![]() Modern differential geometry encompasses a long list of branches or research topics and draws on differential calculus, integral calculus, algebra, and differential equation to study problems in geometry or differentiable manifolds. 18Ĭurvature analysis is an important aspect of differential geometry (DG), which is a fundamental topic in mathematics and its study dates back to the 18th century. Curvature, as a measure on how much a surface is deviated from being flat, 13 is a major player in molecular stereospecificity, 14 the characterization of protein-protein and protein-nucleic acid interaction hot spots and drug binding pockets 15- 17 and the analysis of molecular solvation. For example, lipid spontaneous curvature and BAR domain mediated membrane curvature sensing are all known biophysical effects. 12 Curvature analysis, such as the smoothness and curvedness of a given biomolecular surface, is an important issue in molecular biophysics. ![]() 1- 11 Among them, surface modeling is a low-dimensional representation of biomolecules, an important concept in GDA. Geometric data analysis (GDA) of biomolecules concerns molecular structural representation, molecular surface definition, surface meshing and volumetric meshing, molecular visualization, morphological analysis, surface annotation, pertinent feature extraction, etc at a variety of scales and dimensions. Extensive numerical experiments are carried out to demonstrate that the proposed DG-GL strategy outperforms other advanced methods in the predictions of drug discovery-related protein-ligand binding affinity, drug toxicity, and molecular solvation free energy. These low-dimensional differential geometry representations are paired with a robust machine learning algorithm to showcase their descriptive and predictive powers for large, diverse, and complex molecular and biomolecular datasets. Differential geometry apparatuses are utilized to construct element interactive curvatures in analytical forms for certain analytically differentiable density estimators. We encode crucial chemical, physical, and biological information into 2D element interactive manifolds, extracted from a high-dimensional structural data space via a multiscale discrete-to-continuum mapping using differentiable density estimators. We put forward a differential geometry-based geometric learning (DG-GL) hypothesis that the intrinsic physics of three-dimensional (3D) molecular structures lies on a family of low-dimensional manifolds embedded in a high-dimensional data space.
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