Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions

We investigate how well the graph of a bilinear function \(b{:}\;[0,1]^n\rightarrow \mathbb {R}\) can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most c times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor c cannot be bounded by a constant independent of n. More precisely, we show that for a random bilinear function b we have asymptotically almost surely \(c\geqslant \sqrt{n}/4\). On the other hand, we prove that \(c\leqslant 600\sqrt{n}\), which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions b for which the McCormick relaxation is equal to the convex hull.     

Chemical basis of peptidoglycan discrimination by PrkC, a key kinase involved in bacterial resuscitation from dormancy

Bacterial Ser/Thr kinases modulate a wide number of cellular processes. In Bacillus subtilis , the Ser/Thr kinase PrkC has been shown to induce germination of bacterial spores in response to DAP-type but not Lys-type cell wall muropeptides. Muropeptides are a clear molecular signal that growing conditions are promising, since they are produced during cell wall peptidoglycan remodeling associated with cell growth and division of neighboring bacteria. However, whether muropeptides are able to bind the protein physically and how the extracellular region is able to distinguish the two types of muropeptides remains unclear. Here we tackled the important question of how the extracellular region of PrkC (EC-PrkC) senses muropeptides. By coupling NMR techniques and protein mutagenesis, we exploited the structural requirements necessary for recognition and binding and proved that muropeptides physically bind to EC-PrkC through DAP-moiety-mediated interactions with an arginine residue, Arg500, belonging to the protein C-terminal PASTA domain. Notably, mutation of this arginine completely suppresses muropeptide binding. Our data provide the first molecular clues into the mechanism of sensing of muropeptides by PrkC.     

Minimal cut-generating functions are nearly extreme

We study continuous (strongly) minimal cut generating functions for the model where all variables are integer. We consider both the original Gomory–Johnson setting as well as a recent extension by Y?ld?z and Cornuéjols (Math Oper Res 41:1381–1403, 2016). We show that for any continuous minimal or strongly minimal cut generating function, there exists an extreme cut generating function that approximates the (strongly) minimal function as closely as desired. In other words, the extreme functions are “dense” in the set of continuous (strongly) minimal functions.     

A 3-Slope Theorem for the infinite relaxation in the plane

In this paper we consider the infinite relaxation of the corner polyhedron with 2 rows. For the 1-row case, Gomory and Johnson proved in their seminal paper a sufficient condition for a minimal function to be extreme, the celebrated 2-Slope Theorem. Despite increased interest in understanding the multiple row setting, no generalization of this theorem was known for this case. We present an extension of the 2-Slope Theorem for the case of 2 rows by showing that minimal 3-slope functions satisfying an additional regularity condition are facets (and hence extreme). Moreover, we show that this regularity condition is necessary, unveiling a structure which is only present in the multi-row setting.     

Chemistry of Lipid A: At the Heart of Innate Immunity

In many Gram‐negative bacteria, lipopolysaccharide (LPS) and its lipid A moiety are pivotal for bacterial survival. Depending on its structure, lipid A carries the toxic properties of the LPS and acts as a potent elicitor of the host innate immune system via the Toll‐like receptor 4/myeloid differentiation factor 2 (TLR4/MD‐2) receptor complex. It often causes a wide variety of biological effects ranging from a remarkable enhancement of the resistance to the infection to an uncontrolled and massive immune response resulting in sepsis and septic shock. Since the bioactivity of lipid A is strongly influenced by its primary structure, a broad range of chemical syntheses of lipid A derivatives have made an enormous contribution to the characterization of lipid A bioactivity, providing novel pharmacological targets for the development of new biomedical therapies. Here, we describe and discuss the chemical aspects regarding lipid A and its role in innate immunity, from the (bio)synthesis, isolation and characterization to the molecular recognition at the atomic level.     

Biopolymer Skeleton Produced by Rhizobium radiobacter: Stoichiometric Alternation of Glycosidic and Amidic Bonds in the Lipopolysaccharide O‐Antigen

The lipopolysaccharide (LPS) O‐antigen structure of the plant pathogen Rhizobium radiobacter strain TT9 and its possible role in a plant‐microbe interaction was investigated. The analyses disclosed the presence of two O‐antigens, named Poly1 and Poly2. The repetitive unit of Poly2 constitutes a 4‐α‐l ‐rhamnose linked to a 3‐α‐d ‐fucose residue. Surprisingly, Poly1 turned out to be a novel type of biopolymer in which the repeating unit is formed by a monosaccharide and an amino‐acid derivative, so that the polymer has alternating glycosidic and amidic bonds joining the two units: 4‐amino‐4‐deoxy‐3‐O‐methyl‐d ‐fucose and (2′R,3′R,4′S)‐N‐methyl‐3′,4′‐dihydroxy‐3′‐methyl‐5′‐oxoproline). Differently from the O‐antigens of LPSs from other pathogenic Gram‐negative bacteria, these two O‐antigens do not activate the oxidative burst, an early innate immune response in the model plant Arabidopsis thaliana, explaining at least in part the ability of this R. radiobacter strain to avoid host defenses during a plant infection process.     

Promoter-Controlled Synthesis and Conformational Analysis of Cyclic Mannosides up to a 32-mer

Cyclodextrins are widely used as carriers of small molecules for drug delivery owing to their remarkable host properties and excellent biocompatibility. However, cyclic oligosaccharides with different sizes and shapes are limited. Cycloglycosylation of ultra-large bifunctional saccharide precursors is challenging due to the constrained conformational spaces. Herein we report a promoter-controlled cycloglycosylation approach for the synthesis of cyclic α-(1→6)-linked mannosides up to a 32-mer. Cycloglycosylation of the bifunctional thioglycosides and (Z)-ynenoates was found to be highly dependent on the promoters. In particular, a sufficient amount of a gold(I) complex played a key role in the proper preorganization of the ultra-large cyclic transition state, providing a cyclic 32-mer polymannoside, which represents the largest synthetic cyclic polysaccharide to date. NMR experiments and a computational study revealed that the cyclic 2-mer, 4-mer, 8-mer, 16-mer, and 32-mer mannosides adopted different conformational states and shapes.