Goldman-Hodgkin-Katz Equation Research Articles

Addressing Challenges in Ion-Selectivity Characterization in Nanopores.

Ion selectivity is the basis for designing smart nanopore/channel-based devices, e.g., ion separators and biosensors. Quantitative characterization of ion selectivities in nanopores often employs the Nernst or Goldman-Hodgkin-Katz (GHK) equation to interpret transmembrane potentials. However, the direction of the measured transmembrane potential drop is not specified in these equations, and selectivity values calculated using absolute values of transmembrane potentials do not directly reveal the ion for which the membrane is selective. Moreover, researchers arbitrarily choose whether to use the Nernst or GHK equation and overlook the significant differences between them, leading to ineffective quantitative comparisons between studies. This work addresses these challenges through (a) specifying the transmembrane potential (sign) and salt concentrations in terms of working and reference electrodes and the solutions in which they reside when using the Nernst and GHK equations, (b) reporting of both Nernst-selectivity and GHK-selectivity along with solution compositions and transmembrane potentials when comparing different nanopores/channels, and (c) performing simulations to define an ideal selectivity for nanochannels. Experimental and modeling studies provide significant insight into these fundamental equations and guidelines for the development of nanopore/channel-based devices.

Lee's "Transmembrane Electrostatically-Localized Proton" model does NOT offer a better understanding of neuronal transmembrane potentials.

Recently, in a paper entitled "Protonic conductor: Better understanding [sic] neural resting and action potential," Lee applied his Transmembrane Electrostatically-Localized Protons (TELP) hypothesis to neuronal signaling. He stated that Hodgkin's cable theory "could not fully explain the different conductive patterns in unmyelinated and myelinated nerves," whereas his TELP hypothesis "enables much better understanding of neural resting/action potential and [the biological significance of] axon myelination…" However, Lee's TELP hypothesis predicts that under resting conditions, the neuron "accumulat[es] excess negative charge (anions) inside," whereas resting chloride gradients actually feature excess Cl- outside of the cell. Experiments on the neuron have shown that raising external [K+] and decreasing external [Cl-] cause membrane potential depolarization, which is predicted by the Goldman equation, but opposite to TELP hypothesis predictions. Finally, based on his TELP hypothesis, Lee predicted that the main purpose of myelin is to insulate the axonal plasma membrane specifically against proton permeability. However, he cited literature showing that myelin contains proteins that may "serve as a proton conductor with the localized protons." Thus, we show here that Lee's TELP hypothesis is highly problematic, and does NOT offer a "better understanding" of neuronal transmembrane potentials.NEW & NOTEWORTHY In this manuscript I critique a 2020 J. Neurophysiol. paper by James W. Lee. His TELP hypothesis 1) mispredicts the resting neuron's excess of external chloride; 2) predicts the preponderance of surface H+ over Na+ using ΔG° rather than ΔG; 3) mispredicts the dependence of the neuronal resting potential on external [Na+], [K+], and [Cl-]; 4) neither cites experimental results nor proposes experiments to test his hypothesis; and 5) presents a problematic characterization of the purpose of myelin.

Capturing membrane structure and function in lattice Boltzmann models.

We develop a mesoscopic approach to model the nonequilibrium behavior of membranes at the cellular scale. Relying on lattice Boltzmann methods, we develop a solution procedure to recover the Nernst-Planck equationsand Gauss's law. A general closure rule is developed to describe mass transport across the membrane, which is able to account for protein-mediated diffusion based on a coarse-grained representation. We demonstrate that our model is able to recover the Goldman equationfrom first principles and show that hyperpolarization occurs when membrane charging dynamics are controlled by multiple relaxation timescales. The approach provides a promising way to characterize non-equilibrium behaviors that arise due to the role of membranes in mediating transport based on realistic three-dimensional cell geometries.

Changes in Astroglial K+ upon Brief Periods of Energy Deprivation in the Mouse Neocortex.

Malfunction of astrocytic K+ regulation contributes to the breakdown of extracellular K+ homeostasis during ischemia and spreading depolarization events. Studying astroglial K+ changes is, however, hampered by a lack of suitable techniques. Here, we combined results from fluorescence imaging, ion-selective microelectrodes, and patch-clamp recordings in murine neocortical slices with the calculation of astrocytic [K+]. Brief chemical ischemia caused a reversible ATP reduction and a transient depolarization of astrocytes. Moreover, astrocytic [Na+] increased by 24 mM and extracellular [Na+] decreased. Extracellular [K+] increased, followed by an undershoot during recovery. Feeding these data into the Goldman–Hodgkin–Katz equation revealed a baseline astroglial [K+] of 146 mM, an initial K+ loss by 43 mM upon chemical ischemia, and a transient K+ overshoot of 16 mM during recovery. It also disclosed a biphasic mismatch in astrocytic Na+/K+ balance, which was initially ameliorated, but later aggravated by accompanying changes in pH and bicarbonate, respectively. Altogether, our study predicts a loss of K+ from astrocytes upon chemical ischemia followed by a net gain. The overshooting K+ uptake will promote low extracellular K+ during recovery, likely exerting a neuroprotective effect. The resulting late cation/anion imbalance requires additional efflux of cations and/or influx of anions, the latter eventually driving delayed astrocyte swelling.

Recent Bioelectricity-Related Articles Selected by Ann M. Rajnicek, Media Editor of Bioelectricity

Recent Bioelectricity-Related Articles Selected by Ann M. Rajnicek, Media Editor of <i>Bioelectricity</i>

Gender differences in cell volume fraction (CVF): a structural parameter reflecting the energy efficiency of maintaining the resting membrane potential.

The cell volume fraction (CVF) of the human brain is high (~82%) and is preserved across healthy aging while the brain declines in volume. These two observations, supported by several independent techniques, suggest that CVF is an important structural parameter. A new biophysical model is presented that incorporates CVF into the Goldman equation of classical membrane electrophysiology. The Goldman equation contains few structural constraints beyond two compartments separated by a semipermeable membrane supporting ion gradients. As potassium is the most permeable ion in the resting state, the resting membrane potential is determined by the potassium ion gradient. This biophysical model indicates that the sodium-potassium ion pumps use less energy at high CVF to maintain the resting membrane potential, explaining the high value of CVF and its conservation with healthy aging. CVF is measured to be statistically significantly higher in the brains of males compared with females, suggesting a structural requirement for higher energy efficiency in the larger male brain to support the greater number of neurons and synapses. As CVF can be measured in humans using quantitative sodium MRI and has potential implications for brain health, CVF may be a quantitative parameter that is useful for assessment of brain health, especially in patients with diseases such as dementia and psychiatric disease that do not have anatomical correlates detectable by clinical proton MRI.

The Ionic Selectivity of Lysenin Channels in Open and Sub-Conducting States.

The electrochemical gradients established across cell membranes are paramount for the execution of biological functions. Besides ion channels, other transporters, such as exogenous pore-forming toxins, may present ionic selectivity upon reconstitution in natural and artificial lipid membranes and contribute to the electrochemical gradients. In this context, we utilized electrophysiology approaches to assess the ionic selectivity of the pore-forming toxin lysenin reconstituted in planar bilayer lipid membranes. The membrane voltages were determined from the reversal potentials recorded upon channel exposure to asymmetrical ionic conditions, and the permeability ratios were calculated from the fit with the Goldman–Hodgkin–Katz equation. Our work shows that lysenin channels are ion-selective and the determined permeability coefficients are cation and anion-species dependent. We also exploited the unique property of lysenin channels to transition to a stable sub-conducting state upon exposure to calcium ions and assessed their subsequent change in ionic selectivity. The observed loss of selectivity was implemented in an electrical model describing the dependency of reversal potentials on calcium concentration. In conclusion, our work demonstrates that this pore-forming toxin presents ionic selectivity but this is adjusted by the particular conduction state of the channels.

Proton Quantum Tunneling: Influence and Relevance to Acidosis-Induced Cardiac Arrhythmias/Cardiac Arrest.

Acidosis and its associated pathologies predispose patients to develop cardiac arrhythmias and even cardiac arrest. These arrhythmias are assumed to be the result of membrane depolarization, however, the exact mechanism of depolarization during acidosis is not well defined. In our study, the model of quantum tunneling of protons is used to explain the membrane depolarization that occurs during acidosis. It is found that protons can tunnel through closed activation and inactivation gates of voltage-gated sodium channels Nav1.5 that are present in the membrane of cardiac cells. The quantum tunneling of protons results in quantum conductance, which is evaluated to assess its effect on membrane potential. The quantum conductance of extracellular protons is higher than that of intracellular protons. This predicts an inward quantum current of protons through the closed sodium channels. Additionally, the values of quantum conductance are influential and can depolarize the membrane potential according to the quantum version of the GHK equation. The quantum mechanism of depolarization is distinct from other mechanisms because the quantum model suggests that protons can directly depolarize the membrane potential, and not only through indirect effects as proposed by other mechanisms in the literature. Understanding the pathophysiology of arrhythmias mediated by depolarization during acidosis is crucial to treat and control them and to improve the overall clinical outcomes of patients.

Effectiveness of an active learning strategy using a simple interactive computer-based application in teaching the concept of membrane potential to medical undergraduates

Background: Teaching resting membrane potential (RMP) to medical students is challenging. Various teaching-learning methods have been used earlier by medical teachers to make this concept easier for students to learn. Aim and Objective: This study aims to evaluate the effectiveness of an active learning session using computer-based application in improving the understanding of the concept of generation of RMP, in medical undergraduates. Materials and Methods: The 1st year medical undergraduates (n = 109) attending physiology classes in ESIC Medical College took part in the session. The students were given scenarios with different concentration of ions and permeabilities and were asked to derive the RMP with the help of a computer-based application. The application calculates the membrane potential using modified Goldman equation when the concentrations and relative permeabilities of ions are given as input using a graphical user interface. The students’ understanding of the concept was assessed before and after the session by pre- and post-test questionnaire. Difference in pre- and post-test scores was analyzed using paired t-test. Feedback was obtained after the session. Results: There was a significant increase in the post-test scores when compared to the pre-test scores (P < 0.001). More than 90% of students who took part in the session felt that the session had helped them in understanding the concept well, and they wished to have more such active learning sessions in the future. Conclusion: The study results suggest that the active learning session on generation of membrane potential using a computer-based application, improved the students’ understanding on the concept, and it was well appreciated by the students.

Conservation Law of Cell Bioelectricity Membrane Area and Ion Inequality Equation Based on Potassium Channel “Origami Windmill” Model

At present, there are misunderstandings and even subversive errors in the understanding of cellular bioelectricity phenomena. The theory explaining the mechanism of bioelectricity generation is the ionic theory, and the ionic theory is based on the membrane theory. The membrane theory may be the correct part, and the ionic theory does not give a definite affirmation; the membrane theory may be the wrong part, and ionic theory does not give a clear negative, instead, has been further strengthened in the later period. Whether it is membrane theory or ionic theory, the author has ignored the structure and characteristics of ion channels, and has not found the essence and law of ion exchange inside and outside the cell membrane. Therefore, the GHK equation and the H-H equation based on the ionic theory have congenital deficiencies. Therefore, it is necessary to put forward a new doctrine on this basis and establish a new mathematical model based on the new doctrine to re-explain the mechanism of bioelectricity generation scientifically and rationally. This paper is based on the principle of K+ channel “origami windmill” model, the conservation law of membrane area of cell action potential ion exchange was proposed, and establishes a new mathematical model of cell action potential based on the conservation law of membrane area—ion inequality equation. The expression of conservation law of membrane area: SA (t) + SB (t) = S0, the expression of the ion inequality equation: resting potential N = N0 + Vt; action potential N = N0+ 170t -90t2. The conservation law of membrane area holds that bioelectrical activity is a process in which different cations replace each other on the inner surface of the cell membrane. Cl- is already surrounded by the cell membrane at the beginning of cell membrane formation, which determines the “positive” and “negative” of the cell action potential; The cations such as Na+, K+, etc., the amount of mutual replacement on the inner surface of the cell membrane determines the amplitude of the action potential, and the ratio of mutual replacement complies with the principle that the membrane area is equal and the ions are not equal. The ion inequality equation, cited the kinematics principle, can not only reflect the nature and rule of cation exchange on the inner surface of cell membrane,but also can reasonable avoid cell membrane expansion force and ionic driving force, completed the bioelectricity generation mechanism from the qualitative to the quantitative expression, supports the preexistence of membrane theory, to verify the theory of brain cell activation, the theory of dove-likeparticles and “origami windmill” model, has opened up a scientific and quantitative description cell action potential new ways and means.

Transmembrane potential of physiologically relevant model membranes: Effects of membrane asymmetry.

Transmembrane potential difference (Vm) plays important roles in regulating various biological processes. At the macro level, Vm can be experimentally measured or calculated using the Nernst or Goldman-Hodgkin-Katz equation. However, the atomic details responsible for its generation and impact on protein and lipid dynamics still need to be further elucidated. In this work, we performed a series of all-atom molecular dynamics (MD) simulations of symmetric model membranes of various lipid compositions and cation contents to evaluate the relationship between membrane asymmetry and Vm. Specifically, we studied the impact of the asymmetric distribution of POPS (1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-l-serine), PIP2 (phosphatidylinositol 4,5-bisphosphate), as well as Na+ and K+ on Vm using atomically detailed MD simulations of symmetric model membranes. The results suggest that, for an asymmetric POPC-POPC/POPS bilayer in the presence of NaCl, the presence of the monovalent anionic lipid POPS in the inner leaflet polarizes the membrane (ΔVm < 0). Intriguingly, replacing a third of the POPS lipids by the polyvalent anionic signaling lipid PIP2 counteracts this effect, resulting in a smaller negative membrane potential. We also found that replacing Na+ ions in the inner region by K+ depolarizes the membrane (ΔVm > 0). These divergent effects arise from variations in the strength of cation-lipid interactions and are correlated with changes in lipid chain order and head-group orientation.

Enlightening Point of View Based on Potassium Channel “Origami Windmill” Model

Applying the K+ channel origami windmill model principle, the whole process of action potential generation of nerve fiber cells and cardiomyocytes is explained reasonably. Its core view is: cell action potential decline phase, dominated by influx of potassium ions. This is contrary to the core idea of the traditional theory—ionic theory. According to ionic theory, cell action potential decline phase, dominated by outflow of K+ ions. In the face of such two opposing views, the author traces back to the source, combing and commenting on the basic theoretical research on the production mechanism of cell bioelectricity generation and the results of classical basic experiments. And a series of enlightening viewpoints are produced：1. Ionic theory has its preexistence deficiency; 2. There are principle defects in GHK equation and H-H equation; 3. Ion channels may “same direction sharing”; 4. Sodium-potassium pumps, calcium pumps and so on May not exist. The thesis's Point is falsifiable. If “cell action potential decline stage, dominated by outflow of K+ ions” can be proved, all the points in this paper will not be true. On the contrary, the series of enlightening views in this paper will inevitably lead to a revolution in the field of life science. The viewpoint of this paper is not only closely related to the causes of human major diseases, treatment principles and methods, but also may have a profound impact on the future research direction of life science researchers. So the author suggests that scholars in the field of life science should re sort out the existing bioelectricity knowledge system—eliminate the false, save the true and clear the source, especially focusing on the papers published by Hodgkin and Huxley in 1952; At the same time, it is necessary to reevaluate the scientificity and scientific value of membrane theory, ionic theory, GHK equation and H-H equation, and put forward new theories based on Bernstein membrane theory, and establishing a new mathematical model of cell action potential.

Protonic conductor: better understanding neural resting and action potential.

With the employment of the transmembrane electrostatic proton localization theory with a new membrane potential equation, neural resting and action potential is now much better understood as the voltage contributed by the localized protons/cations at a neural liquid- membrane interface. Accordingly, the neural resting/action potential is essentially a protonic/cationic membrane capacitor behavior. It is now understood with a newly formulated action potential equation: when action potential is <0 (negative number), the localized protons/cations charge density at the liquid-membrane interface along the periplasmic side is >0 (positive number); when the action potential is >0, the concentration of the localized protons and localized nonproton cations is <0, indicating a "depolarization" state. The nonlinear curve of the localized protons/cations charge density in the real-time domain of an action potential spike appears as an inverse mirror image to the action potential. The newly formulated action potential equation provides biophysical insights for neuron electrophysiology, which may represent a complementary development to the classic Goldman-Hodgkin-Katz equation. With the use of the action potential equation, the biological significance of axon myelination is now also elucidated as to provide protonic insulation and prevent any ions both inside and outside of the neuron from interfering with the action potential signal, so that the action potential can quickly propagate along the axon with minimal (e.g., 40 times less) energy requirement.NEW & NOTEWORTHY The newly formulated action potential equation provides biophysical insights for neuron electrophysiology, which may represent a complementary development to the classic Goldman-Hodgkin-Katz equation. The nonlinear curve of the localized protons/cations charge density in the real-time domain of an action potential spike appears as an inverse mirror image to the action potential. The biological significance of axon myelination is now elucidated as to provide protonic insulation and prevent any ions from interfering with action potential signal.

Quantum Electrochemical Equilibrium: Quantum Version of the Goldman–Hodgkin–Katz Equation

The resting membrane voltage of excitable cells such as neurons and muscle cells is determined by the electrochemical equilibrium of potassium and sodium ions. This voltage is calculated by using the Goldman–Hodgkin–Katz equation. However, from the quantum perspective, ions with significant quantum tunneling through closed channels can interfere with the electrochemical equilibrium and affect the value of the membrane voltage. Hence, in this case the equilibrium becomes quantum electrochemical. Therefore, the model of quantum tunneling of ions is used in this study to modify the Goldman–Hodgkin–Katz equation in such a way to calculate the resting membrane voltage at the point of equilibrium. According to the present calculations, it is found that lithium—with its lower mass—shows a significant depolarizing shift in membrane voltage. In addition to this, when the free gating energy of the closed channels decreases, even sodium and potassium ions depolarize the resting membrane voltage via quantum tunneling. This study proposes the concept of quantum electrochemical equilibrium, at which the electrical potential gradient, the concentration gradient and the quantum gradient (due to quantum tunneling) are balanced. Additionally, this concept may be used to solve many issues and problems in which the quantum behavior becomes more influential.

Magnesium Ions Depolarize the Neuronal Membrane via Quantum Tunneling through the Closed Channels

Magnesium ions have many cellular actions including the suppression of the excitability of neurons; however, the depolarization effect of magnesium ions seems to be contradictory. Thus several hypotheses have aimed to explain this effect. In this study, a quantum mechanical approach is used to explain the depolarization action of magnesium. The model of quantum tunneling of magnesium ions through the closed sodium voltage-gated channels was adopted to calculate the quantum conductance of magnesium ions, and a modified version of Goldman–Hodgkin–Katz equation was used to determine whether this quantum conductance was significant in affecting the resting membrane potential of neurons. Accordingly, it was found that extracellular magnesium ions can exhibit a depolarization effect on membrane potential, and the degree of this depolarization depends on the tunneling probability, the channels’ selectivity to magnesium ions, the channels’ density in the neuronal membrane, and the extracellular magnesium concentration. In addition, extracellular magnesium ions achieve a quantum conductance much higher than intracellular ones because they have a higher kinetic energy. This study aims to identify the mechanism of the depolarization action of magnesium because this may help in offering better therapeutic solutions for fetal neuroprotection and in stabilizing the mood of bipolar patients.

Tunable Anion-Selective Transport through Monolayer Graphene and Hexagonal Boron Nitride.

Membranes that selectively filter for both anions and cations are central to technological applications from clean energy generation to desalination devices. 2D materials have immense potential as these ion-selective membranes due to their thinness, mechanical strength, and tunable surface chemistry; however, currently, only cation-selective membranes have been reported. Here we demonstrate the controllable cation and anion selectivity of both monolayer graphene and hexagonal boron nitride. In particular, we measure the ionic current through membranes grown by chemical vapor deposition containing well-known defects inherent to scalably produced and wet-transferred 2D materials. We observe a striking change from cation selectivity with monovalent ions to anion selectivity by controlling the concentration of multivalent ions and inducing charge inversion on the 2D membrane. Furthermore, we find good agreement between our experimental data and theoretical predictions from the Goldman–Hodgkin–Katz equation and use this model to extract selectivity ratios. These tunable selective membranes conduct up to 500 anions for each cation and thus show potential for osmotic power generation.

Mathematical expression of membrane potential based on Ling's adsorption theory is approximately the same as the Goldman-Hodgkin-Katz equation.

The Goldman-Hodgkin-Katz equation (GHK equation), one of the most successful achievements of membrane theory in electrophysiology, can precisely predict the membrane potential. Its conceptual foundation lies in the idea that the transmembrane ion transport across the plasma membrane is responsible for the membrane potential generation. However, the potential virtually equivalent to the membrane potential is generated even across the impermeable membrane. In this work, I discus the membrane potential generation mechanism and find that the potential formula based on the long-dismissed Ling's adsorption theory, which attributes the membrane potential generation to the mobile ion adsorption rather than the transmembrane ion transport, is the same as the GHK equation. Based on this finding, I derive a conclusion that the membrane potential is generated by the ion adsorption against the existing electrophysiological concept.

Another interpretation of the Goldman-Hodgkin-Katz equation based on Ling's adsorption theory.

According to standard membrane theory, the generation of membrane potential is attributed to transmembrane ion transport. However, there have been a number of reports of membrane behavior in conflict with the membrane theory of cellular potential. Putting aside the membrane theory, we scrutinized the generation mechanism of membrane potential from the view of the long-dismissed adsorption theory of Ling. Ling's adsorption theory attributes the membrane potential generation to mobile ion adsorption. Although Ling's adsorption theory conflicts with the broadly accepted membrane theory, we found that it well reproduces experimentally observed membrane potential behavior. Our theoretical analysis finds that the potential formula based on the GHK eq., which is a fundamental concept of membrane theory, coincides with the potential formula based on Ling's adsorption theory. Reinterpreting the permeability coefficient in the GHK eq. as the association constant between the mobile ion and adsorption site, the GHK eq. turns into the potential formula from Ling's adsorption theory. We conclude that the membrane potential is generated by ion adsorption as Ling's adsorption theory states and that the membrane theory of cellular potential should be amended even if not discarded.

Estimating outflow facility through pressure dependent pathways of the human eye.

We develop and test a new theory for pressure dependent outflow from the eye. The theory comprises three main parameters: (i) a constant hydraulic conductivity, (ii) an exponential decay constant and (iii) a no-flow intraocular pressure, from which the total pressure dependent outflow, average outflow facilities and local outflow facilities for the whole eye may be evaluated. We use a new notation to specify precisely the meaning of model parameters and so model outputs. Drawing on a range of published data, we apply the theory to animal eyes, enucleated eyes and in vivo human eyes, and demonstrate how to evaluate model parameters. It is shown that the theory can fit high quality experimental data remarkably well. The new theory predicts that outflow facilities and total pressure dependent outflow for the whole eye are more than twice as large as estimates based on the Goldman equation and fluorometric analysis of anterior aqueous outflow. It appears likely that this discrepancy can be largely explained by pseudofacility and aqueous flow through the retinal pigmented epithelium, while any residual discrepancy may be due to pathological processes in aged eyes. The model predicts that if the hydraulic conductivity is too small, or the exponential decay constant is too large, then intraocular eye pressure may become unstable when subjected to normal circadian changes in aqueous production. The model also predicts relationships between variables that may be helpful when planning future experiments, and the model generates many novel testable hypotheses. With additional research, the analysis described here may find application in the differential diagnosis, prognosis and monitoring of glaucoma.

The role of H+/OH− channels in saline pathology of Chara australis: brief history

The pH banding pattern, observed in Chara and Nitella early last century, suggests that there are H+ or OH− conducting channels in the alkaline bands of the characean plasma membrane. If the cells are exposed to high pH medium, pHo, these channels become the dominant membrane conductance: High pH state. Goldman-Hodgkin-Katz (GHK) equation can model the current-voltage (I/V) characteristics of this state with either H+ or OH− as permeant ion. However, H+ requires unrealistically large increases in channel conductance/number parameter as pHo increases. Recently we discovered a typical spiky noise in the membrane potential difference (PD) of salt sensitive Chara australis, appearing immediately upon exposure to saline. The H+/OH− channels are thought to open transiently in small patches all over the cell, while negative membrane PD does not support their open state/s. As the membrane PD depolarizes due to inhibition of the proton pump by salinity, the H+/OH− channels remain open to dominate the I/V characteristics, which can also be modeled by GHK equation. The addition of 1 mM ZnCl2, the main known blocker of animal H+ channels, abolishes both the PD noise and the H+/OH− I/V characteristics in high pH state and in salt stressed cells. Confocal microscopy and fluorescein isothiocyanate (FITC)-dextran illuminate the pH changes around cells and reveal transient bright spots early in saline exposure, becoming more numerous and fixed in space with time in saline APW.