Computation of Impulse Conduction in Myelinated Fibers; Theoretical Basis of the Velocity-Diameter Relation

For myelinated fibers, it is experimentally well established that spike conduction velocity is proportional to fiber diameter. However no really satisfactory theoretical treatment has been proposed. To treat this problem a theoretical axon was described consisting of lengths of passive leaky cable (internode) regularly interrupted by short isopotential patches of excitable membrane (node). The nodal membrane was assumed to obey the Frankenhaeuser-Huxley equations. The explicit diameter dependencies of the various parameters were incorporated into the equations. The fiber diameter to axon diameter ratio was taken to be constant, and the internode length was taken to be proportional to the fiber diameter. Both these conditions reflect the situation that exists in real, experimental fibers. Dimensional analysis shows that these anatomical conditions are equivalent to Rushton's (1951) assumption of corresponding states. Hence, conduction velocity will be proportional to fiber diameter, in complete agreement with the experimental findings. Digital computer solutions of these equations were made in order to compute a set of actual velocities. Computations made with constant internode length or constant myelin thickness (i.e., nonconstant fiber diameter to axon diameter ratio) did not show linearity of the velocity-diameter relation.     

Multiple membrane systems as biological models

Summary The current-voltage equations for double, triple, and quadruple membrane systems are derived in closed form from the flow equations of irreversible thermodynamics. Numerical examples show that the behavior of these systems is very similar to that of nerve and muscle membranes. Multiple membrane systems exhibit resting potentials which do not have a characteristic Nernst concentration dependence; nonpermeant ions play a significant role in this nonlogarithmic behavior. Furthermore, multiple membrane systems have rectification properties similar to those of biological membranes. The direction of rectification is determined by the polarity of the membrane systems, not by the ionic concentrations in the bathing solutions.     

Nigericin-induced charge transfer across membranes.

The electric properties of the bilayer lecithin membranes have been studied in the presence of the antibiotic nigericin. When the antibiotic concentration is about 10(-7) ohm-1 cm-2. The potassium ion concentration gradient gives rise to a transmembrane potential of the order of 40 mV per 10-fold concentration gradient with the side of the higher potassium concentration negative. The transmembrane potential produced by the hydrogen ion concentration gradient is a function of the potassium ion concentration which is equal on both sides of the membrane. For low potassium ion concentrations the hydrogen potential has the expected polarity with the solution having higher concentration of protons negative. For potassium ion concentrations exceeding 0.03 M the hydrogen potential has the reverse polarity. This unexpected result cannot be accounted for in terms of the available simple hypotheses about the charge transport mechanism for nigericin in BLM. In order to account for the experimental results obtained, a theoretical approach has been developed based on the assumption that charge is transported across the membrane by nigericin dimers. The theoretical predicitons are in satisfactory agreement with the experimental results. The model also yields some predictions which may be verified in future experiments.     

Liquid membrane potential in nonisothermal systems.

Electrical membrane potential equations for liquid ion exchange membranes, characterized by the presence of uncharged associated species and by exclusion of co-ions (no electrolyte uptake) have been derived. The irreversible thermodynamic theories already developed for solid membranes with fixed charged site density have been extended to include the different physicochemical aspects of the liquid membranes. To this purpose the dissipation function has been written with reference to the fluxes of all the species present in the membrane. It has been found that the mobile charged site, the counterions, and the uncharged associated species contribute to the electrical membrane potential through their phenomenological coefficients. The electrical membrane potential equations have been integrated in isothermal and nonisothermal conditions for monoionic and biionic systems. The theoretical predictions have been experimentally tested by studying the electrical potential of liquid membranes formed with solutions of tetraheptylammonium salts in omicron-dichlorobenzene.     

Nonlinear cable equations for axons. II. Computations and experiments with external current electrodes

We have investigated the steady-state potential and current distributions resulting from current injection into a close-fitting channel into which a squid axon is placed. Hybrid computer solutions of the cable equations, using the Hodgkin-Huxley equations to give the membrane current density, were in good agreement with experimental observations. A much better fit was obtained when the Hodgkin-Huxley leakage conductance was reduced fivefold.     

Ion flux across lipid bilayer membranes with charged surfaces

Summary In this paper we derive expressions for the ion flux across lipid bilayer membranes with charged surfaces treating the membrane as a continuous phase interposed between two electrolyte solutions and calculating the ion flux with the Nernst-Planck equations. The theoretical results are compared with experiments of Seufert and Hashimoto on lipid bilayer membranes with charged surface active agents added to the membranes. If the charge of both membrane surfaces has the same sign the flux of the gegenions is greatly increased whereas the flux of the coions decreases to a small amount. For oppositely charged membrane surfaces the membrane behaves like a np semiconductor and typical rectification voltage-current characteristics are obtained.     

Models of active transport of ions in biomembranes of various types of cells

Nonequilibrium statistical models of the active transport of ions in biomembranes have been constructed. Differences of chemical potentials of the ATP-ADP reaction and the electrochemical potential of ions were taken as the thermodynamic forces responsible for the flow of ions through the membrane. The active transport of ions was viewed as a cross phenomenon arising from the chemical reaction of the ATP hydrolysis. These models provide independent calculations of the resting potential at the biomembrane and concentrations of ions in a cell on the assumption the free energy of the ATP-ADP reaction is fully (without the dissipation loss) converted to the free energy of transported ions. They take into account the presence of nonpenetrating ions in a cell. It was shown that different concentrations of nonpenetrating ions have a considerable effect on the resting potential. The proposed models were compared with experimental data obtained for different types of cells including neurons, muscular cells, bacteria, plants, and mitochondria. Calculated values of the membrane potential and ion concentrations were in good qualitative agreement with experimental data.     

Model of Active Transport of Ions in Archaea Cells

A mathematical model of the active transport of main ions in cells of archaebacteria has been constructed. A set of equations has been developed and solved for ion fluxes through the bacterium membrane. The model is based on the principle “one ion—one transport system.” Considering experimental data, the major transport mechanism was determined for each ion and the balance equation was written on the basis of this mechanism in the stationary state. This allowed calculating values of the membrane potential and intracellular concentrations of the ions independently. The calculated values of the intracellular concentrations and resting potential are in qualitative agreement with the corresponding experimental values for cells of extremely halophilic archaea.     

A bridged Mo(VI) complex containing six and five coordinate Mo: Synthesis, 95Mo N.m.r. spectroscopy and X-ray crystal structure of Mo2O5(C17H17N2O)2

The kinetics of the reaction between Carcinus maenas hemocyanin and cyanide has been studied at various KCN concentrations and a different temperatures (21° and 4°C) by following the decrease of the copper-peroxide absorption band, centered at 337 nm, of the copper still bound to the protein and the intrinsic fluorescence changes as functions of time. In all conditions used, the absorption band completely disappears and KCN concentration affects only the rate of the process. The reaction is kinetically homogeneous indicating no site-site interaction. The apparent rate constant increases with the square of cyanide concentration and the inverse of O₂ concentration. The copper still bound decreases at a rate slower than the 337 nm absorption and the process is not kinetically homogeneous. The fluorescence of the protein increases after an induction period showing an inflection point at about 50% of the total effect. A kinetic model has been proposed on the assumption that the two metal ions are removed sequentially from the active site. The experimental data are in agreement with the theoretical equations derived from the model. The equilibrium constants for the formation of the complex between the first and the second copper ion with cyanide and the rate constants of their decomposition have been calculated. The rate-limiting process for the removal of the second copper ion is the formation of the complex with cyanide.     

RECTIFICATION AND INDUCTANCE IN THE SQUID GIANT AXON

Previous measurements have shown that the electrical properties of the squid axon membrane are approximately equivalent to those of a circuit containing a capacity shunted by an inductance and a rectifier in series. Selective ion permeability of a membrane separating two electrolytes may be expected to give rise to the rectification. A quasi-crystalline piezoelectric structure of the membrane is a plausible explanation of the inductance. Some approximate calculations of behavior of an axon with these membrane characteristics have been made. Fair agreement is obtained with the observed constant current subthreshold potential and impedance during the foot of the action potential. In a simple case a formal analogy is found between the calculated membrane potential and the excitability defined by the two factor formulations of excitation. Several excitation phenomena may then be explained semi-quantitatively by further assuming the excitability proportional to the membrane potential. Some previous measurements and subthreshold potential and excitability observations are not consistent with the circuit considered and indicate that this circuit is only approximately equivalent to the membrane.     

Nigericin-induced charge transfer across membranes

Summary The electric properties of the bilayer lecithin membranes have been studied in the presence of the antibiotic nigericin. When the antibiotic concentration is about 10^–6 m the conductivity of the BLM is increased up to 10^–7 ohm^–1 cm^–2. The potassium ion concentration gradient gives rise to a transmembrane potential of the order of 40 mV per 10-fold concentration gradient with the side of the higher potassium concentration negative. The transmembrane potential produced by the hydrogen ion concentration gradient is a function of the potassium ion concentration which is equal on both sides of the membrane. For low potassium ion concentrations the hydrogen potential has the expected polarity with the solution having higher concentration of protons negative. For potassium ion concentrations exceeding 0.03m the hydrogen potential has the reverse polarity. This unexpected result cannot be accounted for in terms of the available simple hypotheses about the charge transport mechanism for nigericin in BLM. In order to account for the experimental results obtained, a theoretical approach has been developed based on the assumption that charge is transported across the membrane by nigericin dimers. The theoretical predictions are in satisfactory agreement with the experimental results. The model also yields some predictions which may be verified in future experiments.     

Nonlinear cable equations for axons. I. Computations and experiments with internal current injection

Steady-state potential and current distributions resulting from internal injection of current in the squid giant axon have been measured experimentally and also computed from nonlinear membrane cable equation models by numerical methods, using the Hodgkin-Huxley equations to give the membrane current density. The solutions obtained by this method satisfactorily reproduce experimental measurements of the steady-state distribution of membrane potential. Computations of the input current-voltage characteristic for a nonlinear cable were in excellent agreement with measurements on axons. Our results demonstrate the power of Cole's equation to extract the nonlinear membrane characteristics simply from measurement of the input resistance.     

THE CONDUCTIVITIES OF AQUEOUS SOLUTIONS OF GLYCINE,d,l-VALINE,AND l-ASPARAGINE

1. The conductivities of aqueous solutions of glycine, d,l-valine, and l-asparagine have been determined, and comparisons have been made with similar data reported in the literature. 2. On the basis of certain theoretical considerations, calculations of the expected conductivities of aqueous solutions of glycine, asparagine, aspartic acid, and glutamic acid have been made and these data have been compared with similar data obtained experimentally. 3. The dissociation constants of the carboxyl groups of aspartic acid and glutamic acid have been calculated from conductivity data. 4. It is shown that alanine has no effect on the ionic atmosphere of solutions of potassium chloride.     

Membrane transport of electrolyte ions and time-dependent membrane potential

Equations are derived for the transport of a symmetrical electrolyte, consisting of cations and anions of equal valency, through a neutral membrane that separates two solutions of finite volume under quasi-steady-state conditions. The time-dependent membrane potential produced by the flow of ions is taken into account. Deviation of the time course of the solute concentrations from that of neutral solutes is found to be determined by the permeability ratio of cations and anions (when this ratio equals unity, the derived membrane transport equations reduce to those for neutral substances). Simple approximate expressions for the solute concentrations and of the membrane potential as functions of time are proposed, which are in excellent agreement with the exact numerical results.     

Capacitance of bilayers in the presence of lipophilic ions

The capacitance of glycerolmonooleate and egg phosphatidylcholine bilayer membranes in the presence of NaCl solutions containing tetraphenylborate, tetraphenylarsonium or dipicrylamine ions has been measured using alternating current techniques over a wide range of frequencies (1–200 kHz). The concentrations of ions corresponded to the lower limits of conductance saturation. Similar determinations were also made with solutions containing no lipophilic ions. The experimental method used in this work requires correction of admittance measurements for the solution resistance in series with the membrane, as well as careful area determinations. In all cases membrane capacitance levels off at sufficiently high frequencies to values which are independent of frequency. The high-frequency capacitance, which is regarded as the ‘geometrical capacitance’ due to dielectric polarization, is practically unaffected by the presence of lipophilic ions. The results support the assumption made in other studies, such as in charge pulse investigations, that the adsorption of lipophilic ions at concentrations up to the saturation range does not have an important effect on the dielectric properties of bilayers.     

A method to relate steady-state ionic currents, conductances, and membrane potential in ion exchange membranes with unknown thermodynamic properties

A method is presented by which the steady-state properties of an homogeneous, permselective membrane at uniform temperature can be predicted without knowledge of its thermodynamic properties other than assuming that they are functions only of local mole fractions in the membrane. By making this assumption, it is shown how the ionic conductances can be calculated at any point in the membrane from two sets of measurements, (a) R(symm), the steady-state resistance of the membrane measured between identical solutions and (b) V(0), the potential difference between nonidentical solutions for zero current. These two parameters are measured at different external solution compositions (e.g. a varying sodium-potassium ratio ranging from zero to infinity). From these measurements it is shown how the flux equations may be integrated without a knowledge of mobilities, activity coefficients, and other interior membrane parameters. The application of the method to fixed site membranes with variable mobilities is described and the theory for this particular case has also been verified experimentally in glass membranes.1 A possible application to biological membranes is discussed and a comparison is made between the present treatment and previous treatments used to calculate the steady-state properties of cell membranes, notably the theory of Teorell, Meyer, and Sievers and the constant field theory.     

Interferometric investigation of near-membrane diffusion layers

This paper presents an interferometric method of the investigation of near-membrane diffusion layers. With the aid of this method a concrete investigation was made of such layers formed in the neighbourhood of a horizontally situated membrane which separates solutions of different concentrations. On the basis of interferograms obtained, a computer analysis of these interferograms is made. This permitted to obtain, among others, curves of the distribution of solution concentrations within these layers. These curves are next compared with curves made on the basis of equations given in the paper [16]. A satisfactory compatibility between the results of the experimental and theoretical investigations are obtained.     

Foundations of modeling in cryobiology—I: Concentration,Gibbs energy,and chemical potential relationships

Mathematical modeling plays an enormously important role in understanding the behavior of cells, tissues, and organs undergoing cryopreservation. Uses of these models range from explanation of phenomena, exploration of potential theories of damage or success, development of equipment, and refinement of optimal cryopreservation/cryoablation strategies. Over the last half century there has been a considerable amount of work in bio-heat and mass-transport, and these models and theories have been readily and repeatedly applied to cryobiology with much success. However, there are significant gaps between experimental and theoretical results that suggest missing links in models. One source for these potential gaps is that cryobiology is at the intersection of several very challenging aspects of transport theory: it couples multi-component, moving boundary, multiphase solutions that interact through a semipermeable elastic membrane with multicomponent solutions in a second time-varying domain, during a two-hundred Kelvin temperature change with multi-molar concentration gradients and multi-atmosphere pressure changes. In order to better identify potential sources of error, and to point to future directions in modeling and experimental research, we present a three part series to build from first principles a theory of coupled heat and mass transport in cryobiological systems accounting for all of these effects. The hope of this series is that by presenting and justifying all steps, conclusions may be made about the importance of key assumptions, perhaps pointing to areas of future research or model development, but importantly, lending weight to standard simplification arguments that are often made in heat and mass transport. In this first part, we review concentration variable relationships, their impact on choices for Gibbs energy models, and their impact on chemical potentials.     

Anomalous Reactances in Electrodiffusion Systems

A frequency response analysis of a constrained diffusion boundary has been made by linearizing the Nernst-Planck equations for a small applied AC current. The number of time constants and their dependence on ionic concentrations and electric field as well as membrane parameters such as dielectric constant, thickness, etc. have been evaluated by this method. Numerical solutions have been carried out for cases when the Planck charging time can be neglected and the results are presented in the form of impedance loci. These impedance loci show that if the membrane separates two univalent electrolytes with a common anion it will exhibit a combined capacitative inductive response with a 90° phase angle. The dependence of these anomalous reactances on ionic concentrations and the electric field is consistent with the behavior of the Hodgkin-Huxley axon suggesting that a homogeneous electrodiffusion regime could be adequate as a basic model for the kinetic behavior of biological membranes.     

Kinetic analysis of enzyme systems with suicide substrate in the presence of a reversible competitive inhibitor, tested by simulated progress curves

The use of suicide substrates remains a very important and useful method in enzymology for studying enzyme mechanisms and designing potential drugs. Suicide substrates act as modified substrates for the target enzymes and bind to the active site. Therefore the presence of a competitive reversible inhibitor decreases the rate of substrate-induced inactivation and protects the enzyme from this inactivation. This lowering on the inactivation rate has evident physiological advantages, since it allows the easy acquisition of experimental data and facilitates kinetic data analysis by providing another variable (inhibitor concentration). However despite the importance of the simultaneous action of a suicide substrate and a competitive reversible inhibition, to date no corresponding kinetic analysis has been carried out. Therefore we present a general kinetic analysis of a Michaelis-Menten reaction mechanism with double inhibition caused by both, a suicide substrate and a competitive reversible inhibitor. We assume rapid equilibrium of the reversible reaction steps involved, while the time course equations for the reaction product have been derived with the assumption of a limiting enzyme. The goodness of the analytical solutions has been tested by comparison with the simulated curves obtained by numerical integration. A kinetic data analysis to determine the corresponding kinetic parameters from the time progress curve of the product is suggested. In conclusion, we present a complete kinetic analysis of an enzyme reaction mechanism as described above in an attempt to fill a gap in the theoretical treatment of this type of system.