RESEARCH PAPER A method for the quantification of biased signalling at constitutively active receptors

Transcription

1 British Journal of Pharmacology British Journal of Pharmacology (208) RESERCH PPER method for the quantification of biased signalling at constitutively active receptors Correspondence David Hall, Fibrosis and Lung Injury DPU, Respiratory herapy rea, GlaxoSmithKline, Gunnels Wood Road, Stevenage, Herts SG 2NY, UK, and Jesús Giraldo, Laboratory of Molecular Neuropharmacology and Bioinformatics, Institut de Neurociències and Unitat de Bioestadística, Universitat utònoma de Barcelona, Bellaterra 0893, Spain. Received 24 October 207; Revised 5 February 208; ccepted 20 February 208 David Hall and Jesús Giraldo 2,3 Fibrosis and Lung Injury DPU, GlaxoSmithKline, Stevenage, UK, 2 Laboratory of Molecular Neuropharmacology and Bioinformatics, Institut de Neurociències and Unitat de Bioestadística, Universitat utònoma de Barcelona, Bellaterra, Spain, and 3 Network Biomedical Research Center on Mental Health (CIBERSM), Bellaterra, Spain BCKGROUND ND PURPOSE Biased agonism, the ability of an agonist to differentially activate one of several signal transduction pathways when acting at a given receptor, is an increasingly recognized phenomenon at many receptors. he Black and Leff operational model lacks a way to describe constitutive receptor activity and hence inverse agonism. hus, it is impossible to analyse the biased signalling of inverse agonists using this model. In this theoretical work, we develop and illustrate methods for the analysis of biased inverse agonism. EXPERIMENL PPROCH Methods were derived for quantifying biased signalling in systems that demonstrate constitutive activity using the modified operational model proposed by Slack and Hall. he methods were illustrated using Monte Carlo simulations. KEY RESULS he Monte Carlo simulations demonstrated that, with an appropriate experimental design, the model parameters are identifiable. he method is consistent with methods based on the measurement of intrinsic relative activity (R i )(ΔΔlogR or ΔΔlog(τ/ )) proposed by Ehlert and Kenakin and their co-workers but has some advantages. In particular, it allows the quantification of ligand bias independently of system bias removing the requirement to normalize to a standard ligand. CONCLUSIONS ND IMPLICIONS In systems with constitutive activity, the Slack and Hall model provides methods for quantifying the absolute bias of agonists and inverse agonists. his provides an alternative to methods based on R i and is complementary to the ΔΔlog(τ/ )methodof Kenakin et al. insystemswhereuseofthatmethodisinappropriateduetothepresenceofconstitutiveactivity. bbreviations D 2 receptor, dopamine D 2 receptor; μ receptor, μ-opioid receptor DOI0./bph he British Pharmacological Society

2 Quantifying biased signalling Introduction Biased agonism is a direct consequence of the ability of GPCRs to signal through more than one pathway. lthough these receptors were originally called GPCRs because it was thought that they only signalled through G proteins, it was later found that they can also signal through other accessory proteins such as β-arrestins. In addition, the signalling through G proteins may not be specific, and one receptor may signal through a variety of G proteins. Because various signalling pathways can be used by a receptor, agonists can be either unbiased or biased depending on whether they do or do not activate each of the pathways with the same efficacy, respectively (Costa-Neto et al., 206). Biased agonism introduces an extra level of complexity in receptor functionality. However, this property is currently at the forefront of drug discovery programmes in order to find ligands that induce GPCR signalling in a biased manner (Rankovic et al., 206). here are many examples of biased agonism with applications on different targets and therapeutic necessities. For simplicity, we will comment on just two the μ-opioid receptor (μ receptor) and the dopamine D 2 receptor (D 2 receptor), which are associated with pain and psychiatric disorders respectively. Opioids, and morphine in particular, are typical analgesics for pain. Morphine is a μ receptor agonist, which signals through both G protein and β-arrestin. It has been shown that the G protein pathway is responsible for pain relief, whereas the β-arrestin pathway is associated with a number of adverse effects, such as tolerance, nausea, vomiting, sedation, constipation and respiratory depression (Rankovic et al., 206). his has led to the quest for ligands with biased μ receptor agonism towards the G protein pathway. Drug discovery efforts succeeded in the discovery of RV30 (DeWire et al., 203), which is currently in phase 2 clinical trials (Viscusi et al., 206), and more recently of PZM2 (Manglik et al., 206). ypically used antipsychotics act as unbiased D 2 receptor antagonists. part from its therapeutically beneficial antipsychotic action, this unbiased D 2 antagonism also has the disadvantage of causing extrapyramidal side effects. Interestingly, drug discovery efforts on structural variations on the low efficacy D 2 agonist, aripiprazole, ledtoβ-arrestin-biased D 2 agonists, which conserved the desirable antipsychotic effects of the unbiased antagonists but without the unwanted effects (Rankovic et al., 206) (see also Urs et al., 207, for a review). hus, biased agonists either by favouring the G protein pathway (μ receptor) or the β-arrestin pathway (D 2 receptor) allow preservation of the therapeutic advantages of unbiased ligands while avoiding their unwanted side effects. Obviously, accurate pharmacological procedures are needed to provide reliable methods to quantify biased signalling parameters. In an effort towards this goal, a scale based on the operational model of agonism has been proposed (Figueroa et al., 2009; Kenakin et al., 202). Quantifying bias by the operational model of agonism he operational model of agonism (Black and Leff, 983) is a two-step model of receptor function. he first step consists of the binding of the ligand to the receptor, while the second step includes all the molecular processes connecting the ligand-bound receptor with the observed final pharmacological effect. By considering that pharmacological effect is a sigmoid function of the concentration of ligand receptor complexes, Equation is obtained. E m τ n n E ð þ Þ n þ τ n n ; () where E is the pharmacological effect, E m is the maximum effect of the system, [] is the concentration of the agonist, τ is the operational efficacy of in the receptor system, is the dissociation constant of for the receptor and n is a parameter related to the slope of the E/[] curves. value of n equal to yields rectangular hyperbolic E/[] curves, whereas values of n greater or less than allow for steeper or flatter curves than the rectangular hyperbola, respectively (Black and Leff, 983; Black et al., 985). he operational efficacy, τ, wasdefined (Black and Leff, 983) as τ =[R] /K E,where[R] represents the total receptor concentration and K E the value of the concentration of agonist receptor complex, [R], which causes half of the maximum possible effect, E m.inotherwords,k E is inversely proportional to the efficacy of the R complex. hus, τ contains both tissue and ligand receptor efficacy parameters. In addition, is a conditional or functional constant because, although it corresponds to the concentration of ligand that occupies 50% of the overall receptor population, it does not, in general, correspond to the affinity for an individual equilibrium step, for example, the binding of the agonist to the inactive receptor conformation, because it incorporates, in addition, the receptor activation conformational change. hus, when [] =, an agonist will occupy more than 50% of the receptors in the active conformation (that is to say, [R * ]/([R * ]+[R * ]) > 0.5). Receptor activation will allow the receptor to be bound by an accessory protein, either G protein or β-arrestin, which, after subsequent activation, will lead to the successful transduction of the initial agonist binding into a functional effect. he mathematical structure of Equation is weak for fitting purposes. It cannot be used to derive estimates of the affinity and efficacy of an agonist from a single E/[]curvebecause there is not a unique set of parameter estimates that provide an optimal fit to the curve. his problem can be solved by generating several curves with different maximal responses without affecting the system functionality at the microscopic molecular level of the receptor. his is known as the irreversible inactivation method, which yields pharmacological systems with different [R] values (Furchgott, 966). When fitting Equation to a set of E/[] curves(τ varies between those curves bearing different [R] values but E m, and n aresharedinallthecases),thecomponentsofthepharmacological activation process are distributed into the and τ parameter estimates. s it was shown in Roche et al. (203b), consideration of agonist-induced receptor activation (R R * ) affects the values of and τ operational parameter estimates a highly efficacious agonist leads to a decrease of (increase of ligand affinity because of agonist binding to active receptor states), whereas a weakly efficacious agonist leads to a decrease of τ. In addition to the agonist-induced receptor activation conformational change, τ incorporates the binding of either the G protein or β-arrestin to the R * binary complex (Kenakin and British Journal of Pharmacology (208)

3 DHallandJGiraldo Christopoulos, 203; Ehlert, 205) and, indeed, subsequent transduction steps. composite parameter (log(τ/ )), defined by combining operational efficacy (τ) and functional affinity ( ) in a single parameter, was identified as a primary index for drug activity (Figueroa et al., 2009; Kenakin et al., 202). his composite parameter denoted as either LogR (Figueroa et al., 2009) or transduction coefficient (Kenakin et al., 202) served to define a scale for comparing drugs within a particular pathway (Δlog(τ/ )) relative to a reference compound. Finally, biased agonism between two signalling pathways can be quantified by ΔΔlog(τ/ ). he ΔΔlog(τ/ ) biased agonism scale was shown to be independent of receptor density ([R] ) notwithstanding the value of the slope parameter (n) (Kenakin et al., 202). his property makes this scale appropriate to quantify biased agonism at different cell types. Despite its proven utility for biased agonism quantification, the ΔΔlog(τ/ ) scale suffers from a lack inherent to the operational model as it was originally proposed (Black and Leff, 983). Equation shows that the pharmacological effect is zero in the absence of agonist. Even if a term to account for background effect is included in Equation, this cannot account for constitutive receptor activity. For positive E m,no combination of values of τ and canresultinnegativevalues of E whatever the value of []. Hence, the response cannot decrease below this background effect. herefore, the operational model as originally formulated is unable to describe the behaviour of inverse agonists. It follows that a new formulation is needed if we want to encompass the quantification of biased agonism to the whole pharmacological space. Since the Black and Leff operational model does not consider constitutive receptor activity, it is impossible to use methods based on the Black and Leff operational model to determine whether inverse agonists exhibit biased signalling along different signal transduction pathways. Similarly, it is impossible to quantify the bias of a ligand that acts as an agonist of one signal transduction pathway and an inverse agonist of another. However, Hall (2006) has described a modification of the operational model, which does include constitutive receptor activity. Subsequently, this model was used by Slack and Hall (202) to demonstrate that, in systems displaying constitutive activity, it is possible to determine the absolute intrinsic efficacy of ligands at that receptor, rather than relative efficacies, if the receptor density can be varied. his model should, therefore, provide a suitable basis for the derivation of a method for determining the bias of ligands that display inverse agonism along some signalling pathways. Ehlert et al. (20b) have also developed an operational model which includes constitutive activity and demonstrated that it can be used to determine R i for inverse agonists and hence ΔΔlogR to quantify ligand bias for this class of ligand; see also Roche et al., 203a, for a review. he ΔΔlogR (ΔΔlog(τ/ )) method of Ehlert and coworkers (Ehlert, 2008; Figueroa et al., 2009) has a number of useful properties, which were highlighted in Kenakin et al. (202). It should be noted here that Kenakin et al. (202) distinguish between Δlog(τ/ )andlogr i as they used the original definition of intrinsic relative activity (Ehlert et al., 999) using parameters of a concentration response curve (R i =E max EC 50B /E maxb EC 50, where and B refer to the test and standard ligand, respectively). Firstly, under certain circumstances (full agonists that have concentration response curves with unit slopes), the reciprocal of the EC 50 of an agonist is a good approximation of τ/ making it trivial to estimate from a single concentration response curve. τ/ can also be derived straightforwardly for partial agonists when a full agonist can also be characterized since τ is a simple function of the intrinsic activity of a partial agonist whatever the slope of the curve (or can be obtained by simultaneous fitting, e.g. Leff et al., 990). In this regard, a complementary scale of biased agonism for partial agonists based on τ values has been recently developed (Burgueño et al., 207). Secondly, normalization to the values of a reference agonist removes the impact of the bias inherent in the different coupling efficiencies of different signalling pathways, which would otherwise form part of an analysis based on the operational model τ parameter. hirdly, the value of ΔΔlog(τ/ ) was shown to be independent of the receptor density in the system in which it is measured (for all values of the slope of the transducer function). Finally, there are circumstances in which the same agonist acting at the same receptor does not exhibit the same value of along two different signalling pathways, and the inclusion of the term is claimed to account for this manifestation of the efficacy of the ligand. In any case, will simply cancel if it is the same. Including implicitly in the analysis also avoids the additional error of measuring and using an independent estimate of ligand affinity especially if the conditions of the functional and binding assays are not precisely matched. here is, however, an argument that changes in measured affinity along different pathways may indicate that the system under consideration is not actually adequately described by a Black and Leff operational model (see ppendix) either because at least one of the measurements has not been taken at equilibrium or because other assumptions inherent in an operational model analysis are violated (but see below). his would, of course, invalidate an analysis based on the operational model. In summary, current approaches for quantification of biased agonism based on operational models may include (Ehlert et al., 20b) or not (Kenakin et al., 202; Burgueño et al., 207) constitutive receptor activity but all require the presence of a reference agonist for within signalling pathway normalization (because they include pathway-dependent coupling efficiency in the operational efficacy, τ). hus, there is value in developing a new analytical methodology in this field, which includes constitutive receptor activity but does not require the normalization to the properties of reference ligands. Methods heory he closest analogue of τ/ in the Slack and Hall operational model is χ/ since the Black and Leff operational model is the special case of the Slack and Hall model with χ << and under these conditions χ τ (see ppendix and Slack and Hall, 202). Here, is the intrinsic efficacy of the ligand, its apparent equilibrium dissociation constant and χ the coupling efficiency of the signal transduction system British Journal of Pharmacology (208)

4 Quantifying biased signalling However, χ/ suffers from the same dependence on coupling efficiency as τ/. It would be preferable, indeed desirable, to derive a parameter that does not involve coupling efficiency since this eliminates the requirement to normalize to a reference ligand. Since χ quantifies coupling efficiency in the Slack and Hall model, the obvious solution is to determine the value of Δlog(/ )orevenδlog() (where the single Δ indicates the lack of normalization). It is worth noting at this point that a ternary complex model only approximates an operational model with unit slope under two conditions [R] << [G] or [R] >> [G], and it is only under that latter condition that such a model also exhibits a receptor reserve (Hall, 2006). Under the former condition, the system is reserveless (EC 50 = ), and hence, the potency of an agonist for any response should be the same under this condition, no matter what signal transduction pathway is engaged (assuming all are reserveless). Only the maximal response can differ under these conditions, and hence, the only indication of bias is a different order of intrinsic activity. Indeed, under this condition, the ratio of ratios of maximal response to basal activity is a measure of absolute bias (see ppendix). Systems in which [R] << [G] also have the potential to exhibit different affinities for the same agonist at a given receptor. o achieve this, it is necessary for the signal transduction systems to be partitioned such that they do not compete for the same pool of receptors. he most obvious way to achieve this is when measurements are made in different cell types or lines that only express one of the signal transduction proteins in question. lternatively, it may be possible for the proteins to be partitioned into distinct cellular domains, which behave independently. Clearly, the latter would require experimental confirmation. When [R] and [G] are within an order of magnitude of each other, the binding isotherm is not a sigmoid or linear rational function of ligand concentration, and a standard operational model is not appropriate to analyse the system. here are also issues with identifying a well-defined transducer function to translate the stimulus (a linear combination of [R], [RG], [R] and [RG]) into the response ([RG] +[RG]) in this case. It is further noteworthy that in this case, the slope of the binding isotherm and that of the activation curve for G cannot be greater than unity (Giraldo et al., 2002). hus, only in exceptional circumstances can concentration response curves have unit Hill coefficients under these conditions. lso, as shown in the ppendix, when [R] >> [G], the value of should be the same when measured for any response so Δlog(/ )andδlog() should be the same (to within experimental error). So, can these parameters be derived? he previous characterization of the Slack and Hall operational model (with arbitrary slope) indicated that it is indeed possible to estimate and independently for any ligand if receptor density, and hence signalling in the absence of ligand, can be varied. his would clearly be a rigorous method for estimating the bias. However, for systems that have linear rational binding and transducer functions (i.e. those that generate concentration response curves with unit slope) and where it is possible to estimate the activity along the signal transduction system which is independent of the receptor under consideration (the background activity), it is possible to estimate / directly from a single concentration response curve. Under these conditions, the Slack and Hall model is given by E max χð þ Þ E ð þ χþþð þ χþ (2) hen, when [] =0,E []=0 = E max χ/( + χ), when [], E [] = E max χ/( + χ), and the potency of the ligand is [] 50 = ( + χ)/( + χ). Hence, E E maxχ 50 E 0 þ χ ð þ χþ E maxχ þ χ þ χ (3) hus, / is accessible from a single concentration response curve. We note that the left-hand side of Equation 3 has previously been shown to be equal to the affinity of a ligand for the active state of the receptor (Ehlert et al., 20b), and hence, this will also be true of /. It is also possible to re-parameterize Equation 2 in terms of / and directly measurable quantities E max χð þ Þ E ð þ χþþð þ χþ χ þ χ þ K a þ χ þ ð þ χþ þ E þ E 0 E max E E max χ þ þ χ þ χ þ E 0 þ ½ Š0 þ E E ½ Š his further underlines the ease with which / can be determined in simple experimental systems. From a practical perspective, a more useful form of Equation 4 for curve fitting is Equation 5 below. E E 0 (4) E E þ 0 Q þ bkgd; (5) þ 0 Q where Q =log(/ ). his accounts for the log-normal distribution of this parameter. he term bkgd has also been added to account for background activity. n important consideration in the context of ligand bias is to understand how multistate mechanistic models translate into macroscopic operational models since, strictly, it is impossible for a ligand to exhibit biased agonism at a receptor that can only adopt two conformations ( active and inactive ). his is considered in some detail in the ppendix where we show that multistate models are consistent with the Slack and Hall operational model and hence that it is meaningful to estimate bias using Slack and Hall intrinsic efficacy ratios. We also demonstrate that such models define physical limits on (4) British Journal of Pharmacology (208)

5 DHallandJGiraldo the macroscopic intrinsic efficacy of agonists and inverse agonists. his has previously been shown for the two-state model (Slack and Hall, 202). Below, we demonstrate the relationship between these parameters and the affinity of the ligand for the active and inactive states of the receptor. In circumstances where the background activity (i.e. activity in the functional assays that would be present in the absence of the receptor in question) cannot be determined independently or where the transducer function (and hence the concentration response curves) has a slope which differs from unity, there is no alternative but to vary the coupling efficiency (e.g. by varying the receptor density) in order to estimate intrinsic efficacy and affinity (we demonstrate that there is no re-parameterization simply in terms of / in this case in the ppendix). In this case, the two parameters will be determined separately so it would be possible to determine whether the affinity differed significantly between functional responses. If not, then Δlog() can be estimated as the measure of bias. It should be noted at this point that the expressions for and / (Equations 2, 4, 9 and 20) are independent of [R], and hence, like ΔΔlog(τ/ ), Δlog() and Δlog(/ ) are independent of the receptor density in the system that they are measured in (barring the requirement for [R] to be high enough for there to be constitutive activity). Finally in this section, we note that, in terms of a twostate model, it can be shown that = α( + L)/( + αl)(thisuses the notation of Hall (2000), where L is the receptor isomerization constant ([R * ]/[R]) and α is the ratio of the affinities of the ligand for the active and inactive states of the receptor the definition of intrinsic efficacy in this model). he derivation of this expression is given in the supporting information in Slack and Hall (202). It is equal to the ratio of the amount of R * at saturating ligand concentrations to that of R * in the absence of ligand. his ratio has subsequently been termed R act by Ehlert (205) who discussed its use as a measure of pathway activation. Now, when L << (i.e. there is relatively little R * in the absence of agonist), the expression for above simplifies to = α/( + αl), and then if α < /L (that is the ligand causes less than 50% of the receptor to be in the active state at saturating concentrations), + αl and α (Slack and Hall, 202). In other words, for inverse agonists and agonists with no more than moderate intrinsic efficacy (by the criterion < α < /L), provides a good estimate of the active state selectivity of the ligand. his supports our earlier assertion that measurement of provides a rigorous basis for measurement of ligand bias. With the same assumptions, / is a good approximation of the affinity constant of the ligand for the inactive state of the receptor (K). Following this logic, / is a measure of αk, theaffinity constant for the active state and hence a weaker (if more easily measurable) basis for a measure of ligand bias, particularly if estimates of differ among responses, since it does not account for receptor activation. his is clearly also an issue with measures of bias based on R i, as noted in Ehlert (205). hus, the Slack and Hall model also provides an alternative method to those published by Ehlert and co-workers (Ehlert et al., 20a; Ehlert et al., 20b) for estimating the affinity of ligands for the inactive and active states of the receptor. Similar arguments applied to the multistate model derived in the ppendix (i.e. that L L ) also indicate that, for a receptor with multiple states, would represent a weighted average measure of ligand selectivity for the active state(s) responsible Figure Simulations of analysis using Equation 5. Responses in two systems were simulated with Equation 2 for four ligands ( D). he seed (true mean) and simulated parameters are summarized in able, and the results of fitting Equation 5 are summarized in able 3. Points are the mean ± SEM of six simulated data sets, and the curves show the mean of the fitted curves. he values of the system-related parameters (bkgd, E 0 )weresharedfor all ligands during fitting. For response, the SD of the normal distribution defining the between-occasion variability was set to 5% of the mean value of the normally distributed parameters. For the log-normally distributed parameters, the SD of the log of the parameter was set to 0.5. he SD of the within-occasion variability was set to 5% of the simulated mean value at each agonist concentration (giving proportional rather than constant errors). For response 2, the SD of the between-occasion variability was set to 25% of the mean value of the normally distributed parameters. For the log-normally distributed parameters, the SD of the log of the parameter was set to he SD of the within-occasion variability was set to 20% of the simulated mean value at each agonist concentration British Journal of Pharmacology (208)

6 Quantifying biased signalling for the response under consideration. With the same assumptions, / is a measure of the affinity constant for the most abundant state in the absence of ligand, presumably an inactive state. Nomenclature of targets and ligands Key protein targets and ligands in this article are hyperlinked to corresponding entries in http//www. guidetopharmacology.org, the common portal for data from the IUPHR/BPS Guide to PHRMCOLOGY (Harding et al., 208), and are permanently archived in the Concise Guide to PHRMCOLOGY 207/8 (lexander et al., 207). Results Simulations he two methods discussed above are illustrated using simulated data in Figures and 2. Input parameters and fitted estimates are summarized in ables 5. For the simulations illustrating analysis using Equation 5, the data were simulated using Equation 2 adding random, normally (E max, bkgd) or log-normally distributed errors to the parameters to simulate between-occasion variability and then adding random, normally distributed error to the individual data points to simulate within-occasion variability (see figure captions). he data were then analysed using Equation 5 using Microsoft Excel by minimizing the residual sum of squares using the Solver dd-in. similar approach was taken for simulations based on Equation 22, but in this case, the simulated data were analysed using Equation 6, below. s can be seen from ables 3 and 4, the simulations indicate that both methods result in good estimates of the input parameters when sufficiently rich data sets are available. s previously noted, the Slack and Hall (202) model is only identifiable when the receptor density is varied sufficiently for the response in the presence of saturating concentrations of ligand to differ between conditions. his may require more than two different receptor densities to be evaluated. lso, the analysis of a reference agonist alongside an inverse agonist provides a much more robust estimate of E max (unless the systems demonstrate high levels of constitutive activity). hus, we have demonstrated that the Slack and Hall operational model, where applicable, provides a method for determining the absolute bias of receptor ligands for different signalling pathways. Figure 2 Simulations of analysis using Equation 6. Responses in two systems were simulated with Equation 22 for four ligands ( D). he seed (true mean) and simulated parameters are summarized in able 2, and the results of fitting Equation 6 are summarized in able 4. Points are the mean ± SEM of six simulated data sets, and the curves show the mean of the fitted curves. he values of the system-related parameters (bkgd, E max, χ, χ 2 and n) were shared for all ligands during fitting. For response, the SD of the normal distribution defining the between-occasion variability was set to 5% of the mean value of the normally distributed parameters. For the log-normally distributed parameters except n, the SD of the log of the parameter was set to 0.5. he SD of the within-occasion variability was set to 5% of the simulated mean value at each agonist concentration (giving proportional rather than constant errors). For response 2, the SD of the between-occasion variability was set to 20% of the mean value of the normally distributed parameters. For the log-normally distributed parameters except n, the SD of the log of the parameter was set to he SD of the within-occasion variability was set to 20% of the simulated mean value at each agonist concentration. In both cases, the SD of n was set to one-third of that of the other parameters. British Journal of Pharmacology (208)

7 DHallandJGiraldo able Summary of the parameters used in generating the data from Equation 2 for the simulations of analysis by Equation 5 Response Response 2 Parameter rue mean Simulated rue mean Simulated E max ± ± 8.7 bkgd ± ±.3 logχ ± ± 0.24 log ± ± 0.23 log B ± ± 0.23 log C ± ± 0. log D ± ± 0.22 p ± ± 0.22 pb ± ± 0.7 pc ± ± 0.2 pd ± ± 0.28 Values are the mean ± SD of the six simulated sets of parameters. Subscripts D refer to the four ligands. able 2 Summary of the parameters used in generating the data from Equation 22 for the simulations of analysis by Equation 6 Response Response 2 Parameter rue mean Simulated rue mean Simulated E max ± ±.8 bkgd ± ±.2 logχ ± ± 0.4 logχ ± ± 0.5 log ± ± 0.8 log B ± ± 0.3 log C ± ± 0.9 log D ± ± 0.24 p ± ± 0.26 pb ± ± 0.23 pc ± ± 0.9 pd ± ± 0.20 log(n) ± ± 0.09 Values are the mean ± SD of the six simulated sets of parameters. Subscripts D refer to the four ligands. Subscripts and 2 refer to the different receptor expression levels. Since this method can only be applied to systems that demonstrate constitutive activity, it is complementary to the ΔΔlog(τ/ ) method of Kenakin et al. (202), which should only be applied to systems that lack measurable constitutive signalling. E max 0 X 0 K þ 0 F 0 N E 0 K 0 N þ þ 0 X 0 K þ 0 F 0 N þ bkgd; (6) where F = log(), K = log( ), N = log(n) and X = log(χ). Discussion Concluding remarks In this report, we have presented novel methods for the analysis of biased signalling in systems with measurable levels of constitutive activity. hese methods have some advantages over those that have previously been published, although Δlog(/ ) carries some of the issues associated with methods based on R i since it too is a measure of relative affinity for the active state of a receptor not relative intrinsic efficacy. key issue occurs when estimates of 2052 British Journal of Pharmacology (208)

8 Quantifying biased signalling able 3 Comparison of the simulated and fitted parameters for analysis by Equation 5 Response Response 2 Parameter Simulated Fitted Simulated Fitted bkgd 0.80± ± ±0. 5.±0.4 E ± ± ±.4 6.9±.4 E.44 ± ± ± ± 3.3 E B 0.05 ± ± ± ± 0.04 E C.3±0.05.2± ± ±0.5 E D.32 ± ± ± ± 3.8 log / 2.49 ± ± ± ± 0.5 log B /B 0.0 ± ±0..08 ± ± 0.20 log C /C.55 ± ± ± ± 0.5 log D /D.47 ± ± ± ± 0.09 Values are the mean ± SEM of the six simulated sets of parameters. Subscripts D refer to the four ligands. able 4 Comparison of the simulated and fitted parameters for analysis by Equation 6 Response Response 2 Parameter Simulated Fitted Simulated Fitted E max.67 ± ± ± ± 5.7 bkgd 0.95 ± ± ± ± 0.83 logχ 0.30 ± ± ± ± 0.06 logχ ± ± ± ± 0.09 log.77 ± ± ± ± 0.2 log B 0.96 ± ± ± ± 0.9 log C 0.93 ± ± ± ± 0.09 log D.39 ± ± ± ± 0.23 p 0.56 ± ± ± ± 0.0 pb 2.07 ± ± ± ± 0.07 pc.04 ± ± ± ± 0.0 pd 0.08 ± ± ± ± 0.2 log(n) 0.6 ± ± ± ± 0.06 log(/ ) 2.33 ± ± ± ± 0.5 log(/ ) B. ± ± ± ± 0.7 log(/ ) C.97 ± ± ± ± 0.3 log(/ ) D.32 ± ± ± ± 0.5 Values are the mean ± SEM of the six simulated sets of parameters. Subscripts D refer to the four ligands. Subscripts and 2 refer to the different receptor expression levels. differ between pathways. In this case, it is possible for a ligand to be biased according to its intrinsic efficacy but not according to Δlog(/ ). For example, if for pathway =0.0and =0 6 while for pathway 2 2 =and 2 =0 4, the ligand s efficacy is biased (see below for a discussion of how this might be classified) but Δlog(/ )=0 indicating no bias. Indeed, it is possible for Δlog() and Δlog(/ )tobecontradictoryif 2, indicating that some care is required in using the latter method since contributions from differences in affinity and intrinsic efficacy are not distinguished. here is also an issue with interpreting the direction of a ligand s bias when comparing agonists and inverse agonists. n agonist is more efficacious along pathway compared with pathway 2 when > 2. However, an inverse agonist is more efficacious (causes more inhibition) along pathway British Journal of Pharmacology (208)

9 DHallandJGiraldo able 5 Bias factors determined from simulations based on fitting Equations 5 and 6 Equation 5 Equation 6 Parameter Simulated Fitted Simulated Fitted Δlog(/ ) 0.4 ± ± ± ± 0.8 Δlog(/ ) B.0 ± ± ± ± 0.27 Δlog(/ ) C.46 ± ± ± ± 0.7 Δlog(/ ) D 0.59 ± ± ± ± 0.2 Δlog 0.7 ± ± 0.5 Δlog B.04 ± ± 0.25 Δlog C.38 ± ± 0.3 Δlog D 0.94 ± 0..0 ± 0.29 when 2 >. his interpretation results in, for example, / 2 = 0 indicating 0-fold bias for pathway for an agonist but 0-fold bias for pathway 2 for an inverse agonist. his also raises a question around how to quantify bias for a ligand that is an agonist of one pathway and inverse agonist of another. he simplest way to resolve these issues is to define bias as the preference to agonize a pathway. In this case, a ligand would be biased towards a given pathway whenever its intrinsic efficacy is numerically greater for that pathway. For an agonist, this is the intuitive definition. It is also intuitively reasonable that a ligand that is an agonist of one pathway and inverse agonist of another is biased towards the pathway it agonizes. By this logic then, an inverse agonist of two pathways is biased towards the pathway it inhibits least, that is, the one for which is greater. cknowledgements his study was supported in part by Ministerio de Economía y Competitividad (SF R). he authors would like to thank Bin Zhou for valuable comments on the manuscript. uthor contributions J.G. conceived the study. D..H. derived the models and performed the simulations. Both J.G. and D..H. drafted, reviewed and approved the final manuscript. Conflict of interest he authors declare no conflicts of interest. Declaration of transparency and scientific rigour his Declaration acknowledges that this paper adheres to the principles for transparent reporting and scientific rigourof preclinical research recommended by funding agencies, publishers and other organisations engaged with supporting research. References lexander SPH, Christopoulos, Davenport P, Kelly E, Marrion NV, Peters J et al. (207). he Concise Guide to PHRMCOLOGY 207/8 G protein-coupled receptors. Br J Pharmacol 74 (Suppl ) S7 S29. Black JW, Leff P (983). Operational models of pharmacological agonism. Proc R Soc Lond B Biol Sci Black JW, Leff P, Shankley NP, Wood J (985). n operational model of pharmacological agonism the effect of E/[] curve shape on agonist dissociation constant estimation. Br J Pharmacol Burgueño J, Pujol M, Monroy X, Roche D, Varela MJ, Merlos M et al. (207). complementary scale of biased agonism for agonists with differing maximal responses. Sci Rep Costa-Neto CM, Parreiras ESL, Bouvier M (206). pluridimensional view of biased agonism. Mol Pharmacol DeWire SM, Yamashita DS, Rominger DH, Liu G, Cowan CL, Graczyk M et al. (203). G protein-biased ligand at the μ-opioid receptor is potently analgesic with reduced gastrointestinal and respiratory dysfunction compared with morphine. J Pharmacol Exp her Ehlert FJ (2008). On the analysis of ligand-directed signaling at G protein-coupled receptors. Naunyn Schmiedebergs rch Pharmacol Ehlert FJ (205). Functional studies cast light on receptor states. rends Pharmacol Sci Ehlert FJ, Griffin M, Sawyer GW, Bailon R (999). simple method for estimation of agonist activity at receptor subtypes comparison of native and cloned M3 muscarinic receptors in guinea pig ileum and transfected cells. J Pharmacol Exp her Ehlert FJ, Griffin M, Suga H (20a). nalysis of functional responses at G protein-coupled receptors estimation of relative affinity constants for the inactive receptor state. J Pharmacol Exp her Ehlert FJ, Suga H, Griffin M (20b). nalysis of agonism and inverse agonism in functional assays with constitutive activity estimation of 2054 British Journal of Pharmacology (208)

10 Quantifying biased signalling orthosteric ligand affinity constants for active and inactive receptor states. J Pharmacol Exp her Figueroa KW, Griffin M, Ehlert FJ (2009). Selectivity of agonists for the active state of M to M4 muscarinic receptor subtypes. J Pharmacol Exp her Furchgott R (966). he use of β-haloalkylamines in the differentiation of receptors and in the determination of dissociation constants of receptor agonist complexes. In Harper NJ, Simmonds B (eds). dvances in Drug Research. cademic Press New York, pp Giraldo J, Vivas NM, Vila E, Badia (2002). ssessing the (a) symmetry of concentration effect curves empirical versus mechanistic models. Pharmacol her Hall D (2000). Modeling the functional effects of allosteric modulators at pharmacological receptors an extension of the twostate model of receptor activation. Mol Pharmacol Hall D (2006). Predicting dose response curve behavior mathematical models of allosteric receptor ligand interactions. In Bowery NG (ed). llosteric Receptor Modulation in Drug argeting. aylor & Francis New York, pp Hall D (203). pplication of receptor theory to allosteric modulation of receptors. Prog Mol Biol ransl Sci Harding SD, Sharman JL, Faccenda E, Southan C, Pawson J, Ireland S et al. (208). he IUPHR/BPS Guide to PHRMCOLOGY in 208 updates and expansion to encompass the new guide to IMMUNOPHRMCOLOGY. Nucl cids Res 46 D09 d06. Kenakin, Christopoulos (203). Signalling bias in new drug discovery detection, quantification and therapeutic impact. Nat Rev Drug Discov Kenakin, Watson C, Muniz-Medina V, Christopoulos, Novick S (202). simple method for quantifying functional selectivity and agonist bias. CS Chem Nerosci Leff P, Prentice DJ, Giles H, Martin GR, Wood J (990). Estimation of agonist affinity and efficacy by direct, operational model-fitting. J Pharmacol Methods Manglik, Lin H, ryal DK, McCorvy JD, Dengler D, Corder G et al. (206). Structure-based discovery of opioid analgesics with reduced side effects. Nature Rankovic Z, Brust F, Bohn LM (206). Biased agonism an emerging paradigm in GPCR drug discovery. Bioorg Med Chem Lett Roche D, Gil D, Giraldo J (203a). Mechanistic analysis of the function of agonists and allosteric modulators reconciling two-state and operational models. Br J Pharmacol Roche D, Gil D, Giraldo J (203b). Multiple active receptor conformation, agonist efficacy and maximum effect of the system the conformation-based operational model of agonism. Drug Discov oday Urs NM, Peterson SM, Caron MG (207). New concepts in dopamine D 2 receptor biased signaling and implications for schizophrenia therapy. Biol Psychiatry Viscusi ER, Webster L, Kuss M, Daniels S, Bolognese J, Zuckerman S et al. (206).randomized,phase2studyinvestigatingRV30,a biased ligand of the μ-opioid receptor, for the intravenous treatment of acute pain. Pain Weiss JM, Morgan PH, Lutz MW, Kenakin P (996a). he cubic ternary complex receptor-occupancy model. III. Resurrecting efficacy. J heor Biol Weiss JM, Morgan PH, Lutz MW, Kenakin P (996b). he cubic ternary complex receptor-occupancy model I. Model description. J heor Biol WeissJM,MorganPH,LutzMW,KenakinP(996c).hecubic ternary complex receptor-occupancy model II. Understanding apparent affinity.jheorbiol ppendix ternary complex model with competing transducer proteins Following our previous approach (Hall, 2006), we will derive a ternary complex model with two competing transducer proteins to derive the required operational model under conditions where it exhibits receptor reserve. his will provide a rigorous mathematical demonstration that we may, in fact, neglect the presence of additional transducer proteins when considering bias in systems whose behaviour is consistent with that of an operational model with receptor reserve. We will begin by deriving a system without constitutive activity to support a classical operational model and then derive the model with constitutive activity. From the perspective of GPCR signalling, we have consciously not included the binding of nucleotide to the G protein. his is in part for mathematical simplicity but also because the key conclusions in terms of pathway independence will be the same although, clearly, the precise mathematical expressions will differ. However, Stein and Ehlert demonstrated that an operational model using a simple two-state model of receptor activation as stimulus (Ehlert et al., 20b) can accurately describe relevant features of a ternary complex model including nucleotide exchange (Stein and Ehlert, 205) suggesting that the model described here is unlikely to be misleading. he first system under consideration is illustrated in the scheme below. Slack RJ, Hall D (202). Development of operational models of receptor activation including constitutive receptor activity and their use to determine the efficacy of the chemokine CCL7 at the CC chemokine receptor CCR4. Br J Pharmacol Stein RS, Ehlert FJ (205). kinetic model of GPCRs analysis of G protein activity, occupancy, coupling and receptor-state affinity constants. J Recept Signal ransduct Res Scheme ternary complex model with competing transducer proteins. British Journal of Pharmacology (208)

11 DHallandJGiraldo Here, R is the receptor, the ligand and G and the two transducer proteins. he conservation equations are as follows ½RŠþ½RŠþ½RGŠþ½RŠ ½GŠþ½RGŠ ½Š ½Šþ½RŠ It will be assumed that [] is in large excess. he equilibrium dissociation constants are as follows described by an operational model with receptor reserve, we may treat each signalling cascade independently. rivially, it must also be the case that ½R ½RŠ ½Š K Š (2) ½RŠ K þ K þ K ½RŠ ½RŠ ; ½GŠ½RŠ ½RGŠ ; K ½Š½RŠ ½RŠ We will consider the equilibrium concentration of the RG complex. By making appropriate substitutions from the definitions of the equilibrium constants, the equation for [R] can be expressed in terms of [RG] as follows K ½RGŠ þ ½RGŠ ½GŠ ½GŠ ½RGŠ þ ½RG Šþ ½Š½RGŠ K ½GŠ K K þ G þ þ ½RGŠ ½RGŠ ½RGŠ ½RGŠ K þ þ ½RGŠ þ ½Š K ½RGŠ 2 ½RGŠ þ þ þk þ K G ½Š K K! ½Š ½RGŠ þ 0 o derive an operational model, we now assume (or, indeed, require) that [R] >> [G] +[] ( /K ). his allows the simplification of the quadratic equation above to ½RGŠ 2 ½RGŠ þ þk þ ½RŠ 0 Note now that [] no longer appears in these expressions. his assumption renders formation of RG independent of formation of R. It is equivalent to assuming that the free concentration of R is not changed by binding to the transducer proteins. Now, since the same assumption makes ([][R] + [] + K ) 2 >> [] 2 [R] [G], the physically relevant solution to the quadratic is very well approximated by ½RGŠ ½GŠ K þ þ his approximation is detailed in Hall (203). his can be rearranged to give ½RG ½RŠ K Š G ; () ½RŠ K þ þ which, taking formation of RG as the effect of the agonist, is an operational model with E max =[G] and τ =[R] /.hus, under conditions when a pharmacological system can be Including constitutive activity he required model is now Scheme 2 ternary complex model with competing transducer proteins and constitutive activity. In this case, the conservation equations are as follows ½RŠþ½RG ½GŠþ½RG ½Š ½ Šþ ½R Šþ½R Šþ½RGŠ Šþ½R Šþ ½RŠ Šþ½RGŠþ½RŠ he equilibrium constants are as follows K ½RŠ ½RŠ ; ½RGŠ ½RGŠ ; βk ½RŠ ½RŠ ; ½GŠ½RŠ ½RGŠ ; α ½GŠ½RŠ ½RGŠ ; K ½Š½RŠ ½RŠ ; βk ½Š½RŠ ½RŠ Now we must consider the equilibrium concentration of thergandrgcomplex.hederivationisalittlemorecomplicated. First, it is necessary to derive an expression for [R] British Journal of Pharmacology (208)

12 Quantifying biased signalling ½RŠþ ½RGŠþ½RŠþ½RŠþ½RGŠþ½RŠ ½RŠ þ þ ½GŠ þ ½Š þ ½GŠ þ ½Š K K βk K ½RŠ þ ½GŠ þ ½Š þ ½GŠ þ ½Š K K βk K þ (3) Now an expression for [G] intermsof[g] and other constants is required. ½GŠþ½RGŠþ½RGŠ ½GŠ þ ½RŠ þ ½RŠ ½GŠ ½GŠ ½GŠ þ ½RŠ þ ½RŠ þ ½RŠ þ ½GŠ þ þ ½RŠ þ þ ½GŠ þ ½Š þ ½GŠ þ ½Š K K βk K þ þ ½GŠ þ þ ½Š þ K K βk þ þ ½GŠ þ þ ½Š þ þ ½RŠ K K βk þ ½GŠ þ þ ½GŠ þ þ ½Š þ þ ½RŠ K K βk ½GŠ þ þ þ ½GŠ þ þ ½Š þ K K βk ½GŠ 2 þ þ ½GŠ þ þ ½Š þ þ ½RŠ K K βk þ þ þ ½Š þ 0 K K βk It is now necessary to assume that >> þ sup a ½Š ; ½Š K βk (where sup indicates the larger of the two values in the braces), which results in the preceding quadratic simplifying to ½GŠ 2 þ þ ½GŠ þ þ ½RŠ K þ þ þ ½Š þ 0 K K βk Further, since þ þ ½RŠ K þ 2 >> þ þ ½Š þ þ ; K K βk the physically meaningful root of the quadratic can beapproximated as ½GŠ þ K þ ½Š K þ βk (4) þ þ þ K Equations 3 and 4 can now be substituted into the expressions for [RG] and[rg] 0 0 ½RGŠ ½RŠ½GŠ þ þ ½Š þ þ þ ½Š þ þ ½GŠ þ C K K βk B þ þ ½RŠ K K βk K K þ 0 0 ½RŠ þ þ ½Š þ þ þ ½Š þ þ þ þ þ ½Š þ K K βk B K B K K βk G þ þ K K βk K þ þ ½RŠ K þ C K þ 0 ½RŠ ½GŠ þ K K þ G þ þ ½RŠ þ þ ½RŠ K þ C K þ ½GŠ þ þ ½RŠ K þ þ þ K British Journal of Pharmacology (208)

13 DHallandJGiraldo But [R] >> [G],so ½RG Š ½GŠ (5) þ þ þ K lso, the midpoint, [] 50, of this expression is K þ ½GŠ 50 þ ½GŠ þ ½Š K α þ ½Š βk lso, ½RG Š ½RGŠ ½RGŠþ½RGŠ þ þ ½RŠ K þ ; (6) þ þ K þ þ ½RŠ þ þ ½RŠ K þ G K þ ½RŠ α (7) his is of the form of the model of Slack and Hall (202) with E max =[G], =/K, χ =[R] / and =/α. lso, note that if Equation 7 is rearranged to ½RG Šþ ½RGŠ K ½RŠ K þ α ; þ ½RŠ þ þ ½RŠ α (7) Note then that the expressions when [] = 0 and when [] >> K are linearly dependent on [R] but the expression for [] 50 is independent of [R].Inotherswords,ifthereceptor density is varied, the EC 50 remains constant while the basal and maximal effect will vary in proportion. With this condition, then, the transducer function is of linear rather than linear rational form, and the response is simply proportional to fractional occupancy. his is equivalent to the condition χ <<, and since for agonists >, this must also imply that χ <<. lso, however, all of these expressions contain terms in []aswellas[g]sotheapparentefficacy under this condition is dependent on activity along and the concentration of the components along the other signalling pathway(s) as well as the one measured. Now when χ <<, the corresponding expressions from the Slack and Hall model are as follows E 0 E maxχ þ χ E maxχ E 0 E maxχ þ χ E maxχ 50 ð þ χþ þ χ and compared with Equation, it can be seen that Equation is a special case of Equation 7 in which [R] / <<, K << []/α, and we must identify in Equation with α in Equation 7. It is then possible to see that with these assumptions τ χ =[R] /α. It is also instructive to consider the alternative simplifying condition, [R] << [G],[].Inthiscase ½RG Šþ½RGŠ þ ½GŠ ½ Š G þ ½Š K From this expression, when [] =0, When [] >> K, ½RGŠ ½RGŠ ½ Š0 ½ Š þ ½ Š þ ½GŠ ½ Š þ K G þ ½GŠ ½ Š þ ½Š K ½ Š α ½ Š α þ ½Š βk G þ G α þ βk Hence, the ratio of the maximal to the basal response is a measure of intrinsic efficacy. pplying this to the ternary complex model gives G ½RGŠ ½RG Š 0 ½GŠ αk G þ ½GŠ þ ½Š α βk α þ ½GŠ þ ½Š K G K þ ½GŠ þ ½Š α βk þ ½GŠ ½GŠ þ ½Š he corresponding expression for the pathway is β þ G ½ Š þ ½Š K α þ ½Š βk þ ½GŠ Hence, it is possible to measure bias by taking the ratio of these two values as follows K 2058 British Journal of Pharmacology (208)