Effect of conductivity interfaces in electrocardiography

This paper develops numerical techniques for two problems of interest in electrocardiography: (1) the determination of infinite-media surface potential gradients from boundedmedia torso potentials, and (2) the effects of lung-torso, blood mass-torso, and torso-air conductivity interfaces on torso surface potentials. This paper is an extension of a previous paper on the effects of conductivity interfaces which also utilized an integral equation formulation of Laplace's equation. Supported in part by USPHS Grants HE 05716 and HE 11307 and American Heart Grant 67-850.     

Effects of the torso boundary and internal conductivity interfaces in electrocardiography: An evaluation of the ‘Infinite medium’ approximation

The analytic, eccentric spheres model of the torso was used to examine the validity of approximating the ‘infinite medium’ potential by integrating ‘finite medium potentials’ over the torso surface. Although idealized, the analytic model is sophisticated enough for all important torso conductivity and geometry parameters to be preserved in the formulation. The model generates both ‘finite medium’ potentials (for which the torso is surrounded by air) and also ‘infinite medium’ potentials (for which the outermost layer of the torso extends outward to infinity). The finite medium torso potentials were integrated over the torso surface in accordance with the approximation used by many investigators in an effort to make the surface distribution more representative of the primary cardiac sources. The resulting potential distribution was compared with the true infinite medium potential, in which the effects of internal inhomogeneities (secondary sources) were taken into account. The difference between the two representations was found to be significant, and caution should be used when interpreting such data.     

Numerical modeling of the formation of steady-state nonequilibrium distributions of particles interacting through a power-law potential

The formation of a steady-state nonequilibrium distribution function of particles interacting through the repulsive potential U ~ α/r ^β(1≤β≤4), which operates at an infinite range, is studied numerically. The collisional particle dynamics in such a system is investigated using a spatially homogeneous nonlinear collision integral in the Landau-Fokker-Planck form, which is a model Boltzmann collision integral for arbitrary potentials of interaction accompanied by little momentum transfer between particles in collisions. Numerical modeling is based on completely conservative difference schemes. It is shown that the principal condition for the existence of steady-state nonequilibrium distributions is the presence of a particle or an energy flux oriented in the proper manner in momentum space. A steady-state local distribution exists inside the momentum interval between the energy source and sink and has the form of a gradually decreasing function. Since a radical change in the distribution function under nonequilibrium conditions leads to an anomalous enhancement of the conduction of a medium and its emission characteristics, the results obtained can be used, e.g., to predict the behavior of semiconductors with an intrinsic or extrinsic conductivity under the action of particle fluxes or electromagnetic radiation.     

Modeling extracellular field potentials and the frequency-filtering properties of extracellular space

Extracellular local field potentials are usually modeled as arising from a set of current sources embedded in a homogeneous extracellular medium. Although this formalism can successfully model several properties of extracellular local field potentials, it does not account for their frequency-dependent attenuation with distance, a property essential to correctly model extracellular spikes. Here we derive expressions for the extracellular potential that include this frequency-dependent attenuation. We first show that, if the extracellular conductivity is nonhomogeneous, there is induction of nonhomogeneous charge densities that may result in a low-pass filter. We next derive a simplified model consisting of a punctual (or spherical) current source with spherically symmetric conductivity/permittivity gradients around the source. We analyze the effect of different radial profiles of conductivity and permittivity on the frequency-filtering behavior of this model. We show that this simple model generally displays low-pass filtering behavior, in which fast electrical events (such as Na(+)-mediated action potentials) attenuate very steeply with distance, whereas slower (K(+)-mediated) events propagate over larger distances in extracellular space, in qualitative agreement with experimental observations. This simple model can be used to obtain frequency-dependent extracellular field potentials without taking into account explicitly the complex folding of extracellular space.     

On shunting inhibition

The interaction between excitation and inhibition is analyzed for nerve cylinders when reversal potentials for synaptic action are included. Both impulsive and sustained conductance changes are employed to model synaptic action.Exact results, in terms of Green's functions are obtained for the solutions of the cable equation with reversal potentials when there are impulsive conductance changes. The amplification factor for an inhibitory input due to a prior excitatory input is found exactly. In the case of an infinite cylinder, the dependence of this factor on the spatial separation of the excitatory and inhibitory synapses is one plus a Gaussian density function. Similar results aply when excitation follows inhibition. There is shunting inhibition even for impulsive conductance changes in the cable, but it is very different from that for sustained conductance changes. The interaction of excitation and inhibition is also studied in the full cable equation with reversal potentials and sustained conductance changes. An exact result is obtained for the potential in response to simultaneous excitation and inhibition at the same space point in an infinite cable. The effects of timing and spatial separation of inputs is analyzed in a finite nerve cylinder by numerically integrating the cable equation by the Crank-Nicolson method. Shunting inhibition is found to be most effective, for the chosen parameter values, when inhibition quickly foolows excitation. The EPSP amplitude at the soma is found to be roughly proportional to the distance from the soma to the site of inhibition when the excitation is at the center of the nerve cylinder.Dedicated to Jane Pauley     

A porous-medium approach for modeling heart mechanics. I. theory

A method is developed for evaluating the distribution of the blood pressure, stresses and strains of the muscle fibers, and motion of the cardiac wall due to the cyclic contractions of the heart. The cardiac system is subdivided into two media: the chambers and the wall; the latter is enclosed by two impermeable surfaces (with one interface separating the two media and the other confining the wall). The momentum balance equation for the blood in the (two) cardiac ventricles is averaged, yielding a modification of Forchheimer's law, namely inclusion of the time derivative of the flux. The contracting muscle provides the driving force for the blood flow, and the endocardial velocity is thus taken as identical to that of the blood in the cavity next to the wall. Translation of the endocardium is governed by the blood pressure gradients in the ventricles. The blood pressure and stress-strain pattern in the myocardium are analyzed by applying concepts of the theory of mixtures to the blood and to the saturated solid matrix. With the blood pressure simulated by a modified Darcy law with relative fluid-solid velocity, fiber stresses and strains can be assessed with the aid of appropriate constitutive and compatibility laws.     

Growth-induced Water Potentials in Plant Cells and Tissues

A physical analysis of water movement through elongating soybean (Glycine max L. Merr.) hypocotyls was made to determine why significant water potentials persist in growing tissues even though the external water potentials were zero and transpiration is virtually zero. The analysis was based on a water transport theory modified for growth and assumed that water for growing cells would move through and along the cells in proportion to the conductivity of the various pathways.

Water potentials calculated for individual cells were nearly in local equilibrium with the water potentials of the immediate cell surroundings during growth. However, water potentials calculated for growing tissue were 1.2 to 3.3 bars below the water potential of the vascular supply in those cells farthest from the xylem. Only cells closest to the xylem had water potentials close to that of the vascular supply. Gradients in water potential were steepest close to the xylem because all of the growth-sustaining water had to move through this part of the tissue. Average water potentials calculated for the entire growing region were −0.9 to −2.2 bars depending on the tissue diffusivity.

For comparison with the calculations, average water potentials were measured in elongating soybean hypocotyls using isopiestic thermocouple psychrometers for intact and excised tissue. In plants having virtually no transpiration and growing in Vermiculite with a water potential of −0.1 bar, rapidly growing hypocotyl tissue had water potentials of −1.7 to −2.1 bars when intact and −2.5 bars when excised. In mature, nongrowing hypocotyl tissue, average water potentials were −0.4 bar regardless of whether the tissue was intact or excised.

The close correspondence between predicted and measured water potentials in growing tissue indicates that significant gradients in water potential are required to move growth-associated water through and around cells over macroscopic distances. The presence of such gradients during growth indicates that cells must have different cell wall and/or osmotic properties at different positions in the tissue in order for organized growth to occur. The mathematical development used in this study represents the philosophy that would have to be followed for the application of contemporary growth theory when significant tissue water potential gradients are present.

     

The gradient between pre-dawn rhizoplane and bulk soil matric potentials, and its relation to the pre-dawn root and leaf water potentials of four species

Uptake of soil water by plants may result in significant gradients between bulk soil and soil in the vicinity of roots. Few experimental studies of water potential gradients in close proximity to roots, and no studies on the relationship of water potential gradients to the root and leaf water potentials, have been conducted. The occurrence and importance of pre-dawn gradients in the soil and their relation to the pre-dawn root and leaf water potentials were investigated with seedlings of four species. Pre-germinated seeds were grown without watering for 7 and lid in a silt loam soil with initial soil matric potentials of -0.02, -0.1 and -0.22 MPa. Significant gradients, independent of the species, were observed only at pre-dawn soil matric potentials lower than -0.25 MPa; the initial soil matric potentials were -0.1 MPa. At an initial bulk soil matric potential of -0.22 MPa, a steep gradient between bulk and rhizoplane soil was observed after 7 d for maize (Zea mays L. cv. Issa) and sunflower (Helianthus annuus L. cv. Nanus), in contrast to barley (Hordeum vulgare L. cv. Athos) and wheat (Triticum aestivum L. cv. Kolibri). Pre-dawn root water potentials were usually about the same as the bulk soil matric potential and were higher than the rhizoplane soil matric potential. Pre-dawn root and leaf water potentials tended to be much higher than rhizoplane soil matric potentials when the latter were lower than -0.5 MPa. It is concluded that plants tend to become equilibrated overnight with the wetter bulk soil or with wetter zones in the bulk soil. Plants can thus circumvent negative effects of localized steep pre-dawn soil matric potential gradients. This may be of considerable importance for water uptake and growth in drying soil.     

Proton permeability of the plasma membrane of rat cortical synaptosomes

1. Transmembrane pH gradients (acidic inside) and electrical gradients (negative inside) were estimated in cortical synaptosomes from the distribution of the weak base methylamine and the lipophilic cation tetraphenylphosphonium, respectively. 2. Acidic interior pH gradients were produced by outwardly directed K+ gradients in Na+-free media. External K+ accelerated the dissipation of preformed H+ gradients. The appearance of H+ in the medium was directly demonstrated by pH-stat titration of a weakly buffered medium. Amiloride failed to inhibit K+-induced H+ release. 3. Elevating K+ in the absence of Na+ did not affect the endogenous contents of noradrenaline, dopamine, and serotonin, as determined by high-performance liquid chromatography with electrochemical detection. 4. H+ diffusion potentials were generated when outwardly directed H+ gradients were imposed onto the plasma membrane indicating an electrogenic H+ efflux which is not coupled to other ions. 5. At low K+ in the Na+-free sucrose medium, the plasma membrane potential Em (derived from distribution of tetraphenylphosphorium cation) did not approach a value for EK, the K+ equilibrium potential (calculated from K+ gradients). The deviation of Em from EK could be quantitatively described by a modified constant-field equation, taking a relative H+/K+ permeability coefficient of 12,400 into consideration. 6. It is concluded that synaptosomes have a H+ conductance pathway in their plasma membrane in addition to the Na+/H+ antiporter. H+ influx is driven by and leads to a reduction of Em. K+/H+ exchange resulted from the electrical coupling of K+ and H+ fluxes via parallel K+ and H+ channels. Since the Na+/H+ antiporter counteracts passive equilibration of H+ under physiological conditions, a continuous cycling of H+ across the plasma membrane will take place. A possible physiological role of the H+ leak in pHi regulation is discussed.     

A technique to measure root tip hydraulic conductivity and root water potential simultaneously

A procedure for the simultaneous measurement of hydraulic conductivityand xylem water potential of roots is presented. Roots remainintact and attached to the transpiring plant during measurement.The rate of water uptake by roots is measured at different waterpotential gradients along the root radial axis, obtained byplacing them in solutions with different osmotic potentials.Hydraulic conductivity and xylem water potential are calculatedby regression analysis of the relationship between water uptakerate and osmotic potential of the bathing solution, assumingthat xylem water potential and reflection coefficient remainconstant during measurement. Results for tomato plants experiencingdrought are presented and discussed. Key words: Root, hydraulic conductivity, water potential     

The effects of heart-cavity blood resistivity and volume on endocardial potentials: theoretical predictions

Theoretical concepts regarding the potential variations within the bloodfilled cavities of the heart caused by excitation of the myocardium are scarce. In an attempt to understand these potential variations, calculations were made of the potentials due to a dipole adjacent to a disk of finite resistivity. The dipole was assumed to be in an infinite two-dimensional medium of higher resistivity. Equations for the potential inside a sphere due to an external source are also given.On the basis of the results, predictions were made on the effects of the intracardiac blood on the endocardial potentials: (1) the potential at the center of the cavity is independent of blood resistivity and equivalent to the free space potential; (2) potentials due to a tangential dipole are uniformly reduced; (3) potentials due to a radial dipole are smaller near the dipole, but greater at the far end of the dipole axis; (4) for a dipole radial to or making an angle with the cavity, potentials are reduced at some points but increased at others; (5) the potential along the transverse axis of the dipole has a constant value depending on conductivity; (6) intracardiac blood tends to smooth out potential variations within the cavity; (7) the potential depends both on dipole distance and direction; and (8) if the cavity volume increases, potentials decrease with distance from the dipole; the potential at the adjacent endocardial wall changes little.     

Selective Electrodiffusion of Zinc Ions in a Zrt-, Irt-like Protein,ZIPB

All living cells need zinc ions to support cell growth. Zrt-, Irt-like proteins (ZIPs) represent a major route for entry of zinc ions into cells, but how ZIPs promote zinc uptake has been unclear. Here we report the molecular characterization of ZIPB from Bordetella bronchiseptica, the first ZIP homolog to be purified and functionally reconstituted into proteoliposomes. Zinc flux through ZIPB was found to be nonsaturable and electrogenic, yielding membrane potentials as predicted by the Nernst equation. Conversely, membrane potentials drove zinc fluxes with a linear voltage-flux relationship. Direct measurements of metal uptake by inductively coupled plasma mass spectroscopy demonstrated that ZIPB is selective for two group 12 transition metal ions, Zn²⁺ and Cd²⁺, whereas rejecting transition metal ions in groups 7 through 11. Our results provide the molecular basis for cellular zinc acquisition by a zinc-selective channel that exploits in vivo zinc concentration gradients to move zinc ions into the cytoplasm.     

Microoxic-Anoxic Niche of Beggiatoa spp.: Microelectrode Survey of Marine and Freshwater Strains

Beggiatoa spp. grow optimally in media containing opposed gradients of oxygen and soluble sulfide, although some strains also require an organic substrate. By using microelectrodes, we characterized oxygen and sulfide gradients during their initial development in uninoculated media and in cultures of marine and freshwater strains. In gradient media, Beggiatoa strains always grew some distance below the air/agar interface as a dense “plate” of constantly gliding filaments with sharply demarcated upper and lower boundaries. Within established plates, the maximum oxygen partial pressure was 0.6 to 6.0% of air saturation and not significantly lower if filaments were fixing nitrogen. Oxygen penetrated only 100 to 300 μm into the plate, and the anoxic fraction increased from less than 10% to approximately 90% during later stages of growth. For lithoautotrophically grown marine strains, the linearity of the oxygen profile above the plate plus its drop to zero therein indicated that oxygen uptake for the entire tube occurred only within the Beggiatoa plate. Consequently, oxygen consumption could be predicted solely from the distance between the air/agar interface and the top of a plate, given the diffusion coefficient for oxygen. By contrast, for freshwater strains grown heterotrophically (with sulfide also in the medium), oxygen profiles were frequently nonlinear because of nonbiological reaction with sulfide which had diffused past the aggregated filaments. For all strains tested, microoxic aggregation also occurred in the absence of sulfide, apparently reflecting a step-up phobic response to oxygen.     

Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling

A passive cable model is presented for a pair of electrotonically coupled neurons in order to investigate the effects of tip-to-tip dendrodendritic gap junctions on the interaction between excitation and either pre or postsynaptic inhibition. The model represents each dendritic tree by a tapered equivalent cylinder attached to an isopotential soma. Analytical solution of the cable equation with synaptic reversal potentials is considered for each neuron to yield a system of Volterra integral equations for the voltage. The solution to the system of linear integral equations (expressed as a Neumann series) is used to determine the current spread within the two coupled neurons, and to re-examine the sensitivity of the soma potentials (in particular) to the coupling resistance for various loci of synaptic inputs. The model is actually posed generally, so that active as well as passive properties could be considered. In the active case, a system of non-linear integral equations is derived for the voltage.     

Secondary transport of amino acids in Nitrosomonas europaea

Nitrosomonas europaea is capable of incorporating exogenously supplied amino acids. Studies in whole cells revealed that at least eight amino acids are actively accumulated, probably by the action of three different transport systems, each with high affinity ( molar range) for several amino acids. Evidence for the action of secondary mechanisms of transport was obtained from efflux, counterflow and exchange experiments. More detailed information was obtained from studies in liposomes in which solubilized integral membrane proteins of N. europaea were incorporated. Uptake of l-alanine in these liposomes could be driven by artificially imposed pH gradients and electrical potentials, but not by chemical sodium-ion gradients. These observations indicate that l-alanine is transported by a H⁺/alanine symport system. The ecological significance of secondary amino acid transport systems in autotrophic ammonium-oxidizing bacteria is discussed.     

Movement of Cations through Cuticles of Citrus aurantium and Acer saccharum: Diffusion Potentials in Mixed Salt Solutions

We examined some biophysical mechanisms of ion migration across leaf cuticles enzymatically isolated from Acer saccharum L. and Citrus aurantium L. leaves. Diffusion potential measurements were used to calculate the permeabilities of Cl^-, Li⁺, Na⁺, and Cs⁺ ions all as a ratio with respect to the permeability of K⁺ in cuticles. In 2 millimolar ionic strength solutions the permeability sequence from high to low was K = Cs > Na > Li » Cl. When the outer and inner surfaces of cuticles were bathed in artificial precipitation and artificial apoplast, respectively, diffusion potentials ranging from −52 to −91 millivolts were measured (inside negative). The Goldman equation predicted that the measured potentials were enough to increase the driving force on the accumulation of heavy metals by a factor of 4 to 7. Other ions migrate with forces 3 to 10 times less than predicted by the Goldman equation for concentration differences alone. Our analysis showed that Ca²⁺, and perhaps Mg²⁺, might even be accumulated against concentration gradients under some circumstances. Their uptake was apparently driven by the diffusion potentials created by the outward migration of monovalent salts. We feel that future models predicting leaching of nutrients from trees during acid rain events must be modified to account for the probable influence of diffusion potentials on ion migration.     

Proton flux in large unilamellar vesicles in response to membrane potentials and pH gradients.

The transport of protons across liposomes composed of phosphatidylcholine in response to electrical potentials or pH gradients has been investigated. The results support three major conclusions. The first of these concerns the need for reliable measurements of electrical potentials and pH gradients. It is shown that the potential probe tetraphenylphosphonium and the pH probe methylamine provide accurate and self consistent measures of electrical potentials and pH gradients respectively in these systems. Second, it is shown by two independent techniques that the pH gradients induced in response to valinomycin and potassium dependent electrical potentials are significantly smaller than would be expected for electrochemical equilibrium. The pH gradients observed are stable over an 8 h time course and are sensitive to the ionic composition of the buffers employed, where the presence of external sodium results in the smallest induced pH gradients. These results are discussed in terms of current models of proton conductance across membranes. In a final area of investigation, it is shown that valinomycin and carbonyl cyanide m-chlorophenyl hydrazone (CCCP) can transport sodium ions in a synergistic manner.     

Motion of nanobeads proximate to plasma membranes during single particle tracking

Drag and torque on nanobeads translating within the pericellular layer while attached to glycolipids of the plasma membrane are calculated by a novel hydrodynamic model. The model considers a bead that translates proximate to a rigid planar interface that separates two distinct Brinkman media. The hydrodynamic resistance is calculated numerically by a modified boundary integral equation formulation, where the pertinent boundary conditions result in a hybrid system of Fredholm integrals of the first and second kinds. The hydrodynamic resistance on the translating bead is calculated for different combinations of the Brinkman screening lengths in the two layers, and for different viscosity ratios. Depending on the bead-membrane separation and on the hydrodynamic properties of both the plasma membrane and the pericellular layer, the drag on the bead may be affected by the properties of the plasma membrane. The Stokes-Einstein relation is applied for calculating the diffusivity of probes (colloidal gold nanobeads attached to glycolipids) in the plasma membrane. This approach provides an alternative way for the interpretation of in vitro observations during single particle tracking procedure, and predicts new properties of the plasma membrane structure.     

Approach for an equation of state for adsorbed protein surfaces.

A surface equation of state for polymer interfaces is obtained by equating the chemical potentials of the solvent in the bulk and surface phases. This equation of state contains the fraction of the surface covered with polymer and the surface activity coefficient as parameters. These parameters may be obtained by measuring the thermodynamic modulus of elasticity. Moreover, if the amount of protein in the interface is known the degree of folding in the surface may be evaluated. The present theory was applied to adsorbed beta-lactoglobulin in surfaces. The results are in agreement with qualitative equations and results from other sources.     

Characterization of H+/OH− currents in phospholipid vesicles

In pure phospholipid vesicles, the conductivity of H⁺/OH^– ions exceeds that for other simple inorganic ions. Protons achieve electrochemical equilibrium across egg phosphatidylcholine vesicles within tens of minutes. When pH gradients are established across vesicles, transmembrane potentials develop. Conversely, the establishment of transmembrane potentials leads to the formation of pH gradients. When the phenomenological permeability of H⁺/OH^– ions in vesicles is estimated, values are obtained that are much greater (six orders of magnitude larger) than those for Na⁺ or K⁺. A wide range in the values for this permeability has been reported; however, much of the discrepancy can be attributed to differences in the vesicle systems and experimental conditions. The H⁺/OH^– current appears to be modulated by changes in membrane dielectric constant. However, the dependence of this current on the pH gradient and on the membrane voltage argues against simple diffusion mechanisms as the source of the H⁺/OH^– current. In addition, in vesicle systems the H⁺/OH^– current shows a surprising invariance to changes in the membrane dipole potential, an observation that argues against the role of simple carriers for H⁺ and OH^– ions.