Zheng Qiong, Chen Duan, Wei Guo-Wei
Department of Mathematics, Michigan State University, MI 48824, USA.
J Comput Phys. 2011 Jun;230(13):5239-5262. doi: 10.1016/j.jcp.2011.03.020.
The Poisson Nernst-Planck (PNP) theory is a simplified continuum model for a wide variety of chemical, physical and biological applications. Its ability of providing quantitative explanation and increasingly qualitative predictions of experimental measurements has earned itself much recognition in the research community. Numerous computational algorithms have been constructed for the solution of the PNP equations. However, in the realistic ion-channel context, no second order convergent PNP algorithm has ever been reported in the literature, due to many numerical obstacles, including discontinuous coefficients, singular charges, geometric singularities, and nonlinear couplings. The present work introduces a number of numerical algorithms to overcome the abovementioned numerical challenges and constructs the first second-order convergent PNP solver in the ion-channel context. First, a Dirichlet to Neumann mapping (DNM) algorithm is designed to alleviate the charge singularity due to the protein structure. Additionally, the matched interface and boundary (MIB) method is reformulated for solving the PNP equations. The MIB method systematically enforces the interface jump conditions and achieves the second order accuracy in the presence of complex geometry and geometric singularities of molecular surfaces. Moreover, two iterative schemes are utilized to deal with the coupled nonlinear equations. Furthermore, extensive and rigorous numerical validations are carried out over a number of geometries, including a sphere, two proteins and an ion channel, to examine the numerical accuracy and convergence order of the present numerical algorithms. Finally, application is considered to a real transmembrane protein, the Gramicidin A channel protein. The performance of the proposed numerical techniques is tested against a number of factors, including mesh sizes, diffusion coefficient profiles, iterative schemes, ion concentrations, and applied voltages. Numerical predictions are compared with experimental measurements.
泊松能斯特 - 普朗克（PNP）理论是一种适用于多种化学、物理和生物应用的简化连续介质模型。它能够对实验测量结果进行定量解释并做出越来越多的定性预测，这使其在研究领域获得了广泛认可。为求解PNP方程已构建了许多计算算法。然而，在实际的离子通道环境中，由于存在许多数值障碍，包括不连续系数、奇异电荷、几何奇点和非线性耦合，文献中从未报道过二阶收敛的PNP算法。本工作引入了一些数值算法来克服上述数值挑战，并在离子通道环境中构建了首个二阶收敛的PNP求解器。首先，设计了一种狄利克雷到诺伊曼映射（DNM）算法来缓解由于蛋白质结构导致的电荷奇点。此外，对匹配界面和边界（MIB）方法进行了重新表述以求解PNP方程。MIB方法系统地强制执行界面跳跃条件，并在存在复杂几何形状和分子表面几何奇点的情况下实现二阶精度。此外，利用两种迭代方案来处理耦合的非线性方程。此外，针对包括球体、两种蛋白质和一个离子通道在内的多种几何形状进行了广泛而严格的数值验证，以检验本数值算法的数值精度和收敛阶数。最后，将其应用于一种实际的跨膜蛋白，即短杆菌肽A通道蛋白。针对包括网格尺寸、扩散系数分布、迭代方案、离子浓度和外加电压等多种因素，测试了所提出数值技术的性能。将数值预测结果与实验测量结果进行了比较。
