A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container

This paper deals with a problem the packing polyhex clusters in a regular hexagonal container. It is a common problem in many applications with various cluster shapes used, but symmetric polyhex is the most useful in engineering due to its geometrical properties. Hence, we concentrate on mathematical modeling in such an application, where using the “bee” tetrahex is chosen for the new Compact Muon Solenoid (CMS) design upgrade, which is one of four detectors used in Large Hadron Collider (LHC) experiment at European Laboratory for Particle Physics (CERN). We start from the existing hexagonal containers with hexagonal cells packed inside, and uniform clustering applied. We compare the center-aligned (CA) and vertex-aligned (VA) models, analyzing cluster rotations providing the increased packing efficiency. We formally describe the geometrical properties of clustering approaches and show that cluster sharing is inevitable at the container border with uniform clustering. In addition, we propose a new vertex-aligned model decreasing the number of shared clusters in the uniform scenario, but with a smaller number of clusters contained inside the container. Also, we describe a non-uniform tetrahex cluster packing scheme in the proposed container model. With the proposed cluster packing solution, it is accomplished that all clusters are contained inside the container region. Since cluster-sharing is completely avoided at the container border, the maximal packing efficiency is obtained compared to the existing models.

Jamming and Geometric Representations of Graphs

We expose a relationship between jamming and a generalization of Tutte's barycentric embedding. This provides a basis for the systematic treatment of jamming and maximal packing problems on two-dimensional surfaces.

On Random Greedy Triangle Packing

The behaviour of the random greedy algorithm for constructing a maximal packing of edge-disjoint triangles on $n$ points (a maximal partial triple system) is analysed with particular emphasis on the final number of unused edges. It is shown that this number is at most $n^{7/4+o(1)}$, "halfway" from the previous best-known upper bound $o(n^2)$ to the conjectured value $n^{3/2+o(1)}$. The more general problem of random greedy packing in hypergraphs is also considered.

Interactions between double-stranded RNA regulators and the protein kinase DAI.

The interferon-induced protein kinase DAI, the double-stranded RNA (dsRNA)-activated inhibitor of translation, plays a key role in regulating protein synthesis in higher cells. Once activated, in a process that involves autophosphorylation, it phosphorylates the initiation factor eIF-2, leading to inhibition of polypeptide chain initiation. The activity of DAI is controlled by RNA regulators, including dsRNA activators and highly structured single-stranded RNAs which block activation by dsRNA. To elucidate the mechanism of activation, we studied the interaction of DAI with RNA duplexes of discrete sizes. Molecules shorter than 30 bp fail to bind stably and do not activate the enzyme, but at high concentrations they prevent activation by long dsRNA. Molecules longer than 30 bp bind and activate the enzyme, with an efficiency that increases with increasing chain length, reaching a maximum at about 85 bp. These dsRNAs fail to activate at high concentrations and also prevent activation by long dsRNA. Analysis of complexes between dsRNA and DAI suggests that at maximal packing the enzyme interacts with as little as a single helical turn of dsRNA (11 bp) but under conditions that allow activation the binding site protects about 80 bp of duplex. When the RNA-binding site is fully occupied with an RNA activator, the complex appears to undergo a conformational change.

Interactions between double-stranded RNA regulators and the protein kinase DAI

The interferon-induced protein kinase DAI, the double-stranded RNA (dsRNA)-activated inhibitor of translation, plays a key role in regulating protein synthesis in higher cells. Once activated, in a process that involves autophosphorylation, it phosphorylates the initiation factor eIF-2, leading to inhibition of polypeptide chain initiation. The activity of DAI is controlled by RNA regulators, including dsRNA activators and highly structured single-stranded RNAs which block activation by dsRNA. To elucidate the mechanism of activation, we studied the interaction of DAI with RNA duplexes of discrete sizes. Molecules shorter than 30 bp fail to bind stably and do not activate the enzyme, but at high concentrations they prevent activation by long dsRNA. Molecules longer than 30 bp bind and activate the enzyme, with an efficiency that increases with increasing chain length, reaching a maximum at about 85 bp. These dsRNAs fail to activate at high concentrations and also prevent activation by long dsRNA. Analysis of complexes between dsRNA and DAI suggests that at maximal packing the enzyme interacts with as little as a single helical turn of dsRNA (11 bp) but under conditions that allow activation the binding site protects about 80 bp of duplex. When the RNA-binding site is fully occupied with an RNA activator, the complex appears to undergo a conformational change.