Computing Electrostatic Potential For An Ion-channel Model

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.