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There is an abundance of deep literature on the use of free resolutions to study modules and vector bundle resolutions to study coherent sheaves. When studying a module over the Cox ring of a smooth projective toric variety X, each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which are amenable to algebraic and combinatorial study and also capture desirable geometric information. In this extended abstract, we continue this program in the combinatorially-rich Stanley–Reisner setting. In particular, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings. After augmenting the simplicial complexes associated to these Stanley–Reisner rings with a coloring that reflects the product structure on X, our primary tool is Reisner’s criterion, whose conclusion we interpret in the virtual setting. We also provide two constructions of short virtual resolutions for use beyond the Stanley–Reisner case. 

We use cellular resolutions of monomial ideals to prove an analog of Hilbert’s syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties. 

Francisella tularensisis an extremely infectious pathogen and a category A bioterrorism agent. It causes the highly contagious zoonosis, Tularemia. Currently, FDA approved vaccines against tularemia are unavailable.F.tularensisouter membrane protein A (FopA) is a well-studied virulence determinant and protective antigen against tularemia. It is a major outer membrane protein (Omp) ofF.tularensis. However, FopA-based therapeutic intervention is hindered due to lack of complete structural information for membrane localized mature FopA. In our study, we established recombinant expression, monodisperse purification, crystallization and X-ray diffraction (~6.5 Å) of membrane localized mature FopA. Further, we performed bioinformatics and biophysical experiments to unveil its structural organization in the outer membrane. FopA consists of 393 amino acids and has less than 40% sequence identity to known bacterial Omps. Using comprehensive sequence alignments and structure predictions together with existing partial structural information, we propose a two-domain organization for FopA. Circular dichroism spectroscopy and heat modifiability assay confirmed FopA has a β-barrel domain consistent with alphafold2’s prediction of an eight stranded β-barrel at the N-terminus. Small angle X-ray scattering (SAXS) and native-polyacrylamide gel electrophoresis revealed FopA purified in detergent micelles is predominantly dimeric. Molecular density derived from SAXS at 31 Å shows putative dimeric N-terminal β-barrels surrounded by detergent corona and connected to C-terminal domains via flexible linker. Disorder analysis predicts N- and C-terminal domains are interspersed by a long intrinsically disordered region and alphafold2 predicts this region to be largely unstructured. Taken together, we propose a dimeric, two-domain organization of FopA in the outer membrane: the N-terminal β-barrel is membrane embedded, provides dimerization interface and tethers to membrane extrinsic C-terminal domain via long flexible linker. Structure determination of membrane localized mature FopA is essential to understand its role in pathogenesis and develop anti-tularemia therapeutics. Our results pave the way towards it. 

A 3D-printed modular droplet injector successfully delivered microcrystals of human NAD(P)H:quinone oxidoreductase 1 (NQO1) and phycocyanin with electrical stimulation in a serial crystallography experiment at 120 Hz repetition rate. 

Here, we illustrate what happens inside the catalytic cleft of an enzyme when substrate or ligand binds on single-millisecond timescales. The initial phase of the enzymatic cycle is observed with near-atomic resolution using the most advanced X-ray source currently available: the European XFEL (EuXFEL). The high repetition rate of the EuXFEL combined with our mix-and-inject technology enables the initial phase of ceftriaxone binding to theMycobacterium tuberculosisβ-lactamase to be followed using time-resolved crystallography in real time. It is shown how a diffusion coefficient in enzyme crystals can be derived directly from the X-ray data, enabling the determination of ligand and enzyme–ligand concentrations at any position in the crystal volume as a function of time. In addition, the structure of the irreversible inhibitor sulbactam bound to the enzyme at a 66 ms time delay after mixing is described. This demonstrates that the EuXFEL can be used as an important tool for biomedically relevant research. 

Phytochromes are red-light photoreceptors that were first characterized in plants, with homologs in photosynthetic and non-photosynthetic bacteria known as bacteriophytochromes (BphPs). Upon absorption of light, BphPs interconvert between two states denoted Pr and Pfr with distinct absorption spectra in the red and far-red. They have recently been engineered as enzymatic photoswitches for fluorescent-marker applications in non-invasive tissue imaging of mammals. This article presents cryo- and room-temperature crystal structures of the unusual phytochrome from the non-photosynthetic myxobacteriumStigmatella aurantiaca(SaBphP1) and reveals its role in the fruiting-body formation of this photomorphogenic bacterium. SaBphP1 lacks a conserved histidine (His) in the chromophore-binding domain that stabilizes the Pr state in the classical BphPs. Instead it contains a threonine (Thr), a feature that is restricted to several myxobacterial phytochromes and is not evolutionarily understood. SaBphP1 structures of the chromophore binding domain (CBD) and the complete photosensory core module (PCM) in wild-type and Thr-to-His mutant forms reveal details of the molecular mechanism of the Pr/Pfr transition associated with the physiological response of this myxobacterium to red light. Specifically, key structural differences in the CBD and PCM between the wild-type and the Thr-to-His mutant involve essential chromophore contacts with proximal amino acids, and point to how the photosignal is transduced through the rest of the protein, impacting the essential enzymatic activity in the photomorphogenic response of this myxobacterium.