arXiv subjects

Bob Eisenberg

Publications and source records attributed to Bob Eisenberg.

At least 19 recordsLinked to original sources

arXiv2026

Multiscale retinal flow on a spherical cap of varying aperture

Modelling retinal haemodynamics is crucial for understanding retinal microcirculation but is computationally demanding because it involves coupling between the vasculature and surrounding tissue across multiple scales. This computational burden has been substantially alleviated by a recent analytic solution on the planar disc that enables lumping the capillary bed and surrounding tissue into an effective resistor. However, that formulation treats the retina as a flat surface, whereas the retina is a curved surface with a finite anterior aperture. In this work, we develop a nontrivial and physiologically necessary extension to spherical-cap tissue domains with varying apertures, where surface curvature and finite-aperture boundaries complicate solving coupled Darcy equations on a curved manifold. Using a stereographic projection and a decoupling transformation, we derive an analytic solution for the capillary-tissue system on the spherical cap that represents flow in both the capillary bed and interstitial tissue more realistically while retaining the efficient resistor formulation, a key advantage of the planar-disc formulation. This solution is coupled to one-dimensional (1D) arteriolar and venular flows to obtain a multiscale description of retinal haemodynamics. Using a vasculature model designed to capture retinal vascular features, we show that the multiscale model's predictions are consistent with experimental data. We further explore aperture effects using both a fixed hemispherical vasculature and aperture-dependent vasculature. The aperture affects retinal haemodynamics mainly through changes in the constructed vasculature itself, whereas the surface-averaged pressures and relative terminal flow distributions remain nearly unchanged. This framework provides a foundation for studying retinal pathophysiology on more anatomically realistic domains.

physics.bio-ph↗

arXiv2021

Setting Boundaries for Statistical Mechanics

Statistical mechanics has grown without bounds in space. Statistical mechanics of point particles in an unbounded perfect gas is commonly accepted as a foundation for understanding many systems, including liquids like the concentrated salt solutions of life and electrochemical technology, from batteries to nanodevices. Liquids, however, are not gases. Liquids are filled with interacting molecules and so the model of a perfect gas is imperfect. Here we show that statistical mechanics without bounds (in space) is impossible as well as imperfect, if the molecules interact as charged particles, as nearly all atoms do. The behavior of charged particles is not defined until boundary structures and values are defined because charges are governed by the Maxwell partial differential equations. Partial differential equations require boundary conditions to be computable or well defined. The Maxwell equations require boundary conditions on finite sized spatial boundaries (i.e., structures). Boundary conditions cannot be defined 'at infinity' in a general (i.e., unique) way because the limiting process that defines infinity includes such a wide variety of behavior, from light waves that never decay, to fields from dipole and multipolar charges that decay steeply, to Coulomb fields that decay but not so steeply. Statistical mechanics involving charges thus involves spatial boundaries and boundary conditions of finite size. Nearly all matter involves charges, thus nearly all statistical mechanics requires structures and boundary conditions on those structures. Boundaries and boundary conditions are not prominent in classical statistical mechanics. Including boundaries is a challenge to mathematicians. Statistical mechanics must describe bounded systems if it is to provide a proper foundation for studying matter.

q-bio.OT↗

arXiv2021

Asking Biological Questions of Physical Systems: the Device Approach to Emergent Properties

Life occurs in concentrated `Ringer Solutions' derived from seawater that Lesser Blum studied for most of his life. As we worked together, Lesser and I realized that the questions asked of those solutions were quite different in biology from those in the physical chemistry he knew. Biology is inherited. Information is passed by handfuls of atoms in the genetic code. A few atoms in the proteins built from the code change macroscopic function. Indeed, a few atoms often control biological function in the same sense that a gas pedal controls the speed of a car. Biological questions then are most productive when they are asked in the context of evolution. What function does a system perform? How is the system built to perform that function? What forces are used to perform that function? How are the modules that perform functions connected to make the machinery of life. Physiologists have shown that much of life is a nested hierarchy of devices, one on top of another, linking atomic ions in concentrated solutions to current flow through proteins, current flow to voltage signals, voltage signals to changes in current flow, all connected to make a regenerative system that allows electrical action potentials to move meters, under the control of a few atoms. The hierarchy of devices allows macroscopic properties to emerge from atomic scale interactions. The structures of biology create these devices. The concentration and electrical fields of biology power these devices, more than anything else. The resulting organisms reproduce. Evolution selects the organisms that reproduce more and thereby selects the devices that allow macroscopic control to emerge from the atomic structures of genes and proteins and their motions.

q-bio.OT↗

arXiv2021

On variational principles for polarization responses in electromechanical systems

Classical electrodynamics uses a dielectric constant to describe the polarization response of electromechanical systems to changes in an electric field. We generalize that description to include a wide variety of responses to changes in the electric field, as found in most systems and applications. Electromechanical systems can be found in many physical and biological applications, such as ion transport in membranes, batteries, and dielectric elastomers. We present a unified, thermodynamically consistent, variational framework for modeling electromechanical systems as they respond to changes in the electric field; that is to say, as they polarize. This framework is motivated and developed using the classical energetic variational approach (EnVarA). The coupling between the electric part and the chemo-mechanical parts of the system is described either by Lagrange multipliers or various energy relaxations. The classical polarization and its dielectrics and dielectric constants appear as outputs of this analysis. The Maxwell equations then become universal conservation laws of charge and current, conjoined to an electromechanical description of polarization. Polarization describes the entire electromechanical response to changes in the electric field and can sometimes be approximated as a dielectric constant or dielectric dispersion.

cond-mat.stat-mech↗

arXiv2021

Differential Capacitance of Electric Double Layers: A Poisson-Bikerman Formula

We propose a Poisson-Bikerman (PBik) formula for calculating the differential capacitance (DC) of electrical double layers (EDLs) in aqueous electrolytes or ionic liquids. The PBik theory is a generalization of the classical Poisson-Boltzmann theory to include different steric energies of different-sized ions and water similar to different electrical energies for different-charged ions. Water and ions with interstitial voids in this molecular mean field theory have their physical volumes as they do in molecular dynamics simulations. The PBik formula derived from Fermi distributions of ions and water in arbitrary shape and volume reduces to the Bikerman-Freise formula derived from the lattice model of equal-sized ions. The DC curves predicted by the Gouy-Chapman formula are U-shaped (for point-like ions with zero volume and very dilute solutions). The curves change from U shape to camel shape (Bactrian) and then to bell shape (for finite size ions) as the volume fraction of ions and water changes from zero to medium value then to large value. The transition is characterized by critical and inflection voltages in terms of the particle volume fraction. These voltages determine steric and electrical energies that describe the space/charge competition and saturation properties of ions and water packed in the condensed layer of EDLs under high field conditions. Steric energy is as important as electrical energy in these conditions. PBik computes symmetric DC curves from delicately balanced steric interactions of asymmetric-size ions and water like the experimental data of KPF6 in aqueous solution. It computes asymmetric curves and captures delicately balanced steric or electrical interactions of ions having different volumes or charges in ionic liquids.

cond-mat.soft↗

arXiv2020

Field Theory of Reaction-Diffusion: Mass Action with an Energetic Variational Approach

We extend the energetic variational approach so it can be applied to a chemical reaction system with general mass action kinetics. Our approach starts with an energy-dissipation law. We show that the chemical equilibrium is determined by the choice of the free energy and the dynamics of the chemical reaction is determined by the choice of the dissipation. This approach enables us to couple chemical reactions with other effects, such as diffusion and drift in an electric field. As an illustration, we apply our approach to a non-equilibrium reaction-diffusion system in a specific but canonical set-up. We show by numerical simulation that the input-output relation of such a system depends on the choice of the dissipation.

physics.chem-ph↗

arXiv2020

Molecular Mean-Field Theory of Ionic Solutions: a Poisson-Nernst-Planck-Bikerman Model

We have developed a molecular mean-field theory -- fourth-order Poisson-Nernst-Planck-Bikerman theory -- for modeling ionic and water flows in biological ion channels by treating ions and water molecules of any volume and shape with interstitial voids, polarization of water, and ion-ion and ion-water correlations. The theory can also be used to study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc. The theory can compute electric and steric energies from all atoms in a protein and all ions and water molecules in a channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium with complex properties that mimic experimental data. The theory has been verified with experiments and molecular dynamics data from the gramicidin A channel, L-type calcium channel, potassium channel, and sodium/calcium exchanger with real structures from the Protein Data Bank. It was also verified with the experimental or Monte Carlo data of electric double-layer differential capacitance and ion activities in aqueous electrolyte solutions. We give an in-depth review of the literature about the most novel properties of the theory, namely, Fermi distributions of water and ions as classical particles with excluded volumes and dynamic correlations that depend on salt concentration, composition, temperature, pressure, far-field boundary conditions etc. in a complex and complicated way as reported in a wide range of experiments. The dynamic correlations are self-consistent output functions from a fourth-order differential operator that describes ion-ion and ion-water correlations, the dielectric response (permittivity) of ionic solutions, and the polarization of water molecules with a single correlation length parameter.

cond-mat.soft↗

arXiv2019

Effects of diffusion coefficients on reversal potentials in ionic channels

In this work, the dependence of reversal potentials and zero-current fluxes on diffusion coefficients are examined for ionic flows through membrane channels. The study is conducted for the setup of a simple structure defined by the profile of permanent charges with two mobile ion species, one positively charged (cation) and one negatively charged (anion). Numerical observations are obtained from analytical results established using geometric singular perturbation analysis of classical Poisson-Nernst-Planck models. For 1:1 ionic mixtures with arbitrary diffusion coefficients, Mofidi and Liu [arXiv:1909.01192] conducted a rigorous mathematical analysis and derived an equation for reversal potentials that, in its particular case, can be compared to Goldman-Hodgkin-Katz equation. We summarize and extend these results with numerical observations for biological relevant situations. The numerical investigations on profiles of the electrochemical potentials, ion concentrations, and electrical potential across ion channels are also presented for the zero-current case. Moreover, the behavior of current and fluxes with respect to voltages and permanent charges are investigated. In the opinion of the authors, many results in the paper are not intuitive, and it is difficult, if not impossible, to see all cases without investigations of this type.

cond-mat.soft↗

arXiv2019

What Current Flows Through a Resistor?

Our digital technology depends on mathematics to compute current flow and design its devices. Mathematics describes current flow by an idealization, Kirchhoff's current law. All the electrons that flow into a node flow out. This idealization describes real circuits only when stray capacitances are included in the circuit design. Motivated by Maxwell's equations, we propose that current in Kirchhoff's law be defined as \[{\varepsilon }_{\mathrm{0\ \ }}\frac{\mathrm{\partial }\boldsymbol{\mathrm{E}}}{\mathrm{\partial }t}\ \mathrm{+}\mathrm{\ }\mathrm{\ }\widetilde{\boldsymbol{\mathrm{J}}},\] the sum of (1) displacement current (2) the flux of charge associated with mass. The flux of charge associated with mass $\widetilde{\boldsymbol{\mathrm{J}}}$ includes, for example, the polarization of dielectrics as well as the movement of electrons. Kirchhoff's law becomes exact and universal when current is defined this way. This current is the source of the magnetic field; it is the source of $\boldsymbol{\mathrm{curl}}\boldsymbol{\mathrm{\ }}\boldsymbol{\mathrm{B}}$ in Maxwell's equations. Kirchoff's laws and Maxwell's equations can use the same definition of current.

physics.class-ph↗

arXiv2019

Multiple Scales in the Simulation of Ion Channels and Proteins

Computation of biological processes creates great promise for everyday life and great challenges for physical scientists. Simulations of molecular dynamics appeal to biologists as a natural extension of structural biology. Once biologists see a structure, they want to see it move. Molecular biology shows that a few atoms, often messenger ions like Ca$^{2+}$, control biological function on the scale of cells, sometimes organisms. Enormously concentrated ions (~20 M) in protein channels and enzymes can control important characteristics of living systems, just as highly concentrated ions near electrodes control important characteristics of electrochemical systems. The scale differences needed to simulate all the atoms of biological cells are $10^7$ in linear dimension, $10^{21}$ in three dimensions, $10^{9}$ in resolution, $10^{11}$ in time, and $10^{13}$ in particle number (to deal with concentrations of Ca$^{2+}$). $\mathbf{These}$ $\mathbf{scales}$ $\mathbf{must}$ $\mathbf{be}$ $\mathbf{dealt}$ $\mathbf{with}$ $\mathbf{simultaneously}$ if the simulation is to deal with most biological functions. We suggest a computational approach using explicit multiscale analysis instead of implicit simulation of all scales. Successful computation of ions concentrated in special places may be a significant step to understanding the defining characteristics of biological and electrochemical systems.

q-bio.BM↗

arXiv2018

Profiles for ion activities are multiphasic, not curvilinear

In agreement with previous findings (Nissen 2017a,b), data (Wilczek-Vera et al. 2004) for the ion activities of a variety of simple inorganic compounds are shown to be precisely represented as multiphasic, i.e. as a series of straight lines separated by discontinuous transitions.

physics.bio-ph↗

arXiv2018

Osmosis through a Semi-permeable Membrane: a Consistent Approach to Interactions

The movement of ionic solutions is an essential part of biology and technology. Fluidics, from nano- to micro- to microfluidics, is a burgeoning area of technology which is all about the movement of ionic solutions, on various scales. Many cells, tissues, and organs of animals and plants depend on osmosis, as the movement of fluids is called in biology. Indeed, the movement of fluids through channel proteins (that have a hole down their middle) is fluidics on an atomic scale. Ionic fluids are complex fluids, with energy stored in many ways. Ionic fluids flow driven by gradients of concentration, chemical and electrical potential, and hydrostatic pressure. Each flow is classically described by its own field theory, independent of the others, but of course, in reality every gradient drives every kind of flow to a varying extent. Combining field equations is tricky and so the theory of complex fluids derives the equations, rather than assumes their interactions. When field equations are derived, rather than assumed, their variables are consistent. That is to say all variables satisfy all equations under all conditions with one set of parameters. Here we treat a classical osmotic cell in this spirit, using a sharp interface method to derive boundary conditions consistent with all flows and fields. We allow volume to change with concentration, since changes of volume are a property of ionic solutions known to all who make them in the laboratory. We consider flexible and inflexible membranes. We show how to combine the energetics of the membrane with the energetics of the surrounding complex fluids. The results seem general but need application to specific situations of technological, biological and experimental importance before the consequences of consistency can be understood.

q-bio.QM↗

arXiv2018

Do Bi-Stable Poisson-Nernst-Planck Models Describe Single Channel Gating?

Experiments measuring currents through single protein channels show unstable currents, a phenomena called the gating of a single channel. Channels switch between an 'open' state with a well defined single amplitude of current and 'closed' states with nearly zero current. The existing mean-field theory of ion channels focuses almost solely on the open state. The physical modeling of the dynamical features of ion channels is still in its infancy, and does not describe the transitions between open and closed states, nor the distribution of the duration times of open states. One hypothesis is that gating corresponds to noise-induced fast transitions between multiple steady (equilibrium) states of the underlying system. In this work, we aim to test this hypothesis. Particularly, our study focuses on the (high order) steric Poisson-Nernst-Planck-Cahn-Hilliard model since it has been successful in predicting permeability and selectivity of ionic channels in their open state, and since it gives rise to multiple steady states. We show that this system gives rise to a gating-like behavior, but that important features of this switching behavior are different from the defining features of gating in biological systems. Furthermore, we show that noise prohibits switching in the system of study. The above phenomena are expected to occur in other PNP-type models, strongly suggesting that one has to go beyond over-damped (gradient flow) Nernst-Planck type dynamics to explain the spontaneous gating of single channels.

q-bio.BM↗

arXiv2017

Poisson-Fermi Modeling of Ion Activities in Aqueous Single and Mixed Electrolyte Solutions at Variable Temperature

The combinatorial explosion of empirical parameters in tens of thousands presents a tremendous challenge for extended Debye-Hückel models to calculate activity coefficients of aqueous mixtures of most important salts in chemistry. The explosion of parameters originates from the phenomenological extension of the Debye-Hückel theory that does not take steric and correlation effects of ions and water into account. In contrast, the Poisson-Fermi theory developed in recent years treats ions and water molecules as nonuniform hard spheres of any size with interstitial voids and includes ion-water and ion-ion correlations. We present a Poisson-Fermi model and numerical methods for calculating the individual or mean activity coefficient of electrolyte solutions with any arbitrary number of ionic species in a large range of salt concentrations and temperatures. For each activity-concentration curve, we show that the Poisson-Fermi model requires only three unchanging parameters at most to well fit the corresponding experimental data. The three parameters are associated with the Born radius of the solvation energy of an ion in electrolyte solution that changes with salt concentrations in a highly nonlinear manner.

cond-mat.soft↗

arXiv2017

An effect of large permanent charge: Decreasing flux to zero with increasing transmembrane potential to infinity

In this work, we examine effects of large permanent charges on ionic flow through ion channels based on a quasi-one dimensional Poisson-Nernst-Planck model. It turns out large positive permanent charges inhibit the flux of cation as expected, but strikingly, as the transmembrane electrochemical potential for anion increases in a particular way, the flux of anion decreases. The latter phenomenon was observed experimentally but the cause seemed to be unclear. The mechanisms for these phenomena are examined with the help of the profiles of the ionic concentrations, electric fields and electrochemical potentials. The underlying reasons for the near zero flux of cation and for the decreasing flux of anion are shown to be different over different regions of the permanent charge. Our model is oversimplified. More structural detail and more correlations between ions can and should be included. But the basic finding seems striking and important and deserving of further investigation.

q-bio.QM↗

arXiv2017

Relative dielectric constants and selectivity ratios in open ionic channels

We investigate the effects of the relative dielectric coefficient on ionic flows in open ion channels, using mathematical analysis of reasonably general Poisson-Nernst-Planck type models that can include the finite sizes of ions. The value of the relative dielectric coefficient is of course a crucial parameter for ionic behavior in general. Using the powerful theory of singularly perturbed problems in applied mathematics, we show that some properties of open channels are quite insensitive to variation in the relative dielectric coefficient, thereby explaining such effects seen unexpectedly in simulations. The ratio between the total number of one ion species and that of another ion species, and the ratio between the flux of one ion species and that of another ion species do not depend significantly on the relative dielectric coefficient.

q-bio.BM↗

arXiv2017

Dynamics of Current, Charge and Mass

Electricity plays a special role in our lives and life. Equations of electron dynamics are nearly exact and apply from nuclear particles to stars. These Maxwell equations include a special term the displacement current (of vacuum). Displacement current allows electrical signals to propagate through space. Displacement current guarantees that current is exactly conserved from inside atoms to between stars, as long as current is defined as Maxwell did, as the entire source of the curl of the magnetic field. We show how the Bohm formulation of quantum mechanics allows easy definition of current. We show how conservation of current can be derived without mention of the polarization or dielectric properties of matter. Matter does not behave the way physicists of the 1800's thought it does with a single dielectric constant, a real positive number independent of everything. Charge moves in enormously complicated ways that cannot be described in that way, when studied on time scales important today for electronic technology and molecular biology. Life occurs in ionic solutions in which charge moves in response to forces not mentioned or described in the Maxwell equations, like convection and diffusion. Classical derivations of conservation of current involve classical treatments of dielectrics and polarization in nearly every textbook. Because real dielectrics do not behave in a classical way, classical derivations of conservation of current are often distrusted or even ignored. We show that current is conserved exactly in any material no matter how complex the dielectric, polarization or conduction currents are. We believe models, simulations, and computations should conserve current on all scales, as accurately as possible, because physics conserves current that way. We believe models will be much more successful if they conserve current at every level of resolution, the way physics does.

physics.gen-ph↗

arXiv2017

Poisson-Fermi Formulation of Nonlocal Electrostatics in Electrolyte Solutions

We present a nonlocal electrostatic formulation of nonuniform ions and water molecules with interstitial voids that uses a Fermi-like distribution to account for steric and correlation effects in electrolyte solutions. The formulation is based on the volume exclusion of hard spheres leading to a steric potential and Maxwell's displacement field with Yukawa-type interactions resulting in a nonlocal electric potential. The classical Poisson-Boltzmann model fails to describe steric and correlation effects important in a variety of chemical and biological systems, especially in high field or large concentration conditions found in and near binding sites, ion channels, and electrodes. Steric effects and correlations are apparent when we compare nonlocal Poisson-Fermi results to Poisson-Boltzmann calculations in electric double layer and to experimental measurements on the selectivity of potassium channels for K+ over Na+. The present theory links atomic scale descriptions of the crystallized KcsA channel with macroscopic bulk conditions. Atomic structures and macroscopic conditions determine complex functions of great importance in biology, nanotechnology, and electrochemistry.

cond-mat.soft↗