On the fitting of binding data when receptor dimerization is suspected
Abstract
Mechanistic and empirical modelling are compared in context of dimeric receptors. In particular, the supposed advantages of the twostate dimer model for fitting of binding data with respect to classical approaches such as the twoindependent sites model are investigated. The two models are revisited from both the mechanistic and empirical point of views. The problem of overparameterized models and the benefits of the concurrent use of mechanistic and empirical models for mechanism analysis are discussed. The pros and cons of mathematical models are examined with special emphasis given to the interpretation of the connection between the shapes of the curves and receptor cooperativity. It is shown that a given pharmacological phenotype (curve shape) can be obtained from different receptor genotypes (as, for instance, noninterconvertible monomeric receptor species, receptorG protein interactions and dimeric receptors), though values of the Hill coefficient greater than one are indicative of receptor oligomerization. The existence of a relationship between the recently defined dimer cooperativity index and the more familiar Hill coefficient is proven.
British Journal of Pharmacology (2008) 155, 17–23; doi:10.1038/bjp.2008.234; published online 9 June 2008
Abbreviations:

 GPCR

 Gproteincoupled receptor
Introduction
The socalled twostate dimer model (^{Franco et al., 2005}, ^{2006}) has been reviewed in two recent articles, with the authors putting a particular emphasis on the advantages of this model for fitting of binding data as compared to classical approaches such as the two independent sites model (^{Casadó et al., 2007}; ^{Franco et al., 2008}). In addition, a new parameter reflecting the molecular communication within the dimer, the cooperativity index, was defined.
The twostate dimer model was developed (^{Franco et al., 2005}, ^{2006}) with the aim of accurately describing the binding and function of Gproteincoupled receptors (GPCRs). GPCRs are of great relevance in pharmacological and therapeutic research, as they represent nearly half of the current targets in the drug discovery pipelines (^{Overington et al., 2006}). Although most of the experimental evidence accumulated over the last decade suggests that GPCRs exist and function as dimers or higherorder oligomers (see ^{Gurevich and Gurevich, 2008}; ^{Milligan, 2008}; ^{Szidonya et al., 2008} for review), there is an open debate on the monomeric/dimeric nature of GPCRs (^{James et al., 2006}; ^{Bouvier et al., 2007}). In particular, the issue on whether there is a requirement for the receptor to be dimeric for Gprotein activation is a key topic of investigation (^{White et al., 2007}; ^{Whorton et al., 2007}). Mathematical models can help to elucidate the complexity of the receptor dynamics, and, accordingly, accurate fitting to the experimental data points is of prime importance.
Here, the twostate dimer model and the two independent sites model are revisited from both the mechanistic and empirical points of view. The pros and cons of both approaches are discussed, and the existence of a relationship between the cooperativity index and the more familiar Hill coefficient is proved.
Empirical and mechanistic models
Curve fitting of experimental data points by mathematical models is a common way to extract relevant information from biological systems (^{Kenakin, 1997}). Mathematical models can be classified either as mechanistic or empirical, depending on whether the equation used derives from an explicitly written chemical process or not. As the equation parameters pose a biophysical meaning only in the mechanistic models, it is in these models that the main features of the mechanism involved are captured by the fitting. In contrast, but not as a minor feature, the empirical approach is just limited to finding the most appropriate function and obtaining the parameter values that best characterize the shape of the curve (^{Giraldo et al., 2002}). In an ideal world, mechanistic models would be the preferred models, as they can lead to new knowledge of the biological process; yet the high number of parameters they may contain and the likely correlation between them often precludes their use in standard curvefitting procedures.
Receptor oligomerization and curve modelling
There are several mechanistic proposals in the literature that include receptor oligomerization to account for the shape of binding and response curves (see ^{Colquhoun, 1973}; ^{Wells, 1992}; ^{Christopoulos and Kenakin, 2002}; ^{Springael et al., 2007} for review). Some illustrative examples follow: (1) An application of a model, originally thought for multisubunit enzymes (^{Monod et al., 1965}), to the acetylcholine receptor (^{Karlin, 1967}), allowed the author to account for responses with Hill coefficient greater than one. (2) ^{Wreggett and Wells (1995)} performed an investigation of the binding properties of purified cardiac muscarinic receptors from porcine atria. The variation of the Hill coefficient for some muscarinic ligands, with values lower and greater than unity, together with the presence of a disparity of curve shapes (biphasic and bellshaped curves were found), led the authors to suggest the presence of a multivalent receptor (at least tetravalent), in which data could be described in terms of cooperative interactions (^{Wreggett and Wells, 1995}). (3) A subsequent study of the same receptors but from Syrian hamsterwashed membranes yielded data that were mechanistically described in terms of a model comprising cooperative and noncooperative forms of the receptor (^{Chidiac et al., 1997}). For the cooperative form, at least trivalent or divalent states were necessary for consideration depending on whether native or alkylated membranes were examined. (4) Armstrong and Strange (2001) studied the binding of two radioligands ([^{3}H]spiperone and [^{3}H]raclopride) to D_{2} dopamine receptors expressed in Chinese hamster ovary cells both in the presence and absence of sodium ions. Data were interpreted in terms of a model where the receptor exists as a dimer, and, in the absence of sodium ions, raclopride exerts negative cooperativity across the dimer both for its own binding and for the binding of spiperone. (5) The crosstalk between protomers within a dimer and the resulting cooperativity properties of the bound ligands were the subject of an article in which the model proposed for a dimeric receptor consisted of two oscillating states: one that enables binding sites to crosstalk and another that does not (^{Durroux, 2005}). (6) An explanation of negative binding cooperativity in terms of receptor dimerization was also used in the investigation of glycoprotein hormone receptors (^{Urizar et al., 2005}). (7) Positive and negative binding cooperativity were also observed for vasopressin and oxytocin receptors (^{Albizu et al., 2006}). As positive cooperative binding cannot be explained without considering receptor as multivalent, the authors proposed a dimeric arrangement for these receptors.
From the above, we see that the cross talk between the protomers within a dimeric receptor can result in positive, negative or absence cooperativity for ligand binding, which is reflected by the shape of the curves and can be quantified by the Hill coefficient at the midpoint (n_{H50}), with values higher, lower or equal to one, respectively. Importantly, although saturation binding curves with n_{H50} > 1 cannot be described without considering the receptors as multivalent complexes (^{Mattera et al., 1985}; ^{Christopoulos and Kenakin, 2002}; ^{Albizu et al., 2006}; ^{Franco et al., 2006}), n_{H50} < 1 can result not only from an oligomeric receptor, but also from either distinct pools of noninterconverting receptor species or a monomeric receptor that recognize accessory cellular proteins (for example, Gproteins) whose concentrations are limited, so they fall as they bind to the receptor (^{Lee et al., 1986}; ^{Green et al., 1997}; ^{Colquhoun, 1998}). It thus appears that different mechanistic models can predict similar behaviours, in many cases not being possible a unique interpretation of a single curve, and, consequently, being necessary to perform some complementary experiments (for instance, binding experiments in the presence and absence of Gpp(NH)p to examine the influence of Gprotein interaction) to exclude wrong explanations nevertheless compatible with individual data sets (^{Wells, 1992}) (Figure 1).
The twostate dimer model
The two independent sites model
The cooperativity property under the twostate dimer model and the two independent sites model
When comparing Equations 2 and 4, we can distinguish between empirical (c parameters) and mechanistic (K constants) approaches. In terms of the empirical parameters, Equations 2 and 4 are identical; fitting data with the two independent sites model, with f fixed to one half, or with the twostates dimer model gives the same accuracy. However, there is a region of the pharmacological space that cannot be accommodated by the two independent sites model when the mechanistic equilibrium constants are used, namely, the positive cooperativity condition (see below). Thus, although both approaches are the same from an empirical point of view, differences appear when a mechanistic analysis is followed. On the other hand, allowing variability in the f parameter will increase the flexibility of the two independent sites model, allowing for a better fitting in situations where f is far from 1/2.
In the two independent sites model, the relationship between the values of c_{1}^{2} and 4c_{2} and the sign of cooperativity remains the same. However, mechanistically speaking, a restriction is found. Thus, although there are no contradictions between absence of apparent cooperativity (c_{1}^{2}=4c_{2} implies K_{D1}=K_{D2}) and apparent negative cooperativity (c_{1}^{2}>4c_{2} implies K_{D1}≠K_{D2}), for the condition of apparent positive cooperativity (c_{1}^{2}<4c_{2}) there is a mathematically impossible outcome ((K_{D1}−K_{D2})^{2}<0). Consequently, when using Equation 7, it can be seen that the two independent sites model cannot produce curves with n_{H50}>1 by using any combination of the mechanistic K constants.
The issue of data fitting
The question arises on which is, in general, the best fitting approach: mechanistic or empirical? Mechanistic models are the proper formulations for the analysis of pharmacologic systems under physicochemical principles. However, the many parameters that these models often include preclude classical fitting by gradient nonlinear procedures. Different strategies are possible to overcome this problem; two studies were chosen as examples. (1) The interaction of the nicotinic acetylcholine receptor from Torpedo marmorata with [^{3}H]acetylcholine and the fluorescent agonist NBD5acylcholine was studied by equilibrium binding and kinetic experiments (^{Prinz and Maelicke, 1992}). The model included two binding sites per receptor molecule, a preexisting equilibrium between two states of the nAChR, and a ligandinduced transition between receptor states (note that this scheme is basically the same as that of the twostate dimer model). In addition, an extra doubly occupied receptor state to account for ion transmission was incorporated. The more complete model contained 16 rate constants, bringing the total to 21 parameters when fluorescence quantum yields were considered. After a careful strategy of parameter reduction, the number of parameters was reduced to 8, but even this was a high number to be determined reliably by classical fitting procedures of a single kinetic experiment. Instead, a simultaneous fit to a total of 128 sets, including binding, rapid filter kinetics and fluorescence kinetics, was used. (2) Recently, a mathematical model has been proposed for the constitutively dimeric metabotropic glutamate receptors, integrating a triple state (openopen, closedopen and closedclosed) for the extracellular (venus flytrap) domain, where orthosteric ligands bind, and a double state (inactive, active) for the heptahelical domain responsible for Gprotein activation (^{Rovira et al., 2008}). The model included nine parameters for binding and 12 for function. To validate the model, a published study (^{Kniazeff et al., 2004}), including functional concentrationresponse curves for both wildtype and mutated receptors, was reanalysed. To avoid the problem of the fitting being trapped in a particular local minimum, the authors used a stochastic evolutionary algorithm (^{Roche et al., 2006}). One hundred independent runs were performed for both wildtype and mutated receptor curves allowing statistical comparisons between parameters and a rational interpretation of the parameters that significantly changed after mutation. In addition, the model allowed a mechanistic distinction between two types of cooperativity for the cross talk between the protomers of the venus flytrap domain: one associated with the successive binding to inactive openopen states (binding cooperativity) and the other to the induction of closure of one of the venus flytrap subunits from the partner protomer (induction cooperativity). This conceptual distinction for the cooperativity property allowed a mechanistic explanation for the apparent negative binding cooperativity (^{Suzuki et al., 2004}) and positive functional cooperativity (^{Kniazeff et al., 2004}) found for mGluR agonists.
What about empirical models? Empirical models are the right choice if one is interested only in the geometric characterization of the shape of the curve or else the mechanistic analysis becomes an impossible task. Another feature that can make an empirical model extremely useful is the possibility of questioning the conclusions drawn by a mechanistic model. Thus, interesting issues may appear when one finds that a particular empirical model fits better than a mechanistic equation. Let us suppose, for example, that an experimental biphasic curve is obtained and that we are sure that dimerization, and not two populations of noninterconvertible monomeric receptors, is present. If, Equation 3 would fit data better than Equation 2, this would imply that in the twostate dimer model some aspects of the biological system would be missing. A similar proposal might be suggested if, for an experimental curve bearing apparent positive cooperativity, Equation 10 provides a better fitting than Equation 2. If this is the case, then the fitting by Equation 10 would be preferred when one is more interested in finding the best adjustment rather than in biologically meaningful parameters. Yet, and this is a secondary but not an irrelevant issue, the significant increase in the fitting accuracy of the empirical models, such as Equations 3 and 10 as compared to Equation 2, might indicate the presence of various receptor oligomerization states in the system.
To illustrate the discussion, the abovementioned models were fitted to a set of data points selected from the literature, which displayed a typical biphasic curve (^{Motulsky and Christopoulos, 2004}). Here, the original response variable was assigned to the ligandbound concentration and analysed accordingly. Figure 3 shows the theoretical curves obtained from the models, and Table 1 shows the fitting comparison between them by the extrasum of squares Ftest; this is a standard statistical procedure commonly used for model comparison and is available in most data analysis programs. As mentioned above, and assuming that data points correspond to a real situation, the improved accuracy provided by the addition of the f parameter and the n_{H1} and n_{H2} Hill coefficients in the two independent sites model, as compared to the twostate dimer receptor model, would suggest that some aspects of the complexity of receptor dimerization are missing in the twostate dimer receptor model. The question as to whether more than one oligomerization state is coexisting in the system would require further work and, most likely, a more complex mechanistic model to account for it.
Model  df^{a}  SSE^{b}  Parameter estimates 

A. Twostate dimer model  18  0.1287  K_{D1}=10^{−7.6}; K_{D2}=10^{−4.9} 
B. Empirically used twostate dimer and two independent sites models  18  0.1287  c_{1}=10^{−4.9}; c_{2}=10^{−12.4} 
C. Two independent sites model with f fixed to ½  18  0.1287  K_{D1}=10^{−7.6}; K_{D2}=10^{−4.9} 
D. Two independent sites model*  17  0.0544  K_{D1}=10^{−8.1}; K_{D2}=10^{−5.1}; f=0.33 
E. Two independent sites model with the empirical inclusion of Hill coefficients* ^{#}  15  0.0186  K_{D1}=10^{−8.0}; K_{D2}=10^{−5.1}; f=0.38; n_{H1}=1.16; n_{H2}=2.25 
 Parameter estimates of a set of models, and statistical comparisons of resulting fittings.
 Comparisons between models by the extrasum of squares Ftest (^{Motulsky and Christopoulos, 2004}): *P<0.05 with either of Models A, B or C; ^{#}P<0.05 with Model D.
 ^{a}Degrees of freedom.
 ^{b}Sum of squares of the error.
It is worth noting that the feature that a particular empirical model (constructed by including extra parameters in a mechanistic one) fits data better than the mechanistic model from which it is derived, does not prove that the latter is wrong, but sets a warning message that some pieces of the puzzle could have been omitted in the former mechanistic formulation. In this line, the extra complexity derived by new evidences suggesting direct interactions between receptors belonging to different GPCR classes as, for example, between α_{2A}adrenergic and μopioid receptors (^{Vilardaga et al., 2008}) or between 5HT_{2A} and metabotropic glutamate receptors (^{GonzálezMaeso et al., 2008}) becomes a challenge for further investigations on mechanistic modelling approaches.
Concluding remarks
Receptor dimerization, the crosstalk between protomers and the ensuing cooperativity property are current topics in pharmacologic research. The shapes of binding and function curves reflect the molecular interactions between the components of the signal transduction machinery, often being the only information available for the experimenter. Mathematical models can be helpful for assessing the receptorligand interactions involved in these processes and the quantification of the magnitude and sign of receptor cooperativity. The latter issue was the main subject of the present article; several mathematical models were compared, and the equivalence between the Hill coefficient and the dimer cooperativity index was shown.
Mathematical models can be classified as either mechanistic or empirical. Mechanistic models represent the real system by a set of equilibrium/kinetic constants that precisely characterize the mechanism of binding or function of the receptor. Because of their biophysical nature, mechanistic models are the ideal formulations for the analysis of experimental curve data points. Regretfully, the numerous parameters often included by these models preclude their use by classical curvefitting procedures such as gradient nonlinear regression. Stochastic approaches can be the right choice when several local minima are present, as these techniques, in contrast to the widely used regression procedures, explore the complete parameter space, avoiding the problem of the fitting being trapped in a particular local minimum. However, if there are too many parameters for the amount of available data, the problem of parameter identifiability can be solved only by either getting more experimental data or using an empirical model that would contain the same amount of information as the available data. Empirical models employ the minimum number of parameters for the determination of the shape of the curve and, accordingly, do not present difficulties for standard curve fitting. In general, empirical models lack physical basis and are limited to obtaining the common geometric descriptors (midpoint location and slope, asymptotes, and so on) of the curves. Yet if the empirical model is a simplification of a mechanistic model, then some physical principles would be reflected in its formulation. Interestingly, empirical and mechanistic models can be used concurrently in ligandbinding data fitting allowing for complementary information to be obtained, with the analysis of accuracy of fitting being an indication for further investigation on the complexity of receptor dynamics.
Acknowledgments
This study was supported in part by Ministerio de Educación y Ciencia (SAF200765913) and Fundació La Marató de TV3 (Ref. 070530). JG is grateful to Arthur Christopoulos and Antonio Guzmán for a critical reading of the paper and to Carmen Castro for technical assistance. The author is also grateful to the anonymous referees for their helpful comments.
Conflict of interest
The author states no conflict of interest.