Geometric and potential driving formation and evolution of biomolecular surfaces

Abstract

This paper presents new geometrical flow equations for the theoretical modeling of biomolecular surfaces in the context of multiscale implicit solvent models. To account for the local variations near the biomolecular surfaces due to interactions between solvent molecules, and between solvent and solute molecules, we propose potential driven geometric flows, which balance the intrinsic geometric forces that would occur for a surface separating two homogeneous materials with the potential forces induced by the atomic interactions. Stochastic geometric flows are introduced to account for the random fluctuation and dissipation in density and pressure near the solvent-solute interface. Physical properties, such as free energy minimization (area decreasing) and incompressibility (volume preserving), are realized by some of our geometric flow equations. The proposed approach for geometric and potential forces driving the formation and evolution of biological surfaces is illustrated by extensive numerical experiments and compared with established minimal molecular surfaces and molecular surfaces. Local modification of biomolecular surfaces is demonstrated with potential driven geometric flows. High order geometric flows are also considered and tested in the present work for surface generation. Biomolecular surfaces generated by these approaches are typically free of geometric singularities. As the speed of surface generation is crucial to implicit solvent model based molecular dynamics, four numerical algorithms, a semi-implicit scheme, a Crank-Nicolson scheme, and two alternating direction implicit (ADI) schemes, are constructed and tested. Being either stable or conditionally stable but admitting a large critical time step size, these schemes overcome the stability constraint of the earlier forward Euler scheme. Aided with the Thomas algorithm, one of the ADI schemes is found to be very efficient as it balances the speed and accuracy.

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

“…(5) recovers the mean curvature flow used in our earlier construction of minimal molecular surfaces [6]. It reproduces the surface diffusion flow [4] when q = 1 and P = 0. It has been shown that the surface generated with the fourth order geometric PDE demonstrates a morphology distinguished from that obtained with the mean curvature flow.…”

“…Although the constraint has little impact to most image analysis, in which the spacing h is unit, it does lead to some difficulties in other applications. Typically, alternating direction implicit (ADI) methods are implemented to by-pass the stability constraint in solving high-order PDEs [4]. Another approach used in previous PDE transform is the Fourier pseudospectral method via the fast Fourier transform (FFT).…”

“…As a result, one can bypass the image preprocessing, i.e., the convolution of the noise image with a smooth mask in the application of the PDE operator to noisy images, which is a very tricky process in the application of geometric flows to noisy iamges. High order geometric PDEs have been widely applied to image and surface analysis [4, 13, 14, 28, 29, 34, 53, 59, 66]. Recently, arbitrarily high-order geometric PDEs have been modified for molecular surface formation and evolution [4] $$\frac{\partial S}{\partial t}={false(-1false)}^q\sqrt{g(|\nabla {\nabla }^{2q}S|)}\nabla ·true(\frac{\nabla false({\nabla }^{2q}Sfalse)}{\sqrt{gfalse(false|\nabla {\nabla }^{2q}Sfalse|false)}}true)+Pfalse(S,false|\nabla Sfalse|false),$$ where S is the hypersurface function, g (|∇∇ 2 q S |) = 1 + |∇∇ 2 q S | 2 is the generalized Gram determinant and P is a generalized potential term, including microscopic interactions in biomolecular surface construction.…”

“…Due to the stiffness of high-order nonlinear PDEs, computational techniques for solving higher order geometric PDEs are important issues. Alternating direction implicit (ADI) schemes are developed in the literature for integrating high-order nonlinear PDEs [4, 63]. However, in the treatment of digital images, signals and data, the mesh sizes can be set to unit.…”

A self-consistent phase-field approach to implicit solvation of charged molecules with Poisson–Boltzmann electrostatics

“…10,12,36,45,[78][79][80] In this work, we develop a phase-field variational implicitsolvent model (PF-VISM) with the PB electrostatics. It is an alternative to the original VISM that uses a sharp-interface formulation, and it extends our previous work 81,82 that only used the Coulomb-field approximation (CFA) of electrostatic free energy.…”