Drosophila Dscam1 (Down Syndrome Cell Adhesion Molecules) and vertebrate clustered protocadherins (Pcdhs) are two classic examples of the extraordinary isoform diversity from a single genomic locus. Dscam1 encodes 38,016 distinct isoforms via mutually exclusive splicing in D. melanogaster, while the vertebrate clustered Pcdhs utilize alternative promoters to generate isoform diversity. Here we reveal a shortened Dscam gene family with tandemly arrayed 5′ cassettes in Chelicerata. These cassette repeats generally comprise two or four exons, corresponding to variable Immunoglobulin 7 (Ig7) or Ig7–8 domains of Drosophila Dscam1. Furthermore, extraordinary isoform diversity has been generated through a combination of alternating promoter and alternative splicing. These sDscams have a high sequence similarity with Drosophila Dscam1, and share striking organizational resemblance to the 5′ variable regions of vertebrate clustered Pcdhs. Hence, our findings have important implications for understanding the functional similarities between Drosophila Dscam1 and vertebrate Pcdhs, and may provide further mechanistic insights into the regulation of isoform diversity.
Abstract. Exterior channel of energy estimates for the radial wave equation were first considered in three dimensions in [6], and for the 5-dimensional case in [12]. In this paper we find the general form of the channel of energy estimate in all odd dimensions for the radial free wave equation. This will be used in the companion paper [11] to establish the soliton resolution for equivariant wave maps in R 3 exterior to the ball B(0, 1) and in all equivariance classes.
Abstract. In this article we consider the initial value problem for the m-equivariant ChernSimons-Schrödinger model in two spatial dimensions with coupling parameter g ∈ R. This is a covariant NLS type problem that is L 2 -critical. We prove that at the critical regularity, for any equivariance index m ∈ Z, the initial value problem in the defocusing case (g < 1) is globally wellposed and the solution scatters. The problem is focusing when g ≥ 1, and in this case we prove that for equivariance indices m ∈ Z, m ≥ 0, there exist constants c = cm,g such that, at the critical regularity, the initial value problem is globally wellposed and the solution scatters when the initial data φ0 ∈ L 2 is m-equivariant and satisfies φ0 2 L 2 < cm,g. We also show that √ cm,g is equal to the minimum L 2 norm of a nontrivial m-equivariant standing wave solution. In the self-dual g = 1 case, we have the exact numerical values cm,1 = 8π(m + 1).
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