In this note we prove that a finite family $\{X_1,\dots,X_d\}$ of real r.v.'s that is exchangeable and such that $(X_1,\dots,X_d)$ is invariant with respect to a subgroup of $SO(d)$ acting irreducibly, is actually invariant with respect to the action of the full group $SO(d)$. Three immediate consequences are deduced: a characterization of isotropic spherical random eigenfunctions whose Fourier coefficients are exchangeable, an extension of Bernstein's characterization of the Gaussian and a characterization of the Lebesgue measure on the sphere.

We introduce the notion of Kähler topologically hyperbolic manifold, as a"topological" generalization of Kähler [Gro91] and weakly Kähler [BDET24] hyperbolic manifolds. Analogously to [BCDT24], we show the birational invariance of this property and then that Kähler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact Kähler manifolds with trivial first real Chern class. Then, we prove spectral gap theorems for positive holomorphic Hermitian vector bundles on Kähler topologically hyperbolic manifolds, obtaining in particular effective non vanishing results à la Kawamata for adjoint line bundles. We finally explore the effects of Kähler topologically hyperbolicity on Ricci and scalar curvature of Kähler metrics. In the appendix, it is given an explicit description of degree 2 hyperbolic classes for finitely presented groups, and an algebro-geometric consequence for Kähler topologically hyperbolic surfaces: they are necessarily of general type.

Turnip yellows virus (TuYV) is a plant virus infecting important crops such as oilseed rape. TuYV is phloem-restricted and transmitted by aphids. The capsid contains two subunit types: the major capsid protein (CP) and a minor component (RTP∗) which arises from the C-terminal cleavage of a readthrough product (RTP). RTP∗ contains the CP sequence fused with a structured domain, denoted NRTD, which is a key determinant of virus transmission. Though both CP and RTP∗ are involved in virus movement and aphid transmission, how RTP∗ is incorporated into the capsid is poorly understood. We present here the structural characterisation, by immunogold labelling and 3D cryo-EM, of the wild-type TuYV and a mutant whose capsid contains the CP only. We show that incorporation of RTP∗ does not impair the capsid structure, and the NRTD does not adopt well-defined positions at the capsid surface. The number of incorporated RTP∗s suggests a random insertion.

We study the Mabuchi functional associated to a big cohomology class. We define an invariant associated to transcendental Fujita approximations, whose vanishing is related to the Yau-Tian Donaldson conjecture. Assuming vanishing (finiteness) of this invariant we establish (almost) convexity along weak geodesics. As an application, we give an explicit expression of the distance d_p in the big setting for finite entropy potentials.

Let $X$ be a compact complex manifold of complex dimension $n$ and $\alpha$ be a smooth closed real form on $X$ such that its cohomology class $\{ \alpha \}\in H^{1,1}(X, \mathbb{R})$ is big. In this paper we prove that, given a bounded function $f$ with bounded distributional laplacian in $X,$ the $\alpha$-psh envelope $P(f)$ is also locally bounded with locally bounded distributional laplacian on the ample locus of $\{\alpha\}.$

The polerovirus turnip yellows virus (TuYV) (formerly beet western yellows virus, BWYV) is a member of the genus Polerovirus in the family Solemoviridae . It is transmitted from plant to plant by aphids according to a circulative and non-propagative mode. TuYV has a wide host range among herbaceous plants and infects important crops such as oilseed rape. Members of the Solemoviridae family are strictly restricted to the three cell types constituting the phloem; the nucleated phloem parenchyma cells and companion cells, where the virus replicates, and the sieve elements, which convey the virus to sites distant from the inoculation point. TuYV consists of a single-stranded positive sense RNA genome of approximately 6 kb encapsidated into an icosahedral capsid of approximately 30 nm in diameter. The capsid is composed of two types of subunits: the major capsid protein (CP, ~23 kDa) and a minor component, denoted RTP* (~54 kDa), which arises from the C-terminal cleavage of a readthrough gene product (RTP, 74 kDa). The minor capsid component contains the CP sequence at its N-terminus fused with a structured domain (denoted NRTD) which has been shown to be a key determinant of virus transmission and necessary for crossing aphid gut epithelial cells during viral uptake. Though both the CP and RTP* structural proteins are involved in virus movement and aphid transmission, how RTP* is incorporated in the viral particles is poorly understood. We present here the structural characterisation of the TuYV wild-type virions (viral capsids containing both CP and RPT*) and a mutant whose capsids contain the CP only. The comparison of the cryo-EM maps of the wild-type and the mutated virus (resolved at 4.1 and 3.5 Å respectively) reveals that the RTP* does not impair the structural organization of the capsid shell. Furthermore, electron cryo-microscopy and immunogold labelling observations of both viruses indicate that the NRTDs do not adopt well-defined positions relative to the capsid surface, and that only four to six NRTDs or NRTD dimers are incorporated into the viral particles, suggesting a random insertion of the RTP* into the TuYV capsid. ### Competing Interest Statement The authors have declared no competing interest.

Abstract We introduce the notion of weakly Kähler hyperbolic manifold which generalizes that of Kähler hyperbolic manifold given in the early 1990s by M. Gromov, and establish its basic features. We then investigate its spectral properties and show a spectral gap result (on a suitable modification). As applications, we prove that weakly Kähler hyperbolic manifolds are of general type and we study the geometry of their subvarieties and entire curves, verifying – among other things – various aspects of the Lang and the Green–Griffiths conjectures for this class of manifolds.

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, \theta, \phi)$ (provided $n>1$), while they have the same singularities of $\phi$ when $n=1$.

We show that a compact Kähler manifold bimeromorphic to a weakly Kähler hyperbolic manifold is weakly Kähler hyperbolic, providing an answer to a problem raised by J. Kollár in his 1995 book "Shafarevic maps and automorphic forms"

Nanoviruses are plant multipartite viruses with a genome composed of six to eight circular single-stranded DNA segments. The distinct genome segments are encapsidated individually in icosahedral particles that measure ≈18 nm in diameter. Recent studies on the model species Faba bean necrotic stunt virus (FBNSV) revealed that complete sets of genomic segments rarely occur in infected plant cells and that the function encoded by a given viral segment can complement the others across neighbouring cells, presumably by translocation of the gene products through unknown molecular processes. This allows the viral genome to replicate, assemble into viral particles and infect anew, even with the distinct genome segments scattered in different cells. Here, we question the form under which the FBNSV genetic material propagates long distance within the vasculature of host plants and, in particular, whether viral particle assembly is required. Using structure-guided mutagenesis based on a 3.2 Å resolution cryogenic-electron-microscopy reconstruction of the FBNSV particles, we demonstrate that specific site-directed mutations preventing capsid formation systematically suppress FBNSV long-distance movement, and thus systemic infection of host plants, despite positive detection of the mutated coat protein when the corresponding segment is agroinfiltrated into plant leaves. These results strongly suggest that the viral genome does not propagate within the plant vascular system under the form of uncoated DNA molecules or DNA:coat-protein complexes, but rather moves long distance as assembled viral particles.

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Amp\`ere mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, \theta, \phi)$ (provided $n>1$), while they have the same singularities of $\phi$ when $n=1$.

We give a precise estimate for the average scalar curvature of the Weil–Petersson metric on the moduli space ℳ_g as g→∞ up to the order 1/g^2 .

Self-assembly of macromolecules into higher-order symmetric structures is fundamental for the regulation of biological processes. Higher-order symmetric structure self-assembly by the gene expression machinery, such as bacterial DNA-dependent RNA polymerase (RNAP), has never been reported before. Here, we show that the stress-response σB factor from the human pathogen, Mycobacterium tuberculosis, induces the RNAP holoenzyme oligomerization into a supramolecular complex composed of eight RNAP units. Cryo-electron microscopy revealed a pseudo-symmetric structure of the RNAP octamer in which RNAP protomers are captured in an auto-inhibited state and display an open-clamp conformation. The structure shows that σB is sequestered by the RNAP flap and clamp domains. The transcriptional activator RbpA prevented octamer formation by promoting the initiation-competent RNAP conformation. Our results reveal that a non-conserved region of σ is an allosteric controller of transcription initiation and demonstrate how basal transcription factors can regulate gene expression by modulating the RNAP holoenzyme assembly and hibernation.

Tuberculosis claims significantly more than one million lives each year. A feasible way to face the issue of drug resistance is the development of new antibiotics. Bacterial uridine 5'-monophosphate (UMP) kinase is a promising target for novel antibiotic discovery as it is essential for bacterial survival and has no counterpart in human cells. The UMP kinase from M. tuberculosis is also a model of particular interest for allosteric regulation with two effectors, GTP (positive) and UTP (negative). In this study, using X-ray crystallography and cryo-electron microscopy, we report for the first time a detailed description of the negative effector UTP-binding site of a typical Gram-positive behaving UMP kinase. Comparison between this snapshot of low affinity for Mg-ATP with our previous 3D-structure of the GTP-bound complex of high affinity for Mg-ATP led to a better understanding of the cooperative mechanism and the allosteric regulation of UMP kinase. Thermal shift assay and circular dichroism experiments corroborate our model of an inhibition by UTP linked to higher flexibility of the Mg-ATP-binding domain. These new structural insights provide valuable knowledge for future drug discovery strategies targeting bacterial UMP kinases.

Arrestins interact with G protein–coupled receptors (GPCRs) to stop G protein activation and to initiate key signaling pathways. Recent structural studies shed light on the molecular mechanisms involved in GPCR-arrestin coupling, but whether this process is conserved among GPCRs is poorly understood. Here, we report the cryo–electron microscopy active structure of the wild-type arginine-vasopressin V2 receptor (V2R) in complex with β-arrestin1. It reveals an atypical position of β-arrestin1 compared to previously described GPCR-arrestin assemblies, associated with an original V2R/β-arrestin1 interface involving all receptor intracellular loops. Phosphorylated sites of the V2R carboxyl terminus are clearly identified and interact extensively with the β-arrestin1 N-lobe, in agreement with structural data obtained with chimeric or synthetic systems. Overall, these findings highlight a notable structural variability among GPCR-arrestin signaling complexes.

Structures of the AVP-V2R-G s protein complex provide molecular insights into the function of this clinically relevant target.

Let $(X, \omega)$ be a compact K\"ahler manifold of complex dimension n and $\theta$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class $\{ \theta \}\in H^{1,1}(X, \mathbb{R})$ is pseudoeffective. Let $\varphi$ be a $\theta$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $\omega$ such that $\varphi \leq f. $ Then the non-pluripolar measure $\theta_\varphi^n:= (\theta + dd^c \varphi)^n$ satisfies the equality: $$ {\bf{1}}_{\{ \varphi = f \}} \ \theta_\varphi^n = {\bf{1}}_{\{ \varphi = f \}} \ \theta_f^n,$$ where, for a subset $T\subseteq X$, ${\bf{1}}_T$ is the characteristic function. In particular we prove that \[ \theta_{P_{\theta}(f)}^n= { \bf {1}}_{\{P_{\theta}(f) = f\}} \ \theta_f^n\qquad {\rm and }\qquad \theta_{P_\theta[\varphi](f)}^n = { \bf {1}}_{\{P_\theta[\varphi](f) = f \}} \ \theta_f^n. \]

Let D be a bounded strongly convex domain in the complex space of dimension n. For a fixed point p epsilon partial derivative D, we consider the solution of a homogeneous complex Monge-Ampere equation with a simple pole at p. We prove that such a solution enjoys many properties of the classical Poisson kernel in the unit disc and thus deserves to be called the pluricomplex Poisson kernel of D with pole at p. In particular we discuss extremality properties (such as a generalization of the classical Phragmen-Lindelof theorem), relations with the pluricomplex Green function of D, uniqueness in terms of the associated foliation and boundary behaviors. Finally, using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions.