Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.

Muscle-Type Nicotinic Receptor Blockade by Diethylamine, the Hydrophilic Moiety of Lidocaine

Lidocaine bears in its structure both an aromatic ring and a terminal amine, which can be protonated at physiological pH, linked by an amide group. Since lidocaine causes multiple inhibitory actions on nicotinic acetylcholine receptors (nAChRs), this work was aimed to determine the inhibitory effects of diethylamine (DEA), a small molecule resembling the hydrophilic moiety of lidocaine, on Torpedo marmorata nAChRs microtransplanted to Xenopus oocytes. Similarly to lidocaine, DEA reversibly blocked acetylcholine-elicited currents (IACh) in a dose-dependent manner (IC50 close to 70 μM), but unlike lidocaine, DEA did not affect IACh desensitization. IACh inhibition by DEA was more pronounced at negative potentials, suggesting an open-channel blockade of nAChRs, although roughly 30% inhibition persisted at positive potentials, indicating additional binding sites outside the pore. DEA block of nAChRs in the resting state (closed channel) was confirmed by the enhanced IACh inhibition when pre-applying DEA before its co-application with ACh, as compared with solely DEA and ACh co-application. Virtual docking assays provide a plausible explanation to the experimental observations in terms of the involvement of different sets of drug binding sites. So, at the nAChR transmembrane (TM) domain, DEA and lidocaine shared binding sites within the channel pore, giving support to their open-channel blockade; besides, lidocaine, but not DEA, interacted with residues at cavities among the M1, M2, M3, and M4 segments of each subunit and also at intersubunit crevices. At the extracellular (EC) domain, DEA and lidocaine binding sites were broadly distributed, which aids to explain the closed channel blockade observed. Interestingly, some DEA clusters were located at the α-γ interphase of the EC domain, in a cavity near the orthosteric binding site pocket; by contrast, lidocaine contacted with all α-subunit loops conforming the ACh binding site, both in α-γ and α-δ and interphases, likely because of its larger size. Together, these results indicate that DEA mimics some, but not all, inhibitory actions of lidocaine on nAChRs and that even this small polar molecule acts by different mechanisms on this receptor. The presented results contribute to a better understanding of the structural determinants of nAChR modulation.