Lipidic cubic phases: A novel concept for the crystallization of membrane proteins

Understanding the mechanisms of action of membrane proteins requires the elucidation of their structures to high resolution. The critical step in accomplishing this by x-ray crystallography is the routine availability of well-ordered three-dimensional crystals. We have devised a novel, rational approach to meet this goal using quasisolid lipidic cubic phases. This membrane system, consisting of lipid, water, and protein in appropriate proportions, forms a structured, transparent, and complex three-dimensional lipidic array, which is pervaded by an intercommunicating aqueous channel system. Such matrices provide nucleation sites (“seeding”) and support growth by lateral diffusion of protein molecules in the membrane (“feeding”). Bacteriorhodopsin crystals were obtained from bicontinuous cubic phases, but not from micellar systems, implying a critical role of the continuity of the diffusion space (the bilayer) on crystal growth. Hexagonal bacteriorhodopsin crystals diffracted to 3.7 Å resolution, with a space group P63, and unit cell dimensions of a = b = 62 Å, c = 108 Å; α = β = 90° and γ = 120°.

Parametrizations of elliptic curves by Shimura curves and by classical modular curves

Fix an isogeny class

Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

A joint hazard and time scaling model to compare survival curves.

To provide a more general method for comparing survival experience, we propose a model that independently scales both hazard and time dimensions. To test the curve shape similarity of two time-dependent hazards, h1(t) and h2(t), we apply the proposed hazard relationship, h12(tKt)/ h1(t) = Kh, to h1. This relationship doubly scales h1 by the constant hazard and time scale factors, Kh and Kt, producing a transformed hazard, h12, with the same underlying curve shape as h1. We optimize the match of h12 to h2 by adjusting Kh and Kt. The corresponding survival relationship S12(tKt) = [S1(t)]KtKh transforms S1 into a new curve S12 of the same underlying shape that can be matched to the original S2. We apply this model to the curves for regional and local breast cancer contained in the National Cancer Institute's End Results Registry (1950-1973). Scaling the original regional curves, h1 and S1 with Kt = 1.769 and Kh = 0.263 produces transformed curves h12 and S12 that display congruence with the respective local curves, h2 and S2. This similarity of curve shapes suggests the application of the more complete curve shapes for regional disease as templates to predict the long-term survival pattern for local disease. By extension, this similarity raises the possibility of scaling early data for clinical trial curves according to templates of registry or previous trial curves, projecting long-term outcomes and reducing costs. The proposed model includes as special cases the widely used proportional hazards (Kt = 1) and accelerated life (KtKh = 1) models.