85 research outputs found
On Enumeration of Conjugacy Classes of Coxeter Elements
In this paper we study the equivalence relation on the set of acyclic
orientations of a graph Y that arises through source-to-sink conversions. This
source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a
Coxeter group. We give a direct proof of a recursion for the number of
equivalence classes of this relation for an arbitrary graph Y using edge
deletion and edge contraction of non-bridge edges. We conclude by showing how
this result may also be obtained through an evaluation of the Tutte polynomial
as T(Y,1,0), and we provide bijections to two other classes of acyclic
orientations that are known to be counted in the same way. A transversal of the
set of equivalence classes is given.Comment: Added a few results about connections to the Tutte polynomia
Equivalences on Acyclic Orientations
The cyclic and dihedral groups can be made to act on the set Acyc(Y) of
acyclic orientations of an undirected graph Y, and this gives rise to the
equivalence relations ~kappa and ~delta, respectively. These two actions and
their corresponding equivalence classes are closely related to combinatorial
problems arising in the context of Coxeter groups, sequential dynamical
systems, the chip-firing game, and representations of quivers.
In this paper we construct the graphs C(Y) and D(Y) with vertex sets Acyc(Y)
and whose connected components encode the equivalence classes. The number of
connected components of these graphs are denoted kappa(Y) and delta(Y),
respectively. We characterize the structure of C(Y) and D(Y), show how delta(Y)
can be derived from kappa(Y), and give enumeration results for kappa(Y).
Moreover, we show how to associate a poset structure to each kappa-equivalence
class, and we characterize these posets. This allows us to create a bijection
from Acyc(Y)/~kappa to the union of Acyc(Y')/~kappa and Acyc(Y'')/~kappa, Y'
and Y'' denote edge deletion and edge contraction for a cycle-edge in Y,
respectively, which in turn shows that kappa(Y) may be obtained by an
evaluation of the Tutte polynomial at (1,0).Comment: The original paper was extended, reorganized, and split into two
papers (see also arXiv:0802.4412
Cycle Equivalence of Graph Dynamical Systems
Graph dynamical systems (GDSs) can be used to describe a wide range of
distributed, nonlinear phenomena. In this paper we characterize cycle
equivalence of a class of finite GDSs called sequential dynamical systems SDSs.
In general, two finite GDSs are cycle equivalent if their periodic orbits are
isomorphic as directed graphs. Sequential dynamical systems may be thought of
as generalized cellular automata, and use an update order to construct the
dynamical system map.
The main result of this paper is a characterization of cycle equivalence in
terms of shifts and reflections of the SDS update order. We construct two
graphs C(Y) and D(Y) whose components describe update orders that give rise to
cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper
bound for the number of cycle equivalence classes one can obtain, and we
enumerate these quantities through a recursion relation for several graph
classes. The components of these graphs encode dynamical neutrality, the
component sizes represent periodic orbit structural stability, and the number
of components can be viewed as a system complexity measure
Coxeter Groups and Asynchronous Cellular Automata
The dynamics group of an asynchronous cellular automaton (ACA) relates
properties of its long term dynamics to the structure of Coxeter groups. The
key mathematical feature connecting these diverse fields is involutions.
Group-theoretic results in the latter domain may lead to insight about the
dynamics in the former, and vice-versa. In this article, we highlight some
central themes and common structures, and discuss novel approaches to some open
and open-ended problems. We introduce the state automaton of an ACA, and show
how the root automaton of a Coxeter group is essentially part of the state
automaton of a related ACA.Comment: 10 pages, 4 figure
Order Independence in Asynchronous Cellular Automata
A sequential dynamical system, or SDS, consists of an undirected graph Y, a
vertex-indexed list of local functions F_Y, and a permutation pi of the vertex
set (or more generally, a word w over the vertex set) that describes the order
in which these local functions are to be applied. In this article we
investigate the special case where Y is a circular graph with n vertices and
all of the local functions are identical. The 256 possible local functions are
known as Wolfram rules and the resulting sequential dynamical systems are
called finite asynchronous elementary cellular automata, or ACAs, since they
resemble classical elementary cellular automata, but with the important
distinction that the vertex functions are applied sequentially rather than in
parallel. An ACA is said to be pi-independent if the set of periodic states
does not depend on the choice of pi, and our main result is that for all n>3
exactly 104 of the 256 Wolfram rules give rise to a pi-independent ACA. In 2005
Hansson, Mortveit and Reidys classified the 11 symmetric Wolfram rules with
this property. In addition to reproving and extending this earlier result, our
proofs of pi-independence also provide significant insight into the dynamics of
these systems.Comment: 18 pages. New version distinguishes between functions that are
pi-independent but not w-independen
Posets from Admissible Coxeter Sequences
We study the equivalence relation on the set of acyclic orientations of an
undirected graph G generated by source-to-sink conversions. These conversions
arise in the contexts of admissible sequences in Coxeter theory, quiver
representations, and asynchronous graph dynamical systems. To each equivalence
class we associate a poset, characterize combinatorial properties of these
posets, and in turn, the admissible sequences. This allows us to construct an
explicit bijection from the equivalence classes over G to those over G' and G",
the graphs obtained from G by edge deletion and edge contraction of a fixed
cycle-edge, respectively. This bijection yields quick and elegant proofs of two
non-trivial results: (i) A complete combinatorial invariant of the equivalence
classes, and (ii) a solution to the conjugacy problem of Coxeter elements for
simply-laced Coxeter groups. The latter was recently proven by H. Eriksson and
K. Eriksson using a much different approach.Comment: 16 pages, 4 figures. Several examples have been adde
Dynamics Groups of Asynchronous Cellular Automata
We say that a finite asynchronous cellular automaton (or more generally, any
sequential dynamical system) is pi-independent if its set of periodic points
are independent of the order that the local functions are applied. In this
case, the local functions permute the periodic points, and these permutations
generate the dynamics group. We have previously shown that exactly 104 of the
possible 256 cellular automaton rules are pi-independent. In this article, we
classify the periodic states of these systems and describe their dynamics
groups, which are quotients of Coxeter groups. The dynamics groups provide
information about permissible dynamics as a function of update sequence and, as
such, connect discrete dynamical systems, group theory, and algebraic
combinatorics in a new and interesting way. We conclude with a discussion of
numerous open problems and directions for future research.Comment: Revised per referee's comment
Editorial: Addressing roles for glycans in immunology using chemical biology
Glycoconjugates, macromolecules containing carbohydrates (glycans) conjugated to proteins or lipids, are a diverse class of biopolymers capable of regulating cell-cell interactions. They are present in a high, natural heterogeneity, which originates from the complex mechanisms involved in their biosynthesis. Genetic and environmental factors determine the ensemble of glycans on any particular cell type, in a non-template encoded manner. As a consequence, the cell surface glycan profile provides a tightly-regulated temporal and spatial signature containing crucial biological information. This information is translated into biological functions by glycan binding proteins (GBPs), also called lectins. Importantly, our immune system is modulated by three major GBP families: C-type lectins, galectins, and Siglecs. The abilities of these GBPs to modulate immune cell function is intimately connected to their ability to differentiate ?self? or ?non-self? glycans from our own cells or pathogens, respectively. Hence, GBP?glycan interactions are critical mediators in immune cell homeostasis. Genetic manipulation of glycan processing enzymes has shed light on the roles of glycans in pathologies such as autoimmune diseases and cancer. However, genetic tools such as genomic manipulation and transgenic animal models have shown to be insufficient to fully untangle the roles of GBP-glycan interactions. Accordingly, recent advances in our understanding of GBPs and how they control immune cell function via glycan recognition has been driven by the development of chemical tools.In this Research Topic, we explore recent work illuminating the various roles of glycans and/or GBPs in controlling immune cell function with special emphasis placed on chemical biology approaches that have been instrumental in such efforts. Potential subjects covered may include:? Immunological roles of Glycan-binding proteins? Glycans as immunomodulators? Development of ligands to probe glycan-binding proteins? Chemical biology approaches to modulate glycan-binding proteins and their glycan ligands? Glycans and synthetic derivatives as novel adjuvants? Glycan-based targeted delivery? Intracellular glycosylation in immune cells? Tissue homing of immune cells mediated by glycans? Glycolipid presentation to immune cells? Glycan-based vaccines? Analytical methods for functional characterization of lectin-glycan interactionsFil: Macauley, Matthew S.. University of Alberta; CanadáFil: Rademacher, Christoph. Max Planck Institute of Colloids and Interfaces; AlemaniaFil: Mariño, Karina Valeria. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Instituto de BiologÃa y Medicina Experimental. Fundación de Instituto de BiologÃa y Medicina Experimental. Instituto de BiologÃa y Medicina Experimental; Argentin
Enzymatic Characterization and Inhibition of the Nuclear Variant of Human O-GlcNAcase
Increasing cellular O-GlcNAc levels through pharmacological inhibition of O-GlcNAcase, the enzyme responsible for removal of the O-GlcNAc post-translational modification, is being increasingly used to aid in discerning the roles played by this form of intracellular glycosylation. Interestingly, two forms of O-GlcNAcase have been studied; a full-length isoform that is better characterized, and a shorter nuclear-localized variant, arising from failure to splice out one intron, which has not been as well characterized. Given the increasing use of O-GlcNAcase inhibitors as research tools, we felt that a clear understanding of how these inhibitors affect both isoforms of O-GlcNAcase is important for proper interpretation of studies making use of these inhibitors in cell culture and in vivo. Here we describe an enzymatic characterization of the nuclear variant of human O-GlcNAcase. We find that this short nuclear variant of O-GlcNAcase, which has the identical catalytic domain as the full-length enzyme, has similar trends in a pH-rate profile and Taft linear free energy analysis as the full-length enzyme. These findings strongly suggest that both enzymes use broadly similar transition states. Consistent with this interpretation, the short isoform is potently inhibited by several previously described inhibitors of full-length O-GlcNAcase including PUGNAc, NAG-thiazoline, and the selective O-GlcNAcase inhibitor NButGT. These findings contrast with earlier studies and suggest that studies using O-GlcNAcase inhibitors in cultured cells or in vivo can be interpreted with the knowledge that both these forms of O-GlcNAcase are inhibited when present
A highly concise preparation of O-deacetylated arylthioglycosides of N-acetyl-D-glucosamine from 2-acetamido-3,4,6-tri-O-acetyl2-deoxy-a-D-glucopyranosyl chloride and aryl thiols or disulfides
An expedient and mild route to a range of aryl 2-acetamido-2-deoxy-1-thio-beta-D-glucopyranosides has been devised from 2-acetamido-3,4,6-tri-O-acetyl-2-deoxy-alpha-D-glucopyranosyl chloride and arylthiols or aryl disulfides using phase transfer catalysis conditions. This simple procedure compresses up to three synthetic steps into a one-pot reaction, obviating the need for tedious workups and chromatography and directly furnishes crystalline materials in good yields. The procedure is compatible with a range of thiols and disulfides and may be amenable for preparing a wide range of thioglycosides with various glycons and aglycons
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