High-order fractional partial differential equation transform for molecular surface construction

Abstract

Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

Energy minimization in medical image analysis: Methodologies and applications

“…The proposed MMS is capable of handling internal as well as open cavities, and is typically free of geometric singularities. As described in , the molecular surfaces can also be created by applying high‐order fractional PDEs. Employing the fast fractional Fourier transform algorithm, the proposed PDE transform provides a robust and efficient approach for molecular surface analysis.…”

Fourier Spectral Method for Solving Fractional-order System

“…A lot of twospecies models with various functional responses or Holling type-II, reaction-diffusions, and delays have been reported by many researchers (see, for instance [2,30,31,34] and references therein). In recent times, researchers have shown a great deal of interest in the study of fractional reaction-diffusion systems [1,3,9,10,12,15,24,26,35,36,39,40,41], and some classical textbooks [13,27,28,42].…”

Multi-core CPU or GPU-accelerated Multiscale Modeling for Biomolecular Complexes

“…The topological boundary of the union of all possible probes is called the SES, with no intersection with atoms. A lot of research has been conducted in approximating the SES, including the the alpha-shapes [1, 13], the beta-shapes [26, 36], the MSMS [37], the advancing front and generalized Delaunay approaches [27], NURBS approximation [6], and PDE-based methods [21, 46, 53]. The biomolecules were also represented as implicit models.…”

“…A variety of algorithms were developed in improving the modeling efficiency. For example, the Fast Fourier Transform was used in [21, 53] to get better performance of the PDE transform, and a Non-uniform Fast Fourier Transform (NFFT) was adopted to improve the polynomial-form summation of the kernel functions [7]. The programmable GPU has brought in a new direction for vast data processing in geometric modeling [24, 30, 35, 42, 44].…”