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The accumulation of potassium in the extracellular space surrounding nerve cells is a fundamental aspect of biophysics that has garnered significant attention in recent research. This phenomenon holds implications for various neurological conditions, including spreading depression, migraine, certain types of epilepsy, and potentially, learning processes. A quantitative analysis is essential for understanding the dynamics of potassium clearance following a series of action potentials. This clearance process involves multiple structures along the nerve, including glia, the extracellular space, axons, and the perivascular space, necessitating a spatially distributed systems approach akin to the cable equations of nerve physiology. In this study, we propose a multi-domain model for the optic nerve to investigate potassium accumulation and clearance dynamics. The model accounts for the convection, diffusion, and electrical migration of fluid and ions, revealing the significant roles of glia and the perivascular space in potassium buffering. Specifically, our findings suggest that potassium clearance primarily occurs through convective flow within the syncytia of glia, driven by osmotic pressure differences. Additionally, the perivascular space serves as a crucial pathway for potassium buffering and fluid circulation, further contributing to the overall clearance process. Importantly, our model's adaptability allows for its application to diverse structures with distinct channel and transporter distributions across the six compartments, extending its utility beyond the optic nerve.

It is certain that electrical properties-whether slow (sec) or fast (nsec), even optical (fsec)-are described by Maxwell's equations, and there are terms that depend on the rate of change of the electric and magnetic fields. In particular, Maxwell's equation for the curl of the magnetic field contains both the steady current and a term depending upon the temporal derivative of the electric displacement field. The latter is referred to as displacement current, and is generally believed to have been included originally by Maxwell himself, although there is evidence it was earlier considered by Kirchhoff. Maxwell's equations and Kirchoff's circuit laws both are important over the wide range of frequencies with which electronics traditionally deals. And, displacement current is an important contribution to these in both classical and quantum mechanics. Here, the development of displacement current, its importance in both classical and quantum mechanics, and some applications are provided to illustrate the fundamental role that it plays in the dynamics of a wide range of systems.

Maxwell defined a true or total current in a way not widely used today. He said "... true electric current ... is not the same thing as the current of conduction but that the time-variation of the electric displacement must be taken into account in estimating the total movement of electricity". We show that true or total current is a universal property of electrodynamics independent of properties of matter. We use mathematics without a dielectric constant. The resulting Maxwell Current Law is a generalization of the Kirchhoff Law of Current used in circuit analysis, that also includes displacement current. The generalization is not a long-time low frequency approximation in contrast to traditional presentation of Kirchhoff's Law. The Maxwell Current Law does not require currents to be in circuits. It has been applied to three dimensional systems like the signaling system of nerve and muscle fibers. The Maxwell Current Law clarifies the flow of electrons, protons, and ions in mitochondria that generate ATP, the molecule that stores chemical energy throughout life. The currents are globally coupled because mitochondria are short. Focusing on Maxwell current reinterprets the classical chemiosmotic hypothesis of ATP production. The conduction current of protons in mitochondria is driven by the protonmotive force including its component electrical potential, just as in the classical chemiosmotic hypothesis. The electrical potential is now the electrical potential as defined in physical sciences by Maxwell partial differential equations. The conduction current is now just a part of the true current analyzed by Maxwell. Details of accumulation of charges do not have to be considered in analysis of true current because true current does not accumulate. It is true total current that provides the energy that generates the ATP, not just the protonmotive force., Comment: Version 4 with typos corrected and further discussion of stray capacitances and chemiosmotic hypothesis

Chemical reactions involve the movement of charges, and this work presents a mathematical model for describing chemical reactions in electrolytes. The model is developed using an energy variational method that aligns with classical thermodynamics principles. It encompasses both electrostatics and chemical reactions within consistently defined energetic and dissipative functionals. Furthermore, the energy variation method is extended to account for open systems that involve the input and output of charge and mass. Such open systems have the capability to convert one form of input energy into another form of output energy. In particular, a two-domain model is developed to study a reaction system with self-regulation and internal switching, which plays a vital role in the electron transport chain of mitochondria responsible for ATP generation crucial process for sustaining life. Simulations are conducted to explore the influence of electric potential on reaction rates and switching dynamics within the two-domain system. It shows that the electric potential inhibits the oxidation reaction while accelerating the reduction reaction., Comment: 25 pages

We applied coherence analysis to molecular dynamics simulations of the plant protein crambin, a thionin storage protein found in Abyssinian cabbage. Coherence analysis was developed by engineers to identify linear interactions, without statistical assumptions. Coherence is greater than 0.9 between the displacement of oxygen and nitrogen atoms of H bonds in alpha helices for frequencies between 0.391 GHz and 5.08 GHz (corresponding reciprocally to times of 2.56 ns and 0.197 ns). These H bonds act much like a linear system. Unrelated atoms have uncorrelated motions and much smaller coherence, say 0.02. Groups of atoms (that form a layer of an alpha helix) were averaged and the coherence function of two groups was evaluated. Layers of the alpha helix form a linear system, suggesting that the harmonic analysis of classical molecular dynamics can successfully describe the allosteric interactions of the layers of an alpha helix., Comment: Version accepted by the journal ACS Omega, including Supplemental Information. PubMed Central (for the NIH) PMC10116620

Kirchhoff's Current Law is an essential tool in the design of circuits that operate very quickly, faster than nanoseconds. But Kirchhoff's current is often identified as the flow of particles. The continuity equation or the Maxwell-Ampere law shows that the sum of displacement current $\textbf{plus}$ particle current is conserved by Maxwell's equations and Kirchhoff's law. Kirchoff included the displacement current in the current of his law, from early on. This Kirchhoff current (including the displacement current) does not vary with spatial location in the ionic channels of biology. Electronic circuits switching in nanoseconds are analyzed using the Bohm representation of quantum mechanics including particle and displacement current., Comment: This work has been submitted to the IEEE for possible publication

A mathematical model for chemical reactions in electrolytes is developed using an Energy variational method consistent with classical thermodynamics. Electrostatics and chemical reactions are included in properly defined energetic and dissipative functionals. The energy variation method is generalized to deal with open systems with inputs of charge, mass, and energy. The open systems can transform input energy of one type into output energy of another type. The generalized method is used to analyze the conversion of electrical current into proton flow by cytochrome c oxidase, an important enzyme in mitochondria that helps generate the 'energetic currency of life' ATP. The structure of the oxidase guides flows of current and mass that interact according to the energetic and dissipative functionals of the generalized theory. Kirchhoff's current law provides important coupling when the enzyme is in its natural setting in the mitochondrial membrane. The natural function of the enzyme is the result. Electron flows are converted into proton flows and gradients., Comment: 45 pages, 14 figures

Voltage-gated Kv channels play fundamental roles in many biological processes, such as the generation of the action potential. The gating mechanism of Kv channels is characterized experimentally by single-channel recordings and ensemble properties of the channel currents. In this work, we propose a bubble model coupled with a Poisson-Nernst-Planck (PNP) system to capture the key characteristics, particularly the delay in the opening of channels. The coupled PNP system is solved numerically by a finite-difference method and the solution is compared with an analytical approximation. We hypothesize that the stochastic behaviour of the gating phenomenon is due to randomness of the bubble and channel sizes. The predicted ensemble average of the currents under various applied voltages across the channels is consistent with experimental observations, and the Cole-Moore delay is captured by varying the holding potential.

Membranes are present in all cells and tissues. Mathematical models of cells and tissues need a compact mathematical description of membranes with a resolution of about 1 nm. Membranes isolate cells because ions have difficulty penetrating the dielectric barrier they create. Here we introduce a dielectric mathematical membrane condition to replace a condition that did not include dielectric properties. Our mathematical membrane condition includes a dielectric lipid bilayer punctured by channels that conduct ions selectively., Comment: 5 pages,1 figure

Complex fluids flow in complex ways in complex structures. Transport of water and various organic and inorganic molecules in the central nervous system are important in a wide range of biological and medical processes [C. Nicholson, and S. Hrab\v{e}tov\'a, Biophysical Journal, 113(10), 2133(2017)]. However, the exact driving mechanisms are often not known. In this paper, we investigate flows induced by action potentials in an optic nerve as a prototype of the central nervous system (CNS). Different from traditional fluid dynamics problems, flows in biological tissues such as the CNS are coupled with ion transport. It is driven by osmosis created by concentration gradient of ionic solutions, which in term influence the transport of ions. Our mathematical model is based on the known structural and biophysical properties of the experimental system used by the Harvard group Orkand et al [R.K. Orkand, J.G. Nicholls, S.W. Kuffler, Journal of Neurophysiology, 29(4), 788(1966)]. Asymptotic analysis and numerical computation show the significant role of water in convective ion transport. The full model (including water) and the electrodiffusion model (excluding water) are compared in detail to reveal an interesting interplay between water and ion transport. In the full model, convection due to water flow dominates inside the glial domain. This water flow in the glia contributes significantly to the spatial buffering of potassium in the extracellular space. Convection in the extracellular domain does not contribute significantly to spatial buffering. Electrodiffusion is the dominant mechanism for flows confined to the extracellular domain., Comment: 61 Pages, 16 figures

The accumulation of potassium in the narrow space outside nerve cells is a classical subject of biophysics that has received much attention recently. It may be involved in potassium accumulation \textcolor{black}{including} spreading depression, perhaps migraine and some kinds of epilepsy, even (speculatively) learning. Quantitative analysis is likely to help evaluate the role of potassium clearance from the extracellular space after a train of action potentials. Clearance involves three structures that extend down the length of the nerve: glia, extracellular space, and axon and so need to be described as systems distributed in space in the tradition used for electrical potential in the `cable equations' of nerve since the work of Hodgkin in 1937. A three-compartment model is proposed here for the optic nerve and is used to study the accumulation of potassium and its clearance. The model allows the convection, diffusion, and electrical migration of water and ions. We depend on the data of Orkand et al to ensure the relevance of our model and align its parameters with the anatomy and properties of membranes, channels, and transporters: our model fits their experimental data quite well. The aligned model shows that glia has an important role in buffering potassium, as expected. The model shows that potassium is cleared mostly by convective flow through the syncytia of glia driven by osmotic pressure differences. A simplified model might be possible, but it must involve flow down the length of the optic nerve. It is easy for compartment models to neglect this flow. Our model can be used for structures quite different from the optic nerve that might have different distributions of channels and transporters in its three compartments. It can be generalized to include a fourth (distributed) compartment representing blood vessels to deal with the glymphatic flow into the circulatory system., Comment: 35 pages, 13 figures

Electrodynamics is usually written with a polarization vector field to describe the response of matter to electric fields, or more specifically, to describe changes in distribution of charge as an electric field is changed. This approach does not allow unique specification of a polarization field from measurements of electric and magnetic fields. Many polarization fields produce the same electric and magnetic fields, because only the divergence of the polarization enters Maxwell's first equation, relating charge and electric field. The curl of any function can be added to a polarization field without changing the electric field at all. The divergence of the curl is always zero. Models of structures that produce polarization cannot be uniquely determined from electrical measurements for the same reason. Models must describe charge distribution not just distribution of polarization to be unique. I propose a different paradigm to describe field dependent charge, i.e., to describe the phenomena of polarization. I propose an operational definition of polarization that has worked well in biophysics where a field dependent, time dependent polarization provides the gating current that makes neuronal sodium and potassium channels respond to voltage. The operational definition has been applied successfully to experiments for nearly fifty years. Estimates of polarization have been computed from simulations, models, and theories using this definition and they fit experimental data quite well. I propose that the same operational definition be used to define polarization charge in experiments, models, computations, theories, and simulations of other systems. Charge movement needs to be computed from a combination of electrodynamics and mechanics because 'everything interacts with everything else'. The classical polarization field need not enter into that treatment at all., Comment: Typos corrected

Ions in channels have been imagined as hard balls in a macroscopic mechanical model, for a very long time. Hard balls interact by collisions in such models, randomly knocking each other on and off `binding' sites in thermal motion. But ions have large charge, and the hard balls of classical models do not. The electrodynamics of charge guarantee strong correlations between the movements of ions on all time scales, even those of atomic scale thermal motion. Correlations are present whenever Maxwell's equations apply, so they are present in individual trajectories, not just averages. Indeed, in a series system like an idealized narrow channel, the correlation is perfect (within the accuracy of Maxwell's equations) because of conservation of total current (that includes Maxwell's displacement current, the ethereal ${\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t}$).The ethereal component of current prevents spatial variation of total current in a series system. The stochastic complexity of spatial thermal motion disappears for current }in a series system. Total current does not depend on location in a series system like a narrow ion channel on any time scale. Spatial variables are not needed in the description of total current in a one dimensional channel on any time scale, according to the Maxwell equations. Removing the spatial dependence of total current should dramatically simplify theories and simulations, one might imagine., Comment: Typos corrected, unfortunate wording improved, and now a comparison with x-independent channel properties in semiconductors. Version 3, typos and infelicities corrected

Mikhail Shapiro's lab investigated a promising new method for non-invasive control of calcium currents in individual cells in the nervous system by the selective heating of nanoparticles and show that simple physical laws, properly applied, explain what is happening, and so can be a foundation for constructing improved methods and techniques. They use custom built instrumentation and detailed quantitative measurement to show that special surface properties are not needed to explain their results. Their work is taken as an example of the need for quantitative measurement in biophysics and biology in general. Classical examples are cited and the argument is made that the success of structural and molecular biology has hidden the need for quantitative measurements and controls. Semiconductor and computational electronics is cited as a science even more successful than structural and molecular biology that depends on quantitative measurement, controls, and analysis., Comment: Biophysical Journal (2020)

Although ligand-binding sites in many proteins contain a high number density of charged side chains that can polarize small organic molecules and influence binding, the magnitude of this effect has not been studied in many systems. Here, we use a quantum mechanics/molecular mechanics (QM/MM) approach in which the ligand is the QM region to compute the ligand polarization energy of 286 protein-ligand complexes from the PDBBind Core Set (release 2016). We observe that the ligand polarization energy is linearly correlated with the magnitude of the electric field acting on the ligand, the magnitude of the induced dipole moment, and the classical polarization energy. The influence of protein and cation charges on the ligand polarization diminishes with the distance and is below 2 kcal/mol at 9 $\unicode{x212B}$ and 1 kcal/mol at 12 $\unicode{x212B}$. Considering both polarization and solvation appears essential to computing negative binding energies in some crystallographic complexes. Solvation, but not polarization, is essential for achieving moderate correlation with experimental binding free energies.

Kirchhoff's current law is thought to describe the translational movement of charged particles through resistors. But Kirchhoff's law is widely used to describe movements of current through resistors in high speed devices. Current at high frequencies/short times involves much more than the translation of particles. Transients abound. Augmentation of the resistors with ad hoc 'stray' capacitances is often used to introduce transients into models like those in real resistors. But augmentation hides the underlying problem, rather than solves it: the location, value and dielectric properties of the stray capacitances are not well determined. Here, we suggest a more general approach, that is well determined. If current is redefined as in Maxwell's equations, independent of the properties of dielectrics, Kirchhoff's law is exact and transients arise automatically without ambiguity. The transients in a particular real circuit-a high density integrated circuit for example-can then be described by measured constitutive equations together with Maxwell's equations without the introduction of arbitrary circuit elements., Comment: Version 4, December 5, 2022. Updated to cite new work. Version 3: Expanded treatment of continuity equation

Maxwell's equations describe the relation of charge and electric force almost perfectly even though electrons and permanent charge were not in his equations, as he wrote them. For Maxwell, all charge depended on electric field. Charge was induced and polarization was described by a single dielectric constant. Electrons, permanent charge, and polarization are important when matter is involved. Polarization of matter cannot be described by a single dielectric constant ${\varepsilon }_{\mathrm{r}}$ with reasonable realism today. Only vacuum is well described by a single dielectric constant ${\varepsilon }_{\mathrm{0}}$. Here, Maxwell's equations are rewritten to include permanent charge and any type of polarization. Rewriting is in one sense petty, and in another sense profound, in either case presumptuous. Rewriting confirms the legitimacy of electrodynamics. One cannot be sure ahead of time that a theory of electrodynamics without electrons or permanent charge (like Maxwell's equations as he wrote them) would be legitimate or not. After all, a theory cannot calculate the fields produced by charges (for example, electrons) that are not in the theory at all! After updating: (1) Maxwell's equations seem universal and exact. (2) Polarization must be described explicitly to use Maxwell`s equations in applications. (3) Conservation of total current (including displacement current ${\varepsilon }_{\mathrm{0\\ }}\frac{\mathrm{\partial }\boldsymbol{\mathrm{E}}}{\mathrm{\partial }t}$) becomes exact, independent of matter, allowing precise definition of electromotive force EMF in circuits.(4) Kirchhoff's current law becomes as exact as Maxwell`s equations themselves. (5) Conservation of total current needs to be satisfied in a wide variety of systems where it has not traditionally received much attention. (6) Classical chemical kinetics is seen to need revision to conserve current., Comment: Version 2 & 3 have typos corrected, minor revisions in text and references. Versions 4-6 have some new ideas. Version 7 has an important typo corrected on p. 13, after eq 1

A dielectric dilemma faces scientists because Maxwell's equations are poor approximations as usually written, with a single dielectric constant. Maxwell's equations are then not accurate enough to be useful in many applications involving ionic solutions and even solids. The dilemma can be partially resolved by a rederivation of conservation of current, where current is defined now to include the 'polarization of the vacuum' $\epsilon_0 \frac{\partial E}{\partial t}$. Conserveration of current becomes Kirchoff's current law with this definition, in the one dimensional circuits of our electronic technology. With this definition, Kirchoff's laws are valid whenever Maxwell's equations are valid, explaining why those laws are able to describe circuits that switch in nanoseconds., Comment: Typos corrected. Cryptic comments decoded

In traditional hydrodynamic theories for ionic fluids, conservation of the mass and linear momentum is not properly taken care of. In this paper, we develop hydrodynamic theories for a viscous, ionic fluid of $N$ ionic species enforcing mass and momentum conservation as well as considering the size effect of the ionic particles. The theories developed are quasi-incompressible in that the mass-average velocity is no longer divergence-free whenever there exists variability in densities of the fluid components, and the models are dissipative. We present several ways to derive the transport equations for the ions, which lead to different rates of energy dissipation. The theories can be formulated in either number densities, volume fractions or mass densities of the ionic fluid components. We show that the theory with the Cahn-Hilliard transport equation for ionic species reduces to the classical Poisson-Nernst-Planck (PNP) model with the size effect for ionic fluids when the densities of the fluid components are equal and the entropy of the solvent is neglected. It further reduces to the PNP model when the size effect is neglected. A linear stability analysis of the model together with two of its limits, which is the extended PNP model (EPNP defined in the text) and the classical PNP model (CPNP) with the finite size effect, on a constant state and a comparison among the three models in 1D space are presented to highlight the similarity and the departure of this model from the EPNP and the CPNP model.

There exists a large body of research on the lens of mammalian eye over the past several decades. The objective of the current work is to provide a link between the most recent computational models to some of the pioneering work in the 1970s and 80s. We introduce a general non-electro-neutral model to study the microcirculation in lens of eyes. It describes the steady state relationships among ion fluxes, water flow and electric field inside cells, and in the narrow extracellular spaces between cells in the lens. Using asymptotic analysis, we derive a simplified model based on physiological data and compare our results with those in the literature. We show that our simplified model can be reduced further to the first generation models while our full model is consistent with the most recent computational models. In addition, our simplified model captures the main features of the full model. Our results serve as a useful link intermediate between the computational models and the first generation analytical models.

The action potential signal of nerve and muscle is produced by voltage sensitive channels that include a specialized device to sense voltage. Gating currents of the voltage sensor are now known to depend on the back-and-forth movements of positively charged arginines through the hydrophobic plug of a voltage sensor domain. Transient movements of these permanently charged arginines, caused by the change of transmembrane potential, further drag the S4 segment and induce opening/closing of ion conduction pore by moving the S4-S5 linker. The ion conduction pore is a separate device from the voltage sensor, linked (in an unknown way) by the mechanical motion and electric field changes of the S4-S5 linker. This moving permanent charge induces capacitive current flow everywhere. Everything interacts with everything else in the voltage sensor so everything must interact with everything else in its mathematical model, as everything does in the whole protein. A PNP-steric model of arginines and a mechanical model for the S4 segment are combined using energy variational methods in which all movements of charge and mass satisfy conservation laws of current and mass. The resulting 1D continuum model is used to compute gating currents under a wide range of conditions, corresponding to experimental situations. Chemical-reaction-type models based on ordinary differential equations cannot capture such interactions with one set of parameters. Indeed, they may inadvertently violate conservation of current. Conservation of current is particularly important since small violations (<0.01%) quickly (<< 10-6 seconds) produce forces that destroy molecules. Our model reproduces signature properties of gating current: (1) equality of on and off charge in gating current (2) saturating voltage dependence in QV curve and (3) many (but not all) details of the shape of gating current as a function of voltage., Comment: 25 pages, 10 figures

Impedance Spectroscopy resolves electrical properties into uncorrelated variables, as a function of frequency, with exquisite resolution. Separation is robust and most useful when the system is linear. Impedance spectroscopy combined with appropriate structural knowledge provides insight into pathways for current flow, with more success than other methods. Biological applications of impedance spectroscopy are often not useful since so much of biology is strongly nonlinear in its essential features, and impedance spectroscopy is fundamentally a linear analysis. All cells and tissues have cell membranes and its capacitance is both linear and important to cell function. Measurements proved straightforward in skeletal muscle, cardiac muscle, and lens of the eye. In skeletal muscle, measurements provided the best estimates of the predominant (cell) membrane system that dominates electrical properties. In cardiac muscle, measurements showed definitively that classical microelectrode voltage clamp could not control the potential of the predominant membranes, that were in the tubular system separated from the extracellular space by substantial distributed resistance. In the lens of the eye, impedance spectroscopy changed the basis of all recording and interpretation of electrical measurements and laid the basis for Rae and Mathias extensive later experimental work. Many tissues are riddled with extracellular space as clefts and tubules, for example, cardiac muscle, the lens of the eye, most epithelia, and of course frog muscle. These tissues are best analyzed with a bidomain theory that arose from the work on electrical structure described here. There has been a great deal of work since then on the bi-domain and this represents the most important contribution to biology of the analysis of electrical structure in my view., Comment: A chapter in "Impedance Spectroscopy Theory, Experiment, and Applications:Solid State, Corrosion, Power sources",3rd Edition Evgenij Barsoukov (ed.), J. Ross Macdonald (ed.), Wiley-Interscience, 2016