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A Finite Element Solution of Lateral Periodic Poisson-Boltzmann Model for Membrane Channel Proteins.
Membrane channel proteins control the diffusion of ions across biological membranes. They are closely related to the processes of various organizational mechanisms, such as: cardiac impulse, muscle contraction and hormone secretion. Introducing a membrane region into implicit solvation models extends the ability of the Poisson-Boltzmann (PB) equation to handle membrane proteins. The use of lateral periodic boundary conditions can properly simulate the discrete distribution of membrane proteins on the membrane plane and avoid boundary effects, which are caused by the finite box size in the traditional PB calculations. In this work, we: (1) develop a first finite element solver (FEPB) to solve the PB equation with a two-dimensional periodicity for membrane channel proteins, with different numerical treatments of the singular charges distributions in the channel protein; (2) add the membrane as a dielectric slab in the PB model, and use an improved mesh construction method to automatically identify the membrane channel/pore region even with a tilt angle relative to the z-axis; and (3) add a non-polar solvation energy term to complete the estimation of the total solvation energy of a membrane protein. A mesh resolution of about 0.25 Ã… (cubic grid space)/0.36 Ã… (tetrahedron edge length) is found to be most accurate in linear finite element calculation of the PB solvation energy. Computational studies are performed on a few exemplary molecules. The results indicate that all factors, the membrane thickness, the length of periodic box, membrane dielectric constant, pore region dielectric constant, and ionic strength, have individually considerable influence on the solvation energy of a channel protein. This demonstrates the necessity to treat all of those effects in the PB model for membrane protein simulations
Between Algorithm and Model: Different Molecular Surface Definitions for the Poisson-Boltzmann based Electrostatic Characterization of Biomolecules in Solution
The definition of a molecular surface which is physically sound and computationally efficient is a very interesting and long standing problem in the implicit solvent continuum modeling of biomolecular systems as well as in the molecular graphics field. In this work, two molecular surfaces are evaluated with respect to their suitability for electrostatic computation as alternatives to the widely used Connolly-Richards surface: the blobby surface, an implicit Gaussian atom centered surface, and the skin surface. As figures of merit, we considered surface differentiability and surface area continuity with respect to atom positions, and the agreement with explicit solvent simulations. Geometric analysis seems to privilege the skin to the blobby surface, and points to an unexpected relationship between the non connectedness of the surface, caused by interstices in the solute volume, and the surface area dependence on atomic centers. In order to assess the ability to reproduce explicit solvent results, specific software tools have been developed to enable the use of the skin surface in Poisson-Boltzmann calculations with the DelPhi solver. Results indicate that the skin and Connolly surfaces have a comparable performance from this last point of view
Proto-Plasm: parallel language for adaptive and scalable modelling of biosystems
This paper discusses the design goals and the first developments of
Proto-Plasm, a novel computational environment to produce libraries
of executable, combinable and customizable computer models of natural and
synthetic biosystems, aiming to provide a supporting framework for predictive
understanding of structure and behaviour through multiscale geometric modelling
and multiphysics simulations. Admittedly, the Proto-Plasm platform is
still in its infancy. Its computational framework—language, model library,
integrated development environment and parallel engine—intends to provide
patient-specific computational modelling and simulation of organs and biosystem,
exploiting novel functionalities resulting from the symbolic combination of
parametrized models of parts at various scales. Proto-Plasm may define
the model equations, but it is currently focused on the symbolic description of
model geometry and on the parallel support of simulations. Conversely, CellML
and SBML could be viewed as defining the behavioural functions (the model
equations) to be used within a Proto-Plasm program. Here we exemplify
the basic functionalities of Proto-Plasm, by constructing a schematic
heart model. We also discuss multiscale issues with reference to the geometric
and physical modelling of neuromuscular junctions
Large-scale parallelised boundary element method electrostatics for biomolecular simulation
Large-scale biomolecular simulations require a model of particle interactions capable of incorporating
the behaviour of large numbers of particles over relatively long timescales. If
water is modelled as a continuous medium then the most important intermolecular forces
between biomolecules can be modelled as long-range electrostatics governed by the Poisson-
Boltzmann Equation (PBE).
We present a linearised PBE solver called the "Boundary Element Electrostatics Program"(BEEP). BEEP is based on the Boundary Element Method (BEM), in combination
with a recently developed O(N) Fast Multipole Method (FMM) algorithm which approximates
the far-�field integrals within the BEM, yielding a method which scales linearly with
the number of particles. BEEP improves on existing methods by parallelising the underlying
algorithms for use on modern cluster architectures, as well as taking advantage of recent
progress in the �field of GPGPU (General Purpose GPU) Programming, to exploit the highly
parallel nature of graphics cards.
We found the stability and numerical accuracy of the BEM/FMM method to be highly
dependent on the choice of surface representation and integration method. For real proteins
we demonstrate the critical level of surface detail required to produce converged electrostatic
solvation energies, and introduce a curved surface representation based on Point-Normal
G1-continuous triangles which we �find generally improves numerical stability compared to a
simpler surface constructed from planar triangles. Despite our improvements upon existing
BEM methods, we �find that it is not possible to directly integrate BEM surface solutions
to obtain intermolecular electrostatic forces. It is, however, practicable to use the total
electrostatic solvation energy calculated by BEEP to drive a Monte-Carlo simulation
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