Tits polygons

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank 2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least 4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank 1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

Structural changes in bacteriophage T7 upon receptor-induced genome ejection

Many tailed bacteriophages assemble ejection proteins and a portal–tail complex at a unique vertex of the capsid. The ejection proteins form a transenvelope channel extending the portal–tail channel for the delivery of genomic DNA in cell infection. Here, we report the structure of the mature bacteriophage T7, including the ejection proteins, as well as the structures of the full and empty T7 particles in complex with their cell receptor lipopolysaccharide. Our near–atomic-resolution reconstruction shows that the ejection proteins in the mature T7 assemble into a core, which comprises a fourfold gene product 16 (gp16) ring, an eightfold gp15 ring, and a putative eightfold gp14 ring. The gp15 and gp16 are mainly composed of helix bundles, and gp16 harbors a lytic transglycosylase domain for degrading the bacterial peptidoglycan layer. When interacting with the lipopolysaccharide, the T7 tail nozzle opens. Six copies of gp14 anchor to the tail nozzle, extending the nozzle across the lipopolysaccharide lipid bilayer. The structures of gp15 and gp16 in the mature T7 suggest that they should undergo remarkable conformational changes to form the transenvelope channel. Hydrophobic α-helices were observed in gp16 but not in gp15, suggesting that gp15 forms the channel in the hydrophilic periplasm and gp16 forms the channel in the cytoplasmic membrane.

On Ramsey numbers of hedgehogs

The hedgehog Ht is a 3-uniform hypergraph on vertices $1, \ldots ,t + \left({\matrix{t \cr 2}}\right)$ such that, for any pair (i, j) with 1 ≤ i < j ≤ t, there exists a unique vertex k > t such that {i, j, k} is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog Ht is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(Ht) = O(t2 ln t).

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M, R)$ be a Morse function, and $\Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h \in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{\infty}$, and by $S(f)={h\in D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $h\in S(f)$ induces an automorphism of the graph $\Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) \cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$. In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere. The present paper is devoted to the case $M = S^2$. In this situation $\Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 \to R$ whose fixed subtree $Fix(G)$ consists of more than one point.

Structure of large dsDNA viruses

Nucleocytoplasmic large dsDNA viruses (NCLDVs) encompass an ever-increasing group of large eukaryotic viruses, infecting a wide variety of organisms. The set of core genes shared by all these viruses includes a major capsid protein with a double jelly-roll fold forming an icosahedral capsid, which surrounds a double layer membrane that contains the viral genome. Furthermore, some of these viruses, such as the members of the Mimiviridae and Phycodnaviridae have a unique vertex that is used during infection to transport DNA into the host.

Adenoviral E2 IVa2 protein interacts with L4 33K protein and E2 DNA-binding protein

Adenovirus (AdV) is thought to follow a sequential assembly pathway similar to that observed in dsDNA bacteriophages and herpesviruses. First, empty capsids are assembled, and then the genome is packaged through a ring-like structure, referred to as a portal, located at a unique vertex. In human AdV serotype 5 (HAdV5), the IVa2 protein initiates specific recognition of viral genome by associating with the viral packaging domain located between nucleotides 220 and 400 of the genome. IVa2 is located at a unique vertex on mature capsids and plays an essential role during genome packaging, most likely by acting as a DNA packaging ATPase. In this study, we demonstrated interactions among IVa2, 33K and DNA-binding protein (DBP) in virus-infected cells by in vivo cross-linking of HAdV5-infected cells followed by Western blot, and co-immunoprecipitation of IVa2, 33K and DBP from nuclear extracts of HAdV5-infected cells. Confocal microscopy demonstrated co-localization of IVa2, 33K and DBP in virus-infected cells and also in cells transfected with IVa2, 33K and DBP genes. Immunogold electron microscopy of purified HAdV5 showed the presence of IVa2, 33K or DBP at a single site on the virus particles. Our results provide indirect evidence that IVa2, 33K and DBP may form a complex at a unique vertex on viral capsids and cooperate in genome packaging.

Building Graphs from Colored Trees

We will explore the computational complexity of satisfying certain sets of neighborhood conditions in graphs with various properties. More precisely, fix a radius $\rho$ and let $N(G)$ be the set of isomorphism classes of $\rho$-neighborhoods of vertices of $G$ where $G$ is a graph whose vertices are colored (not necessarily properly) by colors from a fixed finite palette. The root of the neighborhood will be the unique vertex at the "center" of the graph. Given a set $\mathcal{S}$ of colored graphs with a unique root, when is there a graph $G$ with $N(G)=\mathcal{S}$? Or $N(G) \subset \mathcal{S}$? What if $G$ is forced to be infinite, or connected, or both? If the neighborhoods are unrestricted, all these problems are recursively unsolvable; this follows from the work of Bulitko [Graphs with prescribed environments of the vertices. Trudy Mat. Inst. Steklov., 133:78–94, 274, 1973]. In contrast, when the neighborhoods are cycle free, all the problems are in the class $\mathtt{P}$. Surprisingly, if $G$ is required to be a regular (and thus infinite) tree, we show the realization problem is NP-complete (for degree 3 and higher); whereas, if $G$ is allowed to be any finite graph, the realization problem is in P.