Studies on Reversal Potential Via Classical Poisson-Nernst-Planck Systems. Part I: Multiple Pieces of Nonzero Permanent Charges having the Same Sign

• Yiwei Wang ^[1] ; Jianwei Shen ^[4] ; Lijun Zhang ^[2] ; Mingji Zhang ^[3]
  1. [1] Shandong University of Science and Technology

     Shandong University of Science and Technology

     China

     
  2. [2] Zhejiang University of Science and Technology

     Zhejiang University of Science and Technology

     China

     
  3. [3] New Mexico Institute of Mining and Technology

     New Mexico Institute of Mining and Technology

     Estados Unidos

     
  4. [4] North China University of Water Resources and Electric Power

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 6, 2025
• Idioma: inglés
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• Resumen

  • We investigate ionic flows through a membrane channel under zero-current conditions via the classical Poisson-Nernst-Planck (PNP) model. The channel contains two distinct regions of nonzero permanent charge of the same sign, separated by uncharged regions. Reversal potential−the transmembrane voltage at which net ionic current vanishes−is a fundamental characteristic of ion channels, and it depends on both the channel’s fixed charge distribution and the ions’ diffusion properties. However, previous analyses were limited to simpler charge configurations (single charged segment or equal diffusion coefficients). We employ a geometric singular perturbation approach to reduce the nonlinear PNP boundary value problem to a pair of algebraic governing equations under zero-current conditions. This allows a rigorous analytical study of how the reversal potential Vrev responds to channel structural parameters and ion transport properties. We derive explicit formulas and conditions for Vrev in the presence of two same-sign permanent charge segments. In particular, we prove that a unique reversal potential exists for any given set of permanent charges and diffusion coefficients. Our analysis reveals that the reversal potential is generally nonzero with unequal diffusion coefficients, indicating that an electric field is required to counterbalance mobility differences. Moreover, we characterize how Vrev varies with the magnitude of the fixed charges and with the relative size of the two charged segments. We find monotonic trends−for example, increasing the charge density in the larger segment or increasing the disparity in diffusion coefficients shifts Vrev in a predictable manner (positive or negative), depending on which side of the channel carries the greater fixed charge.

    Analytical asymptotic expansions are also obtained for limiting cases (such as small permanent charge or small diffusion coefficient ratio), providing further insight. These results deepen the theoretical understanding of how multiple same-sign fixed charge regions jointly determine the reversal potential. Our findings highlight the nontrivial interplay between channel charge distribution and ion mobility, which is essential for explaining and predicting ion selective behaviors in biological and synthetic channels.