1521-0111/92/4/414–424$25.00 https://doi.org/10.1124/mol.117.108787MOLECULAR PHARMACOLOGY Mol Pharmacol 92:414–424, October 2017Copyright ª 2017 by The American Society for Pharmacology and Experimental Therapeutics

MINIREVIEW

A Scale of Agonism and Allosteric Modulationfor Assessment of Selectivity, Bias, and Receptor Mutation

Terry KenakinDepartment of Pharmacology, University of North Carolina School of Medicine, Chapel Hill, North Carolina

Received March 11, 2017; accepted June 12, 2017

ABSTRACTAn index of agonism is described that can be used to quantifyagonist receptor selectivity, bias, cell-based agonism, andthe effects of receptor mutation on signaling. The parameter isderived from agonist concentration-response curves and com-prises the maximal response to the agonist (max) and the EC50 inthe form of Dlog(max/EC50). This parameter is derived fromequations describing agonists as positive allosteric facilitatorsof receptor-signaling protein interaction. A similar index is also

derived to quantify the potentiating effects of positive allostericmodulators, which can be used to quantify in situ positiveallosteric modulator activity in vivo. These indices lend them-selves to statistical analysis and are system-independent in thatthe effects of the system processing of agonist response anddifferences in assay sensitivity and receptor expression arecancelled. The various applications of the Dlog(max/EC50) scaleare described for each pharmacologic application.

IntroductionA critical component of the lead optimization process in new

drug discovery for agonists and allosteric modulators is thedetermination of relative measures of activity that are notspecifically linked only to the assays in which they aremeasured; that is, they are system-independent measuresof activity. For full agonists, system-independent measures ofactivity are achieved through relative potency ratios (ratios ofconcentration of agonist producing half-maximal responsevalues [EC50]); however, this scale devolves into a nonlinearscale when comparing full and partial agonists, so it cannot beused for the comparison of these types of agonists. In thisarticle, agonism is developed as a positive allosteric modula-tion of the natural receptor-signaling protein interaction,and the resulting scale is presented as a system-independentmeasure of the relative receptor activation for any set ofagonists (full or partial). Separately, the same approach isapplied to the assessment of the relative activity of any set ofallosteric modulators, positive (PAMs) or negative (NAMs).Although the activity of NAMs can be quantified withstandard methods for antagonists, the affinity of PAMs isuniquely dependent on the cobinding agonist and thus requiresspecial methods to assess. This new scale may be particularlyuseful in that it can be used to furnish system-independent

measures of PAM activity in vivo for the advancement ofcandidate molecules.

Receptor-Signaling Protein InteractionThe discovery of constitutive seven transmembrane recep-

tor activity (Costa and Herz, 1989) and subsequent recastingof the ternary complex model for receptors to the extendedternary complex model (Samama et al., 1993) have revealedthat agonists can be considered simply as PAMs of an alreadyongoing spontaneous association between the receptor and thesignaling protein (Kenakin, 2015). In accordance with thestandard functional allosteric model for receptors (Ehlert,2005; Kenakin, 2005; Price et al., 2005), such modulators (inthis case, agonists) can change the affinity of the receptorfor the signaling protein (through a cooperativity term a)and the efficacy of the receptor-signaling protein complexfor production of cellular response (through a cooperativityterm b). In terms of binding, the relevant protein speciescan be described within the context of the standardStockton-Ehlert allosteric binding model (Stockton et al.,1983; Ehlert, 1988) whereby the agonist (denoted A) andsignaling protein (denoted G) bind to separate but interac-tive sites on the receptor. Thus, both A and G interact withthe receptor with equilibrium association constants K’a andKg, respectively:This binding model (eq. 1) is then placed as the receptorspecies producing unit for the Black-Leff operational modelhttps://doi.org/10.1124/mol.117.108787.

ABBREVIATIONS: EC50, concentration of agonist producing half-maximal response; NAM, negative allosteric modulator; PAM, positive allostericmodulator.

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of agonism (Black and Leff, 1983) to yield the functionalallosteric model. An “allosteric vector” can be described todenote the direction of modulation (Kenakin and Miller,2010), which in this case has the binding of a ligand to thereceptor directing the modification of the interaction be-tween the receptor and a signaling protein within thecytosol; this vector constitutes agonism. It should be notedthat G protein–coupled receptors have been described asallosteric proteins in the literature in early mathematicalmodels (Karlin, 1967; Thron, 1973). The standard func-tional allosteric model for a cytosol-directed vector can beused to show that ratios of the maximal response (max) andEC50 (concentration of agonist producing half-maximalresponse) from agonist concentration-response curves for aset of agonists (to furnish Dlog(max/EC50) values) create asystem independent scale of agonism that can be used toquantify selectivity (see Appendix 1 for derivation). Thissame conclusion can be reached through derivation of thetermwithin the Black-Leff operational model as well (Kenakin,2015) (see Appendix 2).Historically, the relative activity of agonists has been

quantified through potency ratios (e.g., differences in thenegative logarithm of EC50 values denoted as DpEC50). Forfull agonists, DpEC50 values are constant over all ranges ofsystem sensitivity (as long as both agonists produce fullsystem response; i.e., they are full agonists). This relationshipbreaks down, however, when one or both of the agonistsbecome a partial agonist because the impact of tissue sensi-tivity on the potency of full and partial agonists is differ-ent. Specifically, reductions in receptor density for full agonistsproduce defined dextral displacement of concentration-responsecurves, in contrast to the effect produced on curves for partialagonists, which essentially do not change location along theconcentration axis but rather show depressed maxima (Fig.1A), producing distinctly nonlinear changes in the relativepEC50 values of an agonist when it expresses partial agonism

in systems of low sensitivity (see curvature in the relationshipbetween pEC50 and receptor density shown in Fig. 1B). Thisnonlinearity makes DpEC50 values dependent on tissuesensitivity and thus not useful as a system-independentindex of relative agonist activity. For dose-response curveswith Hill coefficients not significantly different from unity,normalization of agonist activity through inclusion of themaximal response [in the form of Dlog(max/EC50) values]corrects this dependence on tissue sensitivity and yields atruly system-independent scale of agonism (see Fig. 1); thiseffect is shown with experimental data in Fig. 2 for theb-adrenoceptor partial agonist prenalterol and full agonistisoproterenol. As shown in Fig. 2, the DpEC50 values pro-duce a distinct curvature with varying tissues, whereasDlog(max/EC50) values remain constant through a range oftissue conditions.Three important points to consider when discussing

log(max/EC50) values are the following: the fact that calcu-lation of max/EC50 reduces agonism to a single number.This is useful from the point of view of allowing statisticalanalyses (i.e., as in the analysis of signaling bias, Kenakinet al., 2012). The various formulae to do this are described inAppendix 3, with the key parameter to enable the calcula-tions being an estimate of pooled variance (Kenakin et al.,2012). Thus, estimates of the agonism produced by anymolecule can be made with mean log(max/EC50) values 695% confidence limit, comparison of the relative base agonistactivity in any one system can be made with Dlog(max/EC50)values 6 95% confidence limits; finally, comparisons ofselectivity, bias, and the effects of mutation made throughDDlog(max/EC50) 6 95% confidence limit estimates withsimple adjustment of formulae based on the pooled variance(see Appendix 3).The second point is that the comparison of log(max/EC50)

values between agonists in a single functional system toproduce Dlog(max/EC50) allows the system independent scal-ing of agonism within a given functional system. When this isdone, the agonism of a test compound is compared with areference agonist and system effects are cancelled (i.e., thesensitivity of the functional system is not an issue), and theDlog(max/EC50) reflects molecular efficacy and affinity in asystem independent manner within a given assay. The thirdpoint involves the cross comparison of different responsesystems, whether they be signaling pathways in the cell ordifferent receptors. Once the power of a test agonist isscaled to the same reference agonist in two systems, thenDDlog(max/EC50) values cancel cross-system effects (includ-ing differences in assay sensitivity) and become independent

Fig. 1. Effect of changing receptor density (and/or tissuesensitivity) on concentration response to an agonist. (A) Notehow less sinistral displacement of EC50 values is observed forthe agonist when it produces partial agonism compared to fullagonism. (B) Dlog(max/EC50) values (solid line) and DpEC50values (dotted line) with changing tissue sensitivity for twoagonists, one of which produces partial agonism at low tissuesensitivities. Note how DpEC50 values vary with tissue sen-sitivity, whereas Dlog(max/EC50) values remain stable.

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measures of the power of the agonist activation across thetwo systems. This can be in terms of different receptors(receptor selectivity), different signaling pathways (biasedsignaling), cell-based agonist selectivity, or measures of theimpact of receptor mutation on a given agonist activity. It isworth considering these settings as they are fundamentalpharmacologic procedures applied to the quantitative mea-surement of agonist effect. As a prerequisite, it is useful toconsider some operational features of the analysis.

Practical Use of the log(Max/EC50)It is important to note that the maximal response for

agonism must be expressed as a fraction of the maximalwindow available in the assay to express agonism and notsimply as themaximal response to themost efficacious agonistin the assay. For example, if direct activation of adenylatecyclase with forskolin in a given functional assay produces amaximal elevation of cAMP greater than the most efficaciousagonist in the assay, then themaximal response to the agonistmust be expressed as a fraction of the maximal effect offorskolin. In addition, the derivation given in Appendix1 assumes that the Hill coefficient of the agonist concentrationresponse curves is not significantly different from unity. In thecomparison of the Black-Leff operationalmodel scale ofDlog(t/KA),to Dlog(max/EC50) values, this is an explicit requirement to equatethe two values (Appendix 2). In some instances in experimentalpharmacology, this is not the case; therefore, it is useful to explorethe effects of slopes differing from unity on the immutability ofthe Dlog(max/EC50) scale with receptor density and tissue sensi-tivity. Specifically, the comparison is made between a curve fit totheHill equation of response5 [A]nmax/([A]n1EC50

n)wheren51and n � 1. Then comparisons of different log(max/EC50) valuescan be made through simulation whereby the actual values of

Dlog(max/EC50) values for concentration-response curves of differ-ent slopes can be compared with the true values of Dlog(t/KA); asshown in eq. 2, this yields an error term w, where

w 5  Dlogðmax=EC50Þ-Dlogðt=KAÞ: (2)

Figure 3 shows a simulation surface of the dependence of w onslope and the maximal response of the agonist concentration-response curves. It can be seen from this figure that slopes.1provide Dlog(max/EC50) values that depend only slightly onslope (as indicated by the small deviation with agonistmaxima). In fact, the main region of deviation occurs withagonist concentration-response curves of low maximal re-sponse and slopes significantly less than unity. This should be

Fig. 2. Ratios of log(max/EC50) values forthe b-adrenoceptor full-agonist isoproter-enol and the partial agonist prenalterol(left ordinate axis; data as open circles)as a function of the maximal response toprenalterol in range of isolated tissues(abscissae). Dotted line shows the ratioof pEC50 values (right ordinate axis; datain filled circles). Data from Kenakin andBeek (1982).

Fig. 3. Effects of efficacy (range of receptor densities) t and slope of theagonist concentration-response curve (slope) on the differences betweenindices of agonism as calculated by log(t/KA) versus log(max/EC50) values.

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considered as a caution to the use of Dlog(max/EC50) values forlow-efficacy agonists (maximal responses,35%) demonstratingconcentration-response curves with slopes significantly lowerthan 0.5.Table 1 shows the sequential procedures required to apply

the DDlog(max/EC50) scale for quantification of selectivity fordifferent receptors, signaling pathways, cell types, and re-ceptor sequence.

DDlog(Max/EC50) Quantification of ReceptorSelectivity

Historically, receptor selectivity has been expressed as theratio of agonist potencies, and for full agonists, this yieldsuseful and system independent measures; however, as dis-cussed, the use of maximal responses extends this scale to allagonists, partial and full, and provides a more inclusive scale.Table 2 shows data describing concentration-response curvesfor four muscarinic agonists onM1 andM4 receptors (Chinesehamster ovary cell GTPgS binding) from Watt et al. (2011).Selectivity can be calculated through conventional po-tency values (EC50) or through log(max/EC50); in the case oftalsaclidine, different outcomes illustrate the effect of ignoringdifferences in maximal response. In cases where partialagonism is produced, EC50 values overestimate the agonismof the ligand because full agonism is assumed in the calcula-tion. The first step is to cancel the effect of the sensitivity ofeach assay by comparing agonism with a reference compound;in this case, acetylcholine is the reference. Considering pEC50

values first, talsaclidine is 0.032-fold as active as acetylcholineon M1 receptors and 0.066-fold as active as acetylcholine onM4 receptors; this leads to an overall selectivity of talsaclidineof 2.1 forM4 receptors (talsaclidine is relativelymore active onM4 thanM1 receptors). Use of log(max/EC50) values leads to adifferent conclusion; talsaclidine is 0.022-fold as active on M1receptors and 0.016-fold as active on M4 receptors, whichleads to an overall selectivity of 1.38-fold for M1 receptors(nonselective). This is in keeping with the lower maximalresponse of talsaclidine for M4 receptors (21.6%), indicatinga lower activity for that receptor subtype. In general, pEC50

values assume amaximal response for M4 receptors, therebyoverestimating M4 activity and erroneously classifyingtalsaclidine as M4-selective. Figure 4 shows the selectivityof the four agonists in these recombinant functional systems,where it can be seen that as the maximal responses to theagonists diminish, the disparity between DDlog(max/EC50)and DDpEC50 increases.

The general tissue selectivity of agonists also can bequantified; Fig. 5 shows the relative activity of eight musca-rinic agonists, compared with that of acetylcholine, in guineapig bladder and ileum. Relative selectivity is calculatedthrough DDlog(max/EC50) values and relative agonist potencyratios as DDpEC50 values. In this calculation, the value foreach agonist within a given tissue is compared with that ofacetylcholine through Dlog(max/EC50) or DpEC50 values, andthen the selectivity between the tissues is assessed throughdifferences in these values in these two tissues to yieldDDlog(max/EC50) or DDpEC50 values for tissue selectivity (tocancel differences in tissue sensitivity between the assays).The result is a measure of how well the agonists activate themuscarinic receptors of guinea pig bladder and ileum. Tissue-selective differences might be seen with varying mixtures ofreceptor type in each tissue- or in a cell-type effect on biasedagonism (Kenakin, 2016); the analysis makes no assumptionsas to the nature of the differences in potency and functionsonly as an operational measure of observed selectivity. It canbe seen that, as expected, estimates are identical when bothagonists produce full agonism. In drug discovery programs,however, where new test molecules of low intrinsic activity arecompared with powerful standard agonists, this scale becomesimportant, illustrated by the deviations in values in Fig. 5, inwhich one of the agonists is a partial agonist in either tissue(see far right column of table showing relative maxima andcompare differences between DDlog(max/EC50) and DDpEC50).

DDlog(Max/EC50) Quantification of Signaling BiasJust as extracellular agonist selectivity can be quantified

using this scale, so too can intracellular selectivity (agonistbias). This occurs when a given agonist that interacts with apleiotropically coupled receptor selectively activates one ormore of the signaling cascades at the expense of others (i.e., itbiases the stimulus and does not distribute activation evenlyamong the available pathways). This is predicted to be theresult of the stabilization of different receptor active states bydifferent agonists (Kenakin and Morgan, 1989; Kenakin,1995) and, irrespective of mechanism, is a widespread phe-nomenon in pharmacology studied by numerous researchgroups and given a variety of names [i.e., stimulus trafficking(Kenakin, 1995), biased signaling (Jarpe et al., 1998), func-tional selectivity (Lawler et al., 1999); collateral efficacy(Kenakin, 2005), functional dissociation (Whistler and vanZastrow, 1999), biased inhibition (Kudlacek et al., 2002), anddifferential engagement (Manning, 2002). Insofar as signaling

TABLE 1Practical application of the DDlog(max/EC50) scale to quantify selectivity

Procedure Rationale

1) Fits dose-response data to function to yield max (maximal response)and EC50

Furnishes individual values for log(max/EC50), a single index ofagonism

2) Choose a reference agonist for comparison of all test agonists; usethe same reference agonist for all systems (receptors, pathways, celltypes)

Ratios to the index for the reference agonist cancels differences inreceptor density, cell sensitivity, assay sensitivity, etc.

3) Calculate Dlog(max/EC50) values for each test agonist(Dlog(max/EC50) = log(max/EC50)ref - log(max/EC50)test)

Scales agonist activity of the test agonist to that of the referenceagonist within a given system (receptor type, signaling pathway,cell type, etc.)

4) Calculate DDlog(max/EC50) values across the two systems beingcompared (different receptors, signaling pathways, cell types,protein sequence, etc.) (DDlog(max/EC50) = Dlog(max/EC50)ref –Dlog(max/EC50)test)

With the individual differences in sensitivity between the two systemscancelled, DDlog(max/EC50) values provide a system-independentmeasure of the relative agonism of each test agonist in both systems

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bias may be a therapeutically exploitable favorable agonistproperty, it is useful to have a quantitative scale to guidemedicinal chemists in efforts to optimize this effect.A theoretically optimal scale for this uses DDlog(t/KA)

values (Kenakin et al., 2012; Kenakin and Christopoulos,2013), and just as DDlog(max/EC50) values can be usefulsurrogates for agonist selectivity, they can also function asthe same for signaling bias. Thus, when the slopes of theconcentration response curves to the agonists are .0.5 ormaximal responses .35%, DDlog(max/EC50) values can pro-vide convenient and rapid assessment of signaling bias. As forreceptor selectivity, log(max/EC50) values are calculated foreach agonist for two signaling pathways and then comparedthrough Dlog(max/EC50) values using a reference agonistin each (the reference agonist must be the same for bothpathways); this cancels the relative effects of assay sensitivityin each assay. This is extremely important because signalingassays, such as effects on second messengers (e.g., cAMP) arehighly coupled and much more sensitive than assays quanti-fying b-arrestin complementation. After this, cross pathway

comparison can be done through comparison of Dlog(max/EC50)values to yield values of DDlog(max/EC50). The bias is thencalculated as the antilog of DDlog(max/EC50) values. It shouldbe noted that the bias is a vector that can be expressed intwo directions. For example, a bias for two agonists A andB showing that agonist A favors the cAMP system (overb-arrestin) by a factor of 5 can also be expressed as agonistA having a bias away from b-arrestin of 0.2. In general,when bias values are reported, this vector orientation mustalways be denoted.Figure 6 shows the bias of five opioid receptor agonists

compared with that of salvinorin A for k-opioid receptorinhibition of cAMP production and b-arrestin signalingpathways (White et al., 2014). As with receptor selectivity, itcan be seen that bias estimates differ when one of the agonistsproduces partial maximal response and simple EC50 values (inthe form of DDpEC50) versus consideration of maxima (in theform of DDlog(max/EC50) estimates are used. In general, bias isunderestimated if only EC50 values are used.

Assessment of Cell-Type SpecificityA well known observation in pharmacology is the imposi-

tion of cell-type effects on receptor selectivity. For example,expression of the same receptor in different host cell types canproduce differences in the relative potency ratios of agonists(e.g., calcitonin) (Christmanson et al., 1994; Watson et al.,2000). Although this is incompatible with a monotonic receptorcoupling scheme for agonists in cells, it can occur if agonistsproduce biased signaling at the receptor and the difference hostcell types emphasize the heterogenous signals in different ways(Kenakin, 2016); in these cases, DDlog(max/EC50) valuescan be used to identify cell-type specificity. Specifically,bias plots, where the response to an agonist in one cell typeis expressed as a function of the response in another celltype, can furnish visual data to indicate where an agonistproduces a unique response in a given cell type over otheragonists. For example, Fig. 7 shows label-free responses tomuscarinic agonists in HT-29 and SF268 cells (Deng et al.,2013). A linear relationship would not necessarily be expectedas different cell types may have differing receptor expressionlevels and efficiency of receptor coupling, but if the agonists

Fig. 4. Radar plot showing muscarinic receptor selectivity for fouragonists activating M1 compared with M4 receptors. Selectivity expressedas DDlog(max/EC50) values (solid line) and DDpEC50 values (dotted line).Data recalculated from Watt et al., 2011.

TABLE 2M1/M4 receptor selectivity for agonists

Max. EC50 (nM) log(max/EC50) pEC50 Dlog(max/EC50) Rel. Aga DpEC50 Rel.Ag.b

Relative agonism at M1 receptorsAcetylcholine

1 25.7 7.59 7.59 0 1 0 1

Sabcom. 0.389 56.2 6.84 7.25 20.75 0.178 20.34 0.457Talsaclidinec 0.693 812.8 5.93 6.09 21.66 0.022 21.5 0.032Xanol 0.637 43.7 7.16 7.36 20.43 0.375 20.23 0.589

Relative agonism at M4 receptorsAcetylcholine 0.87 52.5 7.22 7.28 0 1 0 1Sabcom 0.2 67.6 6.47 7.17 20.75 0.178 2011 0.776Talsaclidine 0.216 794.3 5.43 6.1 21.79 0.016 21.18 0.066Xanol 0.46 63.1 6.86 7.2 20.36 0.44 20.08 0.832

aRelative agonism based on log(max/EC50) values.bRelative agonism based on EC50 values.cTalsaclidine selectivity was calculated as DDlog(max/EC50). DDLog(max/EC50) = 0.13: Talsaclidine is 100.13 = 1.38 selective for M1 receptors.

Talsaclidine selectivity calculated as DDpEC50. DDpEC5 = 20.32: Talsaclidine is 1020.32 = 0.48 selective for M1 receptors (i.e., 2.1 � selective forM4 receptors).

Data from Watt et al., 2011.

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produce a uniform receptor-active state, then a concordance(i.e., no deviations in the relationship for any one agonist) inthis bias plot would be expected for all agonists tested. As seenin Fig. 7, however, although most of the agonists followed afairly uniform pattern, bethanechol shows a distinctly differ-ent bias, being uniquely more active in SF268 cells than inHT-29 cells (compared with the other agonists). This differ-ence can be quantified and statistically estimated throughDDlog(max/EC50) values. For example, the data shown in thetable with Fig. 7 shows that bethanechol is 7.86-fold biasedtoward producing responses in SF268 cells versus HT-29 cells.If, in the example shown in Fig. 7, the mean cell bias towardresponse in SF268 cells for acetylcholine, methacholine, carba-chol, andOxo-M is 1.7, bethanechol produces a 7.86/1.75 4.5-foldselective bias toward SF268 cells compared with these otheragonists. This type of analysis might be applied to the testing ofligands in healthy cells those from disease models (or tumorversus normal cells) to identify unique cell-based activity fortherapeutic applications.

Assessment of Receptor MutationIn the study of the effects of receptor mutations on agonist

function and functional signaling, important considerationsare differences owing to variations in receptor expression.The application of DDlog(max/EC50) values negates thisproblem through comparison of effects to a common standardfor both the wild type and mutated receptor. Just as withthe assessment of signaling bias, the internal comparison ofagonist function to a common reference agonist for both the

wild-type and mutated receptor cancels any effective differ-ences in the disposition of the two types of receptor protein bythe cell. Once the relative agonism of two agonists is quanti-fied for each receptor species (wild-type versus mutation),then comparisons between them can be made that will becorrected for efficiency of transduction and expression withDDlog(max/EC50) values. One possible difference from theprocess used to assess signaling bias is in the choice ofreference agonist. Specifically, when quantifying signalingbias, usually the test agonist is compared with the naturalagonist to yield ameasure of predicted differences in signalingwith the synthetic agonist (as opposed to natural signaling).Whereas natural signaling is sometimes referred to as“unbiased signaling,” this is a misnomer since the naturalagonist will be biased according to the physiologic needs ofthe organ; therefore,what ismeasured as “bias” for the syntheticligand is simply a difference from the bias of the natural ligand.In contrast, when exploring the effects of mutation on receptorfunction, the aim often is to assess the effects of themutation onthe natural wild-type receptor interacting with the naturalagonist. Under these circumstances, a synthetic ligand is chosenas the reference agonist (to cancel systems effects), and theinduced bias on the natural ligand is thus measured as anassessment of the effects of the mutation (Tschammer et al.,2011). Figure 8 shows the comparison of thewild-type dopamineD2L receptor with a D2L H3936.35A receptor mutant throughDDlog(max/EC50) and DDpEC50 values; it can be seen from thisfigure that, as with receptor selectivity and agonist bias, theeffects of mutation are underestimated if DDpEC50 values areused (as opposed to DDlog(max/EC50)).

Fig. 6. Radar plot showing biased signaling of k-opioidagonists (G proteins versus b-arrestin) either throughDDlog(max/EC50) values (solid line) or DDpEC50 values(dotted line); reference agonist is salvinorin A. Data fromWhite et al., 2014.

Fig. 5. Radar plot of receptor selectivity, compared withacetylcholine as a reference agonist, expressed asDDlog(max/EC50) values (solid line) and DDpEC50 values(dotted line) for eight muscarinic agonists for agonismin guinea pig ileum and urinary bladder. Data fromRingdahl (1987). Right-most column indicates the maximaof agonists relative to that of acetylcholine. Note howDDlogpEC50 values deviation from DDlog(max/EC50) in-creases with partial agonism.

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Quantifying PAM EffectsAn important distinction between NAMs and PAMs is that

the effective affinity of the latter species (i.e..,PAMs) dependsmuch more on the co-binding ligand than does the former(NAMs). The reason for this comes from the expression foreffective affinity of allosteric ligands in the Stockton-Ehlertallosteric binding model (Stockton et al., 1983; Ehlert, 1988).This predicts that the effective observed affinity of theallosteric modulator (expressed as Kobs) is given by eq. 3:

Kobs 5KB

�½A�=KA 11

�a½A�=KA 11

; (3)

where KB is the equilibrium dissociation constant of themodulator-receptor complex with no cobinding ligand presentand a is the effect of the modulator on the affinity of thecobinding ligand. It can be seen that for NAMs (where a,, 1),there will be a negative effect of cobinding ligand commensu-rate with standard antagonist experiments (i.e., basically amodified Cheng-Prusoff (Cheng and Prusoff, 1973) relation-ship between observed and microaffinity); however, in vivo,ambient agonist concentrations are probably not high, andthis modification of NAM potency will not be extensive. In

contrast, for PAMs where a .. 1, the cobinding ligand willhave a profound effect on the effective affinity of the modulator,even for low concentrations of agonist. For a NAM with a 5

0.01, and assuming a concentration of agonist 5 KA, thecorrection will be a factor of 1.1, whereas for a PAM with a 5100, the correctionwill be a 50-fold increase in observed affinity.This effect means that a useful estimation of the effectiveaffinity of the PAM cannot be obtained in the absence of thecobinding ligand, a fact implicitly considered in the standardscreening assay for PAMs. In these assays the PAM is added toan assay already partially activated by the endogenous agonist.When this is done, the resulting potentiation of the endogenousagonist effect produces a sigmoidal concentration-responsecurve to the PAM referred to as an R50 curve (see Fig. 9).Analysis of the midpoint and maximal asymptote of this

curve yields an interesting parameter of PAM activity.Specifically, it can be seen that the parameter max/R50 (whereR50 is EC50 of the R50 curve) of this curve (see Fig. 9) furnishesa parameter of agonist potentiation that, when used as a ratio,provides a system-independent measure of the power of thePAMs involved to potentiate agonist response (see Appendix4 for derivation). Specifically, different between log(max/R50)values of R50 curves yield differences between the molecular

Fig. 8. Effects of mutation on dopamine D2L receptor(comparison of wild-type to D2L H3936.35A receptor)through a radar plot showing DDlog(max/EC50) values(solid line) or DDpEC50 values (broken line). Referenceagonist is quinpirole. Data from Tschammer et al., 2011.

Fig. 7. Cell-based agonism: Bias plots showing relativeresponses to muscarinic agonists in HT-29 and SF268cells. The system bias for four of the agonists is relativelyuniform, but bethanechol stands out as being moreactive in SF268 cells than HT-29 cells. This effect canbe quantified through a bias calculation as shown in thetable below the figure. If the mean bias toward SF268cells for acetylcholine, methacholine, Oxo-M, and carba-chol is 1.75, then bethanechol is 7.86/1.75 = 4.5-foldselective for SF268 cells. Data from Deng et al., 2013.

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system-independent parameters describing PAM (eq. 4),namely a, b, and KB:

Dlogðmax=R50Þ 5  Dlogðab=KBÞ: (4)

This parameter has the potential to be extremely useful since,in theory, it can be used to measure the relative effects ofPAMs in vivo. This is important since the effective activity ofPAMs is expressed only in the presence of the natural agonist,and the effect is relatively unknown in vivo. Through standardpharmacologic null experimentation, however, R50 curvesobtained in vivo can be used to compare PAMs in a system-independent manner by simply comparing the effects of thePAMs on natural ambient agonist activity in the in vivosystem. Figure 10 shows two log(max/R50) curves for in vitropotentiation of muscarinic receptor activity of acetylcholine bytwo experimental PAMs (Mistry et al., 2016). In this partic-ular case, the Dlog(max/R50) values indicate comparable PAMeffects, which is confirmed by individual estimation andcalculation of Dlog(ab/KB) values measured from separateexperiments fitting data to the functional allosteric model.Specifically, the Dlog(ab/KB) estimate for the compoundsshown is 0.05 and the Dlog(max/R50) shows a comparablevalue (Dlog(max/R50)5 0.12). This method is based on the nullcancellation of the basal activity level of the system and theisolation of the effect of a PAM on that basal level of response.

ConclusionsThis article proposes that two descriptive parameters for

dose-response curves, namely, the EC50 and maximal re-sponse, can be used to furnish system-independent ratios of

agonist activity in a variety of settings. The inclusion ofmaximal response into the index for agonism takes into accountthe heterogeneous effects of varying system sensitivity on dose-response curves for partial and full agonists. This, in turn, allowsseamless comparisons to be made between full and partialagonists in functional systems. The index, log(max/EC50),embodies agonism into a single number, which then lends itselfto statistical analysis and allows null methods to cancel tissueeffects such as receptor number, receptor coupling efficiency,and amplification within functional assays between agonistsfor any given system. Once this cancellation has been done,Dlog(max/EC50) values become system-independent measuresof the power of the test agonist(s) (compared with a referenceagonist) to induce response in the defined system. Theseindices then can be used to compare different systems; thus,DDlog(max/EC50) values can be used to quantify extracellularreceptor selectivity, intracellular receptor selectivity (biasedsignaling), cell-specific agonism, and the effects of receptormutation on natural signaling.In addition, the same parameters from a different type of

dose-response curve, namely, the potentiation of an ambientagonist response by a PAM, can be used to quantify allostericmodulation both in vitro and in vivo. This may be especiallyuseful for the in vivo comparison of PAM effects since theaffinity and potentiating activity of these types of moleculesare dependent on the presence of the cobinding ligand (inthis case, the endogenous agonist), and this may be variablein vivo. If different PAMs are compared under similar condi-tions in an in vivo system, relative measures of PAM activitybased on molecular parameters of ligand-receptor interactionmay be derived.

Fig. 9. Potentiation of a submaximalagonist effect with 2 PAMs. (A) Effects ofPAM1 with a = 120/b = 0.8/KB= 1 mM;curves shown for control ([PAM1] = 0) and5 nM, 20 nM, 0.1 mM, 0.5 mM, 2 mM, and10mM. (C)Effects of PAM2a =50/b=3/KB=10 mM; curves are shown for control([PAM2] = 0) and 50 nM, 0.2 mM, 1 mM,5 mM, 20 mM, and 100 mM. (B) Dose-response curves for the PAMs (R50 curves)as potentiation of the agonist response.log(max/EC50) value for PAM1 = 6.58 andlog(max/EC50) for PAM2 = 5.76, providing aDlog(max/EC50) value of 0.82.Fromseparateestimates of a, b, and KB used to constructthe curves the value for Dlog(ab/KB) is 0.81.

Fig. 10. R50 curves for two PAMs for muscarinic M1 receptors(filled circles = CMPD 10 d/open circles CMPD 1). Table on rightshows calculation of Dlog(max/R50) values (0.12); log(ab/KB) =0.05 from separate estimations of a, b, and KB (Mistry et al.,2016).

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These approaches are clearly applicable to the advancementof candidate molecules in drug discovery programs (quantify-ing selectivity and bias); however, they can also be used toquantify molecular properties of receptors (differences inreceptor signaling seen with receptor mutation) and evenoperational effects of different cell types on receptor signaling.This latter process could be especially useful in the optimiza-tion of cell type (i.e., pathology-related) agonism throughmedicinal chemistry.

Appendix 1: Agonism as a Positive AllostericModulation of Receptor-Signaling Protein

InteractionThe functional allosteric model (Scheme 1) yields two

receptor species that produce cellular response, namely [RG]and [ARG]. These interact with the cell stimulus responsemechanisms: [RG] with an equilibrium dissociation constantKE to a signaling species [RGE] and [ARG] producing responsewith an equilibrium dissociation constant K’E to a signalingspecies [ARGE].From scheme 1, the system defines the following:

½RG�5 ½ARG��a½A�K’a (5)

½AR�5 ½ARG��a½G�Kg (6)

½R�5 ½ARG��a½A�K’a½G�Kg: (7)

The receptor conservation equation ([Rtot] 5 [R] 1 [AR] 1[RG] 1 [ARG]) can be rewritten using eq. 5–7 as:

½Rtot�5 ½G�.KG

�11a½A�

.K’

A

�1 ½A�

.K’

A 11; (8)

where KG and K9A are equilibrium dissociation constants(K9A5 1/K’a and KG 5 1/Kg).Substituting the term in eq. 8 for [Rtot] and defining the

fraction of receptors RG as rG and ARG as rAG respectivelyyields:

rG 5½RG�½Rtot�5

½G�=KG

½G�.KG

�11a½A�K’

A

�1 ½A�

.K’

A 11(9)

rAG 5½ARG�½Rtot� 5

a½A�.K’

A½G�.KG

½G�.KG

�11a½A�K’

A 1 ½A�.K’

A 1 1�: (10)

The subsequent interaction of the receptor-signaling proteincomplex (either agonist bound or not) is processed through theBlack-Leff operationalmodel (Black andLeff, 1983) as a forcingfunction to generate a response from the agonist. Specifically,these fractional receptor species can be entered into the Black-Leff operational model form for response:

Response5

�½RG��KE 1 ½ARG��K’

E

�Em�

½RG�=KE 1 ½ARG��K’E

�1 1

: (11)

The spontaneous active state receptor has a natural efficacy(denoted tG) for the production of response through coupling tothe signaling protein. Defining the efficacy of the active statereceptor as tG5 [Rtot]/KE and the efficacy of the agonist-boundactive state receptor as tA 5 [Rtot]/KE9 further defines thefactor b as the ratio of the efficacy of the nonagonist-bound

receptor (tG) and agonist-bound receptor. The efficacy of theagonist in terms of the Black-Leff operational model (tA)therefore yields the term b as tA/tG, and the operational modelequation can be rewritten:

Response5ðrGtG 1 rAGbtGÞrGtG 1 rAGbtG 11

: (12)

Substituting for rG and rAG from eq. 9 and eq. 10 yields:

Response5

�abtG½A�

�K’

A½G��KG 1 tG½G�=KG

�Em

½A��K’Að11a½G�=KGð11btGÞÞ1 tG½G�=KG 11

: (13)

Equation 13 defines a sigmoidal curve for the agonist resultsfrom which values of maximal response (denoted max) can bederived:

max5abtG½G�=KGEm

11a½G�=KGð11btGÞ: (14)

It should be noted that for all calculations utilizing theBlack-Leff operationalmodel and these indices of agonist activity,the maximal response to the agonist must be expressed as afraction of the maximal window of response available in theassay. Thus, no agonist can produce a maximal response greaterthan unity (the maximal response window for the assay).Similarly, the EC50 is given as:

EC50 5KAðtG½G�=KG 11Þ

11a½G�=KGð11btGÞ (15)

Combining eq. 14 and eq. 15 yields:

maxEC50

5abtG½G�=KGEm

K‘AðtG½G�=KG 11Þ: (16)

It can be shown that a ratio of the quotients max/EC50

(where max refers to the maximal response to the agonist andthe EC50 (the concentration of agonist producing 50% of theagonist maximal response) results in a system independentparameter quantifying agonism. Used as Dlog(max/EC50)values for two agonists (denoted agonist1 and agonist2), thiscan be shown to be:

Dlogðmax=EC50Þ1-25 log�a1b1

.K’

A-1�-log

�a2b2

.K’

A-2�: (17)

Specifically, eq. 17 reveals that log(max/EC50) is a combi-nation of an assay and tissue term and a strictly agonist term(specifically ab/K9A):

log�maxEC50

�5 log

�tG½G�=KGEm

tG½G�=KG 11

�1 log

"ab

K‘A

#(18)

Therefore, the ratio of max/EC50 values, which subtractsand thus cancels the two log((tG[G]/KGEm)/( tG[G]/KG 11))terms is independent of the assay and tissue effects andbecomes a unique identifier of for the two agonists; for agonist1and agonist2 the Dlog(max/EC50) is Dlog(ab/K’A) which is asystem independent ratio of agonism.The value ab/K9A comprises only drug parameters (a is the

change in the affinity of the receptor for the signaling proteinproduced by the binding of the agonist and reciprocally theaffinity of the agonist when the signaling protein interactswith the receptor), K9A is the equilibrium dissociation of the

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receptor agonist complex when the receptor does not interactwith the signaling protein, andb is the change in the efficacy ofthe receptor for production of response produced by the agonist.

Appendix 2: Relationship betweenDlog(max/EC50) and Dlog(t/KA) Through the

Black-Leff Operational ModelAgonist response is modeled by the Black-Leff operational

model for systems yielding response with a variable Hillcoefficient slope as (Black et al, 1985):

Response5½A�ntnAEm

½A�ntnA 1 ð½A�1KAÞn; (19)

where tA is the efficacy of the agonist, n is theHill coefficientof the agonist concentration-response curv, and Em themaximal response window of the functional assay. It should benoted that the K9A in eq. 18 in terms of the Black-Leff model isthe equilibrium dissociation constant of the agonist-responsecomplex for agonism, with the receptor interacting with thesignaling protein. Therefore, theKA term is the operationalequilibrium dissociation constant of the agonist-receptorcomplex (i.e., agonist binding to the receptor as it interactswith the signaling protein). If the agonist is viewed as a modu-lator of signaling protein interaction, then the operational KA isequal to a/K9A. Black et al. (1985) provided expressions for themaximal response (max) as:

max5tnAEm�11 tnA

� (20)

and for the EC50 for half maximal response as:

EC50 5KA��

21 tnA�1=n-1� (21)

This leads to an expression for max/EC50 of:

maxEC50

5tnA

��21 tnA

�1=n-1�Em

KA�11 tnA

� (22)

For n5 1, max/EC505 t Em/KA; ratios of (max/EC50) valuescancel the tissue Em term and yield a strictly agonist-dependent term t/KA. Therefore, ratios of max/EC50 values(in the form of Dlog(max/EC50) values for systems where theslope of the agonist concentration response curves is notsignificantly different form unity) yield strictly agonistdependent (and system-independent) values for relativeagonism:

DLogðmax=EC50Þ5DlogðtA=KAÞ (23)

Appendix 3: Statistical Assessment of DifferenceUsing DDlog(Max/EC50) Values

If individual estimates of log(max/EC50) are available,then a statistical estimate of mean log(max/EC50) values,Dlog(max/EC50) values, and DDlog(max/EC50) values can becalculated in the form of 95% confidence limits of the estimatedvalues. For a set of k to n values for agonist y activating signalingsystem j, sij

2 is defined as:

S2ij 5

1nij-1

+nj

k5 1

�yijk 2 ymean

�2 (24)

Values for sij2 are calculated for sets of K agonists

and all signaling pathways to yield a pooled variancedefined by:

spooled 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi+K

i5 1+2j51s

2ij

dferror

s; (25)

where dferror is given as:

df error 5 +K

i51+2

j51

�nij 2 1

�: (26)

From these values, a 95% confidence limit with two-tailed Tvalues (T97.5) can be calculated. For a mean log(max/EC50)estimate:

95%  c:I:5T97:5 � spooled

ffiffiffiffiffiffi1nij

s: (27)

Within any one assay, agonist comparison with a stan-dard yields a ratio of log(max/EC50) values denoted asDlog(max/EC50). The 95% c.l. of this ratio is defined as

95%  c:I:5T97:5 � spooled

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nij

11ni΄ j

s: (28)

Once values have been normalized to a reference standardagonist within each group (receptor type, signaling pathway,cell type), then a 95% c.l. can be calculated for selectivity orbias for the DDlog(max/EC50) value as:

95%  c:I:5T97:5 � spooled

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ni1

11ni’1

11ni2

11ni’2

s: (29)

See Figure 11 for the application of equations (24) through(29).

Fig. 11. The application of equations (24) through (29).

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Appendix 4: Application of log(Max/R50) Valuesfrom R50 Curves to Quantify the Effects of PAMsThemodel for allosteric effects in functional systems defines

agonist response as (Ehlert, 2005; Kenakin, 2005; Price et al,2005):

Response5tA½A�=KAð11ab½B�=KBÞ

½A�=KAð11a½B�=KB 1 tAð11ab½B�=KBÞÞ1 ½B�=KB 1 1;

(30)

where a is the effect of the modulator ([B]) on the affinity ofthe agonist for the receptor, and b is the effect of themodulatoron the efficacy of the agonist. This equation can be rewritten interms of the modulator as the active species to:

Response5abtA½B�=KBð½A�=KAÞ1 tA½A�=KA

½B�=KBð11a½A�=KAð11btAÞÞ1 ½A�=KAð11 tAÞ11;

(31)

This defines the R50 curve for a potentiating modulator(PAM) increasing the effect of an ambient agonist responsedue to a presence of agonist acting on the receptor (in the formof [A]/KA).The maximal response of the R50 curve is thus given as:

max5abtA½A�=KA

ð11a½A�=KAð11btAÞÞ; (32)

and the hal-maximal effect of the R50 curve (defined as theR50) is given as:

R50 5KBð½A�=KAð11 tAÞ11Þð11a½A�=KAð11btAÞÞ: (33)

This leads to the ratio of max/R50 as:

maxR50

5abtA=½A�=KA

KBð½A�=KAð11btAÞÞ1 1: (34)

It can be seen that this expression is a mixture of tissuespecific and agonist specific factors:

maxR50

5ab

KB� tA½A�=KA

ð½A�=KAð11btAÞÞ1 1: (35)

Therefore, the ratios of max/R50 values can provide systemindependent estimates of the relative activity of PAMs inpotentiating agonist response:

Dlogðmax=R50ÞA-B5 logðab=KBÞA2 logðab=KBÞB: (36)

Authorship Contributions

Performed data analysis: KenakinWrote or contributed to the writing of the manuscript: Kenakin

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Address correspondence to: Dr. Terry Kenakin, Department of Pharma-cology, University of North Carolina School of Medicine, 120 Mason FarmRoad, Room 4042 Genetic Medicine Building, CB# 7365, Chapel Hill, NC27599-7365. E-mail: kenakin@email.unc.edu

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