Ion channels are proteins with a narrow hole down their middle that control a wide range of biological function by controlling the flow of spherical ions from one macroscopic region to another. Ion channels do not change their conformation on the biological time scale once they are open, so they can be described by a combination of Poisson and drift-diffusion (Nernst–Planck) equations called PNP in biophysics. We use singular perturbation techniques to analyse the steady-state PNP system for a channel with a general geometry and a piecewise constant permanent charge profile. We construct an outer solution for the case of a constant permanent charge density in three dimensions that is also a valid solution of the one-dimensional system. The asymptotical current–voltage (I–V) characteristic curve of the device (obtained by the singular perturbation analysis) is shown to be a very good approximation of the numericalI–Vcurve (obtained by solving the system numerically). The physical constraint of non-negative concentrations implies a unique solution, i.e., for each given applied potential there corresponds a unique electric current (relaxing this constraint yields non-physical multiple solutions for sufficiently large voltages).

中文翻译：

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稳态 Poisson-Nernst-Planck 系统的奇异扰动分析：在离子通道中的应用

离子通道是中间有一个窄孔的蛋白质，通过控制球形离子从一个宏观区域到另一个宏观区域的流动来控制广泛的生物功能。离子通道一旦打开就不会在生物时间尺度上改变其构象，因此它们可以通过生物物理学中称为 PNP 的泊松和漂移扩散 (能斯特-普朗克) 方程的组合来描述。我们使用奇异扰动技术来分析具有一般几何形状和分段恒定永久电荷分布的通道的稳态 PNP 系统。我们为三个维度上恒定永久电荷密度的情况构造了一个外部解，这也是一维系统的有效解。渐近电流 - 电压（一世–五) 器件的特性曲线（通过奇异扰动分析获得）被证明是数值的非常好的近似一世–五曲线（通过数值求解系统获得）。非负浓度的物理约束意味着唯一的解决方案，即，对于每个给定的施加电势，都有一个唯一的电流（放宽该约束会产生足够大电压的非物理多重解决方案）。