Regular Submanifolds in Conformal Space Q_p^n

The authors study the regular submanifolds in the conformal space Qp^n and introduce the submanifold theory in the conformal space Qp^n. The first variation formula of the Willmore volume functional of pseudo-Riemannian submanifolds in the conformal space Qp^n is given. Finally, the conformal isotropic submanifolds in the conformal space

A global solution of the Einstein-Yang-Mills equation on the conformal space

The method used to construct the SU(2) Yang-Mills field on a compactified Minkowski space $\overline M $ (which is equivalent to the conformal space) is generalized to construct an SU(N)(N > 2) Yang-Mills field F jk on $\overline M $ . It is proved that both F jk and the invariant metric tensor g jk of $\overline M $ satisfy the Einstein-Yang-Mills equation. The case of N → ∞ is also discussed.

Recent advances in protein conformation sampling by combining machine learning with molecular simulation

The rapid advancement and broad application of machine learning(ML)have driven a groundbreaking revolution in computational biology.One of the most cutting-edge and important applications of ML is its integration with molecular simulations to improve the sampling efficiency of the vast conformational space of large biomolecules.This review focuses on recent studies that utilize ML-based techniques in the exploration of protein conformational landscape.We first highlight the recent development of ML-aided enhanced sampling methods,including heuristic algorithms and neural networks that are designed to refine the selection of reaction coordinates for the construction of bias potential,or facilitate the exploration of the unsampled region of the energy landscape.Further,we review the development of autoencoder based methods that combine molecular simulations and deep learning to expand the search for protein conformations.Lastly,we discuss the cutting-edge methodologies for the one-shot generation of protein conformations with precise Boltzmann weights.Collectively,this review demonstrates the promising potential of machine learning in revolutionizing our insight into the complex conformational ensembles of proteins.

Towards dynamic structure of biological complexes at atomic resolution by cryo-EM

Cryo-electron microscopy makes use of transmission electron microscopy to image vitrified biological samples and reconstruct their three-dimensional structures from two-dimensional projections via computational approaches. After over40 years of development, this technique is now reaching its zenith and reforming the research paradigm of modern structural biology. It has been gradually taking over X-ray crystallography as the mainstream method. In this review, we briefly introduce the history of cryo-EM, recent technical development and its potential power to reveal dynamic structures. The technical barriers and possible approaches to tackle the upcoming challenges are discussed.

Time-Like Conformal Homogeneous Hypersurfaces with Three Distinct Principal Curvatures

A hypersurface x(M)in Lorentzian space R41 is called conformal homogeneous,if for any two points p,q on M,there exists,a conformal transformation of R41,such that(x(M))=x(M),(x(p))=x(q).In this paper,the authors give a complete classifica-tion for regular time-like conformal homogeneous hypersurfaces in R41 with three distinct principal curvatures.

Kobayashi＇s and Teichmiiller＇s Metrics on the Teichmiiller Space of Symmetric Circle Homeomorphisms

We give a direct proof of a result of Earle, Gardiner and Lakic, that is, Kobayashi＇s metric and Teichmuller＇s metric coincide with each other on the Teichmfiller space of symmetric circle homeomorphisms.

Impact of Stable Protein-protein Interaction on Protein Conformational Space

Backbone dihedral angle based clustering approach was applied to investigate the effect of protein complexation on backbone conformational space and the effect on protein dynamics. Three representative enzyme-inhibitor complexes and their comprised proteins were used as models for small- and moderate-sized globular proteins to compare available backbone conformational space before and after complexation. Microsecond time scale molecular dynamic simulations were generated to ensure sufficient statistics. The result suggests that stable protein-protein interactions lead to redistribution of protein backbone mobility and restriction of the protein backbone conformational space, especially for short time scale motions. Surprisingly, these effects are found to be uncorrelated with protein-protein interaction surface. Consistent with many experimental and computational observations, our results indicate that both induced-fit and conformational selection models play roles in stable protein complexation process, with the dominant role being different for different protein complexes.

Conformally flat Lorentzian hypersurfaces in R_1~4 with three distinct principal curvatures

A three dimensional Lorentzian hypersurface x : M_1~3→ R_1~4 is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of R_1~4. Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.

On AdS/CFT correspondence of EYM fields

In the compactized Minkowski space, which is equivalent to the conformal spaceM 4, we introduced a Lorentz metric d σ2 and a Yang-Mills field θ. Later, we proved that dσ2 and θ together satisfy the EYM (Einstein-Yang-Mills) equation. In this paper, it is proved that θ onM 4 (which is the boundary of the anti-de-Sitter space AdS5) can be extended to be a Yang-Mills field $\hat \theta $ on AdS5 such that Hua’s metric ds2 on AdS5, together with $\hat \theta $ satisfies the EYM equation on AdS5.