Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA.
Math Biosci Eng. 2020 May 22;17(4):3736-3766. doi: 10.3934/mbe.2020210.
We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flows through a membrane channel with three ion species, two positively charged with the same valence and one negatively charged. Bikerman's local hard-sphere potential is included in the model to account for ion sizes. The problem is treated as a boundary value problem of a singularly perturbed differential system. Under the framework of a geometric singular perturbation theory, together with specific structures of this concrete model, the existence and uniqueness of solutions to the boundary value problem for small ion sizes is established. Furthermore, treating the ion sizes as small parameters, we derive an approximation of individual fluxes, from which one can further study the qualitative properties of ionic flows and extract concrete information directly related to biological measurements. Of particular interest is the competition between two cations due to the nonlinear interplay between finite ion sizes, diffusion coefficients and boundary conditions, which is closely related to selectivity phenomena of open ion channels with given protein structures. Furthermore, we are able to characterize the distinct effects of the nonlinear interplays between these physical parameters. Numerical simulations are performed to identify some critical potentials which play critical roles in examining properties of ionic flows in our analysis.
我们研究了一种拟一维稳态泊松-纳斯特-普朗克型模型，用于描述带三种离子的膜通道中的离子流，其中两种带同种电荷且价态相同，一种带负电荷。模型中包含了 Bikerman 的局部硬球势，以考虑离子大小。该问题被处理为奇异摄动微分系统的边值问题。在几何奇异摄动理论的框架下，结合具体模型的结构，我们建立了小离子尺寸边值问题解的存在性和唯一性。此外，将离子尺寸视为小参数，我们得到了单个通量的近似解，从中可以进一步研究离子流的定性性质，并提取与生物测量直接相关的具体信息。特别感兴趣的是由于有限离子尺寸、扩散系数和边界条件之间的非线性相互作用，两种阳离子之间的竞争，这与具有给定蛋白质结构的开放离子通道的选择性现象密切相关。此外，我们能够描述这些物理参数之间的非线性相互作用的显著影响。进行了数值模拟以确定一些关键电位，这些电位在我们的分析中对检查离子流性质起着关键作用。
