Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .     

Chemical Synthesis of Trans 8-Methyl-6-Nonenoyl-CoA and Functional Expression Unravel Capsaicin Synthase Activity Encoded by the Pun1 Locus

Capsaicin, produced by diverse Capsicum species, is among the world’s most popular spices and of considerable pharmaceutical relevance. Although the capsaicinoid biosynthetic pathway has been investigated for decades, several biosynthetic steps have remained partly hypothetical. Genetic evidence suggested that the decisive capsaicin synthase is encoded by the Pun1 locus. Yet, the genetic evidence of the Pun1 locus was never corroborated by functionally active capsaicin synthase that presumably catalyzes an amide bond formation between trans 8-methyl-6-nonenoyl-CoA derived from branched-chain amino acid biosynthesis and vanilloylamine derived from the phenylpropanoid pathway. In this report, we demonstrate the enzymatic activity of a recombinant capsaicin synthase encoded by Pun1, functionally expressed in Escherichia coli, and provide information on its substrate specificity and catalytic properties. Recombinant capsaicin synthase is specific for selected aliphatic CoA-esters and highly specific for vanilloylamine. Partly purified from E. coli, the recombinant active enzyme is a monomeric protein of 51 kDa that is independent of additional co-factors or associated proteins, as previously proposed. These data can now be used to design capsaicin synthase variants with different properties and alternative substrate preferences.