On positivity of Ehrhart polynomials

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic Combinatorics, a volume of the Association for Women in Mathematics Series, Springer International Publishin

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result

On Witten multiple zeta-functions associated with semisimple Lie algebras IV

In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.Comment: 22 pag

Decomposition of semigroup algebras

Let A \subseteq B be cancellative abelian semigroups, and let R be an integral domain. We show that the semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings we obtain an algorithm computing the decomposition. When R[A] is a polynomial ring over a field we explain how to compute many ring-theoretic properties of R[B] in terms of this decomposition. In particular we obtain a fast algorithm to compute the Castelnuovo-Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud-Goto conjecture in a range of new cases. Our algorithms are implemented in the Macaulay2 package MonomialAlgebras.Comment: 12 pages, 2 figures, minor revisions. Package may be downloaded at http://www.math.uni-sb.de/ag/schreyer/jb/Macaulay2/MonomialAlgebras/html

Inducible colitis-associated glycome capable of stimulating the proliferation of memory CD4+ T cells

Immune responses are modified by a diverse and abundant repertoire of carbohydrate structures on the cell surface, which is known as the glycome. In this study, we propose that a unique glycome that can be identified through the binding of galectin-4 is created on local, but not systemic, memory CD4+ T cells under diverse intestinal inflammatory conditions, but not in the healthy state. The colitis-associated glycome (CAG) represents an immature core 1–expressing O-glycan. Development of CAG may be mediated by down-regulation of the expression of core-2 β1,6-N-acetylglucosaminyltransferase (C2GnT) 1, a key enzyme responsible for the production of core-2 O-glycan branch through addition of N-acetylglucosamine (GlcNAc) to a core-1 O-glycan structure. Mechanistically, the CAG seems to contribute to super raft formation associated with the immunological synapse on colonic memory CD4+ T cells and to the consequent stabilization of protein kinase C θ activation, resulting in the stimulation of memory CD4+ T cell expansion in the inflamed intestine. Functionally, CAG-mediated CD4+ T cell expansion contributes to the exacerbation of T cell–mediated experimental intestinal inflammations. Therefore, the CAG may be an attractive therapeutic target to specifically suppress the expansion of effector memory CD4+ T cells in intestinal inflammation such as that seen in inflammatory bowel disease

Blockade of T-cell activation by dithiocarbamates involves novel mechanisms of inhibition of nuclear factor of activated T cells.

Dithiocarbamates (DTCs) have recently been reported as powerful inhibitors of NF-kappaB activation in a number of cell types. Given the role of this transcription factor in the regulation of gene expression in the inflammatory response, NF-kappaB inhibitors have been suggested as potential therapeutic drugs for inflammatory diseases. We show here that DTCs inhibited both interleukin 2 (IL-2) synthesis and membrane expression of antigens which are induced during T-cell activation. This inhibition, which occurred with a parallel activation of c-Jun transactivating functions and expression, was reflected by transfection experiments at the IL-2 promoter level, and involved not only the inhibition of NF-kappaB-driven reporter activation but also that of nuclear factor of activated T cells (NFAT). Accordingly, electrophoretic mobility shift assays (EMSAs) indicated that pyrrolidine DTC (PDTC) prevented NF-kappaB, and NFAT DNA-binding activity in T cells stimulated with either phorbol myristate acetate plus ionophore or antibodies against the CD3-T-cell receptor complex and simultaneously activated the binding of AP-1. Furthermore, PDTC differentially targeted both NFATp and NFATc family members, inhibiting the transactivation functions of NFATp and mRNA induction of NFATc. Strikingly, Western blotting and immunocytochemical experiments indicated that PDTC promoted a transient and rapid shuttling of NFATp and NFATc, leading to their accelerated export from the nucleus of activated T cells. We propose that the activation of an NFAT kinase by PDTC could be responsible for the rapid shuttling of the NFAT, therefore transiently converting the sustained transactivation of this transcription factor that occurs during lymphocyte activation, and show that c-Jun NH2-terminal kinase (JNK) can act by directly phosphorylating NFATp. In addition, the combined inhibitory effects on NFAT and NF-KB support a potential use of DTCs as immunosuppressants

Monoubiquitination of syntaxin 3 leads to retrieval from the basolateral plasma membrane and facilitates cargo recruitment to exosomes

Syntaxin 3 (Stx3), a SNARE protein located and functioning at the apical plasma membrane of epithelial cells, is required for epithelial polarity. A fraction of Stx3 is localized to late endosomes/lysosomes, although how it traffics there and its function in these organelles is unknown. Here we report that Stx3 undergoes monoubiquitination in a conserved polybasic domain. Stx3 present at the basolateral—but not the apical—plasma membrane is rapidly endocytosed, targeted to endosomes, internalized into intraluminal vesicles (ILVs), and excreted in exosomes. A nonubiquitinatable mutant of Stx3 (Stx3-5R) fails to enter this pathway and leads to the inability of the apical exosomal cargo protein GPRC5B to enter the ILV/exosomal pathway. This suggests that ubiquitination of Stx3 leads to removal from the basolateral membrane to achieve apical polarity, that Stx3 plays a role in the recruitment of cargo to exosomes, and that the Stx3-5R mutant acts as a dominant-negative inhibitor. Human cytomegalovirus (HCMV) acquires its membrane in an intracellular compartment and we show that Stx3-5R strongly reduces the number of excreted infectious viral particles. Altogether these results suggest that Stx3 functions in the transport of specific proteins to apical exosomes and that HCMV exploits this pathway for virion excretion

p63 heterozygous mutant mice are not prone to spontaneous or chemically induced tumors

Homology between p63 and p53 has suggested that these proteins might function similarly. However, the majority of data from human tumors have not supported a similar role for p63 in tumor suppression. To investigate this issue, we studied spontaneous tumorigenesis in p63+/- mice in both WT and p53-compromised backgrounds. We found that p63+/- mice were not tumor prone and mice heterozygous for both p63 and p53 had fewer tumors than p53+/- mice. The rare tumors that developed in mice with compromised p63 were also distinct from those of p53+/- mice. Furthermore, p63+/- mice were not prone to chemically induced tumorigenesis, and p63 expression was maintained in carcinomas. These findings demonstrate that, in agreement with data from human tumors, p63 plays a markedly different biological role in cancer than p53