arXiv subjects

Bob Eisenberg

Publications and source records attributed to Bob Eisenberg.

At least 37 records · Page 2Linked to original sources

arXiv2016

Flux ratios and channel structures

We investigate Ussing's unidirectional fluxes and flux ratios of charged tracers motivated particularly by the insightful proposal of Hodgkin and Keynes on a relation between flux ratios and channel structure. Our study is based on analysis of quasi-one-dimensional Poisson-Nernst-Planck type models for ionic flows through membrane channels. This class of models includes the Poisson equation that determines the electrical potential from the charges present and is in that sense consistent. Ussing's flux ratios generally depend on all physical parameters involved in ionic flows, particularly, on bulk conditions and channel structures. Certain setups of ion channel experiments result in flux ratios that are universal in the sense that their values depend on bulk conditions but not on channel structures; other setups lead to flux ratios that are specific in the sense that their values depend on channel structures too. Universal flux ratios could serve some purposes better than specific flux ratios in some circumstances and worse in other circumstances. We focus on two treatments of tracer flux measurements that serve as estimators of important properties of ion channels. The first estimator determines the flux of the main ion species from measurements of the flux of its tracer. Our analysis suggests a better experimental design so that the flux ratio of the tracer flux and the main ion flux is universal. The second treatment of tracer fluxes concerns ratios of fluxes and experimental setups that try to determine some properties of channel structure. We analyze the two widely used experimental designs of estimating flux ratios and show that the most widely used method depends on the spatial distribution of permanent charge so this flux ratio is specific and thus allows estimation of (some of) the properties of that permanent charge, even with ideal ionic solutions. ...

q-bio.OT↗

arXiv2016

Conservation of Charge and Conservation of Current

Conservation of current and conservation of charge are nearly the same thing: when enough is known about charge movement, conservation of current can be derived from conservation of charge, in ideal dielectrics, for example. Conservation of current is enforced implicitly in ideal dielectrics by theories that conserve charge. But charge movement in real materials like semiconductors or ionic solutions is never ideal. We present an apparently universal derivation of conservation of current and advocate using that conservation law explicitly as a distinct part of theories and calculations of charge movement in complex fluids and environments. Classical models using ordinary differential equations rarely satisfy conservation of current, including the chemical kinetic models implementing the law of mass action and Markov models. These models must be amended if they are to conserve current. Strict enforcement of conservation of current is likely to aid numerical analysis by preventing artifactual accumulation of charge.

q-bio.OT↗

arXiv2016

Ionic channels in biological membranes. Electrostatic analysis of a natural nanotube

A simple theory of ion permeation through a channel is presented, in which diffusion occurs according to Fick's law and drift according to Ohm's law, in the electric field determined by all the charges present. This theory accounts for permeation in the channels studied to date in a wide range of solutions. Interestingly, the theory works because the shape of the electric field is a sensitive function of experimental conditions, e.g., ion concentration. Rate constants for flux are sensitive functions of ionic concentration because the fixed charge of the channel protein is shielded by the ions in an near it. Such shielding effects are not included in traditional theories of ionic channels, or other proteins, for that matter.

q-bio.BM↗

arXiv2016

Maxwell Matters

Charges are everywhere because most atoms are charged. Chemical bonds are formed by electrons with their charge. Charges move and interact according to Maxwell's equations in space and in atoms where the equations of electrodynamics are embedded in Schroedinger's equation as the potential. Maxwell's equations are universal, valid inside atoms and between stars from times much shorter than those of atomic motion (0.1 femtoseconds) to years (32 mega-seconds). Maxwell's equations enforce the conservation of current. Analysis shows that the electric field can take on whatever value is needed to ensure conservation of current. The properties of matter rearrange themselves to satisfy Maxwell's equations and conservation of current. Conservation of current is as universal as Maxwell's equations themselves. Yet equations of electrodynamics find little place in the literature of material physics, chemistry, or biochemistry. Kinetic models of chemistry and Markov treatments of atomic motion are ordinary differential equations in time and do not satisfy conservation of current unless modified significantly. Systems at equilibrium, without macroscopic flow, have thermal fluctuating currents that are conserved according to the Maxwell equations although their macroscopic averages are zero. The macroscopic consequences of atomic scale fluctuating thermal currents are not known but are likely to be substantial because of the nonlinear interactions in systems like these, in which 'everything interacts with everything else'. Conclusion: current needs to be conserved in models of material systems, even in simplified versions.

physics.bio-ph↗

arXiv2016

A Nonlocal Poisson-Fermi Model for Ionic Solvent

We propose a nonlocal Poisson-Fermi model for ionic solvent that includes ion size effects and polarization correlations among water molecules in the calculation of electrostatic potential. It includes the previous Poisson-Fermi models as special cases, and its solution is the convolution of a solution of the corresponding nonlocal Poisson dielectric model with a Yukawa-type kernel function. Moreover, the Fermi distribution is shown to be a set of optimal ionic concentration functions in the sense of minimizing an electrostatic potential free energy. Finally, numerical results are reported to show the difference between a Poisson-Fermi solution and a corresponding Poisson solution.

physics.chem-ph↗

arXiv2015

Mass Action and Conservation of Current

The law of mass action is used widely. The law of mass action does not automatically conserve current, as is clear from mathematics of a simple case, chosen to illustrate the issues. The law of mass action does not force a series of chemical reactions to have the same current flow everywhere. Interruption of far-away current does not stop current everywhere in a series of chemical reactions, and so does not obey Maxwell equations. An additional constraint and equation is needed to enforce the global continuity of current flow. The additional constraint is introduced in this paper in the special case that the chemical reaction describes spatial movement through narrow channels. In that case, a fully consistent treatment is possible using a variety of models of charge movement. The general case must be dealt with by variational methods that enforce consistency of all the physical laws involved. Variational methods have only recently been developed to ensure that charge flow is conserved globally, along with mass, in dissipative systems like ions in solution or proteins. The Energy Variational Approach EnVarA should allow the development of more robust models of chemical, biochemical, and biological systems, making practical devices more easy to design and build. These difficulties arise away from equilibrium, when current flows, and the law of mass action is applied to a non-equilibrium situation, different from the systems considered when the law was originally derived. Non-equilibrium systems are important. Almost all of biology occurs away from equilibrium. Almost all devices of our technology function away from equilibrium. I believe robust models and device designs in the chemical world will not be possible until continuity of current is embedded in a generalization of the law of mass action using a consistent variational model of energy and dissipation.

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arXiv2015

Poisson-Fermi Model of Single Ion Activities

A Poisson-Fermi model is proposed for calculating activity coefficients of single ions in strong electrolyte solutions based on the experimental Born radii and hydration shells of ions in aqueous solutions. The steric effect of water molecules and interstitial voids in the first and second hydration shells play an important role in our model. The screening and polarization effects of water are also included in the model that can thus describe spatial variations of dielectric permittivity, water density, void volume, and ionic concentration. The activity coefficients obtained by the Poisson-Fermi model with only one adjustable parameter are shown to agree with experimental data, which vary nonmonotonically with salt concentrations.

physics.chem-ph↗

arXiv2015

Poisson-Nernst-Planck-Fermi Theory for Ion Channels

A Poisson-Nernst-Planck-Fermi (PNPF) theory is developed for studying ionic transport through biological ion channels. Our goal is to deal with the finite size of particle using a Fermi like distribution without calculating the forces between the particles, because they are both expensive and tricky to compute. We include the steric effect of ions and water molecules with nonuniform sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of water molecules in an inhomogeneous aqueous electrolyte. Including the finite volume of water and the voids between particles is an important new part of the theory presented here. Fermi like distributions of all particle species are derived from the volume exclusion of classical particles. The classical Gibbs entropy is extended to a new entropy form --- called Gibbs-Fermi entropy --- that describes mixing configurations of all finite size particles and voids in a thermodynamic system where microstates do not have equal probabilities. The PNPF model describes the dynamic flow of ions, water molecules, as well as voids with electric fields and protein charges. The PNPF results are in good accord with experimental currents recorded in a 10^8-fold range of Ca++ concentrations. The results illustrate the anomalous mole fraction effect, a signature of L-type calcium channels. Moreover, numerical results concerning water density, dielectric permittivity, void volume, and steric energy provide useful details to study a variety of physical mechanisms ranging from binding, to permeation, blocking, flexibility, and charge/space competition of the channel.

physics.bio-ph↗

arXiv2015

Numerical Methods for a Poisson-Nernst-Planck-Fermi Model

Numerical methods are proposed for an advanced Poisson-Nernst-Planck-Fermi (PNPF) model for studying ion transport through biological ion channels. PNPF contains many more correlations than most models and simulations of channels, because it includes water and calculates dielectric properties consistently as outputs. This model accounts for the steric effect of ions and water molecules with different sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of polarized water molecules in an inhomogeneous aqueous electrolyte. The steric energy is shown to be comparable to the electrical energy under physiological conditions, demonstrating the crucial role of the excluded volume of particles and the voids in the natural function of channel proteins. Water is shown to play a critical role in both correlation and steric effects in the model. We extend the classical Scharfetter-Gummel (SG) method for semiconductor devices to include the steric potential for ion channels, which is a fundamental physical property not present in semiconductors. Together with a simplified matched interface and boundary (SMIB) method for treating molecular surfaces and singular charges of channel proteins, the extended SG method is shown to exhibit important features in flow simulations such as optimal convergence, efficient nonlinear iterations, and physical conservation. Two different methods --- called the SMIB and multiscale methods --- are proposed for two different types of channels, namely, the gramicidin A channel and an L-type calcium channel, depending on whether water is allowed to pass through the channel. The PNPF currents are in accord with the experimental I-V (V for applied voltages) data of the gramicidin A channel and I-C (C for bath concentrations) data of the calcium channel.

q-bio.BM↗

arXiv2014

Can we make biochemistry an exact science?

Biochemists know that the law of mass action is not exact and not very useful because we cannot transfer it (with unchanged parameters) from one condition to another. I argue that exact equations require calibrated multiscale analysis to deal with ions. Exact theories in biochemistry must use mathematics of interactions because biological ionic solutions, derived from seawater, are complex (not simple) fluids. The activity of one ion depends on every other ion. Mathematics of conservative interactions is well understood but friction is another matter. Mathematicians now have now an energetic variational calculus dealing with friction. Complex fluids need variational methods because everything interacts with everything else. Mathematics designed to handle interactions is needed to produce exact equations. If interactions are not addressed with variational mathematics, they are bewildering. The mathematics must include the global properties of the electric field. Flow of charge in one place changes the flow everywhere by Kirchoff and Maxwell laws. Charge changes physical nature as it flows through a circuit. It is ions in salt water; it is electrons in a vacuum tube; it is quasi-particles in a semiconductor; and it is nothing much in a vacuum capacitor (i.e., displacement current). Charge is abstract. The physical nature of charge and current is strikingly diverse; yet, the flow of current is exactly the same in every element in a series circuit. The global nature of electric flow prevents the law of mass action from being exact. The law of mass action (with rate constants that are constant) does not know about charge. The law of mass action is about mass conservation. I believe the law of mass action must be modified to be consistent with the Kirchoff current law if biochemistry is to be an exact science.

q-bio.BM↗

arXiv2014

Multiple solutions of steady-state Poisson-Nernst-Planck equations with steric effects

Experiments measuring currents through single protein channels show unstable currents. Channels switch between 'open' or 'closed' states in a spontaneous stochastic process called gating. Currents are either (nearly) zero or at a definite level, characteristic of each type of protein, independent of time, once the channel is open. The steady state Poisson-Nernst-Planck equations with steric effects (PNP-steric equations) describe steady current through the open channel quite well, in a wide variety of conditions. Here we study the existence of multiple solutions of steady state PNP-steric equations to see if they themselves, without modification or augmentation, can describe two levels of current. We prove that there are two steady state solutions of PNP-steric equations for (a) three types of ion species (two types of cations and one type of anion) with a positive constant permanent charge, and (b) four types of ion species (two types of cations and their counter-ions) with a constant permanent charge but no sign condition. The excess currents (due to steric effects) associated with these two steady state solutions are derived and expressed as two distinct formulas. Our results indicate that PNP-steric equations may become a useful model to study spontaneous gating of ion channels. Spontaneous gating is thought to involve small structural changes in the channel protein that perhaps produce large changes in the profiles of free energy that determine ion flow. Gating is known to be modulated by external structures. Both can be included in future extensions of our present analysis.

math.AP↗

arXiv2013

Ionic Interactions in Biological and Physical Systems: a Variational Treatment

Chemistry is about chemical reactions. Chemistry is about electrons changing their configurations as atoms and molecules react. Chemistry studies reactions as if they occurred in ideal infinitely dilute solutions. But most reactions occur in nonideal solutions. Then everything (charged) interacts with everything else (charged) through the electric field, which is short and long range extending to boundaries of the system. Mathematics has recently been developed to deal with interacting systems of this sort. The variational theory of complex fluids has spawned the theory of liquid crystals. In my view, ionic solutions should be viewed as complex fluids. In both biology and electrochemistry ionic solutions are mixtures highly concentrated (~10M) where they are most important, near electrodes, nucleic acids, enzymes, and ion channels. Calcium is always involved in biological solutions because its concentration in a particular location is the signal that controls many biological functions. Such interacting systems are not simple fluids, and it is no wonder that analysis of interactions, such as the Hofmeister series, rooted in that tradition, has not succeeded as one would hope. We present a variational treatment of hard spheres in a frictional dielectric. The theory automatically extends to spatially nonuniform boundary conditions and the nonequilibrium systems and flows they produce. The theory is unavoidably self-consistent since differential equations are derived (not assumed) from models of (Helmholtz free) energy and dissipation of the electrolyte. The origin of the Hofmeister series is (in my view) an inverse problem that becomes well posed when enough data from disjoint experimental traditions are interpreted with a self-consistent theory.

q-bio.BM↗

arXiv2013

Ions in Fluctuating Channels: Transistors Alive

Ion channels are proteins with a hole down the middle embedded in cell membranes. Membranes form insulating structures and the channels through them allow and control the movement of charged particles, spherical ions, mostly Na+, K+, Ca++, and Cl-. Membranes contain hundreds or thousands of types of channels, fluctuating between open conducting, and closed insulating states. Channels control an enormous range of biological function by opening and closing in response to specific stimuli using mechanisms that are not yet understood in physical language. Open channels conduct current of charged particles following laws of Brownian movement of charged spheres rather like the laws of electrodiffusion of quasi-particles in semiconductors. Open channels select between similar ions using a combination of electrostatic and 'crowded charge' (Lennard-Jones) forces. The specific location of atoms and the exact atomic structure of the channel protein seems much less important than certain properties of the structure, namely the volume accessible to ions and the effective density of fixed and polarization charge. There is no sign of other chemical effects like delocalization of electron orbitals between ions and the channel protein. Channels play a role in biology as important as transistors in computers, and they use rather similar physics to perform part of that role. Understanding their fluctuations awaits physical insight into the source of the variance and mathematical analysis of the coupling of the fluctuations to the other components and forces of the system.

q-bio.BM↗

arXiv2013

Interacting Ions in Biophysics: Real is not Ideal

Ions in water are important in biology, from molecules to organs. Classically, ions in water are treated as ideal noninteracting particles in a perfect gas. Excess free energy of ion was zero. Mathematics was not available to deal consistently with flows, or interactions with ions or boundaries. Non-classical approaches are needed because ions in biological conditions flow and interact. The concentration gradient of one ion can drive the flow of another, even in a bulk solution. A variational multiscale approach is needed to deal with interactions and flow. The recently developed energetic variational approach to dissipative systems allows mathematically consistent treatment of bio-ions Na, K, Ca and Cl as they interact and flow. Interactions produce large excess free energy that dominate the properties of the high concentration of ions in and near protein active sites, channels, and nucleic acids: the number density of ions is often more than 10 M. Ions in such crowded quarters interact strongly with each other as well as with the surrounding protein. Non-ideal behavior has classically been ascribed to allosteric interactions mediated by protein conformation changes. Ion-ion interactions present in crowded solutions--independent of conformation changes of proteins--are likely to change interpretations of allosteric phenomena. Computation of all atoms is a popular alternative to the multiscale approach. Such computations involve formidable challenges. Biological systems exist on very different scales from atomic motion. Biological systems exist in ionic mixtures (extracellular/intracellular solutions), and usually involve flow and trace concentrations of messenger ions (e.g., 10-7 M Ca2+). Energetic variational methods can deal with these characteristic properties of biological systems while we await the maturation and calibration of all atom simulations of ionic mixtures and divalents.

q-bio.BM↗

arXiv2013

A Conservative Finite Difference Scheme for Poisson-Nernst-Planck Equations

A macroscopic model to describe the dynamics of ion transport in ion channels is the Poisson-Nernst-Planck(PNP) equations. In this paper, we develop a finite-difference method for solving PNP equations, which is second-order accurate in both space and time. We use the physical parameters specifically suited toward the modelling of ion channels. We present a simple iterative scheme to solve the system of nonlinear equations resulting from discretizing the equations implicitly in time, which is demonstrated to converge in a few iterations. We place emphasis on ensuring numerical methods to have the same physical properties that the PNP equations themselves also possess, namely conservation of total ions and correct rates of energy dissipation. We describe in detail an approach to derive a finite-difference method that preserves the total concentration of ions exactly in time. Further, we illustrate that, using realistic values of the physical parameters, the conservation property is critical in obtaining correct numerical solutions over long time scales.

math.NA↗

arXiv2012

Life's Solutions are Complex Fluids. A Mathematical Challenge

Classical thermodynamics and statistical mechanics describe systems in which nothing interacts with nothing. Even the highly refined theory of simple fluids does not deal very well with electrical interactions, boundary conditions, or flows, if at all. Electrical interactions, boundary conditions, and flows are essential features of living systems. Life without flow is death and so a different approach is needed to study biology alive. The theory of complex fluids deals with interactions, boundary conditions, and flows quite well as can be seen in its successful treatment of liquid crystals. I advocate treating ionic solutions in general as complex fluids, with microelements that are the solutes and components of the solution. Enzyme active sites are a special case where some solutes are reactants. Solutes are crowded into active sites of enzyme by the high density of protein charges. The electric field links chemical reactions to charges in the protein and surrounding solutions. Interactions potentiate catalysis and control biological function. I suspect that most chemical reactions that occur in liquids also need to be treated by the theory of complex fluids. The electron movements of these reactions occur in a temporary highly concentrated fluctuation, a transient spatial inhomogeneity in the bulk solution. The electron movements of these reactions (described by quantum mechanics) are coupled to the electric (and sometimes steric) fields of the bulk solution. I suspect the electron movements, inhomogeneities, and chemical reaction (in the condensed phase) need to be treated by the theory of complex fluids because everything interacts with everything else, in this system, as in so many others.

q-bio.OT↗

arXiv2012

Living Devices

The physiological tradition of biological research analyzes biological systems using reduced descriptions much as an engineer uses a 'black box' description of an amplifier. Simple models have been used by physiologists for a very long time. Physiologists have successfully analyzed a broad range of biological systems using a 'device-oriented' approach similar to the approach an engineer would use to investigate her devices. The present generation views biology through the powerful lenses of structural and (molecular) dynamic analysis, understandably enough because of the beauty and power of the analysis, and the ease of using these structures with present freely available software. The problem is that these powerful lenses offer such magnification that the engineering approach cannot be seen. High magnification means limited field of view, because the (spatial) dynamic range cannot cover everything. The function of the structures and molecular dynamics cannot be seen in the work of many biologists, probably because function cannot be immediately seen in the structures and molecular dynamics they compute. It is just as important for biologists to measure the inputs and outputs of their systems as it is to measure their structures. It seems clear, at least to one physiologist, that this research will be catalyzed by assuming that most biological systems are devices that can be analyzed with the same strategies one would use to analyze engineering devices. Thinking today about your biological preparation as a device tells you what experiments to do tomorrow. An important task for many of us is to transmit the physiological tradition to the next generation of biophysicists to help them adapt traditional questions to the new length scales and techniques of molecular and atomic biology.

q-bio.OT↗

arXiv2012

Ion Channels, Natural Nanovalves

Ion channels are proteins with holes down their middle that control the flow of ions and electric current across otherwise impermeable biological membranes. The flow of sodium, potassium, calcium (divalent), and chloride ions have been central issues in biology for more than a century. The flow of current is responsible for the signals of the nervous system that propagate over long distances (meters). The concentration of divalent calcium ions is a 'universal' signal that controls many different systems inside cells. The concentration of divalent calcium and other messenger ions has a role in life rather like the role of the voltage in different wires of a computer. Ion channels also help much larger solutes (e.g., organic acid and bases; perhaps polypeptides) to cross membranes but much less is known about these systems. Ion channels can select and control the movement of different types of ions because the holes in channel proteins are a few times larger than the (crystal radii of the) ions themselves. Biology uses ion channels as selective valves to control flow and thus concentration of crucial chemical signals. For example, the concentration of divalent calcium ions determines whether muscles contract or not. Ion channels have a role in biology similar to the role of transistors in computers and technology. Ion Channels Control Concentrations Important To Life The Way Computers Control Voltages Important To Computers.

q-bio.BM↗