Biomolecular surface construction by PDE transform

SUMMARY This work proposes a new framework for the surface generation based on the partial differential equation (PDE) transform. The PDE transform has recently been introduced as a general approach for the mode decomposition of images, signals, and data. It relies on the use of arbitrarily high‐order PDEs to achieve the time–frequency localization, control the spectral distribution, and regulate the spatial resolution. The present work provides a new variational derivation of high‐order PDE transforms. The fast Fourier transform is utilized to accomplish the PDE transform so as to avoid stringent stability constraints in solving high‐order PDEs. As a consequence, the time integration of high‐order PDEs can be done efficiently with the fast Fourier transform. The present approach is validated with a variety of test examples in two‐dimensional and three‐dimensional settings. We explore the impact of the PDE transform parameters, such as the PDE order and propagation time, on the quality of resulting surfaces. Additionally, we utilize a set of 10 proteins to compare the computational efficiency of the present surface generation method and a standard approach in Cartesian meshes. Moreover, we analyze the present method by examining some benchmark indicators of biomolecular surface, that is, surface area, surface‐enclosed volume, solvation free energy, and surface electrostatic potential. A test set of 13 protein molecules is used in the present investigation. The electrostatic analysis is carried out via the Poisson–Boltzmann equation model. To further demonstrate the utility of the present PDE transform‐based surface method, we solve the Poisson–Nernst–Planck equations with a PDE transform surface of a protein. Second‐order convergence is observed for the electrostatic potential and concentrations. Finally, to test the capability and efficiency of the present PDE transform‐based surface generation method, we apply it to the construction of an excessively large biomolecule, a virus surface capsid. Virus surface morphologies of different resolutions are attained by adjusting the propagation time. Therefore, the present PDE transform provides a multiresolution analysis in the surface visualization. Extensive numerical experiment and comparison with an established surface model indicate that the present PDE transform is a robust, stable, and efficient approach for biomolecular surface generation in Cartesian meshes. Copyright © 2011 John Wiley & Sons, Ltd.