Phase transitions and ion currents in a model ferroelectric channel unit

The hypothesis of ferroelectric electrodiffusion is examined mathematically. A thermodynamic potential, the elastic Gibbs function, written in polynomial form, provides the dielectric equation of state for the model. The other equations of electrodiffusion theory complete the model. This system reduces to a second-order partial differential equation, which is formally solved by the phase-plane method. This solution, applied to the Na channel, leads to a propagating phase-transition wave accompanied by movement of ionic charge. This may be readily interpreted as a transmembrane wave traveling along a ferroelectric unit within, and transporting ions through, the channel. Comparison of the temperature dependence of axonal conduction velocity with that of the spontaneous polarization of Rochelle salt suggests that the Na channel of squid axon contains a ferroelectric unit having a lower Curie point, but decomposing before reaching its upper Curie point. Comparison with data from reconstitution experiments suggests that the ferroelectric unit is a carbohydrate enclosed in an intrinsic protein structure to form a glycoprotein channel. The value experimentally estimated for the surface charge of the Na channel is within the range of spontaneous polarizations of typical ferroelectric crystals. It is argued that the ferroelectric probably is a single crystal of the order-disorder type, which undergoes a first-order transition between a ferroelectric and a paraelectric state during excitational activity. The hypothesis of ferroelectric channel units is consistent with the existence and directionality of the observed "gating" currents.     

Computational modeling of three-dimensional electrodiffusion in biological systems: application to the node of Ranvier

A computational model is presented for the simulation of three-dimensional electrodiffusion of ions. Finite volume techniques were used to solve the Poisson-Nernst-Planck equation, and a dual Delaunay-Voronoi mesh was constructed to evaluate fluxes of ions, as well as resulting electric potentials. The algorithm has been validated and applied to a generalized node of Ranvier, where numerical results for computed action potentials agree well with cable model predictions for large clusters of voltage-gated ion channels. At smaller channel clusters, however, the three-dimensional electrodiffusion predictions diverge from the cable model predictions and show a broadening of the action potential, indicating a significant effect due to each channel's own local electric field. The node of Ranvier complex is an elaborate organization of membrane-bound aqueous compartments, and the model presented here represents what we believe is a significant first step in simulating electrophysiological events with combined realistic structural and physiological data.     

Kinetic Theory Model for Ion Movement through Biological Membranes: III. Steady-State Electrical Properties with Solution Asymmetry

An electrodiffusion model for plasma membrane ion transport, which takes into account the influence of high electric field strengths and ion-membrane molecule interactions, is presented and analyzed. A generalized Nernst-Planck equation for steady-state situations is derived which has electric field-dependent mobility and diffusion coefficients. Under the assumption of a constant electric field within the membrane, this equation is integrated to give a more general form of the Goldman equation. Based on this equation numerical computations of ionic chord conductance as a function of applied electric field strength were carried out for several permeant ion concentration ratios. The model is capable of yielding significantly larger rectification ratios than is the Goldman equation. Further, high field asymptotes to the current vs. electric field strength curve do not generally intersect at the origin.     

Concentration wave of a solute in an artery: the influence of curvature

Mass transport and diffusion phenomena in the arterial lumen are studied through a mathematical model. Blood flow is described by the unsteady Navier-Stokes equation and solute dynamics by an advection-diffusion equation, the convective field being provided by the fluid velocity. A linearization procedure over the steady state solution is carried out and an asymptotic analysis is used to study the effect of a small curvature with respect to the straight tube. Analytical and numerical solutions are found: the results show the characteristics of the long wave propagation and the role played by the geometry on the solute distribution and demonstrate the strong influence of curvature induced by the fluid dynamics.     

Interpretation of the Ussing flux ratio from the fluctuation theorem

The fluctuation theorem gives a mathematical expression to quantify the probability of observing events violating the second law of thermodynamics in a small system over a short period of time. The theorem predicts the ratio of forward (entropy-producing) runs to the backward (entropy-consuming) runs for a nanometer-sized molecular machine in a nonequilibrium system. However, few experimental verifications of the theorem have been carried out. In this paper, I show that the Ussing flux ratio, the ratio of outward to inward unidirectional ion fluxes across a membrane channel, can be derived from the fluctuation theorem if we consider the ion channel and the contacting solutions as a small nonequilibrium system. The entropy change due to ion electrodiffusion is expressed from the fundamental equation for the entropy change. Thus, the empirical flux ratio equation can be interpreted from the more general fluctuation theorem, and serves as a verification of the theorem.     

Application of the Poisson-Nernst-Planck theory with space-dependent diffusion coefficients to KcsA

The Poisson-Nernst-Planck electrodiffusion theory serves to compute charge fluxes and is here applied to the ion current through a protein channel. KcsA was selected as an example because of the abundance of experimental and theoretical data. The potassium channels MthK and KvAP were used as templates to define two open channel models for KcsA. Channel boundary surfaces and protein charge distributions were defined according to atomic radii and partial atomic charges. To establish the sensitivity of the results to these parameters, two different sets were used. Assigning the potassium diffusion coefficients equal to the value for free-diffusion in water (1.96 x 10(-9) m(2)/s), the computed currents overestimated the experimental data. Ion distributions inside the channel suggest that the overestimate is not due to an excess of charge shielding. A good agreement with the experimental data was achieved by reducing the potassium diffusion coefficient inside the channel to 1.96 x 10(-10) m(2)/s, a value of substantial motility but nonetheless in accord with the intuitive notion that the channel has a high affinity for the ions and therefore slows them down. These results are independent of the open channel model and the parameterization adopted for atomic radii and partial atomic charges. The method offers a reliable estimate of the channel current with low computational effort.     

Population persistence under advection–diffusion in river networks

An integro-differential equation on a tree graph is used to model the time evolution and spatial distribution of a population of organisms in a river network. Individual organisms become mobile at a constant rate, and disperse according to an advection-diffusion process with coefficients that are constant on the edges of the graph. Appropriate boundary conditions are imposed at the outlet and upstream nodes of the river network. The local rates of population growth/decay and that by which the organisms become mobile, are assumed constant in time and space. Imminent extinction of the population is understood as the situation whereby the zero solution to the integro-differential equation is stable. Lower and upper bounds for the eigenvalues of the dispersion operator, and related Sturm-Liouville problems are found. The analysis yields sufficient conditions for imminent extinction and/or persistence in terms of the values of water velocity, channel length, cross-sectional area and diffusivity throughout the river network.     

A new current expression for selective ion permeation across membranes

Classical Nernst-Planck electrodiffusion approach to selective ion permeation through membranes, remains a currently popular approach to the problem. Some attempts have, however, been made to use Eyring's theory to explain selective ion permeation through the trans-membrane channels. The common problem with either approach is that the final expression for ionic current does not have selectivity inbuilt, rather selectivity is introduced in the current equation as an arbitrarily chosen parameter. In addition, the negative conductance region is non-reproduceable. To the best of our knowledge, we describe, for the first time in the literature, a selective ion current equation which, in addition, also has inbuilt negative conductance property within the framework of a kinematic approach—to a level of first degree approximation.     

Sensitivity analysis of the Poisson Nernst–Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model

Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst–Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.

     

Linearity, saturation and blocking in a large multiionic channel: divalent cation modulation of the OmpF porin conductance

Measurement of unitary conductance is a fundamental step in the characterization of a protein ion channel permeabilizing a membrane. We study here the effect of salts of divalent cations on the OmpF channel conductance with a particular emphasis in dissecting the role of the electrolyte itself, the role of the counterion accumulation induced by the protein channel charges and other effects not found in salts of monovalent cations. We show that current saturation and blocking are not exclusive properties of narrow (single-file) ion channels but may be observed in large, multiionic channels like bacterial porins. Single-channel conductance measurements performed over a wide range of salt concentrations (up to 3 M) combined with continuum electrodiffusion calculations demonstrate that current saturation cannot be simply ascribed to ion interaction with protein channel residues.     

Simple model of ion transport through alamethicin channels in lipid membranes

The electrical characteristics of wide membrane channels such as those induced in lipid membranes by alamethicin have been analyzed using an electrodiffusion model. The channel is considered to be a water filled cylinder in which the potential energy barrier is a result of the difference in polarization energy of the ion environment when the ion is located inside as compared to outside of the channel. In addition, an electric field related to the channel structure is assumed. It is shown that without postulating any specific chemical ion-channel interaction one can reproduce experimental membrane potentials for NaCl, KCl, and CaCl₂ concentration gradients with a single set of channel parameters. The calculations also yield experimental J-V characteristics of discrete conduction states. In addition, a simple mechanism of interchannel coupling based on the above model is discussed. The model suggests a unifying approach to the problem of the origin of interionic selectivity of membrane channels induced by polyene antibiotics.     

Steady-state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane.

This is the first of two papers dealing with electrodiffusion theory (the Nernst-Planck equation coupled with Gauss's law) and its application to the current-voltage behavior of squid axon. New developments in the exact analysis of the steady-state electrodiffusion problem presented here include (a) a scale transformation that connects a given solution to an infinity of other solutions, suggesting the po-sibility of direct comparison of electrical data for membranes with different thicknesses and other properties; (b) a first-integral relation between the electric field and ion densities more general than analogous relations previously reported, and (c) an exact solution for the homovalent system, i.e., a membrane system permeated by various ion species of the same charge. The latter is a generalization of the known one-ion solution. The properties of the homovalent solution are investigated analytically and graphically. In particular we study the phase-plane curves, which reduce to the parabolas discussed by K. S. Cole in the special case in which the current-density parameter (a linear combination of the ionic current densities) is zero.     

Improved 3D continuum calculations of ion flux through membrane channels

A continuum model, based on the Poisson–Nernst–Planck (PNP) theory, is applied to simulate steady-state ion flux through protein channels. The PNP equations are modified to explicitly account (1) for the desolvation of mobile ions in the membrane pore and (2) for effects related to ion sizes. The proposed algorithm for a three-dimensional self-consistent solution of PNP equations, in which final results are refined by a focusing technique, is shown to be suitable for arbitrary channel geometry and arbitrary protein charge distribution. The role of the pore shape and protein charge distribution in formation of basic electrodiffusion properties, such as channel conductivity and selectivity, as well as concentration distributions of mobile ions in the pore region, are illustrated by simulations on model channels. The influence of the ionic strength in the bulk solution and of the externally applied electric field on channel properties are also discussed.     

Bistable nerve conduction

It has been demonstrated experimentally that slow and fast conduction waves with distinct conduction velocities can occur in the same nerve system depending on the strength or the form of the stimulus, which give rise to two modes of nerve functions. However, the mechanisms remain to be elucidated. In this study, we use computer simulations of the cable equation with modified Hodgkin-Huxley kinetics and analytical solutions of a simplified model to show that stimulus-dependent slow and fast waves recapitulating the experimental observations can occur in the cable, which are the two stable conduction states of a bistable conduction behavior. The bistable conduction is caused by a positive feedback loop of the wavefront upstroke speed, mediated by the sodium channel inactivation properties. Although the occurrence of bistable conduction only requires the presence of the sodium current, adding a calcium current to the model further promotes bistable conduction by potentiating the slow wave. We also show that the bistable conduction is robust, occurring for sodium and calcium activation thresholds well within the experimentally determined ones of the known sodium and calcium channel families. Since bistable conduction can occur in the cable equation of Hodgkin-Huxley kinetics with a single inward current, i.e., the sodium current, it can be a generic mechanism applicable to stimulus-dependent fast and slow conduction not only in the nerve systems but also in other electrically excitable systems, such as cardiac muscles.     

Recovery of the parameters of a lightning current pulse from measurements of the electromagnetic field in the far wave zone

A numerical model of the return stroke of a downward lightning is used to quantitatively analyze the error in recovering the parameters of a lightning current pulse from the electromagnetic field recorded in the far wave zone. It is shown that the error is mainly caused by the deformation of the current wave propagating along the plasma channel from the ground to the cloud and the dependence of the wave phase velocity on the current, rather than by the deviation of lighting from the vertical. The recovery error caused by the nonlinear dependence of the recorded magnetic field on the excitation current can reach a few hundred percent. Results of direct measurements of the lightning current at high-rise buildings and during trigger lightnings cannot be used to calibrate systems for remote monitoring of thunderstorm activity, because radiation from current-carrying metal structures substantially reduces the error caused by the damping of the return stroke wave in the plasma channel.     

Frequency relations of capacitance and conductance of the system electrolyte--membrane--electrolyte in a model framework]

An electrodiffusion model of immediate permeability of ions through the lipid membrane was considered. The model suggests the existence of finite immovable layers of the electrolyte near the membrane. A technique is given for linearization and obtaining analytical solutions of the time-dependent electrodiffusion equations for one ion species. The expression for admittence of the system was obtained. The model allowed to obtain the curves of c (omega) and g (omega) which agreed with the empiric ones.     

Particle transport modeling in pulmonary airways with high-order elements

Inhaled particles can be either harmful (e.g., smoke, exhaust, viruses) or beneficial (e.g., a therapeutic drug). The accurate and computationally efficient simulation of particle transport and deposition remains a challenge because it requires the simultaneous solution of the Navier-Stokes equations and multiple advection-diffusion mass transport equations when the particles are modeled as multiple mono-dispersed populations. The solution of these equations requires that multiple length scales be resolved since the ratio of advection to diffusion varies among the different equations. Here, the spectral element method is examined because the high-order approximation provides greater flexibility in resolving multiple length scales. The problem geometry is based on the Weibel model A of the human airway for convergence tests and the first three generations of a typical rat airway for experimental validation. Particles in the size range 1 to 100 nm are simulated for deposition results. The particle concentration and flux were determined using meshes of varying coarseness to represent the geometry along with basis polynomials of order 5 to 11. The higher-order elements accurately propagate the short wavelengths contained in the advection-diffusion solution without sacrificing efficiency for the more computationally expensive Navier-Stokes solution. As the diffusion coefficient in the advection-diffusion equation decreases (i.e., particle size increases) the advantages of the spectral elements become apparent for the coupled system.     

Different ionic selectivities for connexins 26 and 32 produce rectifying gap junction channels

The functional diversity of gap junction intercellular channels arising from the large number of connexin isoforms is significantly increased by heterotypic interactions between members of this family. This is particularly evident in the rectifying behavior of Cx26/Cx32 heterotypic channels (. Proc. Natl. Acad. Sci. USA. 88:8410-8414). The channel properties responsible for producing the rectifying current observed for Cx26/Cx32 heterotypic gap junction channels were determined in transfected mouse neuroblastoma 2A (N2A) cells. Transfectants revealed maximum unitary conductances (gamma(j)) of 135 pS for Cx26 and 53 pS for Cx32 homotypic channels in 120 mM KCl. Anionic substitution of glutamate for Cl indicated that Cx26 channels favored cations by 2.6:1, whereas Cx32 channels were relatively nonselective with respect to charge. In Cx26/Cx32 heterotypic cell pairs, the macroscopic fast rectification of the current-voltage relationship was fully explained at the single-channel level by a rectifying gamma(j) that increased by a factor of 2.9 as the transjunctional voltage (V(j)) changed from -100 to +100 mV with the Cx26 cell as the positive pole. A model of electrodiffusion of ions through the gap junction pore based on Nernst-Planck equations for ion concentrations and the Poisson equation for the electrical potential within the junction is developed. Selectivity characteristics are ascribed to each hemichannel based on either pore features (treated as uniform along the length of the hemichannel) or entrance effects unique to each connexin. Both analytical GHK approximations and full numerical solutions predict rectifying characteristics for Cx32/Cx26 heterotypic channels, although not to the full extent seen empirically. The model predicts that asymmetries in the conductance/permeability properties of the hemichannels (also cast as Donnan potentials) will produce either an accumulation or a depletion of ions within the channel, depending on voltage polarity, that will result in rectification.