Entropy Measures as Geometrical Tools in the Study of Cosmology.

Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by S = ln χ ( x ) with χ ( x ) being the distance between two nearby geodesics. We derive an equation for the entropy, which by transformation to a Riccati-type equation becomes similar to the Jacobi equation. We further show that the geodesic equation for a null geodesic in a double-warped spacetime leads to the same entropy equation. By applying a Robertson-Walker metric for a flat three-dimensional Euclidean space expanding as a function of time, we again reach the entropy equation stressing the connection between the chosen entropy measure and time. We finally turn to the Raychaudhuri equation for expansion, which also is a Riccati equation similar to the transformed entropy equation. Those Riccati-type equations have solutions of the same form as the Jacobi equation. The Raychaudhuri equation can be transformed to a harmonic oscillator equation, and it has been shown that the geodesic deviation equation of Jacobi is essentially equivalent to that of a harmonic oscillator. The Raychaudhuri equations are strong geometrical tools in the study of general relativity and cosmology. We suggest a refined entropy measure applicable in cosmology and defined by the average deviation of the geodesics in a congruence.

Novel changes in discoidal high density lipoprotein morphology: a molecular dynamics study.

ApoA-I is a uniquely flexible lipid-scavenging protein capable of incorporating phospholipids into stable particles. Here we report molecular dynamics simulations on a series of progressively smaller discoidal high density lipoprotein particles produced by incremental removal of palmitoyloleoylphosphatidylcholine via four different pathways. The starting model contained 160 palmitoyloleoylphosphatidylcholines and a belt of two antiparallel amphipathic helical lipid-associating domains of apolipoprotein (apo) A-I. The results are particularly compelling. After a few nanoseconds of molecular dynamics simulation, independent of the starting particle and method of size reduction, all simulated double belts of the four lipidated apoA-I particles have helical domains that impressively approximate the x-ray crystal structure of lipid-free apoA-I, particularly between residues 88 and 186. These results provide atomic resolution models for two of the particles produced by in vitro reconstitution of nascent high density lipoprotein particles. These particles, measuring 95 angstroms and 78 angstroms by nondenaturing gradient gel electrophoresis, correspond in composition and in size/shape (by negative stain electron microscopy) to the simulated particles with molar ratios of 100:2 and 50:2, respectively. The lipids of the 100:2 particle family form minimal surfaces at their monolayer-monolayer interface, whereas the 50:2 particle family displays a lipid pocket capable of binding a dynamic range of phospholipid molecules.