Allosteric modulation of muscarinic receptors: model building and proof of internal consistency

Jan Proska, Dept. Physical Electronics, FNSPE, Czech Technical University in Prague, V Holesovickach 2, 180 00 Prague 8, Czech Republic.

Allosteric sites on muscarinic acetylcholine receptors represent novel drug targets. Numbers of compounds displaying high structural diversity were found to modulate the orthosteric ligand binding at the muscarinic receptor in positive or negative manner (Christopoulos et al., 2002). For the purposes of the work, common allosteric ligands were grouped to two categories: alcuronium-like and gallamine-like ligands (AGL); obidoxime-like ligands (OL).

The AGL decrease both rate of association and rate of dissociation. There are the seeming paradoxes: the increasing concentration of the positive modulator slows gradually more even the association rate for the competitive antagonist N-methyl-scopolamine at M2 subtype of muscarinic receptor. Similarly, the negative modulator slows even the dissociation rate in a concentration-dependent fashion. To overcome the problem, it was hypothesized that a compulsory order of ligand binding proceeded (Proska et al., 1994). A four-state cyclic kinetic scheme is used to describe the ternary complex model (TCM). However, this intuitive reaction scheme was proposed for the description of allosteric modulation without detailed mathematical analysis (Stockton et al., 1983).

Up to now, there is lack of studies, which would analyse the suggested model with respect to the qualitative behaviour. The formalism of linear algebra proved useful for comparative analyses of these two reaction systems (Smith et al., 1982). Thus, it has been estimated that the number of independent reactions equals three at the four-state TCM. It means the basic situation is described with opened reaction scheme, and the cyclic model is over-determined. The fourth reaction path should not be included in the model without direct experimental evidence that the reversible reaction proceeds. Application of the Laplace transformation to the relevant systems of differential equations gives simple expressions under the appropriate initial conditions. The central dogma of chemical kinetics states that it is not possible to prove that a proposed reaction mechanism is correct.

The model can only be disproved. Using proof by contradiction, a physical inconsistency of the model with cyclic reaction scheme can be simply demonstrated when applied to description of allosteric interactions of AGL at muscarinic receptors.

Unlike the AGL, potency of the OL to decelerate orthosteric ligand association and dissociation kinetics asymptotically converges to the characteristic level (Ellis, 1997). It is in exact accordance with the cyclic four-state model.

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Work was partially supported by Investment project “Multilicense Mathematica” No.2013010003, (CTU Prague).