽ [L]cold ⫽ [L*] ⫹ KD. Clearly, use of a high [L*] will not provide sensitivity to KD. [L*] is, in practice,

Consider a single type of receptor binding site competing for two ligands, L and I, with equilibrium dissociation constants of KD and Ki, respectively. The total concentration of binding sites is Bmax. At equilibrium the concentrations of the two different free ligands are [L] and [I]. In the experiment, the bound-to-free ratio of labeled ligand, [L*R]/[ L*], is measured after variable concentrations of labeled ligand are added to a fixed concentration, Bmax, of receptor binding sites. This is exactly the experiment depicted in Fig. 3 and the graph is linear with slope and intercept as indicated for that figure. In the competition experiment, the same protocol can be followed but with addition of a fixed total concentration of inhibitor. The assumption is that the equilibrium dissociation constant for the inhibitor is different from that for the labeled ligand, Ki ⫽ KD. In analyzing these experiments, investigators often assume that [I] at equilibrium is constant and is independent of KD. Other assumptions implicit in the following equations are that all components of the system are at equilibrium, that [L*R] can be measured without changing the equilibrium, that both ligands interact independently with the receptor and that there are no interactions between ligands. While it is tempting to equate KD, the equilibrium dissociation constant, with IC50 from an inhibition curve, this simple graphical analysis can lead to errors, as shown in this section. In their classic analysis, Cheng and Prusoff [5] showed algebraically that Ki⫽IC50 under some specific conditions, such as noncompetitive kinetics, but the equality fails when competitive inhibition kinetics applies. They developed an analysis for a competitive inhibition system that is reversible and rapidly reaches equilibrium. Their Eq. 3 is our Eq. 7 and shows that Ki is less than IC50 by a factor related to the ratio [S]/KS. IC50 thus depends on the substrate concentration [S] for each experiment and this value cannot be compared between laboratories. The reader should consult the primary reference [5] for derivation of the equations for a noncompetitive inhibitor or an uncompetitive inhibitor. In this special case Eq. 6 becomes v [S] (9) ⫽ Vmax [I] [I] KS 1 ⫹ ⫹ [S] 1 ⫹ Ki KiES

冉

冊 冉

冊

In this case inhibitor has an affinity for the free enzyme, Ki, and for the ES complex, KiES. IC50 is only independent of [S] provided that [S]⬎⬎KS. Recalling that when [I]⫽IC50, v0 ⫺ vi ⫽0.5 , the rearrangement used to arrive at Eq. 7 now v0

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K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) 819 – 826

KS ⫹ [S] . The algebra requires more steps [S] KS ⫹ Ki KiES but the outcome is often simpler, especially when the affinity is equal for free E and ES complex, Ki⫽KiES, in which case IC50⫽Ki. Cheng and Prusoff [5] showed other situations where Ki⫽IC50 but, for the reversible competitive equilibrium experiment, which is most commonly encountered by the radiopharmaceutical chemist, this equality fails. The Cheng-Prusoff equation derives from Eq. 7 and has been used to obtain a KS for enzymes or KD for receptor binding from the measured IC50 for a competitive inhibition experiment. For enzymes and receptors, respectively, yields IC50 ⫽

Ki ⫽

IC50 IC50 and Ki ⫽ [S] [I] 1⫹ 1⫹ KS KD

(10)

This equation should be used with appropriate caution, however, as discussed below. The problem comes about when the Scatchard graph is not linear and this occurs whenever the inhibitor concentration is an appreciable fraction of the receptor concentration, Bmax. Depending on the range of conditions over which the competition experiment was done, it may be difficult to judge the validity of the Cheng-Prusoff equation by visual inspection of the Scatchard graph, Fig. 6. The Cheng-Prusoff equation is widely applied, albeit sometimes without appreciation for assumptions that may break down in some radiopharmaceutical experiments. The assumptions are that free and total radioligand concentrations are approximately equal and that this same condition applies to the unlabeled ligand, the inhibitor. While the algebra developed above refers to free concentration as a carryover from the language of enzyme kinetics, the initial total concentration of reactants is more experimentally accessible. Inclusion of total concentration complicates the algebra and has been ignored in Eq. 10, although other investigators have developed exact equations relating Ki and IC50 [6,7]. The interested reader should refer to the appendix provided in [6] for an exact solution. Because this reference uses different nomenclature to arrive at their final equation 17, it is translated below to the abbreviations used in this review. Ki ⫽

冉冉 冉冉 冉 冉

[L*]tot 1⫹ 2KD

⫹ KD

IC50 [L*R] ⫹2 [L*] 0 [L*R] ⫹ [L*] [L*R] ⫹1 [L*] 0

冢

冊 冊 冉 冊 冊 冊 冊

冣

[L*R] [L*] 0 [L*R] ⫹2 [L*] 0

冊

Fig. 6. Simulated binding curves for a competitive inhibitor experiment. KD was 0.1 nM and Ki was 0.2 nM. This small difference might be typical of a minor decrement in affinity caused by labeling. Bmax was 2 nM and two values were used for [I]. Note that the curvature is greatest when [I] approaches that of Bmax.

冉 冊

[L*R] denotes the experimental binding [L*] 0 ratio when the predetermined total amount of radioligand is used with no competitive inhibitor present. While this equation appears considerably more intimidating than the previous Eq. 10, it only adds some correction factors to the earlier equation and is still easily programmed as a spread[L*R] is sheet function. It reduces to Eq. 10 when [L*] 0 negligible. Under what conditions might the Cheng-Prusoff equation give unacceptable errors? A few generalizations can be made. The correct equation always results in a lower Ki, increased affinity. There may even be conditions that give rise to negative values for Ki, suggesting that the data require an alternative binding mechanism. The error is min[L*R] imized when is small and it is increased when [L*] 0 IC50 is small. These errors can be order-of-magnitude or greater, leading the prudent investigator to use the exact Eq. 11 rather than the simplified version, Eq. 10. where the ratio

冉 冊

冉 冊

8. Analysis of pharmacological endpoints 0

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In all of the analyses developed above, inhibition is measured after a chemical separation and using an assay that is quantified in molar or mass units. However, an analogous experiment is done in pharmacology where the ordinate is measured in units of a pharmacodynamic response. Even though the connection between agonist bind-

K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) 819 – 826

ing and response can be complex and indirect, the same mathematical analysis with the Cheng-Prusoff equation is frequently applied. The curves of bound/free or fractional increment in pharmacodynamic response versus competitor concentration generally appear sigmoidal and the Scatchard plots appear linear over the range of observations. If the analysis program is readily available, what might deter the radiopharmaceutical chemist from using it? We saw with the enzyme models that every condition that followed the law of mass action resulted in a hyperbolic velocity curve, which translates to the sigmoidal receptor binding curve when plotted on a log[L] format, Fig. 7. The sigmoidal curve requires just four parameters: high and low asymptotes, slope and 50% response. In these dose-response experiments, the concentration of drug that results in an effect half way between the background signal and the maximum possible response is called the 50% effective concentration, EC50. The illustration in Fig. 7 is a simulated dose response curve to cartoon the pharmacodynamic response and EC50. Black and Leff [2,8] present a thorough development of agonist-receptor, A-R, interactions using a parallel approach to that derived from enzymology. A ligand that elicits a pharmacodynamic response is called an agonist. [R]tot[A] [AR] ⫽ , similar to Eqs. 1 and 4, where A and [A] ⫹ KD R refer to the agonist ligand and the receptor, respectively, and KD is the equilibrium dissociation constant. In pharmacodynamic response measurements, Effect is not directly proportional to [AR]. Black and Leff [8] introduced the concept of “transducer function” to develop a response equation Effect ⫽

Effectmax [AR] [AR] ⫹ EC50

(12)

where EC50 is the agonist concentration, [A], that produces half-maximal effect. This results in a second hyperbolic function so now we have two hyperbolic functions operating sequentially and the result is determined by both EC50 and [R]tot, which are combined in a ratio [R]tot/EC50, called tau, ␶.

Effect ⫽

Effectmax ␶[A] ⫽ (KD ⫹ [A]) ⫹ ␶ [A]

冉

[A] Effectmax

␶ ␶⫹1

冊

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Fig. 7. Simulated pharmacodynamic response function for an agonist. In this hypothetical experiment the response variable is no longer measured in concentration units. It simply ranges from a background level without any added agonist drug to the maximal response that can be observed in a control experiment.

9. Summary and conclusions The hyperbolic parabola is ubiquitous in describing enzyme velocities, ligand binding rates and pharmacodynamic responses to an agonist. The prudent investigator will graph data directly in this format and apply nonlinear regression analysis to infer the asymptote and equilibrium constants from the non-transformed data. Depending on the experimental conditions and validity of assumptions, the resulting equilibrium constant may have different meaning. A graph in the Scatchard format provides a useful visual test of the data and is quick to identify the presence of multiple types of binding sites. However, this transformed data should not be analyzed by a linear regression model to infer rate parameters such as Bmax and KD because the linear model applied to transformed data will not describe errors appropriately [2]. Competitive inhibition experiments are critical tests for evaluating receptor-binding radiopharmaceuticals, both in vitro and in vivo. While Eq. 10 is more appealing to program than the complete Eq. 11, only a bit more effort is required to program the exact equation and it should be used to evaluate all radiopharmaceuticals. Eq. 11 avoids further testing of the validity of assumptions required to use the Cheng-Prusoff equation with confidence.

KD ⫹ [A] ␶⫹1

The maximum effect is modified by the tau ratio, which is a useful measure of efficacy for a drug. Again, the Cheng-Prusoff equation is widely applied, but the same limitations as described above apply and should lead to caution in interpreting pharmacodynamic inhibition curves [9 –11].

References [1] Segel IH. Enzyme kinetics: behavior and analysis of rapid equilibrium and steady-state enzyme systems. New York: Wiley-Interscience, 1975. [2] Motulsky HJ, Christopoulos A. Fitting models to biological data using linear and nonlinear regression. A practical quide to curve

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[3] [4]

[5]

[6]

K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) 819 – 826 fitting, GraphPad Software, Inc., San Diego, CA 2003, www. graphpad.com. Scatchard G. The attractions of proteins for small molecules and ions. Annal NY Acad Sci 1949;51:660 –72. Reiman EM, Soloff MS. The effect of radioactive contaminants on the estimation of binding parameters by Scatchard analysis. Biochim Biophys Acta 1978;533:130 –9. Cheng Y, Prusoff WH. Relationship between the inhibition constant (Ki) and the concentration of inhibitor which causes 50 per cent inhibition (IC50) of an enzymatic reaction. Biochem Pharmacol 1973; 22:3099 –108. Munson PJ, Rodbard D. An exact correction to the “Cheng-Prusoff” correction. J Receptor Res 1988;8:533– 46.

[7] Linden J. Calculating the dissociation constant of an unlabeled compound from the concentration required to displace radiolabel binding by 50%. J Cyclic Nucleotide Res 1988;8:163–72. [8] Black JW, Leff P. Operational models of pharmacological agonism. Proc Royal Soc London B 1983;220:141– 62. [9] Craig DA. The Cheng-Prusoff relationship: something lost in the translation. Trends Pharmacol Sci 1993;14:89 –91. [10] Leff P, Dougall IG. Further concerns over Cheng-Prusoff analysis. Trends Pharmacol Sci 1993;14:110 –2. [11] Lazareno S, Birdsall NJM. Estimation of antagonist Kb from inhibition curves in functional experiments: alternatives to the ChengPrusoff equation. Trends Pharmacol Sci 1993;14:237–9.