Fast Algorithms for General Spin Systems on Bipartite Expanders

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ( n 2 ) time, where n is the size of the graph.

High-Resolution, Multidimensional Phylogenetic Metrics Identify Class I Aminoacyl-tRNA Synthetase Evolutionary Mosaicity and Inter-modular Coupling

The provenance of the aminoacyl-tRNA synthetases (aaRS) poses challenging questions because of their role in the emergence and evolution of genetic coding. We investigate evidence about their ancestry from curated structure-based multiple sequence alignments of a structurally invariant “scaffold” shared by all 10 canonical Class I aaRS. Three uncorrelated phylogenetic metrics—residue-by-residue conservation, its variance, and row-by-row cladistic congruence—imply that the Class I scaffold is a mosaic assembled from distinct, successive genetic sources. These data are especially significant in light of: (i) experimental fragmentations of the Class I scaffold into three partitions that retain catalytic activities in proportion to their length; and (ii) evidence that two of these partitions arose from an ancestral Class I aaRS gene encoding a Class II ancestor in frame on the opposite strand. Phylogenetic metrics of different modules vary in accordance with their presumed functionality. A 46-residue Class I “protozyme” roots the Class I molecular tree prior to the adaptive radiation of the Rossmann dinucleotide binding fold that refined substrate discrimination. Such rooting is consistent with near simultaneous emergence of genetic coding and the origin of the proteome, resolving a conundrum posed by previous inferences that Class I aaRS evolved long after the genetic code had been implemented in an RNA world. Further, pinpointing discontinuous enhancements of aaRS fidelity establishes a timeline for the growth of coding from a binary amino acid alphabet.

Cytotoxic T-cell-based vaccine against SARS-CoV2: a hybrid immunoinformatic approach

This paper presents an alternative vaccination platform that provides long-term cellular immune protection mediated by cytotoxic T-cells. The immune response via cellular immunity creates superior resistance to viral mutations, which are currently the greatest threat to the global vaccination campaign. Furthermore, we also propose a safer, more facile and physiologically appropriate immunization method using either intra-nasal or oral administration. The underlying technology is an adaptation of synthetic long peptides (SLPs) previously used in cancer immunotherapy. SLPs comprising HLA class I and class II epitopes are used to stimulate antigen cross-presentation and canonical class II presentation by dendritic cells. The result is a cytotoxic T cell-mediated prompt and specific immune response against the virus-infected epithelia and a rapid and robust virus clearance. Peptides isolated from COVID-19 convalescent patients were screened for the best HLA population coverage and were tested for toxicity and allergenicity. 3D peptide folding followed by molecular docking studies provided positive results, suggesting a favourable antigen presentation.

Serre–Tate theory for Calabi–Yau varieties

Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo p 2 {p^{2}} of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse–Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration. We also extend Nygaard’s approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.

Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

Given a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

Novel fluorescent GPCR biosensor detects retinal equilibrium binding to opsin and active G protein and arrestin signaling conformations

Rhodopsin is a canonical class A photosensitive G protein–coupled receptor (GPCR), yet relatively few pharmaceutical agents targeting this visual receptor have been identified, in part due to the unique characteristics of its light-sensitive, covalently bound retinal ligands. Rhodopsin becomes activated when light isomerizes 11-cis-retinal into an agonist, all-trans-retinal (ATR), which enables the receptor to activate its G protein. We have previously demonstrated that, despite being covalently bound, ATR can display properties of equilibrium binding, yet how this is accomplished is unknown. Here, we describe a new approach for both identifying compounds that can activate and attenuate rhodopsin and testing the hypothesis that opsin binds retinal in equilibrium. Our method uses opsin-based fluorescent sensors, which directly report the formation of active receptor conformations by detecting the binding of G protein or arrestin fragments that have been fused onto the receptor's C terminus. We show that these biosensors can be used to monitor equilibrium binding of the agonist, ATR, as well as the noncovalent binding of β-ionone, an antagonist for G protein activation. Finally, we use these novel biosensors to observe ATR release from an activated, unlabeled receptor and its subsequent transfer to the sensor in real time. Taken together, these data support the retinal equilibrium binding hypothesis. The approach we describe should prove directly translatable to other GPCRs, providing a new tool for ligand discovery and mutant characterization.

Central Extensions of Lie Groups Preserving a Differential Form

Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega $, and let $G$ be a Fréchet–Lie group acting on $(M,\omega )$. As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of ${\mathfrak{g}}$ by ${\mathbb{R}}$, indexed by $H^{k-1}(M,{\mathbb{R}})^*$. We show that the image of $H_{k-1}(M,{\mathbb{Z}})$ in $H^{k-1}(M,{\mathbb{R}})^*$ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of $G$ by the circle group ${\mathbb{T}}$. The idea is to represent a class in $H_{k-1}(M,{\mathbb{Z}})$ by a weighted submanifold $(S,\beta )$, where $\beta $ is a closed, integral form on $S$. We use transgression of differential characters from $ S$ and $ M $ to the mapping space $ C^\infty (S, M) $ and apply the Kostant–Souriau construction on $ C^\infty (S, M) $.

ON THE POSITION OF NODES OF PLANE CURVES

The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^{2}$ of degree $n$ . This map assigns to every curve $C\in V_{d,n}$ its nodes. For some $n$ , we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in $\text{Pic}(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz–Harris [‘Geometry of the Severi variety’, Trans. Amer. Math. Soc.309 (1988), 1–34] about whether the canonical class of the Severi variety is effective.

A description of Rasmussen’s invariant from the divisibility of Lee’s canonical class

We give a description of Rasmussen’s [Formula: see text]-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram [Formula: see text], for any choice of an integral domain [Formula: see text] and a non-zero, non-invertible element [Formula: see text], we define the [Formula: see text]-divisibility [Formula: see text] of Lee’s canonical class of [Formula: see text], and prove that a combination of [Formula: see text] and some elementary properties of [Formula: see text] yields a link invariant [Formula: see text]. Each [Formula: see text] possesses properties similar to [Formula: see text], which in particular reproves the Milnor conjecture. If we restrict to knots and take [Formula: see text], then our invariant coincides with [Formula: see text].