离子通道模型的静电势计算
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摘要
离子通道（Ion-Channel）是生物体在生命活动中表现出的各种电现象的基础，它与细胞分化，肌肉运动，神经递质释放等多项生命活动密切相关，并且对于维持细胞内环境稳定有着重要的作用。离子通道研究在探索和理解生物现象及其发生机制方面有着重要意义。当前对离子通道的研究已经形成了一套初步的理论体系，相应的实验技术和数值模拟方法也在逐步地完善。分子动力学模拟作为研究离子通道的一类有效数值模拟手段，不仅可以给出被模拟体系的结构和热力学性质，还可以提供原子运动中的各种微观细节以供分析研究，是对理论和实验的有力补充。静电相互作用的计算效率和精度是影响分子动力学模拟实用性的主要因素，本文主要研究混合溶剂模型下，离子通道中静电相互作用的计算方法。
离子通道的混合溶剂模型选取包含通道蛋白的有限长的圆柱，圆柱内部的溶质和溶液分子采用具体的原子表示，圆柱外部的细胞膜和离子溶液分别看成均一的连续介质，其对圆柱内部原子的影响由Poisson-Boltzmann理论下的极化反应场给出。本文的第一部分工作给出了离子通道混合溶剂模型中静电势求解的一类新型边界积分方法。在离子通道的混合溶剂模型中，由于有限长圆柱表面的特殊性（法向导数具有不连续性），直接推导和给出有限长圆柱表面上的边界积分方程及其具有较高计算精度的计算方法是一件困难的事情。为了克服这一困难，本文引入与圆柱光滑相接的虚拟半球（形成一个“胶囊体”），并在胶囊体内外利用不同的分层格林函数精确地处理体系的边界条件，推导给出了原问题在胶囊体表面上的新型边界积分方程。基于该新型边界积分方程及其在胶囊体表面的贴体曲边三角形网格上的有限元离散，本文给出了离子通道混合溶剂模型中静电势计算的边界积分方法，它避开了有限长圆柱表面角点（法向导数不连续的点）带来的计算困难和误差。数值算例检验了该边界积分方法的精度和有效性。本文的第二部分工作给出了分层介质构型和离子通道混合溶剂模型的镜像电荷方法。在分层介质构型下，本文运用傅立叶变换导出了体系静电势的解析表达式，并在此基础上给出静电势计算的镜像电荷方法。在推导镜像电荷公式时，本文分别运用了最小二乘和Prony拟合方法确定镜像电荷信息（电量和位置）与源电荷（电量和位置）间的关系式。在离子通道的混合溶剂模型中，由于体系结构（有限长圆柱）的复杂性，不仅体系静电势的解析表达式难以给出，而且有限长圆柱构型下的镜像电荷（电量和位置）也很难确定。为此，本文将有限长圆柱近似地看成分层介质和无限长圆柱的耦合，把原问题转化成分层介质和无限长圆柱两种构型下的镜像电荷问题后分别处理，再综合给出了离子通道混合溶剂模型的镜像电荷公式。数值算例及分析表明，本文提出的镜像电荷方法也具有好的数值精度。由于镜像电荷方法给出的静电势写成一些点电荷的库伦势之和的形式，因而在模拟大规模体系时，它可以和快速多极子法相结合以加速计算。边界积分方法和镜像电荷方法在静电势的计算效率和精度上具有各自的优越性，实际应用中可根据模拟体系对计算的要求适当选取。
Ionic channels are protein molecules that conduct ions through a narrow tunnel offixed charge formed by the amino acid residues of the protein. Ion channels are respon-sible for many important functions like signaling in the nervous system, coordination ofmuscle contraction, transport in all tissues and so on, corresponding research has signif-icant meaning in exploring and understanding biological phenomenon and underlyingmechanism. Research on ion channel has developed a primary theory system, dazzlingexperimental techniques and effective numerical simulation methods. Molecular dy-namics simulation is a widely used numerical technique for ion channel research, as astrong compensation for theory and experiment, it is able to give the system structureinformation, thermodynamic properties and detailed information of atom’s movementtrace. Currently, the efficiency and accuracy of the computation of electrostatic poten-tial is a key factor in the utility of molecular dynamics simulation.
In this paper, we will investigate the numerical accuracy and efficiency in com-puting the electrostatic potential for hybrid solvation model of ion channel, in which afinite-height cylinder including the channel proteins is embedded in a layered dielectricmedium representing the biological membrane and ionic solvents. Inside the cylindercavity, the ion channel protein and ions are given as explicit atomistic representations,while outside the cylinder cavity, the surrounding mediums are considered as continu-ous dielectric. Two numerical techniques, a specially designed boundary integral equa-tion method and image charge method, will be investigated and compared in terms ofaccuracy and efficiency for computing the electrostatic potential.
The main issue of applying boundary integral equation method to the hybrid sol-vation model of ion channel arises from the corner/edge singularities of the finite-height cylinder surface. It is because the solution of boundary integral equation onfinite-height cylinder surface will be introduced serious error due to the corner/edgesingularities. In order to avoid such difficulty, an artificially constructed smooth sur-face round-top cylinder, which is formed by two virtual semi-spheres attaching to thefinite-height cylinder, and the layered Green’s functions, which are used to deal the boundary/interface condition exactly, are introduced to deduce a new boundary integralequation hold on the smooth round-top cylinder surface. Base on the new boundaryintegral equation and its body-fitted curved-line triangular finite element discretization,the proposed new boundary integral equation method effectively avoid the error pollu-tion from the geometry irregularities in computing the electrostatic potential for hybridion channel model. The numerical examples support the result with high efficiency andaccuracy.
Image charge method is a semi-analytical method, which calculates the electro-static potential by approximating the analytical form of the electrostatic potential asa sum of point charges’ Coulomb potential. While for the hybrid solvation model ofion channel, due to the complexity of system configuration (the finite-height cylinder),the analytical form of electrostatic potential is hard to be obtained. To solve this is-sue, we simplify the problem by decomposing the finite-height cylinder configurationinto a layered structure and an infinite-height cylinder, and then consider the imagecharge method under these two configurations separately. For the layered structure, theFourier expansion is introduced to deduce an analytical form of the electrostatic poten-tial, in addition the least square method and the Prony fitting method are used to fixthe relation formula of image charges’ information and the source charges’ informa-tion. Combining the image charge method for layered structure and the image chargemethod for infinite-height cylinder developed in [75], we propose the image chargemethod for hybrid solvation model, it is found to give reasonable accuracy meanwhilebeing highly efficient and viable for using the fast multipole method for interactions ofa large number of charges in the atomistic region of the hybrid solvation model.
The proposed boundary integral equation method and image charge method forhybrid solvation model have their own advantages in terms of accuracy and efficiency,thus can be selected for molecular dynamic simulation according to specific computingrequirement.
离子通道的混合溶剂模型选取包含通道蛋白的有限长的圆柱，圆柱内部的溶质和溶液分子采用具体的原子表示，圆柱外部的细胞膜和离子溶液分别看成均一的连续介质，其对圆柱内部原子的影响由Poisson-Boltzmann理论下的极化反应场给出。本文的第一部分工作给出了离子通道混合溶剂模型中静电势求解的一类新型边界积分方法。在离子通道的混合溶剂模型中，由于有限长圆柱表面的特殊性（法向导数具有不连续性），直接推导和给出有限长圆柱表面上的边界积分方程及其具有较高计算精度的计算方法是一件困难的事情。为了克服这一困难，本文引入与圆柱光滑相接的虚拟半球（形成一个“胶囊体”），并在胶囊体内外利用不同的分层格林函数精确地处理体系的边界条件，推导给出了原问题在胶囊体表面上的新型边界积分方程。基于该新型边界积分方程及其在胶囊体表面的贴体曲边三角形网格上的有限元离散，本文给出了离子通道混合溶剂模型中静电势计算的边界积分方法，它避开了有限长圆柱表面角点（法向导数不连续的点）带来的计算困难和误差。数值算例检验了该边界积分方法的精度和有效性。本文的第二部分工作给出了分层介质构型和离子通道混合溶剂模型的镜像电荷方法。在分层介质构型下，本文运用傅立叶变换导出了体系静电势的解析表达式，并在此基础上给出静电势计算的镜像电荷方法。在推导镜像电荷公式时，本文分别运用了最小二乘和Prony拟合方法确定镜像电荷信息（电量和位置）与源电荷（电量和位置）间的关系式。在离子通道的混合溶剂模型中，由于体系结构（有限长圆柱）的复杂性，不仅体系静电势的解析表达式难以给出，而且有限长圆柱构型下的镜像电荷（电量和位置）也很难确定。为此，本文将有限长圆柱近似地看成分层介质和无限长圆柱的耦合，把原问题转化成分层介质和无限长圆柱两种构型下的镜像电荷问题后分别处理，再综合给出了离子通道混合溶剂模型的镜像电荷公式。数值算例及分析表明，本文提出的镜像电荷方法也具有好的数值精度。由于镜像电荷方法给出的静电势写成一些点电荷的库伦势之和的形式，因而在模拟大规模体系时，它可以和快速多极子法相结合以加速计算。边界积分方法和镜像电荷方法在静电势的计算效率和精度上具有各自的优越性，实际应用中可根据模拟体系对计算的要求适当选取。
Ionic channels are protein molecules that conduct ions through a narrow tunnel offixed charge formed by the amino acid residues of the protein. Ion channels are respon-sible for many important functions like signaling in the nervous system, coordination ofmuscle contraction, transport in all tissues and so on, corresponding research has signif-icant meaning in exploring and understanding biological phenomenon and underlyingmechanism. Research on ion channel has developed a primary theory system, dazzlingexperimental techniques and effective numerical simulation methods. Molecular dy-namics simulation is a widely used numerical technique for ion channel research, as astrong compensation for theory and experiment, it is able to give the system structureinformation, thermodynamic properties and detailed information of atom’s movementtrace. Currently, the efficiency and accuracy of the computation of electrostatic poten-tial is a key factor in the utility of molecular dynamics simulation.
In this paper, we will investigate the numerical accuracy and efficiency in com-puting the electrostatic potential for hybrid solvation model of ion channel, in which afinite-height cylinder including the channel proteins is embedded in a layered dielectricmedium representing the biological membrane and ionic solvents. Inside the cylindercavity, the ion channel protein and ions are given as explicit atomistic representations,while outside the cylinder cavity, the surrounding mediums are considered as continu-ous dielectric. Two numerical techniques, a specially designed boundary integral equa-tion method and image charge method, will be investigated and compared in terms ofaccuracy and efficiency for computing the electrostatic potential.
The main issue of applying boundary integral equation method to the hybrid sol-vation model of ion channel arises from the corner/edge singularities of the finite-height cylinder surface. It is because the solution of boundary integral equation onfinite-height cylinder surface will be introduced serious error due to the corner/edgesingularities. In order to avoid such difficulty, an artificially constructed smooth sur-face round-top cylinder, which is formed by two virtual semi-spheres attaching to thefinite-height cylinder, and the layered Green’s functions, which are used to deal the boundary/interface condition exactly, are introduced to deduce a new boundary integralequation hold on the smooth round-top cylinder surface. Base on the new boundaryintegral equation and its body-fitted curved-line triangular finite element discretization,the proposed new boundary integral equation method effectively avoid the error pollu-tion from the geometry irregularities in computing the electrostatic potential for hybridion channel model. The numerical examples support the result with high efficiency andaccuracy.
Image charge method is a semi-analytical method, which calculates the electro-static potential by approximating the analytical form of the electrostatic potential asa sum of point charges’ Coulomb potential. While for the hybrid solvation model ofion channel, due to the complexity of system configuration (the finite-height cylinder),the analytical form of electrostatic potential is hard to be obtained. To solve this is-sue, we simplify the problem by decomposing the finite-height cylinder configurationinto a layered structure and an infinite-height cylinder, and then consider the imagecharge method under these two configurations separately. For the layered structure, theFourier expansion is introduced to deduce an analytical form of the electrostatic poten-tial, in addition the least square method and the Prony fitting method are used to fixthe relation formula of image charges’ information and the source charges’ informa-tion. Combining the image charge method for layered structure and the image chargemethod for infinite-height cylinder developed in [75], we propose the image chargemethod for hybrid solvation model, it is found to give reasonable accuracy meanwhilebeing highly efficient and viable for using the fast multipole method for interactions ofa large number of charges in the atomistic region of the hybrid solvation model.
The proposed boundary integral equation method and image charge method forhybrid solvation model have their own advantages in terms of accuracy and efficiency,thus can be selected for molecular dynamic simulation according to specific computingrequirement.
引文
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