Is the Goldman-Hodgkin-Katz equation universally true?

No. Eighty years ago, the two seminal works by Goldman [J. Gen. Phys. 27, 37 (1943)] and by Hodgkin-Katz [J. of Physio 108, 37 (1949)] derived the foundational framework for interpreting electro-physiological measurements in what is commonly termed the Goldman-Hodgkin-Katz (GHK) theory for the membrane potential. Both seminal papers postulate a constant/uniform electric field within the ion channel. Using a uniform electric field allows for a simple, straightforward calculation of the ionic fluxes and the transmembrane potential, which yields the famous GHK potential. The use of this framework is so widely accepted that one can find a plethora of works that no longer cite the original works and GHK has perhaps become the universal and indisputable descriptor of the underlying physics and biology. In recent works [Phys. Rev. Lett. 134, 228401 (2025) and Phys. Rev. E 111, 064408 (2025)], we revisited GHK and its assumption of a uniform field. Non-approximated numerical simulations showed that the electric field is not always uniform. To understand this discrepancy, it is important to understand that the governing equations can be solved using two different approaches: the GHK approach of assuming a constant electric field or postulating that the system is electroneutral. Each approach yields drastically different non-commutative results. The purpose of this report is to provide a non-mathematical summary of the results and to inform the broader community that GHK is not as universal as previously thought. We will discuss these two newer works, with some emphasis on how they can potentially revolutionize the interpretation of electro-physiological measurements. Importantly, we show that this new framework, which utilizes the mathematical tools developed by the electrodialysis community, also serves as a bridge linking the electrophysiology community and the electrodialysis community.

💡 Research Summary

The paper revisits the long‑standing Goldman‑Hodgkin‑Katz (GHK) equation, which has been the cornerstone of electrophysiology for nearly a century. The authors emphasize that the original derivations by Goldman (1943) and Hodgkin‑Katz (1949) rest on two critical assumptions: (1) the electric field inside the membrane or ion channel is uniform (i.e., the potential varies linearly across the channel) and (2) the system does not enforce local electroneutrality. By inserting the uniform‑field assumption into the Nernst‑Planck equations, the classic GHK voltage expression (Eq. 1) is obtained, linking membrane potential to the ratio of extracellular and intracellular ion concentrations.

Recent work by the same authors (Phys. Rev. Lett. 134, 228401 (2025); Phys. Rev. E 111, 064408 (2025)) performed fully resolved Poisson‑Nernst‑Planck (PNP) simulations using COMSOL. They imposed asymmetric bulk concentrations (c_left ≠ c_right) and a zero‑current condition (i = 0). The numerical results reveal a logarithmic potential profile rather than the linear profile predicted by GHK, indicating that the electric field is not constant but varies spatially. This discrepancy directly challenges the uniform‑field hypothesis.

The authors argue that the root cause lies in how the Poisson equation is treated. In the GHK approach, the influence of space charge on the electric field is neglected, effectively decoupling the field from the charge distribution. Consequently, the resulting space‑charge density is non‑zero, violating both local and global electroneutrality. In contrast, the alternative approach adopted here enforces local electroneutrality (the electrolyte is neutral at every point). This condition yields a relationship between the concentrations of the two ionic species and any fixed surface charge density Σ_s. The Poisson equation then reduces to a simple expression for the potential, which, even when Σ_s = 0, leads to a logarithmic voltage law consistent with the numerical simulations.

Importantly, the paper highlights that Σ_s is not merely a mathematical artifact but a physical parameter representing surface charge on the channel walls. In classical GHK theory Σ_s is assumed to be zero, effectively treating the channel as uncharged. In electrodialysis (ED) and reverse electrodialysis (RED) literature, Σ_s is non‑zero and is essential for generating a net voltage or harvestable current. By allowing Σ_s to take arbitrary values, the electroneutral model bridges electrophysiology with the well‑established RED framework, offering a unified description of ion‑selective transport across both biological and synthetic membranes.

The authors also critique the common practice of treating diffusion coefficients as fitting parameters within GHK. While diffusion coefficients are bulk properties, assuming they remain unchanged inside a confined, charged channel is questionable. The electroneutral model avoids this ambiguity by using bulk diffusion coefficients directly and letting the fixed charge term account for deviations in transport behavior.

In summary, the paper demonstrates that the GHK equation is not a universal solution of the PNP system; it is contingent on the uniform‑field assumption, which fails under realistic conditions where space charge and surface charge are present. The electroneutral approach, grounded in local charge neutrality, yields a distinct, physically consistent voltage–concentration relationship that aligns with high‑fidelity numerical simulations and with the theory of RED. This new framework calls for a re‑examination of past electrophysiological data, suggests revisions to textbook presentations of membrane potential, and opens pathways for more accurate modeling of ion channels, drug interactions, and bio‑inspired energy harvesting devices.