Interpreting enzyme and receptor kinetics: keeping it simple, but not too simple 1

Transcription

1 Nuclear Medicine and Biology 30 (2003) Interpreting enzyme and receptor kinetics: keeping it simple, but not too simple 1 Kenneth A. Krohn *, Jeanne M. Link Department of Radiology, Uniersity of Washington, Seattle, WA , 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 alid but others might be iolated 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 reiew compares enzymeinduced transformations and receptor binding in terms of the appropriate assumptions. The second topic considers the complication of adding a competitie 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, IC 50, of an inhibition cure. Receptor binding may be measured directly by a concentration assay or as a pharmacodynamic response ariable Elseier Inc. All rights resered. Keywords: Receptor kinetics; Enzyme kinetics; Kinetic modeling; Pharmacodynamic models; IC 50 ; Cheng-Prusoff equation 1. The equilibrium model for enzyme kinetics inoling only two reactants Radiopharmaceutical chemists borrow heaily from the rich literature of enzyme kinetics [1]. Howeer, there are pitfalls in a direct translation to receptor studies that warrant a reiew of some experimental details and assumptions that influence both data analysis and mechanistic interpretation. k 1 k P E S ES 3 E P ^k1 E denotes an enzyme and S is a substrate and they form a reersible intermediate, ES, which then breaks up at a rate k p 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 Contribution to the Receptors Meeting, San Diego, February * Corresponding author. address: kkrohn@u.washington.edu (K. A. Krohn). 4. E, S and ES are at equilibrium. ES returns to ES faster than it goes to EP 5. [E] so that formation of ES does not alter 6. The oerall rate of catalysis is goerned by breakup of ES to form EP The rate of product formation is d[p]/dt, which is also -d/dt and defines the elocity,, of the enzyme reaction. We can define the equilibrium dissociation constant, k 1 k 1 [E] [ES], and inoke conseration, [E] t [E][ES], so that d[p] k dt p [ES] which, when diided by the conseration equation, gies [E] t k p[es] [E] [ES]. If we define V max k p [E] t, then k p [E] t [ES]. It is common to parameterize [ES] [E][ES] in terms of, which allows cancellation of [E] and can be rearranged to Eq. 1. (1) 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 /03/$ see front matter 2003 Elseier Inc. All rights resered. doi: /s (03)00132-x

2 820 K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) 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 irreersible. Steady-state only requires that d[es] 0, but ES can dt decompose by two routes: k p 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]. Starting with Eq. 1, one can derie 1, which leads to a linear plot of ersus. This plot is particularly helpful to isually identify when two enzymes catalyze the same reaction, which is identified by a break in the slope of the line. Linearization processes were deeloped before computer programs were aailable to ealuate 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 iolates 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 inoling only two reactants Van Slyke has suggested an alternatie analysis of enzyme kinetic experiments [1]. The enzyme reaction inoles two irreersible steps and the rate equation describes the time required for the oerall reaction, written as the sum of the times for each step. k 1 k p E S 3 ES 3 E P The time required is expressed as t 1 1 k 1 k p k p k 1 ; the rate constant is 1 and 1 k 1 k p t t [E] tot. Again, k p [E] tot but now Kk p /k 1. Algebraic arrangement yields (2) K but now the K is a ratio of two forward rate constants Steady-state is close to equilibrium wheneer k p k 1, but when k p k 1, then [ES] SS [ES] eq.if[p]0, any backreaction can be neglected and d[es] k dt 1 [E]. The rate of decomposition of ES is d[es] k dt 1 [ES] k p [ES]. At steady-state d[es] d[es], and so dt dt k 1 [E] k 1 k p )[ES]. By combining elocity and conseration, k p[es] and, taking [ES] from [E] t [E] [ES] k 1 k 1 k p aboe, 1 k. 1 k 1 k p The cluster of rate constants, termed the Michaelis constant, is k 1 k p K k m [E],soK 1 [ES] m is K ss, not K eq. Again the equation is hyperbolic, K m, but now the assumptions are different. The Ks are also different; k 1 K k m only when k p 30 and K m k p 1 k 1 when k p 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 itro are inariably analyzed at equilibrium. 3. Enzyme kinetics. Adanced concepts From a complete kinetics perspectie, 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 E tot, thus aoiding any requirements about steady-state or equilibrium. Howeer, we cannot write a general integral equation for time or [P] time unless we inoke the condition that d[es] 0, the steady-state requirement. If 0 is high, dt [E] tot the time to reach steady-state is short and the equation is alid.

3 K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) Conditions where the kinetic equations become more complicated: If k p is a reersible process; k p 0. As product builds up, we can no longer assume [P]0. When a sequence of steps is inoled: ES 1 3ES 2 33E P. For example, consider the rate equation that includes the reerse reaction of E P to make ES. For parallel nomenclature, use k 2 and k 2 rather than k p, so that net k 2 [ES] k 2 [E] [P] and the conseration equation is [E] tot [E][ES][EP]. The algebra follows the original [P] f V K max r ms K mp scheme and results in net 1 [P]. K ms K mp Note that the general hyperbolic parabola form of both Eq. 1 and 2 persists and we can still interpret a simple experiment. When K m, V K m K m K max 1 m 2. For an efficient enzyme system, K m intracellular. Under this condition, a molecule-induced change in K m proides effectie 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 elocity cures; n molecules of one site are indistinguishable from one molecule of n sites. The algebra will still simplify to. Alternatiely, if S at one K site influences the binding of additional S to open sites, either by actiation or inhibition, then the kinetic cures no longer follow hyperbolic parabolas. These are called allosteric enzymes and they yield sigmoidal elocity cures. One classic example inoles the O 2 saturation cure 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 plot, [LR] B max [L]. The reader may show that nonspecific binding, NSB, results in [L-NSB]k nsb [L]. Both [L] of these functions are graphed hypothetically in Fig. 2. The same table of numbers was used here as for the preious enzyme graphs. This hyperbolic function is often transformed to a linear form for iewing: [LR] [LR][L] B max [L] leads to the classical Scatchard Eq. 3: [LR] [L] B max 1 [LR] (3) which suggests a plot of [LR] ersus [LR], where the slope [L] is 1. The simulated data of Fig. 2 is transformed to this equation in Fig How does enzyme kinetics relate to receptor models? The ligand-receptor reersible reaction L R ^koff looks much like our initial E S reaction and both generally behae according to second-order rate laws that lead to a steady state. In the receptor case, k off k on [L][R] at equilibrium. When [R] [LR], then [L] [LR]. The fractional occupancy of receptor is described by [R][L] [LR] [R][LR] [R] [R][L] [L]. To consere mass, [L] [R] tot [R][LR] and is often denoted as B max, so that Fig. 3. A Scatchard plot of the simulated receptor binding data. k on LR,

4 822 K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) k p [ES] by the equilibrium model. [E] tot [E] [ES] [EI] Substitute for [ES] and [EI] and recall k p [E] tot,so that 1 [I], which can be rearranged to K i 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 B max and, can also be modeled with the hyperbolic function. [LR] tot n B max[l] (4) [L] A radioligand that binds to two specific sites, with different associated B max and, can also be modeled with the hyperbolic equation but the two-site property is best appreciated when the data is iewed as a Scatchard plot [3]. [LR] tot B max1[l] 1 [L] B max2[l] (5) 2 [L] A simulated example is shown in Fig. 4, which is plotted as a Scatchard graph to isually 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 deelopment inoles two ligands competing for a single binding site. Note that radioactie contaminants can produce a Scatchard plot with a similar shape, as described in [4]. 5. Competitie 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 equialent situation exists in receptor binding studies. k P E S 7 ES 3 E P E I 7 Ki EI where K i [E][I] and K [EI] s [E] [ES]. As before, k p[es] but now [E] tot [E][ES][EI]. 1 [I] K i. (6) The steady-state model gies the same equation, but with K m rather than in the denominator. These equations hae the general form of Eq. 1 except that is now modified by the term1 [I] K i. is unchanged but the apparent is increased. In this model EI does not interact with S; the limited supply of enzyme is simply shared between two substrates, I and S, and, as Segel [1] explains1 [I] is K I, the distribution of enzyme aailable for binding S. Consider the fractional difference in elocity in the presence of inhibitor, i / 0, where 0 is the reaction elocity K S1 [I] i K when [I]0: i 0 1 [I]. It is conenient to conert from K i 0 i relatie elocities to fractional inhibition: 0 [I] K i1, which has a alue of 0.5 when [I] has a [I] concentration of IC 50. Thus, 0.5 or 2 1 K i1 [I] 0.5 and [I] 0.5 K i1, [I] 0.5 [I] 0.5 K i1 IC. 50 (7) Reciprocal plots are also used to interpret competitie inhibition data. Eq. 6 can be inerted to gie 1 [I] 1 K i 1 1. For each [I], a new 1 straight line can be drawn, but the y-intercept stays at.

5 K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) kept as low as possible while still maintaining acceptable counting statistics for accurately assaying separated 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 IC 50 and Fig. 5. Plot of simulated radioligand binding assay where a fixed amount of L* and R are competed with arying amounts of additional unlabeled drug L. The slope changes, related to [I] 1 K i, as does the x-intercept. 6. Competitie inhibition experiments: receptors In a competitie radioligand binding study, the experimenter selects an amount of and [R] and then makes measurements of total radioligand binding, [L*R], in the presence of arying amounts of unlabeled ligand. The algebra for this analysis follows the principles deeloped for competitie enzyme inhibition by two substrates. In this case L* and L may hae the same rate parameters if the labeling procedure did not change the forward or reerse 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, IC 50. If the experiment inoles a single class of binding sites, it will follow the law of mass action with a sigmoidal dose-response. [LR] tot B max[l] [L], where [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 [L*R] B max( [L] cold ) [L] cold [L] cold B max. In receptor binding assays where [L] cold L and L* are the same drug with a single, and [LR]0.5[LR] tot, 0.5 B max B max (8) [L] cold Experiments measure [L] cold for half-maximum inhibition: IC 50 [L] cold. Clearly, use of a high will not proide sensitiity to. is, in practice, Consider a single type of receptor binding site competing for two ligands, L and I, with equilibrium dissociation constants of and K i, respectiely. The total concentration of binding sites is B max. 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 ariable concentrations of labeled ligand are added to a fixed concentration, B max,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, K i. In analyzing these experiments, inestigators often assume that [I] at equilibrium is constant and is independent of. 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, the equilibrium dissociation constant, with IC 50 from an inhibition cure, this simple graphical analysis can lead to errors, as shown in this section. In their classic analysis, Cheng and Prusoff [5] showed algebraically that K i IC 50 under some specific conditions, such as noncompetitie kinetics, but the equality fails when competitie inhibition kinetics applies. They deeloped an analysis for a competitie inhibition system that is reersible and rapidly reaches equilibrium. Their Eq. 3 is our Eq. 7 and shows that K i is less than IC 50 by a factor related to the ratio /. IC 50 thus depends on the substrate concentration for each experiment and this alue cannot be compared between laboratories. The reader should consult the primary reference [5] for deriation of the equations for a noncompetitie inhibitor or an uncompetitie inhibitor. In this special case Eq. 6 becomes 1 [I] K i 1 [I] K ies In this case inhibitor has an affinity for the free enzyme, K i, and for the ES complex, K ies. IC 50 is only independent of proided that. Recalling that when [I]IC 50, 0 i 0.5, the rearrangement used to arrie at Eq. 7 now 0 (9)

6 824 K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) yields IC 50. The algebra requires more steps K i K ies but the outcome is often simpler, especially when the affinity is equal for free E and ES complex, K i K ies, in which case IC 50 K i. Cheng and Prusoff [5] showed other situations where K i IC 50 but, for the reersible competitie equilibrium experiment, which is most commonly encountered by the radiopharmaceutical chemist, this equality fails. The Cheng-Prusoff equation deries from Eq. 7 and has been used to obtain a for enzymes or for receptor binding from the measured IC 50 for a competitie inhibition experiment. For enzymes and receptors, respectiely, K i IC 50 1 and K i IC 50 1 [I] (10) This equation should be used with appropriate caution, howeer, as discussed below. The problem comes about when the Scatchard graph is not linear and this occurs wheneer the inhibitor concentration is an appreciable fraction of the receptor concentration, B max. Depending on the range of conditions oer which the competition experiment was done, it may be difficult to judge the alidity of the Cheng-Prusoff equation by isual 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 deeloped aboe refers to free concentration as a carryoer 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 inestigators hae deeloped exact equations relating K i and IC 50 [6,7]. The interested reader should refer to the appendix proided in [6] for an exact solution. Because this reference uses different nomenclature to arrie at their final equation 17, it is translated below to the abbreiations used in this reiew. K i IC [L*R] 1 [L*R] tot [L*R] 0 D [L*R] 2 0 K [L*R] (11) 0 Fig. 6. Simulated binding cures for a competitie inhibitor experiment. was 0.1 nm and K i was 0.2 nm. This small difference might be typical of a minor decrement in affinity caused by labeling. B max was 2 nm and two alues were used for [I]. Note that the curature is greatest when [I] approaches that of B max. where the ratio [L*R] denotes the experimental binding 0 ratio when the predetermined total amount of radioligand is used with no competitie inhibitor present. While this equation appears considerably more intimidating than the preious Eq. 10, it only adds some correction factors to the earlier equation and is still easily programmed as a spreadsheet function. It reduces to Eq. 10 when [L*R] is 0 negligible. Under what conditions might the Cheng-Prusoff equation gie unacceptable errors? A few generalizations can be made. The correct equation always results in a lower K i, increased affinity. There may een be conditions that gie rise to negatie alues for K i, suggesting that the data require an alternatie binding mechanism. The error is minimized when [L*R] is small and it is increased when 0 IC 50 is small. These errors can be order-of-magnitude or greater, leading the prudent inestigator to use the exact Eq. 11 rather than the simplified ersion, Eq Analysis of pharmacological endpoints In all of the analyses deeloped aboe, inhibition is measured after a chemical separation and using an assay that is quantified in molar or mass units. Howeer, an analogous experiment is done in pharmacology where the ordinate is measured in units of a pharmacodynamic response. Een though the connection between agonist bind-

7 K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) ing and response can be complex and indirect, the same mathematical analysis with the Cheng-Prusoff equation is frequently applied. The cures of bound/free or fractional increment in pharmacodynamic response ersus competitor concentration generally appear sigmoidal and the Scatchard plots appear linear oer the range of obserations. If the analysis program is readily aailable, what might deter the radiopharmaceutical chemist from using it? We saw with the enzyme models that eery condition that followed the law of mass action resulted in a hyperbolic elocity cure, which translates to the sigmoidal receptor binding cure when plotted on a log[l] format, Fig. 7. The sigmoidal cure 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% effectie concentration, EC 50. The illustration in Fig. 7 is a simulated dose response cure to cartoon the pharmacodynamic response and EC 50. Black and Leff [2,8] present a thorough deelopment of agonist-receptor, A-R, interactions using a parallel approach to that deried from enzymology. A ligand that elicits a pharmacodynamic response is called an agonist. [AR] [R] tot[a] [A], similar to Eqs. 1 and 4, where A and R refer to the agonist ligand and the receptor, respectiely, and 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 deelop a response equation Effect Effect max [AR] [AR] EC 50 (12) where EC 50 is the agonist concentration, [A], that produces half-maximal effect. This results in a second hyperbolic function so now we hae two hyperbolic functions operating sequentially and the result is determined by both EC 50 and [R] tot, which are combined in a ratio [R] tot /EC 50, called tau,. Effect Effect [A]Effect max [A] max ( [A]) [A] [A] 1 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 aboe apply and should lead to caution in interpreting pharmacodynamic inhibition cures [9 11]. Fig. 7. Simulated pharmacodynamic response function for an agonist. In this hypothetical experiment the response ariable is no longer measured in concentration units. It simply ranges from a background leel without any added agonist drug to the maximal response that can be obsered in a control experiment. 9. Summary and conclusions The hyperbolic parabola is ubiquitous in describing enzyme elocities, ligand binding rates and pharmacodynamic responses to an agonist. The prudent inestigator 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 alidity of assumptions, the resulting equilibrium constant may hae different meaning. A graph in the Scatchard format proides a useful isual test of the data and is quick to identify the presence of multiple types of binding sites. Howeer, this transformed data should not be analyzed by a linear regression model to infer rate parameters such as B max and because the linear model applied to transformed data will not describe errors appropriately [2]. Competitie inhibition experiments are critical tests for ealuating receptor-binding radiopharmaceuticals, both in itro and in io. 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 ealuate all radiopharmaceuticals. Eq. 11 aoids further testing of the alidity of assumptions required to use the Cheng-Prusoff equation with confidence. References [1] Segel IH. Enzyme kinetics: behaior and analysis of rapid equilibrium and steady-state enzyme systems. New York: Wiley-Interscience, [2] Motulsky HJ, Christopoulos A. Fitting models to biological data using linear and nonlinear regression. A practical quide to cure

8 826 K. A. Krohn and J. M. LinkNuclear Medicine and Biology 30 (2003) fitting, GraphPad Software, Inc., San Diego, CA 2003, www. graphpad.com. [3] Scatchard G. The attractions of proteins for small molecules and ions. Annal NY Acad Sci 1949;51: [4] Reiman EM, Soloff MS. The effect of radioactie contaminants on the estimation of binding parameters by Scatchard analysis. Biochim Biophys Acta 1978;533: [5] Cheng Y, Prusoff WH. Relationship between the inhibition constant (K i ) and the concentration of inhibitor which causes 50 per cent inhibition (IC 50 ) of an enzymatic reaction. Biochem Pharmacol 1973; 22: [6] Munson PJ, Rodbard D. An exact correction to the Cheng-Prusoff correction. J Receptor Res 1988;8: [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: [8] Black JW, Leff P. Operational models of pharmacological agonism. Proc Royal Soc London B 1983;220: [9] Craig DA. The Cheng-Prusoff relationship: something lost in the translation. Trends Pharmacol Sci 1993;14: [10] Leff P, Dougall IG. Further concerns oer Cheng-Prusoff analysis. Trends Pharmacol Sci 1993;14: [11] Lazareno S, Birdsall NJM. Estimation of antagonist K b from inhibition cures in functional experiments: alternaties to the Cheng- Prusoff equation. Trends Pharmacol Sci 1993;14:237 9.