Osculating Circle with Microscopes Within Microscopes

Classically, an osculating circle at a point of a planar curve is introduced technically, often with formula giving its radius and the coordinates of its center. In this note, we propose a new and intuitive definition of this concept: among all the circles which have, on the considered point, the same tangent as the studied curve and thus seem equal to the curve through a microscope, the osculating circle is this that seems equal to the curve through a microscope within microscope.     

Germline mutations of the gene encoding bone morphogenetic protein receptor 1A in juvenile polyposis

Juvenile polyposis (JP; OMIM 174900) is an autosomal dominant gastrointestinal hamartomatous polyposis syndrome in which patients are at risk for developing gastrointestinal cancers. Previous studies have demonstrated a locus for JP mapping to 18q21.1 (ref. 3) and germline mutations in the homolog of the gene for mothers against decapentaplegic, Drosophila, (MADH4, also known as SMAD4) in several JP families. However, mutations in MADH4 are only present in a subset of JP cases, and although mutations in the gene for phosphatase and tensin homolog (PTEN) have been described in a few families, undefined genetic heterogeneity remains. Using a genome-wide screen in four JP kindreds without germline mutations in MADH4 or PTEN, we identified linkage with markers from chromosome 10q22-23 (maximum lod score of 4.74, straight theta=0.00). We found no recombinants using markers developed from the vicinity of the gene for bone morphogenetic protein receptor 1A (BMPR1A), a serine-threonine kinase type I receptor involved in bone morphogenetic protein (BMP) signaling. Genomic sequencing of BMPR1A in each of these JP kindreds disclosed germline nonsense mutations in all affected kindred members but not in normal control individuals. These findings indicate involvement of an additional gene in the transforming growth factor-beta (TGF-beta) superfamily in the genesis of JP, and document an unanticipated function for BMP in colonic epithelial growth control.