The thin string limit of Cosmic Strings coupled to gravity

The thin string limit of Cosmic Strings is investigated using a description in terms of Colombeau's theory of nonlinear generalised functions. It is shown that in this description the energy-momentum tensor has a well defined thin string limit. Furthermore the deficit angle of the conical spacetime that one obtains in the limit may be given in terms of the distributional energy-momentum tensor. On the other hand it is only in the special case of critical coupling that the energy-momentum tensor defined in the Colombeau algebra is associated to a conventional distribution. The asymptotics of both the matter and gravitational field are investigated in the thin string limit and it is shown how this leads to the `conical approximation' which is valid outside the inner core of the string.Comment: 29 pages, 2 figure

The Kinetics of Sodium Extrusion in Striated Muscle As Functions of the External Sodium and Potassium Ion Concentrations

After a 20 min initial washout, the rate of loss of radioactively labeled sodium ions from sodium-enriched muscle cells is sensitive to the external sodium and potassium ion concentrations. In the absence of external potassium ions, the presence of external sodium ions increases the sodium efflux. In the presence of external potassium ions, the presence of external sodium ions decreases the sodium efflux. In the absence of external potassium ions about one-third of the Na+ efflux that depends upon the external sodium ion concentration can be abolished by 10-5 M glycoside. The glycoside-insensitive but external sodium-dependent Na+ efflux is uninfluenced by external potassium ions. In the absence of both external sodium and potassium ions the sodium efflux is relatively insensitive to the presence of 10-5 M glycoside. The maximal external sodium-dependent sodium efflux in the absence of external potassium ions is about 20% of the magnitude of the maximal potassium-dependent sodium efflux. The magnitude of the glycoside-sensitive sodium efflux in K-free Ringer solution is less than 10% of that observed when sodium efflux is maximally activated by potassium ions. The inhibition of the potassium-activated sodium efflux by external sodium ions is of the competitive type. Reducing the external sodium ion concentration displaces the plots of sodium extrusion rate vs. [K]o to the left and upwards

Long Duration Responses in Squid Giant Axons Injected with 134Cesium Sulfate Solutions

Giant axons from the squid were injected with 1.5 M cesium sulfate solutions containing the radioactive isotopes 42K and 134Cs. These axons, when stimulated, gave characteristic long duration action potentials lasting between 5 and 45 msec. The effluxes of 42K and 134Cs were measured both under resting conditions and during periods of repetitive stimulation. During the lengthened responses there were considerable increases in potassium efflux but only small increases in cesium efflux. The selectivity of the delayed rectification process was about 9 times greater for potassium ions than for cesium ions. The data suggest that internal cesium ions inhibit the outward potassium movement occurring during an action potential. The extra potassium effluxes taking place during excitation appear to be reduced in the presence of cesium ions to values between 7 and 22% of those expected in the absence of cesium inhibition

Some Cation Interactions in Muscle

It has been possible to treat potassium, rubidium, and cesium ion entry into frog sartorius muscle by the use of a model which assumes a limited number of sites at the cell surface. The ion concentration in an outer surface layer is regarded as the main factor determining the rate of inward movement. It is supposed that the concentration of ions in the external solution is effective in promoting inward movement only to an extent determined by the fraction of sites occupied. Equations are derived from the model which fit the inward flux versus applied concentration curves experimentally determined for the three ions. The ions were found to compete for the postulated sites in various bi-ionic mixtures, the competition being satisfactorily described by equations derived from the model. The constants assigned to each ion remain invariant and independent of gradients in electrochemical potential. The order of decreasing exchange rate found is K > Rb > Cs. The order of decreasing site affinity found is Rb > K > Cs which is the same order as that observed for the ion selectivity deduced from analytical measurements of cation preference after equilibration in various equimolal mixtures (Lubin and Schneider (21)). The manner in which such a model might affect the application of a theory which assumes electrical driving forces as well is discussed

The Potassium Flux Ratio in Skeletal Muscle As a Test for Independent Ion Movement

The flux ratio of potassium ions was measured on frog sartorius muscle under conditions in which a substantial net potassium loss occurs. Muscle fiber membrane potentials were measured under identical conditions. The observed flux ratios were compared with values calculated from a theoretical relation derived on the assumptions that the unidirectional fluxes are both passive and occur independently. The results favor the conclusion that the potassium fluxes across skeletal muscle membrane occur along passive electrochemical gradients and obey the independence principle

Strophanthidin-Sensitive Components of Potassium and Sodium Movements in Skeletal Muscle As Influenced by the Internal Sodium Concentration

"Low sodium" muscles were prepared which contained around 5 mmoles/kg fiber of intracellular sodium. "High sodium" muscles containing between 15 and 30 mmoles/kg fiber of intracellular sodium were also prepared. In low sodium muscles application of 10-5 M strophanthidin reduced potassium influx by about 5%. Potassium efflux was unaffected by strophanthidin under these conditions. In high sodium muscles, 10-5 M strophanthidin reduced potassium influx by 45% and increased potassium efflux by 70%, on the average. In low sodium muscles sodium efflux was reduced by 25% during application of 10-5 M strophanthidin while in high sodium muscles similarly treated, sodium efflux was reduced by about 60%. Low sodium muscles showed a large reduction in sodium efflux when sodium ions in the Ringer solution were replaced by lithium ions. The average reduction in sodium efflux was 4.5-fold. Of the amount of sodium efflux remaining in lithium. Ringer's solution, 40% could be inhibited by application of 10-5 M strophanthidin. The total sodium efflux from low sodium muscles exposed to Ringer's solution in which lithium had been substituted for sodium ions for a period of 1 hr can be fractionated as 78% Na-for-Na interchange, 10% strophanthidin-sensitive sodium pump, and 12% residual sodium efflux. It is concluded that large strophanthidin-sensitive components of sodium and potassium flux can be expected only at elevated sodium concentrations within the muscle cells

The Influence of Potassium- and Sodium-Free Solutions on Sodium Efflux from Squid Giant Axons

The sensitivity of sodium efflux to the removal of potassium ions from the external solution and the change in sodium efflux occurring when sodium ions are also removed were observed to be related. When Tris was used to replace external sodium ions, increases in sodium efflux were always observed whether the sensitivity of sodium efflux to external potassium ions was weak or strong. Greater percentage increases in sodium efflux occurred, however, the greater the sensitivity of sodium efflux to external potassium ions. When lithium ions were used to replace external sodium ions, increases in sodium efflux occurred if the sensitivity of efflux to external potassium ions was strong whereas decreases in sodium efflux took place if the sensitivity of efflux to external potassium ions was weak. Intermediate sensitivities of efflux to external potassium resulted in no change in efflux upon substitution of lithium ions for external sodium ions. In the presence of 10-5 M ouabain, substitution of Tris for external sodium ions always resulted in a small decrease in sodium efflux. The data can be described in terms of a model which assumes the presence of efflux stimulation sites that are about 98% selective to potassium ions and about 2% selective to sodium or lithium ions

Tracer and Non-Tracer Potassium Fluxes in Frog Sartorius Muscle and the Kinetics of Net Potassium Movement

Experiments were performed to test the applicability of permeability kinetics to whole frog sartorius muscle using K42 ions as tracers of potassium flux. The whole muscle was found to obey closely the kinetic laws expected to hold for single cellular units in which the potassium fluxes are membrane-limited and intracellular mixing is rapid enough not to introduce serious error. In a 5 mM K Ringer's solution, potassium efflux was very nearly equal to influx when the rate constant for K42 loss was applied to the whole of the muscle potassium. Over a fairly wide range of external potassium concentration, the assumed unidirectional fluxes measured with tracer K42 showed good agreement with net potassium changes determined analytically. The specific activity of potassium lost from labeled muscles to an initially K-free Ringer's solution was measured as a test of the adequacy of intracellular mixing. The results were those expected for a population of cells with uniformly distributed intracellular K42. A small deviation was encountered which can be attributed either to a dispersion of fiber sizes in the sartorius or to a possible small additional cellular compartment in each individual fiber. The additional cellular compartment, should it exist, contains from 0.5 to 1 per cent of the muscle potassium. This is evidently not large enough to interfere seriously with the applicability of permeability kinetics to the whole muscle

An Analysis of the Leakages of Sodium Ions into and Potassium Ions out of Striated Muscle Cells

Net sodium influx under K-free conditions was independent of the intracellular sodium ion concentration, [Na]i, and was increased by ouabain. Unidirectional sodium influx was the sum of a component independent of [Na]i and a component that increased linearly with increasing [Na]i. Net influx of sodium ions in K-free solutions varied with the external sodium ion concentration, [Na]o, and a steady-state balance of the sodium ion fluxes occurred at [Na]o = 40 mM. When solutions were K-free and contained 10-4 M ouabain, net sodium influx varied linearly with [Na]o and a steady state for the intracellular sodium was observed at [Na]o = 13 mM. The steady state observed in the presence of ouabain was the result of a pump-leak balance as the external sodium ion concentration with which the muscle sodium would be in equilibrium, under these conditions, was 0.11 mM. The rate constant for total potassium loss to K-free Ringer solution was independent of [Na]i but dependent on [Na]o. Replacing external NaCl with MgCl2 brought about reductions in net potassium efflux. Ouabain was without effect on net potassium efflux in K-free Ringer solution with [Na]o = 120 mM, but increased potassium efflux in a medium with NaCl replaced by MgCl2. When muscles were enriched with sodium ions, potassium efflux into K-free, Mg++-substituted Ringer solution fell to around 0.1 pmol/cm2·s and was increased 14-fold by addition of ouabain

The Dual Effect of Lithium Ions on Sodium Efflux in Skeletal Muscle

Sartorius muscle cells from the frog were stored in a K-free Ringer solution at 3°C until their average sodium contents rose to around 23 mM/kg fiber (about 40 mM/liter fiber water). Such muscles, when placed in Ringer's solution containing 60 mM LiCl and 50 mM NaCl at 20°C, extruded 9.8 mM/kg of sodium and gained an equivalent quantity of lithium in a 2 hr period. The presence of 10-5 M strophanthidin in the 60 mM LiCl/50 mM NaCl Ringer solution prevented the net extrusion of sodium from the muscles. Lithium ions were found to enter muscles with a lowered internal sodium concentration at a rate about half that for entry into sodium-enriched muscles. When sodium-enriched muscles labeled with radioactive sodium ions were transferred from Ringer's solution to a sodium-free lithium-substituted Ringer solution, an increase in the rate of tracer sodium output was observed. When the lithium-substituted Ringer solution contained 10-5 M strophanthidin, a large decrease in the rate of tracer sodium output was observed upon transferring labeled sodium-enriched muscles from Ringer's solution to the sodium-free medium. It is concluded that lithium ions have a direct stimulating action on the sodium pump in skeletal muscle cells and that a significantly large external sodium-dependent component of sodium efflux is present in muscles with an elevated sodium content. In the sodium-rich muscles, about 23% of the total sodium efflux was due to strophanthidin-insensitive Na-for-Na interchange, about 67% being due to strophanthidin-sensitive sodium pumping