On Two-Path Convexity in Multipartite Tournaments

Abstract In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds hold for the Helly number, Radon number, and hull number. We classify all clone-free multipartite tournaments of maximum Helly number, Radon number, hull number, and rank. Finally we determine all convexly independent sets of clone-free multipartite tournaments of maximum rank

On two-path convexity in multipartite tournaments

AbstractIn the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds hold for the Helly number, Radon number, and hull number. We classify all clone-free multipartite tournaments of maximum Helly number, Radon number, hull number, and rank

The Cytoplasmic Hsp70 Chaperone Machinery Subjects Misfolded and Endoplasmic Reticulum Import-incompetent Proteins to Degradation via the Ubiquitin–Proteasome System

The mechanism of protein quality control and elimination of misfolded proteins in the cytoplasm is poorly understood. We studied the involvement of cytoplasmic factors required for degradation of two endoplasmic reticulum (ER)-import–defective mutated derivatives of carboxypeptidase yscY (ΔssCPY* and ΔssCPY*-GFP) and also examined the requirements for degradation of the corresponding wild-type enzyme made ER-import incompetent by removal of its signal sequence (ΔssCPY). All these protein species are rapidly degraded via the ubiquitin–proteasome system. Degradation requires the ubiquitin-conjugating enzymes Ubc4p and Ubc5p, the cytoplasmic Hsp70 Ssa chaperone machinery, and the Hsp70 cochaperone Ydj1p. Neither the Hsp90 chaperones nor Hsp104 or the small heat-shock proteins Hsp26 and Hsp42 are involved in the degradation process. Elimination of a GFP fusion (GFP-cODC), containing the C-terminal 37 amino acids of ornithine decarboxylase (cODC) directing this enzyme to the proteasome, is independent of Ssa1p function. Fusion of ΔssCPY* to GFP-cODC to form ΔssCPY*-GFP-cODC reimposes a dependency on the Ssa1p chaperone for degradation. Evidently, the misfolded protein domain dictates the route of protein elimination. These data and our further results give evidence that the Ssa1p-Ydj1p machinery recognizes misfolded protein domains, keeps misfolded proteins soluble, solubilizes precipitated protein material, and escorts and delivers misfolded proteins in the ubiquitinated state to the proteasome for degradation