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The Problems of Applying Classical Pharmacology Analysis to Modern In Vitro Drug Discovery Assays: Slow Binding Kinetics and High Target Concentration

      Abstract

      The analysis framework used to quantify drug potency in vitro (e.g., Kd or Ki) was initially developed for classical pharmacology bioassays, for example, organ bath experiments testing moderate-affinity natural products. Modern drug discovery can infringe the assumptions of the classical pharmacology analysis equations, owing to the reduction of assay volume in miniaturization, target overexpression, and the increase of compound–target affinity in medicinal chemistry. These assumptions are that (1) the compound concentration greatly exceeds the target concentration (i.e., minimal ligand depletion), and (2) the compound is at equilibrium with the receptor (i.e., rapid ligand binding kinetics). Unappreciated infringement of these assumptions can lead to substantial underestimation of compound affinity, which negatively impacts the drug discovery process, from early-stage lead optimization to prediction of human dosing. This study evaluates the real-world impact of these factors on the target interaction assays used in drug discovery using literature examples, database searches, and simulations. The ranges of compound affinity and the assay types that are prone to depletion and equilibration artifacts are identified. Importantly, the highest-affinity compounds, usually the highest value chemical matter in drug discovery, are the most affected. Methods and simulation tools are provided to enable investigators to evaluate, manage, and minimize depletion or equilibration artifacts. This study enables the correct application of pharmacological data analysis to accurately quantify affinity using modern drug discovery assay technology.

      Keywords

      Introduction

      Accurate quantification of drug–target interaction is critical for the discovery of new therapeutic ligands.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      • Kenakin T.
      Quantifying Biological Activity in Chemical Terms: A Pharmacology Primer to Describe Drug Effect.
      • Rang H.P.
      The Receptor Concept: Pharmacology’s Big Idea.
      • Hall D.A.
      • Langmead C.J.
      Matching Models to Data: A Receptor Pharmacologist’s Guide.
      • Kenakin T.P.
      A Pharmacology Primer.
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      The metric used to quantify ligand–target interaction is the affinity, defined by the equilibrium dissociation constant (Kd), which represents the concentration of ligand required to occupy 50% of the target population. In modern drug discovery, affinity (Kd or Ki) or related measurements (e.g., EC50 or IC50) are determined using high-throughput in vitro assays of ligand–target interaction or target function. These measurements are used to establish the structure–activity relationships (SARs) used by medicinal chemists to optimize new therapeutic candidates.
      • Kenakin T.P.
      A Pharmacology Primer.
      At later stages of drug discovery, the affinity of candidate molecules is used in the prediction of human drug effect in clinical trial design. Accurately measuring affinity is therefore critical for drug discovery.
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      Determining target binding affinity involves the application of data analysis frameworks to the assay data. Classical pharmacology is the foundation of the data analysis framework used to quantify affinity for most drug target classes (with the notable exception of enzymes; see below). In this framework, affinity is determined by applying equations to the familiar concentration–effect curve (from experiments in which the ligand concentration is titrated and the drug effect measured). When this framework was being developed in the early to mid-20th century, the assay modalities and status of medicinal chemistry were very different from today. Organ bath experiments on isolated tissues were the assays of choice for measuring drug effect,
      • Rang H.P.
      The Receptor Concept: Pharmacology’s Big Idea.
      ,
      • Clark A.J.
      The Antagonism of Acetyl Choline by Atropine.
      • Clark A.J.
      The Reaction between Acetyl Choline and Muscle cells.
      • Gaddum J.H.
      The Action of Adrenalin and Ergotamine on the Uterus of the Rabbit.
      and the affinity of test molecules was typically quite low (Kd > 10 nM). The equations used today to quantify drug effect were derived during this classical era, including the familiar sigmoid concentration–effect equation, which is based on the mass action equation
      • Kenakin T.
      The Mass Action Equation in Pharmacology.
      (see below). In the development of any data analysis framework, assumptions are made regarding the nature of the experimental system, which are used in the mathematical derivation of the equations. Assumptions were made in deriving the classical pharmacology equations that were appropriate for the assays and ligands of the time. However, the technological advances of modern drug discovery can potentially invalidate the assumptions underlying these classical equations that we still use to analyze target interaction data. When the assumptions no longer hold, the assay will not report the affinity correctly.
      There are two assumptions of classical pharmacology analysis that are potentially problematic in modern drug discovery:
      • 1
        The concentration of target is much less than the total concentration of ligand added to the assay.
        • Hulme E.C.
        • Trevethick M.A.
        Ligand Binding Assays at Equilibrium: Validation and Interpretation.
        ,
        • Jarmoskaite I.
        • AlSadhan I.
        • Vaidyanathan P.P.
        • et al.
        How to Measure and Evaluate Binding Affinities.
        ,
        • Carter C.M.
        • Leighton-Davies J.R.
        • Charlton S.J.
        Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
        ,
        • Jacobs S.
        • Chang K.J.
        • Cuatrecasas P.
        Estimation of Hormone Receptor Affinity by Competitive Displacement of Labeled Ligand: Effect of Concentration of Receptor and of Labeled Ligand.
        In the organ bath experiments of the classical era, this assumption was reasonable for two reasons. First, the target density on isolated tissues is usually low, at physiological density. Second, the volume of organ baths is high, in the tens or hundreds of milliliters.
        • Clark A.J.
        The Antagonism of Acetyl Choline by Atropine.
        • Clark A.J.
        The Reaction between Acetyl Choline and Muscle cells.
        • Gaddum J.H.
        The Action of Adrenalin and Ergotamine on the Uterus of the Rabbit.
        This combination of low density and high volume results in a very low concentration of target. By contrast, in the modern era the target expression level can be much higher (e.g., in cell lines engineered to overexpress the target) and the assay volume is much lower.
        • Carter C.M.
        • Leighton-Davies J.R.
        • Charlton S.J.
        Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
        High target expression is the norm for target systems used in most assays today. Assay miniaturization with the introduction of microtiter plates has dramatically reduced assay volume to the microliter range. As a result, the concentration of target can often be an appreciable fraction of or, in some cases, can be higher than the concentration of test ligand in the assay.
      • 2
        Equilibrium of the target–ligand binding interaction is closely approached within the incubation time of the assay.
        • Hulme E.C.
        • Trevethick M.A.
        Ligand Binding Assays at Equilibrium: Validation and Interpretation.
        ,
        • Jarmoskaite I.
        • AlSadhan I.
        • Vaidyanathan P.P.
        • et al.
        How to Measure and Evaluate Binding Affinities.
        ,
        • Aranyi P.
        Kinetics of the Glucocorticoid Hormone-Receptor Interaction. False Association Constants Determined in Slowly Equilibrating Systems.
        ,
        • Motulsky H.J.
        • Mahan L.C.
        The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
        The low- to moderate-affinity ligands of the classical era equilibrated rapidly with the target. By contrast, some high-affinity ligands of modern drug discovery equilibrate much more slowly with the target. This means that within the time frame of the assay, target occupancy does not closely approach the equilibrium value, which results in underestimation of target occupancy and affinity.
        • Aranyi P.
        Kinetics of the Glucocorticoid Hormone-Receptor Interaction. False Association Constants Determined in Slowly Equilibrating Systems.
        ,
        • Motulsky H.J.
        • Mahan L.C.
        The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
        This equilibration issue is a matter of the kinetics of target–ligand interaction (see below).
      Obviously, incorrect estimation of affinity has negative consequences for drug development. Unfortunately, the two assumption infringements above disproportionately impact the highest-value chemical matter in drug discovery—the optimized, high-affinity development and clinical candidates. Incorrectly quantifying affinity at this stage can result in high-impact problems, including incorrect prediction of human dosing and difficulties and ambiguities in candidate selection.
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      In addition, lead optimization can also be impacted, resulting in artifactual assay ceilings, which are artificial limits of apparent affinity that cannot be breached owing to the limitations of the assay and analysis rather than the true physicochemical characteristics of the ligand–target interaction. The purpose of this article is to enable investigators to identify and manage the target concentration and equilibration issues in their drug discovery projects. Fortunately, this is reasonably straightforward once the assumptions of pharmacology data analysis are understood.

      The Overlooked Technical Assumptions of Pharmacological Theory

      The familiar basic mechanism of drug–target interaction is shown in Figure 1. This mechanism is easy to understand. The drug molecule (the ligand) binds to the target in a single-site reversible interaction. The strength of binding is defined by the affinity constant Kd, which is the concentration of ligand required to occupy 50% of the target population. The equation defining the occupancy of the drug by the target is familiar to most drug discovery pharmacologists and is shown below:
      Figure 1
      Figure 1Basic mechanism of target–ligand interaction used as the foundation for pharmacological data analysis. A single-ligand molecule interacts reversibly with a single site on the target molecule. The affinity of the interaction, indicative of the potency or strength of the interaction, is defined by the free concentration of drug required to occupy 50% of the targets at equilibrium. This is the parameter Kd. The interaction can also be defined kinetically. Specifically, the rate of association of ligand with target is governed by the association rate constant kon, and the rate of dissociation of the ligand–target complex is defined by the dissociation rate constant koff. From this mechanism, a fundamental equation, the mass action equation,
      • Kenakin T.
      The Mass Action Equation in Pharmacology.
      can be derived that can be used to analyze pharmacological data to estimate the ligand affinity (eq 1). Application of this equation assumes the target concentration is low and the assay is at equilibrium, assumptions often infringed in modern drug discovery assays.
      [RL]=[R]TOT[L]FREEKd+[L]FREE
      (1)


      where [RL] is the concentration of the target–ligand complex, [L]FREE is the free concentration of the ligand (that not bound to the target; see below), and [R]TOT is the total concentration of the target. This is the mass action equation, described in detail in Kenakin.
      • Kenakin T.
      The Mass Action Equation in Pharmacology.
      This equation is the foundation of pharmacological theory, employed by Gaddum, Hill, and Clark in the context of drug binding to a target,
      • Clark A.J.
      The Reaction between Acetyl Choline and Muscle cells.
      ,
      • Gaddum J.H.
      The Action of Adrenalin and Ergotamine on the Uterus of the Rabbit.
      ,
      • Hill A.V.
      The Mode of Action of Nicotine and Curari, Determined by the Form of the Contraction Curve and the Method of Temperature Coefficients.
      and utilized by Langmuir in the definition of particle binding to a surface.
      • Langmuir I.
      The Adsorption of Gases on Plane Surface of Glass, Mica and Platinum.
      The equation has been extended to accommodate functional assay modalities (the measurement of target function, e.g., signaling, rather than target binding) and more complex interactions (requiring the introduction of slope factors, e.g., the Hill slope).
      • Kenakin T.
      Quantifying Biological Activity in Chemical Terms: A Pharmacology Primer to Describe Drug Effect.
      ,
      • Rang H.P.
      The Receptor Concept: Pharmacology’s Big Idea.
      ,
      • Kenakin T.
      The Mass Action Equation in Pharmacology.
      The assumptions of the mass action equation also apply to these additional modalities and interactions and so impact pharmacological data analysis generally. In this study, target interaction is defined in terms of affinity (Kd or Ki), but the principles apply to other metrics of the strength of target–ligand interaction (EC50, IC50, etc.).
      Two assumptions are made in the application of this equation.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      First, it is assumed that the ligand concentration term is the free concentration. This is the concentration of ligand that is not bound to the target. In principle, when we add a ligand to an assay, it is either bound to or not bound to the target. Consequently, the total concentration of ligand, [L]TOT, is defined by
      [L]TOT=[L]FREE+[RL]
      In these pharmacological assays, it is assumed that only a small concentration of the ligand is bound to the target, because it is assumed the target concentration is much lower than the concentration of the ligand added to the assay. Under this condition, the free concentration is effectively equal to the total concentration, that is,
      [L]FREE[L]TOT
      This means we can simply enter the total concentration of the ligand into eq 1 when we analyze pharmacological data. This condition is often referred to as zone A, a condition where less than 10% of the total ligand concentration is bound to the target.
      • Auld D.S.
      • Farmen M.W.
      • Kahl S.D.
      • et al.
      Receptor Binding Assays for HTS and Drug Discovery.
      Zone A was almost always satisfied in the classical era. However, for high-affinity ligands or for assay systems in which the target is highly expressed, this assumption can be violated—the concentration of ligand bound to the target can be a substantial fraction, or even the majority, of the total ligand added to the assay. This condition is referred to as ligand depletion or, more precisely, depletion of the free ligand concentration by the ligand–target complex. Under this condition, we cannot assume that the total concentration is equal to the free ligand concentration.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      ,
      • Jacobs S.
      • Chang K.J.
      • Cuatrecasas P.
      Estimation of Hormone Receptor Affinity by Competitive Displacement of Labeled Ligand: Effect of Concentration of Receptor and of Labeled Ligand.
      Either the free ligand concentration needs to be known, or the assay needs to be adjusted to reduce [RL] so that it is less than 10% of [L]TOT, or a different equation is required to analyze the data. Importantly, for high-affinity ligands employed at low concentrations, nonspecific binding of the ligand to assay surfaces and biological material can also contribute to ligand depletion, especially for very low-volume microplates where the ratio of surface area to volume is high.
      It is also important to note that Kd is defined in terms of the free concentration of ligand, being the free concentration of ligand required to occupy 50% of the target population. This emerges from the thermodynamic foundations of receptor theory, which is beyond the scope of this article. Kd is not defined in terms of the total concentration of ligand added to the assay.
      Second, it is assumed that the equilibrium of the target–ligand interaction has been closely approached when the pharmacological measurement is made.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      ,
      • Aranyi P.
      Kinetics of the Hormone-Receptor Interaction. Competition Experiments with Slowly Equilibrating Ligands.
      More precisely, it is assumed the equilibrium level of target occupancy by the ligand has been closely approached. It takes time for equilibrium to be approached. The time course of ligand binding is dependent on the kinetics of the ligand–target interaction. Surprisingly, the time it takes for equilibrium to be closely approached is dependent on the dissociation rate, the time it takes the ligand to dissociate from the target (see “Impact of Equilibration/Binding Kinetics on Measurements of Affinity” below). Compounds that dissociate slowly from the target take a long time to approach equilibrium, and if the incubation time is not long enough, the affinity of these compounds can be underestimated.
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      ,
      • Aranyi P.
      Kinetics of the Glucocorticoid Hormone-Receptor Interaction. False Association Constants Determined in Slowly Equilibrating Systems.
      These issues are applicable to targets that are analyzed using pharmacological data analysis approaches, including G-protein-coupled receptors (GPCRs), other classes of cell surface receptors such as receptor tyrosine kinases, and therapeutics analyzed using ligand binding assays such as therapeutic antibodies. It should be noted that for enzymes, these two issues have long been recognized as potentially confounding in the estimation of affinity. Enzyme drug targets are analyzed using a different, albeit related, analysis framework—enzyme kinetics. In this field, the implications of target concentration and equilibration have been broadly recognized since the 1970s. Enzyme inhibitors that bind with such high affinity that the target can deplete the free ligand concentration are known as tight binding inhibitors, and equations have been developed to accurately quantify their affinity.
      • Cha S.
      Tight-Binding Inhibitors—I. Kinetic Behavior.
      ,
      • Copeland R.A.
      Tight Binding Inhibition.
      Enzyme inhibitors that dissociate slowly and so take time to equilibrate are known as slow binding inhibitors, and analysis frameworks are available to quantify their interaction with the enzyme.
      • Copeland R.A.
      Slow Binding Inhibitors.
      ,
      • Morrison J.F.
      • Walsh C.T.
      The Behavior and Significance of Slow-Binding Enzyme Inhibitors.
      These analyses for enzymes will not be considered in this article.

      Impact of Target Concentration on Drug–Target Affinity Measurements

      In this section, the effect of target concentration on affinity estimation is assessed in the context of the target interaction assays used in drug discovery. Practical and theoretical guidance is provided to enable investigators to evaluate and manage this potential artifact. Simulator spreadsheets are provided in the Supplemental Files to enable calculations and simulations of the extent of ligand depletion and its effect on affinity estimation.

      Simulating the Effect of Ligand Depletion on Affinity Measurements

      Simulations can be employed to identify the conditions under which target concentration affects the measurement of affinity.
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      ,
      • Goldstein A.
      • Barrett R.W.
      Ligand Dissociation Constants from Competition Binding Assays: Errors Associated with Ligand Depletion.
      • Chang K.J.
      • Jacobs S.
      • Cuatrecasas P.
      Quantitative Aspects of Hormone-Receptor Interactions of High Affinity. Effect of Receptor Concentration and Measurement of Dissociation Constants of Labeled and Unlabeled Hormones.
      • Seeman P.
      • Ulpian C.
      • Wreggett K.A.
      • et al.
      Dopamine Receptor Parameters Detected by [3H]Spiperone Depend on Tissue Concentration: Analysis and Examples.
      To aid the investigator, a simulator is provided in the Supplemental Files called “Target Concentration and Ligand Depletion Simulator”; this is a Microsoft Excel file in which the conditions are entered into cells and the target occupancy by the ligand is calculated and graphed. An equation is required that takes into account ligand depletion.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      This is eq 2 below:
      [RL]=[R]TOT+[L]TOT+Kd([R]TOT+[L]TOT+Kd)24[R]TOT[L]TOT2
      (2)


      This equation conveniently employs the total concentration of ligand added to the assay, [L]TOT, in defining the degree of target occupancy, which is useful because this is a known quantity. (The classical equation, eq 1, by contrast, employs the free concentration [L]FREE, which is usually unknown unless it can be measured.) Note that Kd here is the free concentration of ligand giving 50% target occupancy, not the total concentration of ligand.
      The effect of target concentration on measurements of target occupancy by ligand is shown in Figure 2. Here the ligand concentration is titrated and the total concentration of ligand added to the assay, [L]TOT, is shown on the x axis. Target occupancy by ligand is shown on the y axis. Target occupancy is expressed here as the percent occupancy, that is, the percentage of the total target population bound by ligand ([RL]/[R]TOT × 100). The target occupancy curve is plotted for a series of target concentrations. We first consider a ligand with high affinity (Kd of 0.1 nM) (Fig. 2A). This graph clearly shows that as the concentration of target is increased, the compound appears less potent; that is, there is a rightward shift of the target occupancy curve. This is because more total ligand is required to occupy the target when the target concentration is increased. This is illustrated by the occupancy curve for the high target concentration of 100 nM (open circles in Fig. 2A). By contrast, when the target concentration is low (0.001 nM), much less total ligand is required to occupy the targets (closed circles in Fig. 2A).
      Figure 2
      Figure 2Impact of target concentration on measurement of target occupancy by ligand. Simulations of target occupancy by ligand were performed using eq 2 to mimic standard pharmacological assays in which the ligand concentration is titrated and the target occupancy at the various ligand concentrations determined. (A) High-affinity ligand occupancy curves (Kd = 0.1 nM) at varying target concentrations. The x value is the total concentration of ligand added to the assay. Note that as the target concentration increases, the occupancy curve shifts to the right; that is, higher concentrations of ligand are required to occupy the target, even though the affinity for the target is the same. Note also the steep slopes at the highest target densities. (B) Low-affinity ligand occupancy curves (Kd = 100 nM) at varying target concentrations. Note that as the target concentration increases, there is little effect on the ligand occupancy curve, until the highest concentration (100 nM) is reached, where there is a slight rightward shift. (C) Apparent affinity versus true affinity at various target densities. Apparent Kd was estimated as the total concentration of ligand required to occupy 50% of the target population ([L]50), by fitting simulated occupancy curves to the four-parameter logistic equation with variable slope.

      Motulsky, H. J. Equation: Log(agonist) vs. Response—Variable Slope. https://www.graphpad.com/guides/prism/latest/curve-fitting/REG_DR_stim_variable_2.htm (accessed May 14, 2021).

      Note that when the target concentration is high (100 nM, circles) the apparent affinity does not match the true affinity; for the highest affinity ligands, the apparent Kd is much higher than the true Kd. Under this condition, the apparent Kd approaches a ceiling—the apparent Kd of the high-affinity ligands (true Kd, 0.01–10 nM) is very similar (apparent Kd ranging from 43–55 nM).
      Now let us consider how this distorts the assessment of the compound’s affinity. The target occupancy curves can be used to determine the apparent Kd. This is the total concentration of ligand required to occupy 50% of the target population ([L]50). This is the value we would measure in the experiment. (Note the apparent Kd is not the same as the true Kd, which is the free concentration of ligand required to occupy 50% of the targets rather than the total concentration.) For the example in Figure 2A, the true affinity, or Kd, is 0.1 nM. For a low target concentration (0.001 nM), the occupancy curve faithfully reflects this true affinity, the midpoint ligand concentration of the curve being 0.1 nM. However, for the high target concentration, affinity is dramatically underestimated since the occupancy curve is shifted so far to the right—the [L]50 is 43 nM, which represents a 430-fold underestimate of the ligand’s affinity for the target. Note also that the slope of the curve is substantially increased at the high target concentration compared with the low concentration (Hill slopes of 2.2 and 1.0, respectively). These observations are consistent with systematic experimental studies of receptor concentration effects in binding assays.
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      ,
      • Jacobs S.
      • Chang K.J.
      • Cuatrecasas P.
      Estimation of Hormone Receptor Affinity by Competitive Displacement of Labeled Ligand: Effect of Concentration of Receptor and of Labeled Ligand.
      This effect of target concentration is dependent on the affinity of the ligand. The impact on high-affinity ligands is greater than the effect on low-affinity ligands. A lower-affinity ligand is shown in Figure 2B (Kd = 100 nM). In this case, the effect of target concentration is much smaller than that on the high-affinity ligand (compare with Fig. 2A). The highest target concentration (100 nM) produces only a small rightward shift of the occupancy curve; the [L]50 is 150 nM, representing only a 1.5-fold shift of the true affinity (100 nM). This is a much smaller effect than that on the high-affinity ligand (430-fold shift; see above). This makes sense when we consider the difference of Kd between the high- and low-affinity ligands. Since the ligand concentration at the Kd is so much higher for the low-affinity ligand (100 nM), the target will deplete this concentration to a much smaller extent than it will for the Kd concentration of the high-affinity ligand (1 nM). The extent of ligand depletion can be calculated for these conditions using eq 2. (A calculator spreadsheet is provided in the Supplemental Files, “Ligand Depletion Calculator.”) At 100 nM target, there is 99.9% depletion of the ligand at its Kd concentration for the high-affinity ligand but only 38% for the low-affinity ligand.

      Simulating the Effect on SAR Evaluation and Observation of Assay Ceiling

      Let us suppose the assay in Figure 2 is being used to determine the affinity of a series of compounds in a medicinal chemistry ligand optimization project. The range of true affinities is given on the x axis of Figure 2C, with Kd values ranging from 0.01 nM to 10 µM. The target occupancy is measured and the apparent Kd is quantified as the total concentration of drug required to occupy 50% of the targets ([L]50), as described in the legend to Figure 2. This measured apparent Kd is plotted against the true Kd in Figure 2C. First, we consider ideal conditions, where the target concentration is low (0.01 nM), resulting in minimal ligand depletion. This results in the measured apparent Kd closely matching the true Kd, as shown in the graph (diamonds in Fig. 2C). Now we consider nonoptimal conditions, a target concentration of 100 nM (circles in Fig. 2C). Under this condition, the assay reliably quantifies the affinity for the low-affinity ligands. However, for the high-affinity ligands the Kd is not quantified correctly. Instead, we observe a ceiling of affinity measurement—the apparent Kd of the highest-affinity ligands is approximately the same. The correlation plot is a curve rather than a straight line, and the plateau of the curve represents the assay ceiling. This ceiling Kd value is approximately half of the target concentration in the assay (i.e., 50 nM for the 100 nM target concentration assay).
      This effect has real-world implications on drug discovery. High target concentration assays are often employed in high-throughput screening (HTS). Increasing the target concentration can offer the means to maximize the signal and so increase the robustness of the HTS assay (i.e., high Z′ values), and sometimes the target is maximized because it costs less than the detection reagents used to generate the signal. The low assay volume of HTS assays also increases the target concentration. For the low-affinity compounds sought in the high-throughput screen, target concentration and ligand depletion minimally affect detection and the quantification of ligand–target interaction. However, as the project proceeds and higher- and higher-affinity ligands are being generated, this assay can result in erroneous estimates of affinity. The highest-value chemical matter, the high-affinity leads, is not being characterized correctly—the affinity becomes underestimated. Moreover, the SAR predicted by medicinal chemists used to optimize the series no longer materializes—the compounds approach the assay ceiling, determined by the artifact of the target concentration rather than by a physicochemical limit of the binding interaction.

      Managing Target Concentration and Ligand Depletion Artifacts in Drug Discovery

      Fortunately, approaches are available to manage the target concentration issue in drug discovery campaigns. The first approach is simply awareness. In the simplest case, if a purified target binding assay is being used and it is known that the target concentration is 100 nM, the investigators should be aware that there will be an assay ceiling of the measured Kd values at approximately half this concentration (i.e., 50 nM). For elaborating SAR beyond this limit, either the target concentration will need to be reduced, for example, by using a 96-well plate instead of a 1536-well plate, or, if that is not possible, an alternative assay will be required. Investigators should be aware that certain assay modalities are particularly susceptible to high target concentration effects, and these assays are listed in Table 1. Fluorescence polarization binding assays
      • Auld D.S.
      • Farmen M.W.
      • Kahl S.D.
      • et al.
      Receptor Binding Assays for HTS and Drug Discovery.
      ,
      • Lea W.A.
      • Simeonov A.
      Fluorescence Polarization Assays in Small Molecule Screening.
      and low-volume microtiter plate assays are particularly susceptible to high target concentration artifacts. In addition, very high-affinity radioligands can be depleted because they are used at low concentrations to maximize the signal-to-background ratio.
      • Seeman P.
      • Ulpian C.
      • Wreggett K.A.
      • et al.
      Dopamine Receptor Parameters Detected by [3H]Spiperone Depend on Tissue Concentration: Analysis and Examples.
      By contrast, certain assay modalities are notably less prone to depletion, such as resonance energy transfer fluorescent binding assays, which typically employ lower-affinity tracers.
      • Soave M.
      • Briddon S.J.
      • Hill S.J.
      • et al.
      Fluorescent Ligands: Bringing Light to Emerging GPCR Paradigms.
      ,
      • Sykes D.A.
      • Stoddart L.A.
      • Kilpatrick L.E.
      • et al.
      Binding Kinetics of Ligands Acting at GPCRs.
      Table 1Assay Modalities Prone to Ligand Depletion Artifacts in Drug Discovery.
      Assay TypeComment
      High-density microtiter plates (e.g., 1536 or 3456 wells/plate)Assay volumes are low (<10 µL), which increases the concentration of target, which increases the potential for ligand depletion.
      Fluorescence polarization (FP) binding assaysFP detects the fraction bound of the tracer (if the tracer is the fluorescent partner) rather than that fraction bound of the target.
      • Auld D.S.
      • Farmen M.W.
      • Kahl S.D.
      • et al.
      Receptor Binding Assays for HTS and Drug Discovery.
      ,
      • Lea W.A.
      • Simeonov A.
      Fluorescence Polarization Assays in Small Molecule Screening.
      Since the assay modality detects depletion of free tracer concentration by target, it is by design susceptible to ligand depletion artifacts. FP assays often require high concentrations of target (e.g., 370 nM
      • Duckworth B.P.
      • Aldrich C.C.
      Development of a High-Throughput Fluorescence Polarization Assay for the Discovery of Phosphopantetheinyl Transferase Inhibitors.
      ) to deplete the ligand, resulting in high Kd assay ceilings.
      High-affinity radioligandsRadioligands are usually employed at or around their Kd concentration to maximize the signal-to-background ratio. When the affinity is very high, Kd is very low (e.g., 16 pM
      • Hoare S.R.
      • Strange P.G.
      Regulation of D2 Dopamine Receptors by Amiloride and Amiloride Analogs.
      ). Such low concentrations can exceed the concentration of target required for a robust signal, leading to ligand depletion.
      Approaches are available for detecting ligand depletion. Certain observations are suggestive of depletion in the assay, such as Kd ceilings and steep slopes (Fig. 2 and Carter et al.
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      ). Depletion can be indicated by the slope increasing as the apparent affinity of compounds within a series increases. Tracking the slope value in drug discovery can therefore be useful to identify the depletion artifact. However, other artifacts can produce this effect (such as lack of equilibration). Experimentally, one method to detect depletion is to increase the volume. If depletion is an issue, increasing the volume will result in a decrease of measured Kd. Large increases of volume (>10-fold) are often required to detect this effect,
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      ,
      • Hoare S.R.
      • Strange P.G.
      Regulation of D2 Dopamine Receptors by Amiloride and Amiloride Analogs.
      which can require using a different plate format or tubes to enable higher volumes to be used. Alternatively, the target concentration can be reduced and the effect on Kd determined,
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      ,
      • Jacobs S.
      • Chang K.J.
      • Cuatrecasas P.
      Estimation of Hormone Receptor Affinity by Competitive Displacement of Labeled Ligand: Effect of Concentration of Receptor and of Labeled Ligand.
      although that is often challenging owing to the resulting reduction of signal.
      In radioligand binding assays, ligand depletion of the labeled ligand can be accommodated by simply calculating the free concentration of ligand.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Claro E.
      Analyzing Ligand Depletion in a Saturation Equilibrium Binding Experiment.
      ,
      • Munson P.J.
      • Rodbard D.
      Ligand: A Versatile Computerized Approach for Characterization of Ligand-Binding Systems.
      This is possible because both the bound ligand [RL] and the total ligand [L]TOT are quantified, and they are quantified in the same units—radioactive counts. Free ligand ([L]FREE) is then calculated by subtracting [RL] from [L]TOT.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Claro E.
      Analyzing Ligand Depletion in a Saturation Equilibrium Binding Experiment.
      [RL] is then plotted versus [L]FREE in the saturation analysis to determine the radioligand affinity. The calculation to determine [L]FREE also involves subtraction of nonspecific binding, and if different radioactive counters are used to determine the counts of [RL] and [L]TOT, conversion of counts per minute (cpm) to disintegration per minute (dpm) is required to accommodate different counting efficiencies.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Claro E.
      Analyzing Ligand Depletion in a Saturation Equilibrium Binding Experiment.
      A useful detailed example is provided in Claro.
      • Claro E.
      Analyzing Ligand Depletion in a Saturation Equilibrium Binding Experiment.
      Ligand depletion in radioligand binding assays has been discussed extensively,
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      ,
      • Chang K.J.
      • Jacobs S.
      • Cuatrecasas P.
      Quantitative Aspects of Hormone-Receptor Interactions of High Affinity. Effect of Receptor Concentration and Measurement of Dissociation Constants of Labeled and Unlabeled Hormones.
      ,
      • Claro E.
      Analyzing Ligand Depletion in a Saturation Equilibrium Binding Experiment.
      • Munson P.J.
      • Rodbard D.
      Ligand: A Versatile Computerized Approach for Characterization of Ligand-Binding Systems.
      • Wells J.W.
      • Birdsall N.J.
      • Burgen A.S.
      • et al.
      Competitive Binding Studies with Multiple Sites. Effects Arising from Depletion of the Free Radioligand.
      and large errors of Kd determination (>10-fold) have been observed experimentally when depletion is not considered, that is, when [L]TOT is used instead of [L]FREE in the saturation analysis.
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      ,
      • Jacobs S.
      • Chang K.J.
      • Cuatrecasas P.
      Estimation of Hormone Receptor Affinity by Competitive Displacement of Labeled Ligand: Effect of Concentration of Receptor and of Labeled Ligand.
      ,
      • Seeman P.
      • Ulpian C.
      • Wreggett K.A.
      • et al.
      Dopamine Receptor Parameters Detected by [3H]Spiperone Depend on Tissue Concentration: Analysis and Examples.
      ,
      • Hoare S.R.
      • Strange P.G.
      Regulation of D2 Dopamine Receptors by Amiloride and Amiloride Analogs.
      Importantly, if there is an error in Kd of the radioligand, this will result in a corresponding error in the Ki of unlabeled ligands when the binding assay is used in compound profiling in drug discovery.
      • Carter C.M.
      • Leighton-Davies J.R.
      • Charlton S.J.
      Miniaturized Receptor Binding Assays: Complications Arising from Ligand Depletion.
      Fortunately, certain newer, fluorescence-based binding assays are much less susceptible to ligand depletion. Resonance energy transfer fluorescent binding assays
      • Soave M.
      • Briddon S.J.
      • Hill S.J.
      • et al.
      Fluorescent Ligands: Bringing Light to Emerging GPCR Paradigms.
      ,
      • Sykes D.A.
      • Stoddart L.A.
      • Kilpatrick L.E.
      • et al.
      Binding Kinetics of Ligands Acting at GPCRs.
      ,
      • Zwier J.M.
      • Roux T.
      • Cottet M.
      • et al.
      A Fluorescent Ligand-Binding Alternative Using Tag-lite® Technology.
      ,
      • Georgi V.
      • Schiele F.
      • Berger B.T.
      • et al.
      Binding Kinetics Survey of the Drugged Kinome.
      provide an unusually high signal-to-background ratio, and this enables detection of much lower-affinity target binding interactions (Kd approaching 100 nM in some cases) than is typically achievable with radioligands (typically requiring Kd < 10 nM and ideally <1 nM). The fluorescently labeled small-molecule tracers in such assays usually bind with lower affinity than the parent molecule because the fluorescent moiety substantially affects the SAR, being as large as the parent molecule itself in most cases.
      • Vernall A.J.
      • Hill S.J.
      • Kellam B.
      The Evolving Small-Molecule Fluorescent-Conjugate Toolbox for Class A GPCRs.
      In some cases, the quantum yield of the resonance energy transfer event is markedly high,
      • Zwier J.M.
      • Roux T.
      • Cottet M.
      • et al.
      A Fluorescent Ligand-Binding Alternative Using Tag-lite® Technology.
      and this enables unusually low concentrations of target to be used.
      Measuring the target concentration is highly desirable if this is feasible. For purified protein assays, this can be relatively straightforward. However, for whole-cell assays or for purified proteins of unknown specific activity, it can be challenging to quantify the mass of active protein unless a radioligand binding assay is available. One alternative approach is to titrate the target and keep the ligand concentration constant (using a high-affinity ligand). Equation 2 can be applied to the resulting data to estimate the target concentration.
      In principle, depletion can also be managed by using an equation that accommodates depletion to analyze the data, for example, eq 2, rather than using eq 1. The target concentration does not necessarily need to be known to employ eq 2—instead, it can be a fitted parameter in the analysis. However, there are statistical problems with this approach—large variability of the fitted value of Kd is likely when [R]TOT is an appreciable fraction or exceeds Kd or [L]TOT.

      Comparison between Historical and Modern Conditions

      In the classical era, drug activity was measured using isolated tissues and organ baths, and under these conditions the target concentration is low and so there is minimal ligand depletion. The density of the targets in the tissues being tested in these experiments was much lower compared with modern target expression systems. In addition, the volume of the assay was much higher. Using a literature example, we can calculate the target concentration in a typical assay system of the time, muscarinic receptor-mediated muscle contraction in an isolated tissue strip (longitudinal smooth muscle).
      • Paton W.D.
      • Rang H.P.
      The Uptake of Atropine and Related Drugs by Intestinal Smooth Muscle of the Guinea-Pig in Relation to Acetylcholine Receptors.
      The density of muscarinic acetylcholine receptors, quantified using tritiated atropine, was 180 pmol/g wet weight tissue.
      • Paton W.D.
      • Rang H.P.
      The Uptake of Atropine and Related Drugs by Intestinal Smooth Muscle of the Guinea-Pig in Relation to Acetylcholine Receptors.
      Given the mass of tissue in the experiment (approximately 20 mg) and the volume of the organ bath (10 mL), the calculated target concentration in the assay is 36 fM. This concentration is far below the Kd of almost all drugs, certainly of those moderate-affinity drugs in use at the time, so it was entirely reasonable to assume the free concentration of ligand was effectively equal to the total concentration. In other words, it was reasonable to assume minimal ligand depletion and to use the total ligand concentration in the application of eq 1.
      In the present era of drug discovery, both the density of the target in the target preparation and the volume of the assay are very different from the classical era. These changes have been driven by the need to reduce cost, owing to the enormous increase in throughput of modern drug discovery. Rather than testing a single compound per day on isolated tissues in organ baths, in present-day experiments, tens or hundreds of compounds are tested on engineered cells grown in batch and easily dispensed in microtiter plates. These changes have dramatically increased the target concentration in the assay. For example, for a cell surface receptor such as a GPCR overexpressed in mammalian cells,
      • Andrell J.
      • Tate C.G.
      Overexpression of Membrane Proteins in Mammalian Cells for Structural Studies.
      the number of receptors per cell can be as high as ten million, compared with only 10,000–20,000 for receptors expressed endogenously in tissues.
      • Abbott A.J.
      • Nelsestuen G.L.
      The Collisional Limit: An Important Consideration for Membrane-Associated Enzymes and Receptors.
      In addition, a typical assay volume in a 384-well plate is 25 µL, three orders of magnitude less than the volumes used in an organ bath. If we assume the assay employs 10,000 cells with 107 targets per cell, the target concentration is calculated to be 7 nM. This concentration is approximately 200,000-fold higher than the calculated target concentration in the organ bath experiment (36 fM; see above). More importantly, this target concentration is higher than the Kd of many drug molecules and is in the range of the desired affinity (low nM) for drug candidates. For a 1 nM Kd ligand in this assay, the free concentration of ligand would be substantially less than the total concentration when the ligand is applied at its Kd concentration (i.e., 1 nM). In other words, ligand depletion conditions would apply to such an experiment. Consequently, it would be erroneous to input the total concentration of ligand when employing eq 1 to analyze the data and affinity estimates from the analysis would be inaccurate.

      Impact of Equilibration/Binding Kinetics on Measurements of Affinity

      This second aspect concerns the timing of ligand–target interaction, which is a matter of the kinetics of binding (Fig. 1). This affects the incubation time required for the assay to approach the equilibrium condition. Here the basics of binding kinetics are introduced, as this relates to the required incubation time and practical guidance is provided for how long to incubate target interaction assays, based on surveys and simulations. Tools are provided in the Supplemental Files to enable investigators to explore equilibration.

      Principles of Equilibration and Binding Kinetics

      The pharmacological equations we use to analyze drug discovery data (e.g., eq 1) assume that the target–ligand interaction has closely approached equilibrium when the measurement of drug effect is made.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      It takes time for the ligand to associate with the target and for equilibrium to be approached.
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      ,
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      ,
      • Aranyi P.
      Kinetics of the Hormone-Receptor Interaction. Competition Experiments with Slowly Equilibrating Ligands.
      ,
      • Gray H.E.
      • Luttge W.G.
      A Comment on the Estimation of Times Required for the Attainment of Equilibrium by Noncooperative, Single Site Ligand-Receptor Systems.
      This is illustrated by the association graph shown in Figure 3. In the early phase of the time course, the level of target–ligand complex is increasing rapidly. Association then slows and then finally approaches a plateau. This plateau represents the equilibrium level of the target–ligand complex. For assays to correctly report affinity, it is necessary to make the measurements close to this equilibrium plateau. If measurements are made before this plateau, that is, if the incubation is not long enough, we will underestimate target occupancy by the ligand, and this results in underestimation of affinity. Surprisingly, the time it takes to equilibrate is dependent on the dissociation rate constant (target residence time) of the ligand,
      • Hulme E.C.
      • Trevethick M.A.
      Ligand Binding Assays at Equilibrium: Validation and Interpretation.
      ,
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      ,
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      ,
      • Aranyi P.
      Kinetics of the Hormone-Receptor Interaction. Competition Experiments with Slowly Equilibrating Ligands.
      and the reason for this is explained below. The recommended incubation time is three times the dissociation half-time. Two hours is sufficient for most ligands in drug discovery to closely approach equilibrium. If the dissociation rate of the ligand is particularly slow, relative to the incubation time, affinity can be dramatically underestimated, as described in early studies on steroid receptor ligand binding,
      • Aranyi P.
      Kinetics of the Glucocorticoid Hormone-Receptor Interaction. False Association Constants Determined in Slowly Equilibrating Systems.
      and as recently elaborated in case studies,
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      one of which is described below.
      Figure 3
      Figure 3The time required to approach equilibrium is dependent on the dissociation rate constant. This surprising fact is illustrated by this simulated association time course. Two ligands are considered with different dissociation rates, one that dissociates rapidly (2 min dissociation t1/2, circles) and one that dissociates slowly (3 h t1/2, squares). Both ligands are applied at their Kd concentration (9 nM and 0.1 nM for the rapidly and slowly dissociating ligands, respectively). The rapidly dissociating ligand associates and approaches the equilibrium plateau much more rapidly than the slowly dissociating ligand. The half-time for association, determined as 0.693/kobs, with kobs determined by fitting the data to eq 3, is 1 min for the rapidly dissociating ligand and 1.5 h for the rapidly dissociating ligand. Data were simulated using eq 3. The association rate constant for both ligands is the same (3.85 × 107 M–1 min–1).
      How long do we need to incubate the assay for? To determine this, an understanding of the kinetics of target–ligand interaction is required. As shown in Figure 1, the kinetics is defined by the two binding processes occurring in a reversible binding interaction. These are the processes of ligand association with the target and dissociation of the ligand–target complex, defined by the microscopic association and dissociation rate constants (kon and koff), respectively. kon is a measure of the rate of recognition of ligand by the target and is in the rather abstract units of inverse concentration multiplied by inverse time (e.g., M–1 min–1). koff defines the stability of the ligand complex, or how long the complex stays together. The units of koff are not intuitive either, being inverse time (e.g., min–1). Fortunately, dissociation can be expressed using an intuitive parameter describing how long the ligand remains bound to the target (in units of time, e.g., minutes). Two such parameters are used, the half-time of the complex (dissociation t1/2 = 0.693/koff)
      • Dahl G.
      • Akerud T.
      Pharmacokinetics and the Drug-Target Residence Time Concept.
      or the residence time (1/koff).
      • Copeland R.A.
      • Pompliano D.L.
      • Meek T.D.
      Drug-Target Residence Time and Its Implications for Lead Optimization.
      Here the dissociation t1/2 will be used.
      To determine the time it takes to approach equilibrium, we need to consider the timing of the association process. Association is rather complex; for a more detailed discussion, see Motulsky and Mahan.
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      The association time course curve is defined by three parameters: the association rate constant kon; the ligand concentration; and, surprisingly, the dissociation rate constant koff. The equation defining the association time course is
      [RL]=[RL]eq(1ekobst)
      (3)


      where
      kobs=[L]FREEkon+koff
      (4)


      ([RL]eq is the equilibrium concentration of target–ligand complex and [L]FREE is the free concentration of ligand.) kobs is the parameter that determines the amount of time required for equilibrium to be approached. (It is often described as the observed association rate constant.) Note that it contains the terms [L]FREE, kon, and koff. When we examine the equation defining kobs (eq 4), something rather remarkable emerges. When the concentration of ligand is low, kobs is defined largely by koff. In other words, the observed rate of association, and so the time required to approach equilibrium, is largely dependent on the dissociation rate constant. Understanding this issue can be simplified by rewriting eq 4 in terms of the ligand Kd rather than the rather abstract kon. Since Kd = koff/kon, eq 4 can be rewritten as
      kobs=koff(1+[L]FREEKd)
      (5)


      If we enter a ligand concentration equal to the Kd in eq 5, then we observe the following relationship:
      kobs,[L]FREE=Kd=koff×2
      (6)


      In other words, the observed association rate is close to the dissociation rate, being only twofold higher at the Kd concentration. This demonstrates how the equilibration time is largely dependent on the dissociation rate constant at the concentration equal to the Kd of the ligand.
      The equilibration time is ligand dependent because koff varies between ligands. Ligands that dissociate rapidly from the target also associate rapidly with the target and so equilibrate rapidly. By contrast, ligands that dissociate slowly also associate slowly, and so more time is required for such compounds to approach equilibrium.
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      ,
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      ,
      • Aranyi P.
      Kinetics of the Hormone-Receptor Interaction. Competition Experiments with Slowly Equilibrating Ligands.
      This is illustrated by the association time course graph in Figure 3. This shows the association of two compounds, one that dissociates rapidly (dissociation t1/2 of 2 min) and one that dissociates slowly (dissociation t1/2 of 3 h). The graphs show the association time course at a ligand concentration equal to the Kd. The rapidly dissociating compound associates much more rapidly with the target than the slowly dissociating compound (Fig. 3). The half-time for association for the rapidly dissociating compound, calculated as 0.693/kobs, is 1 min, whereas the half-time for the slowly dissociating compound is much longer at 1.5 h (Fig. 3).

      Recommended Assay Incubation Time

      How long does it take to closely approach equilibrium? Strictly speaking, equilibrium is never reached because the equilibrium plateau is an asymptote, being the level of target occupancy as time approaches infinity. We need to incubate long enough for equilibrium to be reasonably approximated. The recommendation is for the incubation time to be triple the dissociation t1/2. Here this is termed the effective equilibration time (EET). At this time, the level of binding for very low concentrations, well below the Kd, will have reached 88% of the equilibrium value, and binding at the Kd concentration will have reached 98% of the equilibrium value (calculated using eq 3). (Other articles recommend a longer time interval, for example, five times the dissociation half-time,
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      but this can lead to impractically long incubation times for the workflow of modern drug discovery.) An incubation time of 2 h is long enough for most compounds, as detailed in the survey below. Short incubation times (e.g., 15 min) can be problematic and should be avoided for assays used to define SAR in ligand optimization.
      The timing can be more complicated in competition experiments where there are two ligands in the assay: (1) the test ligand and (2) a tracer ligand in a binding assay or an agonist or substrate in a signaling or enzymatic assay. Useful simulations have been performed to guide investigators,
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      ,
      • Aranyi P.
      Kinetics of the Hormone-Receptor Interaction. Competition Experiments with Slowly Equilibrating Ligands.
      and fortunately the findings are straightforward—the EET is simply defined by the slowest dissociating of the two ligands.
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      By the criteria here, EET is approximated when the incubation time is three times the dissociation t1/2 of the slowest-dissociating ligand in the assay.

      Effective Equilibration Time Survey

      In this study, a binding kinetic database is used to survey the EET of ligands in drug discovery. This provides a risk assessment for lack of equilibration at different stages of therapeutic discovery and optimization. Recently, a comprehensive survey of the kinetics of 3812 ligand–target interactions was published and the data set made available.
      • Schuetz D.A.
      • Richter L.
      • Martini R.
      • et al.
      A Structure–Kinetic Relationship Study Using Matched Molecular Pair Analysis.
      In the present study, the EET of the interactions was determined using the dissociation rate constant of the interactions provided in the data set. The EET was calculated as the dissociation t1/2 × 3.
      Here the survey data are used to determine what percentage of target–ligand interactions fail to approach equilibrium for different incubation times (15 min, 1 h, and 2 h). This was defined as the percentage of entries having an EET exceeding the incubation time. This is done for different ranges of affinity of the interactions, providing an assessment of how much the equilibration time could be an issue at different stages of ligand optimization (>100, 10–100, 1–10, 0.1–1, and <0.1 nM). This was done for all target classes combined. These results are shown in Figure 4 and Table 2. The Excel spreadsheet used to perform the survey is provided in the Supplemental Files (“Effective Equilibration Time Survey”) so investigators can perform their own searches. This spreadsheet enables the survey to be broken down by target class (e.g., kinases or GPCRs).
      Figure 4
      Figure 4EET survey of ligand–target interactions in drug discovery. A recently compiled, comprehensive database of target–ligand interaction kinetics
      • Schuetz D.A.
      • Richter L.
      • Martini R.
      • et al.
      A Structure–Kinetic Relationship Study Using Matched Molecular Pair Analysis.
      was used to calculate the EET for 3812 interactions. Here the values are used to assess the potential for lack of equilibration by calculating the percentage of entries where EET exceeds the incubation time. This was done for three incubation times: (A) 15 min, (B) 1 h, and (C) 2 h. The survey was performed for different ranges of affinity, representing the ranges encountered during ligand identification and optimization in drug discovery. The percentage values, number of entries, and other metrics are shown in . The survey spreadsheet used is provided in the (“Effective Equilibration Time Survey”).
      Table 2EET for Various Ranges of Target–Ligand Binding Affinity.
      Kd Range (nM)
      >10010–1001–100.1–1<0.1
      % EET > 15 min510324897
      % EET > 1 h13132768
      % EET > 2 h1291944
      Median EET47 s1.2 min5.1 min14 min93 min
      Median Kd1.3 µM35 nM4.0 nM0.45 nM36 pM
      Median dissociation t1/216 s24 s1.7 min4.8 min31 min
      Number of entries241875546816034
      Data were obtained from the kinetic binding database file in Schuetz et al.,
      • Schuetz D.A.
      • Richter L.
      • Martini R.
      • et al.
      A Structure–Kinetic Relationship Study Using Matched Molecular Pair Analysis.
      using the Excel spreadsheet provided in the Supplemental Files, “Effective Equilibration Time Survey.” The % EET values are the percentage of entries with an EET longer than the specified incubation time (15 min, 1 h, or 2 h). The % EET data are presented graphically in Figure 4. EET is calculated as three times the dissociation t1/2.
      First, we consider the lower-affinity ligands with Kd greater than 100 nM. This range represents HTS hits and early-stage hit-to-lead compounds. The large majority of these ligands equilibrate rapidly. Few of the entries in this range have EET values exceeding 15 min (5%), and almost none have EETs exceeding 1 h (1%) (Fig. 4, Table 2). The median EET is 47 s (Table 2). This result implies that short- or moderate-duration assays will report the affinity correctly (in the context of equilibration) in the large majority of cases for screening hits and low-affinity leads.
      Next, we consider successive ranges increasing in affinity by 10-fold. These are 10–100, 1–10, 0.1–1, and <0.1 nM. These ranges represent successive improvements in affinity resulting from optimization in medicinal chemistry campaigns. The two highest-affinity ranges represent unusually high-affinity ligands that are sometimes encountered in drug discovery. Figure 4 shows that as the affinity increases, equilibration becomes an increasing liability in quantifying affinity correctly. This is especially evident for the short incubation time of 15 min. For the ranges of 10–100, 1–10, and 0.1–1 nM, the percentage of compounds with EET >15 min rises to 10%, 32%, and 48%, respectively (Fig. 4, Table 2). For very high-affinity ligands (Kd < 0.1 nM), almost all ligands have EETs exceeding 15 min (97%). This finding suggests that short incubation should be avoided for assays used to optimize high-affinity ligands in drug discovery. For the longer incubations of 1 and 2 h, equilibration is not a major concern until very high affinity is reached. Few entries have EET values greater than 1 or 2 h in the 1–10 and 10–100 nM ranges (2%–13%), and the median EET is 1.2 and 5.1 min, respectively (Fig. 4, Table 2). Based on the results, a 2 h incubation time is recommended for the assays used to identify high-affinity advanced leads and development candidates. Unfortunately, for the highest-affinity ligands (Kd ranges of 0.1–1 and <0.1 nM) equilibration is an issue for even the longer incubation times. For example, 44% of the very high-affinity ligands (Kd < 0.1 nM) have EET values exceeding 2 h and the median EET is 93 min. When operating in this affinity range, conventional equilibrium binding assays used in drug discovery can no longer be considered reliable for quantifying the affinity accurately, and alternative approaches are recommended to properly quantify the strength of ligand interaction with the target (see below).
      Finally, we consider a specialist case. In some ligand–target interaction assays, the incubation time is very short (≤1 min) because the drug effect starts immediately upon addition of ligand and because the drug effect is rapid and transient. The classic example is calcium signaling, for example, via a GPCR. The lack of equilibration in such assays is well known and the effect of this on the ligand pharmacology well described.
      • Charlton S.J.
      • Vauquelin G.
      Elusive Equilibrium: The Challenge of Interpreting Receptor Pharmacology Using Calcium Assays.
      • Christopoulos A.
      • Parsons A.M.
      • Lew M.J.
      • et al.
      The Assessment of Antagonist Potency under Conditions of Transient Response Kinetics.
      • Bdioui S.
      • Verdi J.
      • Pierre N.
      • et al.
      Equilibrium Assays Are Required to Accurately Characterize the Activity Profiles of Drugs Modulating Gq-Protein-Coupled Receptors.
      The binding kinetic survey
      • Schuetz D.A.
      • Richter L.
      • Martini R.
      • et al.
      A Structure–Kinetic Relationship Study Using Matched Molecular Pair Analysis.
      indicated that 54% of entries have an EET exceeding 1 min, regardless of the Kd, rising to 85% for ligands with Kd ≤10 nM. This finding indicates that equilibrium cannot be assumed in transient response assays for ligands encountered in drug discovery.

      Comparison between Historical and Modern Conditions

      The ligands used in the classical era when pharmacological analysis was being developed were usually isolated natural products, rather than ligands that had been optimized for the target by medicinal chemists. Most of the compounds developed in this era bound with moderate to low affinity (Kd > 10 nM), and the survey above indicates that equilibration is usually not a major concern, even when using short incubations, for this range of affinity. The EET of some of the historical compounds can be determined since the dissociation rate constant has since been measured. For example, the EET is 33 s for acetylcholine interaction with the M3 muscarinic receptor, an interaction studied extensively in the classical era.
      • Sykes D.A.
      • Dowling M.R.
      • Charlton S.J.
      Exploring the Mechanism of Agonist Efficacy: A Relationship between Efficacy and Agonist Dissociation Rate at the Muscarinic M3 Receptor.
      One of the highest-affinity ligands of the era, the muscarinic antagonist atropine, has an EET of 7.7 min.
      • Dowling M.R.
      • Charlton S.J.
      Quantifying the Association and Dissociation rates of Unlabelled Antagonists at the Muscarinic M3 Receptor.
      It was therefore reasonable to suppose the ligands had equilibrated with the target in these experiments, and so the use of an equilibrium binding equation like eq 1 was appropriate for analyzing the data. In the modern era, ligands are optimized to bind with high affinity to the target, and for some target classes, very high-affinity ligands have been developed (Kd < 0.1 nM). The survey above indicates that equilibration is a concern for the higher-affinity ligands and that equilibration cannot be assumed, especially when the affinity is very high. Under these conditions, application of eq 1, which assumes equilibrium has been reached, can result in erroneous estimates of affinity.
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      ,
      • Aranyi P.
      Kinetics of the Glucocorticoid Hormone-Receptor Interaction. False Association Constants Determined in Slowly Equilibrating Systems.
      The magnitude of the error in the affinity estimate is considered in the next section.

      Effect of Lack of Equilibration on Affinity Estimation

      How large an effect does lack of equilibration have on measurement of affinity? Here this was simulated by generating ligand–target saturation curves for various incubation times. A simulator is provided in the Supplemental Files called “Saturation Curve at Selected Time Point Simulator.” The magnitude of affinity underestimation is dependent on the dissociation rate constant and incubation time.
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Motulsky H.J.
      • Mahan L.C.
      The Kinetics of Competitive Radioligand Binding Predicted by the Law of Mass Action.
      ,
      • Aranyi P.
      Kinetics of the Hormone-Receptor Interaction. Competition Experiments with Slowly Equilibrating Ligands.
      ,
      • Gray H.E.
      • Luttge W.G.
      A Comment on the Estimation of Times Required for the Attainment of Equilibrium by Noncooperative, Single Site Ligand-Receptor Systems.
      ,
      • Heise C.E.
      • Sullivan S.K.
      • Crowe P.D.
      Scintillation Proximity Assay as a High-Throughput Method to Identify Slowly Dissociating Nonpeptide Ligand Binding to the GnRH Receptor.
      Figure 5 illustrates a simulation of three ligands with dissociation rate constant values of 0.001, 0.01, and 0.1 min–1 (corresponding to dissociation t1/2 values of 12 h, 69 min, and 6.9 min). The affinity values of the ligands are 0.1, 1, and 10 nM, but note that the correspondence between affinity and the equilibration artifact is dependent on the association rate constant (see below). In this case, the association rate constant was 107 M–1 min–1 for all three ligands.
      Figure 5
      Figure 5Effect of incubation time on affinity estimation. The degree of target occupancy at various incubation times was simulated for three ligands with varying dissociation rates and affinities, using eq 3. The resulting saturation curves were analyzed with the four-parameter logistic equation

      Motulsky, H. J. Equation: Log(agonist) vs. Response—Variable Slope. https://www.graphpad.com/guides/prism/latest/curve-fitting/REG_DR_stim_variable_2.htm (accessed May 14, 2021).

      to determine the apparent Kd (the [L]50 of the curve, that is, concentration of ligand required to occupy 50% of the targets). (A) For the slowly dissociating, highest-affinity ligand, affinity was dramatically underestimated when using a short incubation time (42-fold, at 15 min) and appreciably underestimated at 2 h (5.6-fold). By 12 h, the affinity was reasonably well estimated (1.5-fold difference from true affinity). (B) The effect was lessened for a ligand dissociating more rapidly. (C) For the most rapidly dissociating ligand, the affinity was estimated accurately at all time points. The simulation values used were koff of (A) 0.001 min–1, (B) 0.01 min–1, and (C) 0.1 min–1; kon of 107 M–1 min–1 for all three ligands; and [L]FREE of 0.001, 0.0063, 0.040, 0.25, 1.6, 10, 63, 400, 2500, and 15,000 nM. A simulator is provided in the , called “Saturation Curve at Selected Time Point Simulator,” to enable investigators to explore their conditions of interest.
      For the slowest-dissociating and highest-affinity ligand, the underestimation of affinity was large at incubation times used routinely in drug discovery (Fig. 5A). At 15 min, the apparent Kd ([L]50 of the four-parameter logistic equation fit

      Motulsky, H. J. Equation: Log(agonist) vs. Response—Variable Slope. https://www.graphpad.com/guides/prism/latest/curve-fitting/REG_DR_stim_variable_2.htm (accessed May 14, 2021).

      ) was 4.2 nM, 42-fold higher than the true Kd of 0.1 nM. At 2 h, the difference was reduced but was still 5.6-fold. However, at 12 h, an incubation time feasible for more stable assays, the affinity estimate was reasonably close to the true affinity (0.15 nM vs 0.1 nM, respectively). For the ligand with an affinity of 1 nM and dissociation t1/2 of 69 min, the underestimation of affinity was reduced (Fig. 5B)—at 15 min, the affinity underestimation was 4.6-fold. For the 2 and 12 h incubation times the measured affinity was in good agreement with the true affinity (apparent Kd of 1.2 and 1.0 nM, respectively). Finally, for the 10 nM affinity ligand with dissociation t1/2 of 6.9 min, equilibration was not an issue for the incubation times evaluated (Fig. 5C)—the apparent Kd values were 11, 10, and 10 nM for 15 min, 2 h, and 12 h incubations, respectively. An alternative way of making the assessment is to compare the different ligands at a specific incubation time. This demonstrates an assay floor effect when the incubation time is too short—at 15 min, the apparent Kd of the three ligands is similar, varying only 2.6-fold (4.2–11 nM), even though the true affinity varies by 100-fold (0.1–10 nM). This highlights the fact that lack of equilibration, particularly for short incubations, can result in distortion of SAR owing to the assay floor effect, as detailed in numerous articles.
      • Jarmoskaite I.
      • AlSadhan I.
      • Vaidyanathan P.P.
      • et al.
      How to Measure and Evaluate Binding Affinities.
      ,
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      ,
      • Aranyi P.
      Kinetics of the Glucocorticoid Hormone-Receptor Interaction. False Association Constants Determined in Slowly Equilibrating Systems.
      ,
      • Charlton S.J.
      • Vauquelin G.
      Elusive Equilibrium: The Challenge of Interpreting Receptor Pharmacology Using Calcium Assays.
      • Christopoulos A.
      • Parsons A.M.
      • Lew M.J.
      • et al.
      The Assessment of Antagonist Potency under Conditions of Transient Response Kinetics.
      • Bdioui S.
      • Verdi J.
      • Pierre N.
      • et al.
      Equilibrium Assays Are Required to Accurately Characterize the Activity Profiles of Drugs Modulating Gq-Protein-Coupled Receptors.
      It is important to note that the range of affinity over which the underestimation occurs is dependent on the association rate constant of the ligands. For example, if the kon value is 10-fold lower than that used in Figure 5 (106 M–1 min–1 instead of 107 M–1 min–1), the range of Kd over which the changes occur would be 10-fold higher (1, 10, and 100 nM). Reciprocally, if the kon value is 10-fold higher (108 M–1 min–1), the changes would occur over a Kd range 10-fold lower (0.01, 0.1, and 1 nM). The situation is further complicated by the reality that, for many targets, kon can vary within a chemical series.
      • Schuetz D.A.
      • Richter L.
      • Martini R.
      • et al.
      A Structure–Kinetic Relationship Study Using Matched Molecular Pair Analysis.
      Recently, a case study of this effect was reported, including a description of the effect of the affinity underestimation on the drug development process.
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      Antagonists for a GPCR, the corticotropin-releasing factor receptor, were being developed as potential treatments for endocrine and psychiatric disorders. A first-generation compound had been developed with efficacy demonstrated in animal models and clinical biomarker studies (NBI 30775).
      • Chen C.
      • Wilcoxen K.M.
      • Huang C.Q.
      • et al.
      Design of 2,5-Dimethyl-3-(6-Dimethyl-4-methylpyridin-3-yl)-7-Dipropylaminopyrazolo[1,5-a]py Rimidine (NBI 30775/R121919) and Structure–Activity Relationships of a Series of Potent and Orally Active Corticotropin-Releasing Factor Receptor Antagonists.
      This compound was discontinued for toxicological reasons, and so new molecules were being developed. In the in vitro target interaction assay, the criterion for compound advancement was an affinity equivalent to NBI 30775, which bound with a Kd of 10 nM in the assay. Numerous compounds were identified that met this criterion and had improved pharmacokinetics and drug-like characteristics. However, none of the new compounds displayed appreciable efficacy in the in vivo animal model.
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      It was subsequently discovered that the dissociation rate of the original molecule, NBI 30775, was very slow (dissociation t1/2 of 46 h).
      • Fleck B.A.
      • Hoare S.R.
      • Pick R.R.
      • et al.
      Binding Kinetics Redefine the Antagonist Pharmacology of the Corticotropin-Releasing Factor Type 1 Receptor.
      This indicated that the in vitro assay, with an incubation time of 1.5 h, was very far from equilibrium for NBI 30775 and so was dramatically underestimating the affinity of this ligand.
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      ,
      • Fleck B.A.
      • Hoare S.R.
      • Pick R.R.
      • et al.
      Binding Kinetics Redefine the Antagonist Pharmacology of the Corticotropin-Releasing Factor Type 1 Receptor.
      ,
      • Ramsey S.J.
      • Attkins N.J.
      • Fish R.
      • et al.
      Quantitative Pharmacological Analysis of Antagonist Binding Kinetics at CRF1 Receptors In Vitro and In Vivo.
      By contrast, the new molecules dissociated much more rapidly, and so the assay was providing reasonable estimates of their affinity.
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      Consequently, in reality the new molecules bound with much lower affinity than NBI 30775 (by up to 180-fold), but the molecules could not be distinguished in the assay owing to a detection ceiling resulting from lack of equilibration. The lower affinity of the new molecules explained their lower efficacy in the in vivo model.
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      Once this realization had been made, assays were put in place to estimate the affinity correctly (kinetic binding assays; see below)
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      and new molecules were identified with affinity similar to NBI 30775 that displayed efficacy in the in vivo model, ultimately leading to the development of clinically effective compounds.

      Managing Equilibration Artifacts in Drug Discovery

      In principle, the simplest way to determine whether an assay is at equilibrium is to extend the incubation time and determine whether there is a reduction of the apparent Kd at the later time point (Fig. 5). This is colloquially referred to as a Kd (or Ki) shift assay and can be used to diagnose whether a ligand dissociates and equilibrates slowly from the target. An example is provided by a study of the gonadotropin-releasing hormone receptor,
      • Heise C.E.
      • Sullivan S.K.
      • Crowe P.D.
      Scintillation Proximity Assay as a High-Throughput Method to Identify Slowly Dissociating Nonpeptide Ligand Binding to the GnRH Receptor.
      in which test compounds were competed against a radiolabeled ligand at two time points, 30 min and 10 h. Compounds with a Ki(30 min)/Ki(10 h) ratio of 6 were found to dissociate, and so equilibrate, markedly slowly (dissociation t1/2 ≥ 1 h, EET ≥ 3 h). An alternative experimental approach is to assess the dissociation rate constant for the ligand–target interaction. Qualitative methods are available to diagnose whether the ligand–target complex dissociates slowly, including (1) for receptor antagonists, testing whether the ligand reduces the Emax of agonist responses in functional assays (insurmountable antagonism), which can be diagnostic of a slow antagonist dissociation rate;
      • Ramsey S.J.
      • Attkins N.J.
      • Fish R.
      • et al.
      Quantitative Pharmacological Analysis of Antagonist Binding Kinetics at CRF1 Receptors In Vitro and In Vivo.
      • Kenakin T.
      • Jenkinson S.
      • Watson C.
      Determining the Potency and Molecular Mechanism of Action of Insurmountable Antagonists.
      • Vanderheyden P.M.
      • Fierens F.L.
      • De Backer J.P.
      • et al.
      Distinction between Surmountable and Insurmountable Selective AT1 Receptor Antagonists by Use of CHO-K1 Cells Expressing Human Angiotensin II AT1 Receptors.
      (2) also for antagonists, preincubating with the ligand and then using washout methods to remove free antagonist and measuring the return of responsiveness to an agonist;
      • Bosma R.
      • Witt G.
      • Vaas L.A.I.
      • et al.
      The Target Residence Time of Antihistamines Determines Their Antagonism of the G Protein-Coupled Histamine H1 Receptor.
      ,
      • Sahlholm K.
      • Zeberg H.
      • Nilsson J.
      • et al.
      The Fast-Off Hypothesis Revisited: A Functional Kinetic Study of Antipsychotic Antagonism of the Dopamine D2 Receptor.
      and (3) the kinetic rate index method, applied to kinetic competition binding assays to identify slowly dissociating test ligands.
      • Guo D.
      • van Dorp E.J.
      • Mulder-Krieger T.
      • et al.
      Dual-Point Competition Association Assay: A Fast and High-Throughput Kinetic Screening Method for Assessing Ligand-Receptor Binding Kinetics.
      If the ligand of interest dissociates very slowly, it can be difficult to quantify affinity using conventional equilibrium assays because it can be impractical to incubate the assay for long enough for equilibrium to be approached. For example, for the CRF1 antagonist NBI 30775, the dissociation t1/2 at room temperature was 46 h, which would require an incubation time approaching 6 days. For such ligands, one approach to quantify affinity accurately is to measure the association and dissociation rate constants. From these values, the affinity can be determined since the affinity is related to the binding rate constants by the equation Kd = koff/kon. This method was used to quantify the affinity for CRF1 antagonists.
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      ,
      • Fleck B.A.
      • Hoare S.R.
      • Pick R.R.
      • et al.
      Binding Kinetics Redefine the Antagonist Pharmacology of the Corticotropin-Releasing Factor Type 1 Receptor.
      ,
      • Ramsey S.J.
      • Attkins N.J.
      • Fish R.
      • et al.
      Quantitative Pharmacological Analysis of Antagonist Binding Kinetics at CRF1 Receptors In Vitro and In Vivo.
      It should noted that while measuring the rate constants is an effective way to measure affinity, kinetic binding assays require significant resources for assay development and data analysis.
      • Vauquelin G.
      • Huber H.
      • Swinney D.C.
      Experimental Methods to Determine Binding Kinetics.
      • Georgi V.
      • Dubrovskiy A.
      • Steigele S.
      • et al.
      Considerations for Improved Performance of Competition Association Assays Analysed with the Motulsky-Mahan’s “Kinetics of Competitive Binding” Model.
      • Sykes D.A.
      • Jain P.
      • Charlton S.J.
      Investigating the Influence of Tracer Kinetics on Competition-Kinetic Association Binding Assays: Identifying the Optimal Conditions for Assessing the Kinetics of Low-Affinity Compounds.

      Conclusions and Recommendations

      The mass action equation used routinely to quantify the affinity of drug–target interaction assumes minimal ligand depletion and a close approach to equilibrium, conditions that were met almost universally when the equation was developed but which are frequently infringed in modern drug discovery. Ligand depletion and equilibration artifacts result in underestimation of affinity, particularly for the most valuable chemical matter in drug discovery, the high-affinity ligands from which clinical and development candidates are selected. Underestimation of affinity can have damaging consequences to drug development,
      • Hoare S.R.J.
      • Fleck B.A.
      • Williams J.P.
      • et al.
      The Importance of Target Binding Kinetics for Measuring Target Binding Affinity in Drug Discovery: A Case Study from a CRF1 Receptor Antagonist Program.
      including incorrect predictions of human dosing, confusion in selecting candidate molecules for progression, inconsistencies between in vivo and in vitro assessments of ligand activity, erroneous conclusions of flat SAR when assay ceilings are encountered, failure to identify the highest-affinity molecules, and incorrect assessments of target selectivity. Consequently, identifying assay artifacts that can distort measurements of ligand affinity is important for effective ligand optimization in drug discovery. In this study, the theoretical and practical dimensions of ligand depletion and equilibration have been described. Below are some general recommendations for managing these issues.

      Awareness

      The simplest recommendation is to be aware of the potential for ligand depletion and lack of equilibration to impact the drug discovery project. If an assay is being used with a high target concentration or a very short incubation time, investigators can reasonably assume that there will be a limit of detection of the Kd. If flat SAR is identified, it is recommended to check first whether this is an artifact of the assay before concluding there is a true physicochemical limit to the target–ligand interaction affinity. If very high-affinity ligands are identified (Kd < 0.1 nM), it is reasonable to suspect that the assay is not at equilibrium if a conventional incubation time is being used (e.g., 1 h), and it is advisable to extend the incubation time to assess equilibration.

      Be Amenable to Changing the Assay from That Used for HTS

      The requirements for an HTS assay can result in ligand depletion and lack of equilibration. This usually does not affect affinity estimates of the low-affinity ligands the HTS assay is designed to detect. In HTS assays, ligand depletion can result from miniaturization (i.e., reducing the volume) in order to reduce reagent and consumable costs, and from increasing the target concentration in order to maximize the signal. Sometimes short incubation times are used for HTS assays for workflow reasons or because the response being measured is transient. If the HTS assay continues to be used for ligand optimization, quantifying the affinity of the high-affinity ligands that emerge from medicinal chemistry can be dramatically impacted by the assay conditions. It is recommended that the conditions of the assay be assessed before medicinal chemistry starts and that the assay be adjusted as necessary. For example, assay volumes can be increased, and more sensitive detection systems can be used in order to decrease the target concentration. Such action can be justified by the lower number of compounds being tested and by the high cost of misestimating the affinity of candidate molecules. If a short, transient response is being measured, it is sometimes feasible to detect more long-lasting intermediates of the signaling pathway. A good example is provided by the replacement of the very rapid calcium response assay with the more prolonged inositol phosphate accumulation assay,
      • Bdioui S.
      • Verdi J.
      • Pierre N.
      • et al.
      Equilibrium Assays Are Required to Accurately Characterize the Activity Profiles of Drugs Modulating Gq-Protein-Coupled Receptors.
      which enables more meaningful quantification of the ligand’s pharmacological activity.

      Perform Calculations and Experiments to Realize the Limitations of the Assay

      It is recommended to calculate the target concentration either precisely (e.g., when using a purified target) or approximately (e.g., when using a whole-cell assay). This allows the Kd limit to be determined, as the target concentration divided by 2. If a target concentration artifact is suspected, increase the assay volume to test whether this reduces the Kd value estimate. If equilibration is potentially an issue, increase the incubation time as long as feasible. Finally, it is recommended to assess the limits of sensitivity by performing simulations. The simulators provided in the Supplemental Files are designed for this purpose.
      These considerations will enable investigators to assess the risk of these artifacts impacting the quantification of ligand–target interaction in their drug discovery projects, and to design assays and data analysis to correctly estimate the affinity of target–ligand interaction.
      Declaration of Conflicting Interests
      The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

      Funding

      The author received no financial support for the research, authorship, and/or publication of this article.
      Supplemental material is available online with this article.

      Supplemental Material

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