Convex polytopes and quantization of symplectic manifolds

Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. The space Q(M) is the quantum mechanical analogue of the classical phase space M. We discuss here relations between the volume of M and the dimension of the vector space Q(M). Analogues for convex polyhedra are considered.

Time-Lapse Video Microscopy of Gliding Motility in Toxoplasma gondii Reveals a Novel, Biphasic Mechanism of Cell LocomotionV⃞

Toxoplasma gondii is a member of the phylum Apicomplexa, a diverse group of intracellular parasites that share a unique form of gliding motility. Gliding is substrate dependent and occurs without apparent changes in cell shape and in the absence of traditional locomotory organelles. Here, we demonstrate that gliding is characterized by three distinct forms of motility: circular gliding, upright twirling, and helical rotation. Circular gliding commences while the crescent-shaped parasite lies on its right side, from where it moves in a counterclockwise manner at a rate of ∼1.5 μm/s. Twirling occurs when the parasite rights itself vertically, remaining attached to the substrate by its posterior end and spinning clockwise. Helical gliding is similar to twirling except that it occurs while the parasite is positioned horizontally, resulting in forward movement that follows the path of a corkscrew. The parasite begins lying on its left side (where the convex side is defined as dorsal) and initiates a clockwise revolution along the long axis of the crescent-shaped body. Time-lapse video analyses indicated that helical gliding is a biphasic process. During the first 180o of the turn, the parasite moves forward one body length at a rate of ∼1–3 μm/s. In the second phase, the parasite flips onto its left side, in the process undergoing little net forward motion. All three forms of motility were disrupted by inhibitors of actin filaments (cytochalasin D) and myosin ATPase (butanedione monoxime), indicating that they rely on an actinomyosin motor in the parasite. Gliding motility likely provides the force for active penetration of the host cell and may participate in dissemination within the host and thus is of both fundamental and practical interest.

Nas–Walras equilibria of a large economy

Individuals exchange contracts for the delivery of commodities in competitive markets and, simultaneously, act strategically; actions affect utilities across individuals directly or through the payoffs of contracts. This encompasses economies with asymmetric information. Nash–Walras equilibria exist for large economies, even if utility functions are not quasi-concave and choice sets are not convex, which is the case in standard settings; the separation of the purchase from the sale of contracts and the pooling of the deliveries on contracts guarantee that the markets for commodities clear.

Packing at the protein-water interface.

We have determined the packing efficiency at the protein-water interface by calculating the volumes of atoms on the protein surface and nearby water molecules in 22 crystal structures. We find that an atom on the protein surface occupies, on average, a volume approximately 7% larger than an atom of equivalent chemical type in the protein core. In these calculations, larger volumes result from voids between atoms and thus imply a looser or less efficient packing. We further find that the volumes of individual atoms are not related to their chemical type but rather to their structural location. More exposed atoms have larger volumes. Moreover, the packing around atoms in locally concave, grooved regions of protein surfaces is looser than that around atoms in locally convex, ridge regions. This as a direct manifestation of surface curvature-dependent hydration. The net volume increase for atoms on the protein surface is compensated by volume decreases in water molecules near the surface. These waters occupy volumes smaller than those in the bulk solvent by up to 20%; the precise amount of this decrease is directly related to the extent of contact with the protein.