Biomolecular surface construction by PDE transform

Zheng, Qiong; Yang, Siyang; Guo, Wei

doi:10.1002/cnm.1469

Cited by 34 publications

(68 citation statements)

References 84 publications

(228 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since variational approaches have found their success in a variety of scientific and engineering fields [6, 16, 18–20, 45, 60, 62], a variational derivation of the PDE transform has also been presented [55, 76]. Here we briefly review the variational derivation of the PDE transform.…”

Section: Theory and Algorithmmentioning

confidence: 99%

“…In our previous work, high-order nonlinear evolution PDEs have made vital contributions to mode decomposition [58], image analysis [55], and molecular surface generation [76]. However, solving high-order PDE in Eq.…”

Section: Theory and Algorithmmentioning

confidence: 99%

“…The second type of initial values is defined by the maximum of Gaussian functions [25, 76] as follows

u false(boldr, 0 false) = {max}_{i} true(s exp true(- \frac{false| boldr - r_{i} |^{2} - r_{i}^{2}}{r_{e}^{2}} true) true) .

…”

Section: Numerical Test and Validationmentioning

confidence: 99%

“…The initial molecular surface can be obtained as the isosurface by setting u ( r , 0) = S , where S is a pre-designated isovalue of u , for example, u ( r , 0) = 1 [72, 76]. Although the two types of Gaussian initial data are able to offer smoother isosurfaces, they may still result in isosurfaces with singularities as demonstrated in the following sections.…”

Section: Numerical Test and Validationmentioning

confidence: 99%

“…Typical tasks include edge detection, feature extraction, trend estimation, enhancement, denoising, texture analysis, segmentation, pattern recognition, etc [55, 56, 58]. In addition to these tasks, the PDE transform has also been applied to the surface construction of proteins and viruses [76]. Although all of the important apparatuses for the PDE transform were developed in our earlier work, namely, arbitrarily high-order nonlinear PDE filters [59] and nonlinear PDE based band-pass or high-pass filters [61], the PDE tranform offers a renewed understanding about geometric PDE based methods and a new tool for signal, image and data analysis.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Chen

Wei

2012

Computational and Mathematical Biophysics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Theory and Algorithmmentioning

confidence: 99%

Section: Theory and Algorithmmentioning

confidence: 99%

“…The second type of initial values is defined by the maximum of Gaussian functions [25, 76] as follows

u false(boldr, 0 false) = {max}_{i} true(s exp true(- \frac{false| boldr - r_{i} |^{2} - r_{i}^{2}}{r_{e}^{2}} true) true) .

…”

Section: Numerical Test and Validationmentioning

confidence: 99%

Section: Numerical Test and Validationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Chen

Wei

2012

Computational and Mathematical Biophysics

Self Cite

View full text Add to dashboard Cite

show abstract

Interface methods for biological and biomedical problems

Wei¹

2012

Numer Methods Biomed Eng

View full text Add to dashboard Cite

This special issue contains eight papers on interface methods for biological and biomedical problems. Interface problems arise frequently in science and engineering because of the spatial separation of different material properties between regions or because of a force exerted on the materials of differential physical, chemical, and/or biological properties. Theoretical modeling of interface problems often involves partial differential equations (PDEs) with discontinuous coefficients and singular sources. Mathematical techniques, such as the immersed boundary method, the immersed interface method, and the match interface and boundary method, have emerged in the last few decades as suitable candidates for interface modeling and computations. Some of the recent technical development emphasizes high-order schemes, geometric complexity, stability analysis, geometric singularities, and level set approaches for interface tracking. Indeed, interface techniques have been instrumental to a wide range of advances in real-world biological and biomedical problems. The focus of this special issue is on recent advances in computational methods and theoretical models for interface problems and their applications. A broad range of topics are addressed, including the theoretical, analytical, and computational aspects of interface problems and related biomolecular surfaces.The work of Zheng, Yang, and Wei [1] offers a new framework for the biomolecular surface generation based on the PDE transform. Biomolecular surfaces serve as interfaces in elliptic interface problems, such as those in the Poisson-Boltzmann model for electrostatic analysis and the Poisson-Nernst-Planck model for charge permeation. The PDE transform has been shown to be efficient, stable, and robust for the surface modeling of large proteins and virus molecules.The article by Griffith [2] presents the application of an adaptive, staggered-grid version of the immersed boundary method to the three-dimensional simulation of the fluid dynamics of the aortic heart valve.The paper by Zhao [3] provides an improvement to the differential-geometry-based multiscale solvation model for biomolecular surface representation and solvation analysis originally introduced by Wei. A time-dependent Poisson-Boltzmann equation is proposed to make it easier to deal with the nonlinear term in the original Poisson-Boltzmann equation, which is coupled to the Laplace-Beltrami equation for the surface modeling. A filtering process is employed to stabilize the time integration. The new model is validated by biological properties of macromolecules.Sohn, Li, Li, and Lowengrub [4] present an energy variation approach to investigate the role of hydrodynamic forces on three-dimensional axisymmetric multicomponent vesicles. A set of governing equations for the system is derived by the energy variation. Numerical methods are developed to solve governing PDEs, including fourth-order nonlinear and nonlocal ones. Research described in this article may shed light on the role of vesicles in biological...

show abstract

Geometric modeling of subcellular structures, organelles, and multiprotein complexes

Feng

Xia

Tong

et al. 2012

Numer Methods Biomed Eng

Self Cite

View full text Add to dashboard Cite

SUMMARY Recently, the structure, function, stability, and dynamics of subcellular structures, organelles, and multi-protein complexes have emerged as a leading interest in structural biology. Geometric modeling not only provides visualizations of shapes for large biomolecular complexes but also fills the gap between structural information and theoretical modeling, and enables the understanding of function, stability, and dynamics. This paper introduces a suite of computational tools for volumetric data processing, information extraction, surface mesh rendering, geometric measurement, and curvature estimation of biomolecular complexes. Particular emphasis is given to the modeling of cryo-electron microscopy data. Lagrangian-triangle meshes are employed for the surface presentation. On the basis of this representation, algorithms are developed for surface area and surface-enclosed volume calculation, and curvature estimation. Methods for volumetric meshing have also been presented. Because the technological development in computer science and mathematics has led to multiple choices at each stage of the geometric modeling, we discuss the rationales in the design and selection of various algorithms. Analytical models are designed to test the computational accuracy and convergence of proposed algorithms. Finally, we select a set of six cryo-electron microscopy data representing typical subcellular complexes to demonstrate the efficacy of the proposed algorithms in handling biomolecular surfaces and explore their capability of geometric characterization of binding targets. This paper offers a comprehensive protocol for the geometric modeling of subcellular structures, organelles, and multiprotein complexes.

show abstract

Biomolecular surface construction by PDE transform

Cited by 34 publications

References 84 publications

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

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

Interface methods for biological and biomedical problems

Geometric modeling of subcellular structures, organelles, and multiprotein complexes

Contact Info

Product

Resources

About