We present a new Matched Interface and Boundary (MIB) regularization method for treating charge singularity in solvated biomolecules whose electrostatics are described by the Poisson–Boltzmann (PB) equation. In a regularization method, by decomposing the potential function into two or three components, the singular component can be analytically represented by the Green’s function, while other components possess a higher regularity. Our new regularization combines the efficiency of two-component schemes with the accuracy of the three-component schemes. Based on this regularization, a new MIB finite difference algorithm is developed for solving both linear and nonlinear PB equations, where the nonlinearity is handled by using the inexact-Newton’s method. Compared with the existing MIB PB solver based on a three-component regularization, the present algorithm is simpler to implement by circumventing the work to solve a boundary value Poisson equation inside the molecular interface and to compute related interface jump conditions numerically. Moreover, the new MIB algorithm becomes computationally less expensive, while maintains the same second order accuracy. This is numerically verified by calculating the electrostatic potential and solvation energy on the Kirkwood sphere on which the analytical solutions are available and on a series of proteins with various sizes.
In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy model is discussed in great detail. Mathematically, unlike the previous persistent entropy or topological entropy, a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our multiscale persistent entropy can be used in conformation entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of a multiscale rigidity function built from bond and/or dihedral angle distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the mediate range. Moreover, our method can be successfully used in protein classification. For a test database with around nine hundred proteins, a clear separation between all-alpha and all-beta proteins can be achieved, using only the dihedral and pseudo-bond angle information. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the "regularity" of protein structures. Essentially, PSI can be used to describe the "regularity" information in any systems.
SZU-CHI CHUNGINSTITUTE OF STATISTICSHAO-HSUAN WANGINSTITUTE OF STATISTICPO-YAO NIUINSTITUTE OF STATISTICSU-YUN HUANGINSTITUTE OF STATISTICWEI-HAU CHANGINSTITUTE OF STATISTICI-PING TUINSTITUTE OF STATISTIC
Annals of Mathematical Sciences and Applicaitons , 5, (2), 2020
Principal component analysis (PCA) is arguably the most widely used
dimension-reduction method for vector-type data. When applied to a
sample of images, PCA requires vectorization of the image data, which
in turn entails solving an eigenvalue problem for the sample covariance matrix. We propose herein a two-stage dimension reduction (2SDR)
method for image reconstruction from high-dimensional noisy image
data. The first stage treats the image as a matrix, which is a tensor of
order 2, and uses multilinear principal component analysis (MPCA) for
matrix rank reduction and image denoising. The second stage vectorizes
the reduced-rank matrix and achieves further dimension and noise reduction. Simulation studies demonstrate excellent performance of 2SDR, for
which we also develop an asymptotic theory that establishes consistency
of its rank selection. Applications to cryo-EM (cryogenic electronic microscopy), which has revolutionized structural biology, organic and medical chemistry, cellular and molecular physiology in the past decade, are
also provided and illustrated with benchmark cryo-EM datasets. Connections to other contemporaneous developments in image reconstruction
and high-dimensional statistical inference are also discussed.