Curvature Research Articles

Intrinsically disordered region amplifies membrane remodeling to augment selective ER-phagy

Intrinsically disordered regions (IDRs) play a pivotal role in organellar remodeling. They transduce signals across membranes, scaffold signaling complexes, and mediate vesicular traffic. Their functions are regulated by constraining conformational ensembles through specific intra- and intermolecular interactions, physical tethering, and posttranslational modifications. The endoplasmic reticulum (ER)-phagy receptor FAM134B/RETREG1, known for its reticulon homology domain (RHD), includes a substantial C-terminal IDR housing the LC3 interacting motif. Beyond engaging the autophagic machinery, the function of the FAM134B-IDR is unclear. Here, we investigate the characteristics of the FAM134B-IDR by extensive modeling and molecular dynamics simulations. We present detailed structural models for the IDR, mapping its conformational landscape in solution and membrane-anchored configurations. Our analysis reveals that depending on the membrane anchor, the IDRs collapse onto the membrane and induce positive membrane curvature to varying degrees. The charge patterns underlying this Janus-like behavior are conserved across other ER-phagy receptors. We found that IDRs alone are sufficient to sense curvature. When combined with RHDs, they intensify membrane remodeling and drive efficient protein clustering, leading to faster budding, thereby amplifying RHD remodeling functions. Our simulations provide a perspective on IDRs of FAM134B, their Janus-like membrane interactions, and the resulting modulatory functions during large-scale ER remodeling.

Periodic Bonding Behaviors of Actinide Atoms (An = Th-Cm) with Negatively Curved Nanographene.

The nanographene with negative curvature has been extensively studied due to its interesting properties and potential applications. In the present work, we have performed all-electron scalar relativistic density functional theory (DFT) calculations to understand the periodic interaction mechanisms of actinide atoms (An = Th, Cm) with the TB8C nanographene. The encapsulated complexes (An@TB8C) were formed due to the octagonal vacancy in the TB8C nanographene. TB8C shows fairly high affinity toward An atoms, especially for Th and Pa. AIMD simulations further confirmed the effective trapping of An atom with TB8C. The partial covalent characters of An-C bonds in An@TB8C were revealed through various bond analysis methods. The 6d electrons of An play an important role in the participation of chemical bonds. The delocalization index (DI) is proposed as a useful descriptor in the study of bond strength involving the actinides. Electronic absorption spectra were simulated for further identification in the experiments. The current work has expanded the potential molecular properties and applications of nanographene.

Effects of Spontaneous Curvature on Interfacial Adsorption and Collapse of Phospholipid Monolayers.

To function effectively, pulmonary surfactant must adsorb rapidly to the alveolar air/water interface, but avoid collapse from the interface when compressed to low surface tensions. Prior studies show that phospholipids in the cylindrical monolayers of the inverse hexagonal (HII) phase adsorb quickly. The monolayers have negative curvature, defined by the concave shape of the hydrophilic face. Formation of the HII structures, however, involves significant disruption of optimal chain-packing. Samples with significant spontaneous curvature, formed in the absence of applied force, may nonetheless have lamellar structures. The experiments here tested whether planar lamellar bilayers formed by phospholipids with negative spontaneous curvature might adsorb rapidly but collapse slowly. Prior studies have shown that binary mixtures of dioleoyl phosphatidylcholine-dioleoyl phosphatidylethanolamine (DOPC-DOPE) with higher mol fractions of DOPE (XPE) have more negative spontaneous curvature. Samples with higher XPE adsorbed more rapidly but also collapsed more quickly. Over that range of XPE, small angle X-ray scattering showed only lamellar structures. The HII phase was undetectable. The results suggest that the innate tendency of the phospholipids to form curved structures has primary importance for adsorption rather than the presence of HII structures. Planar structures are insufficient to minimize the tendency of spontaneous curvature to promote collapse. These findings are consistent with adsorption and collapse that occur via rate-limiting structures with significant negative curvature.

Rigidities of isoperimetric inequality under nonnegative Ricci curvature

The sharp isoperimetric inequality for non-compact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions culminated by Balogh and Kristály (2023) also covering metric-measure spaces satisfying the nonnegative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterization of the equality cases is present in the literature. The scope of this paper is to settle this problem by proving, in the same generality as Balogh and Kristály (2023), that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extends the rigidity results of Brendle (2023) for weighted Riemannian manifolds and the rigidity results of Antonelli et al. (2023) for general \mathsf{RCD} spaces. It also applies to the Euclidean setting by proving that optimizers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

Special homogeneous surfaces

Abstract We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.

Convergence analysis of incremental quasi-subgradient method on Riemannian manifolds with lower bounded curvature

We study the convergence analysis of the incremental quasi-subgradient method for solving the sum of geodesic quasi-convex functions on Riemannian manifolds whose sectional curvature is bounded below. The convergence result of the method with diminishing stepsize is established. An application of the studied algorithm to the sum of ratio problems in the setting of Riemannian manifolds with negative sectional curvature is given.

Dicarbatriphyrin(2.1.1) and its Carbacalix[1]phyrin Analogue: Structure-Property Relationships and Application as a Fe(III) Chemosensor.

The investigation of unsaturated or π-conjugated hydrocarbons, particularly within the macrocyclic framework, has garnered significant interest due to their fundamental properties and practical applications. This research focuses on the synthesis and characterization of a novel stable dicarbatriphyrin(2.1.1) which was constructed by amending the [14]triphyrin(2.1.1) framework by, replacing two pyrrole rings with meta-connected phenyl rings and introducing an o-benzene bridge at the C2 position. The structural modification resulted in a saddle-shaped macrocyclic core with negative Gaussian curvature, as confirmed by crystal structure analysis. This unique molecular topology offers acumens into the structure-property relationships in the curved locally π-conjugated contracted porphyrinoids. The macrocycle exhibits fluorescent emission in solution, which is selectively quenched by Fe(III) ions, highlighting its potential as a chemosensor for Fe(III) cations. Additionally, the formation of carbacalix[1]phyrin(2.1.1), a phlorin analogue of dicarbatriphyrin(2.1.1), which exhibits a chair conformation in contrast to the saddle-shaped structure of its oxidized counterpart. Remarkably, the emergence of a mono-pyrrolic calixbenziphyrin marks an unprecedented advancement in the field of calixphyrin chemistry. Moreover, theoretical studies provide strong support for the experimental findings, reinforcing the conclusions drawn from the structural and functional analyses.

Equivariant min-max hypersurface in G-manifolds with positive Ricci curvature

Equivariant min-max hypersurface in G-manifolds with positive Ricci curvature

Study on the flame structure and flow field of hydrogen-enriched combustion in array micro-tube

Addressing climate change and reducing greenhouse gas emissions are critical priorities. Utilizing hydrogen-rich methane or pure hydrogen as fuels within gas turbines, facilitated by array micro-tube premixed combustion technology, is anticipated to markedly accelerate the decarbonization process of the energy sector. In this study, the flame structure of the array micro-tube premixed burner under various fuel compositions was examined using OH-Planar Laser-Induced Fluorescence and Particle Image Velocimetry measurement techniques. The effects of the equivalence ratio (φ) and the hydrogen power ratio (HPR) on the characteristics of the flame front, including its curvature, density, volume, and the associated flow field properties, were discussed. As φ and HPR increase, the wrinkled structure of the flame front is significantly enhanced, with a more pronounced effect on smaller scales. This enhancement leads to the separation of the unburned pockets from the main flame. Concurrently, both the flame length and the flame area decrease with the augmentation of φ and HPR, indicating a more concentrated combustion process and increased combustion intensity under hydrogen-enriched and pure hydrogen conditions. The study also observed a slight increase in both the negative and positive curvatures of the flame front, with a more notable increase in the negative curvature. The increased negative curvature results in an elevated degree of wrinkling and a higher value of Σ (flame surface density), reaching a maximum of 0.876 mm−1 under the conditions where φ is 0.8 and ⟨c⟩ (mean progress variable) is 0.5, resulting in the smallest observed flame volume of 100.6 mm3. Upon coupling the flame with the flow field, it was discovered that the exit flow field of the array micro-tube exhibits symmetry and a characteristic conical flame shape. The burning velocity of the side flame brushes increases, and the velocity peak shifts upstream. The aforementioned findings confirm that the addition of hydrogen increases the laminar flame velocity, enabling the flame to stably anchor to the microtube outlet and thereby enhance the flame's robustness and stability.

MADE: Multicurvature Adaptive Embedding for Temporal Knowledge Graph Completion.

Temporal knowledge graphs (TKGs) are receiving increased attention due to their time-dependent properties and the evolving nature of knowledge over time. TKGs typically contain complex geometric structures, such as hierarchical, ring, and chain structures, which can often be mixed together. However, embedding TKGs into Euclidean space, as is typically done with TKG completion (TKGC) models, presents a challenge when dealing with high-dimensional nonlinear data and complex geometric structures. To address this issue, we propose a novel TKGC model called multicurvature adaptive embedding (MADE). MADE models TKGs in multicurvature spaces, including flat Euclidean space (zero curvature), hyperbolic space (negative curvature), and hyperspherical space (positive curvature), to handle multiple geometric structures. We assign different weights to different curvature spaces in a data-driven manner to strengthen the ideal curvature spaces for modeling and weaken the inappropriate ones. Additionally, we introduce the quadruplet distributor (QD) to assist the information interaction in each geometric space. Ultimately, we develop an innovative temporal regularization to enhance the smoothness of timestamp embeddings by strengthening the correlation of neighboring timestamps. Experimental results show that MADE outperforms the existing state-of-the-art TKGC models.

Novel structures and collapse of solitons in nonminimally gravitating dark matter halos

Ultralight dark matter simulations predict condensates with short-range correlation, known as solitons or boson stars, at the centers of dark matter halos. This paper investigates the formation and collapse of dark matter solitons influenced by nonminimal gravitational effects, characterized by gradient-dependent self-interactions of dark matter and an additional source in Poisson's equation for gravity. Our simulations suggest that the initial evolution of dark matter resembles that without nonminimal gravitational effects. However, regions with negative potential curvature may develop, and solitons will collapse when their densities reach certain critical values for both positive and negative coupling constants. With strong nonminimal gravitational effects, we verify that linear density perturbations could grow on both large and small scales, potentially enhancing structure formation.

Rate-dependent mechanical and self-monitoring behaviors of 3D printed continuous carbon fiber composites

3D printed continuous carbon fiber reinforced polymer (CFRP) composites offer great advantages in structural health monitoring (SHM) owing to their flexibility in complex structure fabrication. Considering the many prospective aerospace applications, tensile experiments were designed to study their mechanical and self-sensing behaviors under a wide range of strain rates. The turning points in the resistance vs strain curves reveal the damage evolution of the specimens and divide the deformations into linear straining and damage evolution regions. Fiber elongation and fiber contact reduction dominate the resistance increase in the linear straining region and the resistance curve behaves linearly. But fiber breakage is the predominant factor in the damage evolution region, yielding a concave resistance curve. Strength, fracture strain, and resistance variation are found to display significant strain rate dependencies that increase with increasing strain rate. Numerous microcracks are formed and evolved into secondary cracks under dynamic loading. This process absorbs more strain energy and produces more carbon fiber breaks, sustaining a higher fracture strain and resistance variations. A model is developed to describe the strain- and strain rate- dependent resistance behaviors, and the predicted results agree well with experimental data. The outcomes of this work contribute to the application of 3D printed continuous CFRP composites in SHM.

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

Adaptive indefinite kernels in hyperbolic spaces

Learning embeddings in hyperbolic space has gained increasing interest in the community, due to its property of negative curvature, as a way of encoding data hierarchy. Recent works investigate the improvement of the representation power of hyperbolic embeddings through kernelization. However, existing developments focus on defining positive definite (pd) kernels, which may affect the intriguing property of hyperbolic spaces. This is due to the structures of hyperbolic spaces being modeled in indefinite spaces (e.g., Kreĭn space). This paper addresses this issue by developing adaptive indefinite kernels, which can better utilize the structures in the Kreĭn space. To this end, we first propose an adaptive embedding function in the Lorentz model and define indefinite Lorentz kernels (iLks) via the embedding function. Due to the isometric relationship between the Lorentz model and the Poincaré ball, these iLks are further extended to the Poincaré ball, resulting in the development of what are termed indefinite Poincaré kernels (iPKs). We evaluate the proposed indefinite kernels on a diversity of learning scenarios, including image classification, few-shot learning, zero-shot learning, person re-identification, knowledge distillation, etc. We show that the proposed indefinite kernels can bring significant performance gains over the baselines and enjoy better representation power from RKKSs than pd kernels.

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

Riemannian metrics with positive scalar curvature play an important role in differential geometry and general relativity. To investigate these metrics, it is necessary to employ concepts and techniques from global analysis, geometric topology, metric geometry, index theory, and general relativity. This workshop brought together researchers from a variety of backgrounds to combine their expertise and promote cross-disciplinary exchange.

Enumeration and Growth Functions for Uniform Tilings of the Hyperbolic Plane

The geometrical properties of a plane determine the tilings that can be built on it. Because of the negative curvature of the hyperbolic plane, we may find several types of groups of symmetries in patterns built on such a surface, which implies the existence of an infinitude of possible tiling families. Using generating functions, we count the vertices of a uniform tiling from any fixed vertex. We count vertices for all families of valence 5 and for general vertices with valence 6 , with even-sized faces. We also give some general results about the behavior of the vertices and edges of the tilings under consideration.

Some Properties of the Potential Field of an Almost Ricci Soliton

In this article, we are interested in finding necessary and sufficient conditions for a compact almost Ricci soliton to be a trivial Ricci soliton. As a first result, we show that positive Ricci curvature and, for a nonzero constant c, the integral of Ric(cξ,cξ) satisfying a generic inequality on an n-dimensional compact and connected almost Ricci soliton (Mn,g,ξ,σ) are necessary and sufficient conditions for it to be isometric to the n-sphere Sn(c). As another result, we show that, if the affinity tensor of the soliton vector field ξ vanishes and the scalar curvature τ of an n-dimensional compact almost Ricci soliton (Mn,g,ξ,σ) satisfies τnσ−τ≥0, then (Mn,g,ξ,σ) is a trivial Ricci soliton. Finally, on an n-dimensional compact almost Ricci soliton (Mn,g,ξ,σ), we consider the Hodge decomposition ξ=ξ¯+∇h, where divξ¯=0, and we use the bound on the integral of Ricξ¯,ξ¯ and an integral inequality involving the scalar curvature to find another characterization of the n-sphere.

Lysophospholipid headgroup size, and acyl chain length and saturation differentially affect vacuole acidification, Ca 2+ transport, and fusion.

SNARE-mediated membrane fusion is regulated by the lipid composition of the engaged bilayers. Lipid composition impacts fusion through direct protein lipid interactions or through modulating the physical properties of membranes at the site of contact, including the induction of positive curvature by lysophospholipids (LPLs). The degree of positive curvature induced is due to the length and saturation of the single acyl chain in addition to the size of the head group. Here we examined how yeast vacuole fusion and ion transport were differentially affected by changes in lysolipid properties. We found that lysophosphatidylcholine (LPC) with acyl chains containing 14-18 carbons all inhibited fusion with IC 50 values ranging from ∼40-120 µM. The monounsaturation of LPC-18:1 had no effect when compared to its saturated counterpart LPC-18:0. On the other hand, head group size played a more significant role in blocking fusion as lysophosphatidic acid (LPA)-18:1 failed to fully inhibit fusion. We also show that both Ca 2+ uptake and SNARE-dependent Ca 2+ efflux was sensitive to changes in the acyl chain length and saturation of LPCs, while LPA only affected Ca 2+ efflux. Finally, we tested these LPLs on vacuole acidification by the V-ATPase. This showed that LPC-18:0 could fully inhibit acidification whereas other LPCs had moderate effects. Again, LPA had no effect. Together these data suggest that the effects of LPLs were due to a combination of head group size and acyl chain length leading to a range in degree of positive curvature.

Deep-Saddle-Shaped Nanographene Induced by Four Heptagons: Efficient Synthesis and Properties.

The construction of multiple heptagonal rings in nanographene is the key step for obtaining exotic carbon nanostructures with a negative curvature and intriguing properties. Herein, a novel saddle-shaped nanographene (1) with four embedded heptagons is synthesized via a highly efficient one-shot Scholl reaction from a predesigned oligophenylene precursor. Notably, a quadruple [6]helicene intermediate was also obtained and isolated by controlling the Scholl reaction conditions. Interestingly, the single crystal structures of 1 display a saddle geometry induced by the four embedded heptagons, resulting in a deep curvature with a width of 16.5 Å and a depth of 8.0 Å. Theoretical calculations at the molecular level suggest a weak antiaromatic character of the heptagons in 1. Remarkably, compound 1 exhibits dual fluorescence from S1 and S2. The deep-saddle-shaped geometry in 1 defines host-guest interactions with fullerenes, which were explored in titration experiments and by theoretical methods. The resulting 1@C60 are stable and are subject to an electron transfer from photoexcited 1 to C60. Our current study underscores the influence of heptagon rings on the photophysical, self-assembly, and electron-donating properties of NGs.

Halofilins as emerging bactofilin families of archaeal cell shape plasticity orchestrators

Bactofilins are rigid, nonpolar bacterial cytoskeletal filaments that link cellular processes to specific curvatures of the cytoplasmic membrane. Although homologs of bactofilins have been identified in archaea and eukaryotes, functional studies have remained confined to bacterial systems. Here, we characterize representatives of two families of archaeal bactofilins from the pleomorphic archaeon Haloferax volcanii, halofilin A (HalA) and halofilin B (HalB). HalA and HalB polymerize in vitro, assembling into straight bundles. HalA polymers are highly dynamic and accumulate at positive membrane curvatures in vivo, whereas HalB forms more static foci that localize in areas of local negative curvatures on the outer cell surface. Gene deletions and live-cell imaging show that halofilins are critical in maintaining morphological integrity during shape transition from disk (sessile) to rod (motile). Morphological defects in ΔhalA result in accumulation of highly positive curvatures in rods but not in disks. Conversely, disk-shaped cells are exclusively affected by halB deletion, resulting in flatter cells. Furthermore, while ΔhalA and ΔhalB cells imprecisely determine the future division plane, defects arise predominantly during the disk-to-rod shape remodeling. The deletion of halA in the haloarchaeon Halobacterium salinarum, whose cells are consistently rod-shaped, impacted morphogenesis but not cell division. Increased levels of halofilins enforced drastic deformations in cells devoid of the S-layer, suggesting that HalB polymers are more stable at defective S-layer lattice regions. Our results suggest that halofilins might play a significant mechanical scaffolding role in addition to possibly directing envelope synthesis.