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Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations

100%
Xie Y., Cheng J., Lu B., Zhang L.
Molecular Based Mathematical Biology
|
2013
|
tom 1
90-108
EN
rithms for solving the 3D electro-diffusion equations such as the Poisson-Nernst-Planck equations and the size-modified Poisson-Nernst-Planck equations in simulations of biomolecular systems in ionic liquid. A set of transformation methods based on the generalized Slotboom variables is used to solve the coupled equations. Calculations of the diffusion-reaction rate coefficients, electrostatic potential and ion concentrations for various systems verify the method’s validity and stability. The iterations between the Poisson equation and the Nernst- Planck equations in the primitive method and in the transformation method are compared to illustrate how the new method accelerates the convergence of the solution. To speed up the convergence, we introduce the DIIS (direct inversion of the iterative subspace) method including Simple Mixing and Anderson Mixing as under-relaxation techniques, the effectiveness of which on acceleration is shown by numerical tests. It is worth noting that the primitive method fails to solve the size-modified Poisson-Nernst-Planck equations for real protein systems but the transformation method succeeds in the simulations of the ACh-AChE reaction system and the DNA fragment. To improve the accuracy of the solution, we introduce high order elements and mesh adaptation based on an a posteriori error estimator. Numerical results indicate that our mesh adaptation process leads to quasi-optimal convergence. We implement our algorithms using the parallel adaptive finite element package PHG [53] and high parallel efficiency is obtained.
2
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Membrane-Channel Protein System Mesh Construction for Finite Element Simulations

88%
Liu T., Bai S., Tu B., Chen M., Lu B.
Molecular Based Mathematical Biology
|
2015
|
tom 3
|
nr 1
EN
We present a method of constructing the volume meshes of the membrane-channel protein system for finite element simulation of ion channels. The membrane channel system consists of the solvent region and the membrane-protein region. Our method focuses on labeling the tetrahedra in the solvent and membrane-protein regions and collecting the interface triangles between different regions. It contains two stages. Firstly, a volume mesh conforming the surface of the channel protein is generated by the surface and volume mesh generation tools: TMSmesh and TetGen. Then a walk-and-detect algorithm is used to identify the pore region to embed the membrane correctly. This method is shown to be robust because of its independence of the pore structure of the ion channels. In addition, we can also get the information of whether the ion channel is open or closed by the walk-and-detect algorithm. An on-line meshing procedure will be available at our website www.continuummodel.org.
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