Hox cluster genomics in the horn shark, Heterodontus francisci

Reconstructing the evolutionary history of Hox cluster origins will lead to insights into the developmental and evolutionary significance of Hox gene clusters in vertebrate phylogeny and to their role in the origins of various vertebrate body plans. We have isolated two Hox clusters from the horn shark, Heterodontus francisci. These have been sequenced and compared with one another and with other chordate Hox clusters. The results show that one of the horn shark clusters (HoxM) is orthologous to the mammalian HoxA cluster and shows a structural similarity to the amphioxus cluster, whereas the other shark cluster (HoxN) is orthologous to the mammalian HoxD cluster based on cluster organization and a comparison with noncoding and Hox gene-coding sequences. The persistence of an identifiable HoxA cluster over an 800-million-year divergence time demonstrates that the Hox gene clusters are highly integrated and structured genetic entities. The data presented herein identify many noncoding sequence motifs conserved over 800 million years that may function as genetic control motifs essential to the developmental process.

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.