Bridge Decomposition of Restriction Measures

Motivated by Kesten's bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of SLE(8/3), and as a result can be generalized to the wider class of restriction measures. Specifically we show that the restriction hulls with index less than one can be decomposed into a Poisson Point Process of irreducible bridges in a way that is similar to Ito's excursion decomposition of a Brownian motion according to its zeros.Comment: 24 pages, 2 figures. Final version incorporates minor revisions suggested by the referee, to appear in Jour. Stat. Phy

Growth Based Morphogenesis of Vertebrate Limb Bud

Many genes and their regulatory relationships are involved in developmental phenomena. However, by chemical information alone, we cannot fully understand changing organ morphologies through tissue growth because deformation and growth of the organ are essentially mechanical processes. Here, we develop a mathematical model to describe the change of organ morphologies through cell proliferation. Our basic idea is that the proper specification of localized volume source (e.g., cell proliferation) is able to guide organ morphogenesis, and that the specification is given by chemical gradients. We call this idea “growth-based morphogenesis.” We find that this morphogenetic mechanism works if the tissue is elastic for small deformation and plastic for large deformation. To illustrate our concept, we study the development of vertebrate limb buds, in which a limb bud protrudes from a flat lateral plate and extends distally in a self-organized manner. We show how the proportion of limb bud shape depends on different parameters and also show the conditions needed for normal morphogenesis, which can explain abnormal morphology of some mutants. We believe that the ideas shown in the present paper are useful for the morphogenesis of other organs

Endothelin-1 and Endothelin Receptor mRNA Expression in Normal and Atherosclerotic Human Arteries

Endothelin-1 (ET-1) is a potent vasoconstrictor peptide implicated in a number of human diseases including atherosclerosis. ET-1 binds to two distinct G protein-coupled receptors, known as the ETA and ETB receptor subtypes. In this study, we have examined ET-l, ETA and ETB mRNA expression levels in human vascular cells cultured in vitro and in normal and atherosclerotic human arteries. The results indicate that (a) ET-1 mRNA is constitutively expressed by endothelial cells but not by smooth muscle cells, (b) endothelial cells express only ETB mRNA but smooth muscle cells co-express ETA and ETB mRNA, and (c) in comparison to normal aorta, ET-l mRNA expression is elevated and endothelin receptor mRNA expression is repressed in atherosclerotic lesions