Multiscale Differential Geometry Learning for Protein Flexibility Analysis.

Protein structural fluctuations, measured by Debye-Waller factors or B-factors, are known to be closely associated with protein flexibility and function. Theoretical approaches have also been developed to predict B-factor values, which reflect protein flexibility. Previous models have made significant strides in analyzing B-factors by fitting experimental data. In this study, we propose a novel approach for B-factor prediction using differential geometry theory, based on the assumption that the intrinsic properties of proteins reside on a family of low-dimensional manifolds embedded within the high-dimensional space of protein structures. By analyzing the mean and Gaussian curvatures of a set of low-dimensional manifolds defined by kernel functions, we develop effective and robust multiscale differential geometry (mDG) models. Our mDG model demonstrates a 27% increase in accuracy compared to the classical Gaussian network model (GNM) in predicting B-factors for a dataset of 364 proteins. Additionally, by incorporating both global and local protein features, we construct a highly effective machine-learning model for the blind prediction of B-factors. Extensive least-squares approximations and machine learning-based blind predictions validate the effectiveness of the mDG modeling approach for B-factor predictions.