Geometric and potential driving formation and evolution of biomolecular surfaces |
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Authors: | P W Bates Zhan Chen Yuhui Sun Guo-Wei Wei Shan Zhao |
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Institution: | (1) Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA;(2) Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA;(3) Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA |
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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.
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Keywords: | Biomolecular surface formation and evolution Mean curvature flow Potential driven geometric flows High order geometric flows Stochastic geometric flows Computational algorithm |
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