Knapsack Problems in Groups

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure

Structure of the AAA protein Msp1 reveals mechanism of mislocalized membrane protein extraction.

The AAA protein Msp1 extracts mislocalized tail-anchored membrane proteins and targets them for degradation, thus maintaining proper cell organization. How Msp1 selects its substrates and firmly engages them during the energetically unfavorable extraction process remains a mystery. To address this question, we solved cryo-EM structures of Msp1-substrate complexes at near-atomic resolution. Akin to other AAA proteins, Msp1 forms hexameric spirals that translocate substrates through a central pore. A singular hydrophobic substrate recruitment site is exposed at the spiral's seam, which we propose positions the substrate for entry into the pore. There, a tight web of aromatic amino acids grips the substrate in a sequence-promiscuous, hydrophobic milieu. Elements at the intersubunit interfaces coordinate ATP hydrolysis with the subunits' positions in the spiral. We present a comprehensive model of Msp1's mechanism, which follows general architectural principles established for other AAA proteins yet specializes Msp1 for its unique role in membrane protein extraction

Assessment of the nucleotide modifications in the high-resolution cryo-electron microscopy structure of the Escherichia coli 50S subunit.

Post-transcriptional ribosomal RNA (rRNA) modifications are present in all organisms, but their exact functional roles and positions are yet to be fully characterized. Modified nucleotides have been implicated in the stabilization of RNA structure and regulation of ribosome biogenesis and protein synthesis. In some instances, rRNA modifications can confer antibiotic resistance. High-resolution ribosome structures are thus necessary for precise determination of modified nucleotides' positions, a task that has previously been accomplished by X-ray crystallography. Here, we present a cryo-electron microscopy (cryo-EM) structure of the Escherichia coli 50S subunit at an average resolution of 2.2 Å as an additional approach for mapping modification sites. Our structure confirms known modifications present in 23S rRNA and additionally allows for localization of Mg2+ ions and their coordinated water molecules. Using our cryo-EM structure as a testbed, we developed a program for assessment of cryo-EM map quality. This program can be easily used on any RNA-containing cryo-EM structure, and an associated Coot plugin allows for visualization of validated modifications, making it highly accessible

The angular momentum of two collided rarefied preplanetesimals and the formation of binaries

This paper studies the mean angular momentum associated with the collision of two celestial objects in the earliest stages of planet formation. Of primary concern is the scenario of two rarefied preplanetesimals (RPPs) in circular heliocentric orbits. The theoretical results are used to develop models of binary or multiple system formation from RPPs, and explain the observation that a greater fraction of binaries originated farther from the Sun. At the stage of RPPs, small-body satellites can form in two ways: a merger between RPPs can have two centers of contraction, or the formation of satellites from a disc around the primary or the secondary. Formation of the disc can be caused by that the angular momentum of the RPP formed by the merger is greater than the critical angular momentum for a solid body. One or several satellites of the primary (moving mainly in low-eccentricity orbits) can be formed from this disc at any separation less than the Hill radius. The first scenario can explain a system such as 2001 QW322 where the two components have similar masses but are separated by a great distance. In general, any values for the eccentricity and inclination of the mutual orbit are possible. Among discovered binaries, the observed angular momenta are smaller than the typical angular momenta expected for identical RPPs having the same total mass as the discovered binary and encountering each other in circular heliocentric orbits. This suggests that the population of RPPs underwent some contraction before mergers became common.Comment: 12 pages, Monthly Notices of Royal Astron. Society, in pres