321 research outputs found
Domain Decomposition Based Hybrid Methods of Finite Element and Finite Difference and Applications in Biomolecule Simulations
The dielectric continuum models, such as Poisson Boltzmann equation (PBE), size modified PBE (SMPBE), and nonlocal modified PBE (NMPBE), are important models in predicting the electrostatics of a biomolecule in an ionic solvent. To solve these dielectric continuum models efficiently, in this dissertation, new finite element and finite difference hybrid methods are constructed by Schwartz domain decomposition techniques based on a special seven-box partition of a cubic domain. As one important part of these methods, a finite difference optimal solver --- the preconditioned conjugate gradient method using a multigrid V-cycle preconditioner --- is described in details and proved to have a convergence rate independent of mesh size in solving a symmetric positive definite linear system. These new hybrid algorithms are programmed in Fortran, C, and Python based on the efficient finite element library DOLFIN from the FEniCS project, and are well validated by test models with known analytical solutions. Comparison numerical tests between the new hybrid solvers and the corresponding finite element solvers are done to show the improvement in efficiency. Finally, as applications, solvation free energy and binding free energy calculations are done and then compared to the experiment data
Mathematics at the eve of a historic transition in biology
A century ago physicists and mathematicians worked in tandem and established
quantum mechanism. Indeed, algebras, partial differential equations, group
theory, and functional analysis underpin the foundation of quantum mechanism.
Currently, biology is undergoing a historic transition from qualitative,
phenomenological and descriptive to quantitative, analytical and predictive.
Mathematics, again, becomes a driving force behind this new transition in
biology.Comment: 5 pages, 2 figure
A Computational Model of Protein Induced Membrane Morphology with Geodesic Curvature Driven Protein-Membrane Interface
Continuum or hybrid modeling of bilayer membrane morphological dynamics
induced by embedded proteins necessitates the identification of
protein-membrane interfaces and coupling of deformations of two surfaces. In
this article we developed (i) a minimal total geodesic curvature model to
describe these interfaces, and (ii) a numerical one-one mapping between two
surface through a conformal mapping of each surface to the common middle
annulus. Our work provides the first computational tractable approach for
determining the interfaces between bilayer and embedded proteins. The one-one
mapping allows a convenient coupling of the morphology of two surfaces. We
integrated these two new developments into the energetic model of
protein-membrane interactions, and developed the full set of numerical methods
for the coupled system. Numerical examples are presented to demonstrate (1) the
efficiency and robustness of our methods in locating the curves with minimal
total geodesic curvature on highly complicated protein surfaces, (2) the
usefulness of these interfaces as interior boundaries for membrane deformation,
and (3) the rich morphology of bilayer surfaces for different protein-membrane
interfaces
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