Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3.
Department of Applied Mathematics, Feng Chia University, Taichung 40724, Taiwan and National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan 10617.
Phys Rev E. 2019 Aug;100(2-1):022406. doi: 10.1103/PhysRevE.100.022406.
Ion channels regulate the flux of ions through cell membranes and play significant roles in many physiological functions. Most of the existing literature focuses on computational approaches based on molecular dynamics simulation or numerical solution of the modified Poisson-Nernst-Planck (PNP) system. In this paper, we present an analytical and computational study of a mathematical model of the KcsA potassium channel, including the effects of ion size (Bikerman model) and solvation energy (Born model). Under equilibrium conditions, we obtain an analytical solution of our modified PNP system, which is used to explain selectivity of KcsA of various ions (K^{+}, Na^{+}, Cl^{-}, Ca^{2+}, and Ba^{2+}) due to negative permanent charges inside the filter region and the effect of ion sizes. Our results show that K^{+} is always selected over Na^{+}, as smaller Na^{+} ions have larger solvation energy. As the amount of negative charges in the filter exceeds a critical value, divalent ions (Ca^{2+} and Ba^{2+}) can enter the filter region and block the KcsA channel. For the nonequilibrium cases, due to difficulties associated with a pure analytical or numerical approach, we use a hybrid analytical-numerical method to solve the modified PNP system. Our predictions of selectivity of KcsA channels and saturation phenomenon of the current-voltage (I-V) curve agree with experimental observations.
离子通道调节离子通过细胞膜的通量，在许多生理功能中发挥着重要作用。大多数现有的文献都集中在基于分子动力学模拟或修正后的泊松-纳斯特-普朗克（PNP）系统数值解的计算方法上。在本文中，我们对 KcsA 钾通道的数学模型进行了分析和计算研究，包括离子大小（Bikerman 模型）和溶剂化能（Born 模型）的影响。在平衡条件下，我们得到了修正后的 PNP 系统的解析解，该解用于解释由于过滤区域内的负永久电荷和离子大小的影响，KcsA 对各种离子（K+、Na+、Cl-、Ca2+和 Ba2+）的选择性。我们的结果表明，K+总是优先于 Na+，因为较小的 Na+离子具有较大的溶剂化能。当过滤区域内的负电荷数量超过一个临界值时，二价离子（Ca2+和 Ba2+）可以进入过滤区域并阻断 KcsA 通道。对于非平衡情况，由于纯解析或数值方法的困难，我们使用混合解析-数值方法来求解修正后的 PNP 系统。我们对 KcsA 通道选择性和电流-电压（I-V）曲线饱和现象的预测与实验观察结果一致。
