<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$V_{rev}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mrow> <mi mathvariant="italic">rev</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> responds to channel structural parameters and ion transport properties. We derive explicit formulas and conditions for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$V_{rev}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mrow> <mi mathvariant="italic">rev</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V_{rev}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mrow> <mi mathvariant="italic">rev</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$V_{rev}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mrow> <mi mathvariant="italic">rev</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> 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.</p>

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

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发表日期：

2025年11月21日

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Yiwei Wang,
Jianwei Shen,
Lijun Zhang,
Mingji Zhang

摘要

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 $V_{rev}$ $V_{rev}$ responds to channel structural parameters and ion transport properties. We derive explicit formulas and conditions for $V_{rev}$ $V_{rev}$ 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 $V_{rev}$ $V_{rev}$ 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 $V_{rev}$ $V_{rev}$ 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.