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 two-state dimer model for fitting of binding data with respect to classical approaches such as the two-independent 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, non-interconvertible monomeric receptor species, receptor-G 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.

Figure 1

Relationship between receptor structure (genotype) and curve shape (phenotype) for receptor oligomerization in receptor-ligand binding studies. A dimeric receptor has been supposed for one of the genotypes, although, more generally, an oligomeric receptor or a multivalent receptor can be assumed. Hill coefficients at the midpoint , both lower and equal to one, can be obtained from either of the genotypes (non-interconvertible monomeric receptor species; receptor-G protein interaction, where the concentration of the G protein is limited; and a dimeric receptor); however, values greater than one are indicative of receptor oligomerization.

Figure 2

Mechanistic and empirical equations of the two-state dimer and the two independent sites models. (a) The two-state dimer receptor model (Franco et al., 2005, 2006): The receptor is a dimeric construct displaying two conformations, inactive (RR) and active (RR)^*. The model includes seven equilibrium constants, with five of them being independent because of the chemical relationships within the thermodynamic cycles. The equilibrium constants are defined as: The concentration of bound agonist and the total concentration of receptor species are: (b) The two independent sites model (see Motulsky and Christopoulos, 2004) for review): The system is composed of two non-interconvertible receptor species, R₁ and R₂. The model is characterized by two ligand-receptor dissociation constants (K_D1 and K_D2) and the proportion of one receptor species relative to the other (f). The equilibrium constants are defined as: The concentration of bound agonist and the total concentration of receptor species are: The two models differ in the mechanistic equations where five and three independent constants are included for the two-state dimer receptor model and the two-independent sites model, respectively. However, the same empirical expressions are obtained (c₁ and c₂ empirical parameters) if f=1/2 in the two independent sites model (see Equation 3 in the main text). Importantly, the condition c₁²<4c₂ characteristic of positive cooperativity in the two-state dimer model leads to an impossible result ((K_D1−K_D2)²<0) in the two independent sites model indicating that, in the latter model, apparent positive cooperativity cannot be mimicked.

Figure 3

Collection of data points following a biphasic curve extracted from Motulsky and Christopoulos (2004). The curve has been chosen solely because of its shape, and the meaning of the original ordinate has been changed here to concentration of bound ligand. Equation parameters and comparison between models are listed in Table 1. Blue line: theoretical curve from models A, B or C. Red line: theoretical curve from model D. Green line: theoretical curve from model E.