Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

Mutation R120G in αB-crystallin, which is linked to a desmin-related myopathy, results in an irregular structure and defective chaperone-like function

αB-crystallin, a member of the small heat shock protein family, possesses chaperone-like function. Recently, it has been shown that a missense mutation in αB-crystallin, R120G, is genetically linked to a desmin-related myopathy as well as to cataracts [Vicart, P., Caron, A., Guicheney, P., Li, A., Prevost, M.-C., Faure, A., Chateau, D., Chapon, F., Tome, F., Dupret, J.-M., et al. (1998) Nat. Genet. 20, 92–95]. By using α-lactalbumin, alcohol dehydrogenase, and insulin as target proteins, in vitro assays indicated that R120G αB-crystallin had reduced or completely lost chaperone-like function. The addition of R120G αB-crystallin to unfolding α-lactalbumin enhanced the kinetics and extent of its aggregation. R120G αB-crystallin became entangled with unfolding α-lactalbumin and was a major portion of the resulting insoluble pellet. Similarly, incubation of R120G αB-crystallin with alcohol dehydrogenase and insulin also resulted in the presence of R120G αB-crystallin in the insoluble pellets. Far and near UV CD indicate that R120G αB-crystallin has decreased β-sheet secondary structure and an altered aromatic residue environment compared with wild-type αB-crystallin. The apparent molecular mass of R120G αB-crystallin, as determined by gel filtration chromatography, is 1.4 MDa, which is more than twice the molecular mass of wild-type αB-crystallin (650 kDa). Images obtained from cryoelectron microscopy indicate that R120G αB-crystallin possesses an irregular quaternary structure with an absence of a clear central cavity. The results of this study show, through biochemical analysis, that an altered structure and defective chaperone-like function of αB-crystallin are associated with a point mutation that leads to a desmin-related myopathy and cataracts.

TectoRNA: modular assembly units for the construction of RNA nano-objects

Structural information on complex biological RNA molecules can be exploited to design tectoRNAs or artificial modular RNA units that can self-assemble through tertiary interactions thereby forming nanoscale RNA objects. The selective interactions of hairpin tetraloops with their receptors can be used to mediate tectoRNA assembly. Here we report on the modulation of the specificity and the strength of tectoRNA assembly (in the nanomolar to micromolar range) by variation of the length of the RNA subunits, the nature of their interacting motifs and the degree of flexibility of linker regions incorporated into the molecules. The association is also dependent on the concentration of magnesium. Monitoring of tectoRNA assembly by lead(II) cleavage protection indicates that some degree of structural flexibility is required for optimal binding. With tectoRNAs one can compare the binding affinities of different tertiary motifs and quantify the strength of individual interactions. Furthermore, in analogy to the synthons used in organic chemistry to synthesize more complex organic compounds, tectoRNAs form the basic assembly units for constructing complex RNA structures on the nanometer scale. Thus, tectoRNA provides a means for constructing molecular scaffoldings that organize functional modules in three-dimensional space for a wide range of applications.

Statistical analysis of shape of objects based on landmark data.

Two objects with homologous landmarks are said to be of the same shape if the configurations of landmarks of one object can be exactly matched with that of the other by translation, rotation/reflection, and scaling. The observations on an object are coordinates of its landmarks with reference to a set of orthogonal coordinate axes in an appropriate dimensional space. The origin, choice of units, and orientation of the coordinate axes with respect to an object may be different from object to object. In such a case, how do we quantify the shape of an object, find the mean and variation of shape in a population of objects, compare the mean shapes in two or more different populations, and discriminate between objects belonging to two or more different shape distributions. We develop some methods that are invariant to translation, rotation, and scaling of the observations on each object and thereby provide generalizations of multivariate methods for shape analysis.