A Theoretical Analysis of the Capacitance of Muscle Fibers Using a Distributed Model of the Tubular System

A model is developed to predict the changes in total capacitance (i.e. total charge stored divided by surface membrane potential) of the tubular system of muscle fibers. The tubular system is represented as a punctated disc and the area of membrane across which current flows is represented as a punctated annulus, the capacitance of the muscle fiber being proportional to this area. The area can be determined from a distributed model of the tubular system, in which the only resistance to radial current flow is presumed to be in the lumen of the tubules. Calculations are made of the variation of capacitance expected as the conductivity of the bathing solution is varied. These calculations include the effects of fixed charge in the tubular lumen and the effects of changes in the shape and volume of the tubular system in solutions of low conductivity. The calculated results fail to fit comparable experimental data, although they do qualitatively account for the known variation of the radial spread of contraction with conductivity of the bathing medium. It is pointed out that the existence of a significant "access resistance" at the mouth of the tubules might explain the discrepancy between theory and experiment.     

Transverse Impedance of Single Frog Skeletal Muscle Fibers

The transverse electrical impedance of single frog skeletal muscle fibers was measured at 31 frequencies that ranged from 1 to 100,000 Hz. Each fiber was bathed entirely in Ringer's solution, but it was positioned so that a central length of 5 mm was in a hollow plastic disk and was electrically isolated from the ends of the fiber. The diameter of the segment of the fiber in the disk was measured and then the segment was pressed from opposite sides by two insulating wedges. Electrical current was passed transversely through the segment between two platinum-platinum black electrodes that were located in the pools of Ringer's solution within the disk. The results were corrected for stray parallel capacitance, series resistance of the Ringer's solution between the fiber and the electrodes, parallel shunt resistance around the fiber, and the phase shift of the measuring apparatus. A nonlinear least-squares routine was used to fit a lumped equivalent circuit to the data from six fibers. The equivalent circuit that was chosen for the fibers contained three parallel branches; each branch was composed of a resistor and a capacitor in series. The model also included a seventh adjustable parameter that was designed to account for the degree of compression of the fibers by the insulating wedges. The branches of the equivalent circuit were assumed to represent the electrical properties of: (a) the myoplasm in series with the membrane capacitance that was exposed directly to the pools of Ringer's solution; (b) the capacitance and series resistance of the transverse tubules that were exposed directly to the pools of Ringer's solution; (c) the membrane capacitance in series with the shunt resistance between the fibers and the insulating wedges. The results gave no indication that current entered the sarcoplasmic reticulum.     

Electrical properties of frog skeletal muscle fibers interpreted with a mesh model of the tubular system.

This paper presents the construction, derivation, and test of a mesh model for the electrical properties of the transverse tubular system (T-system) in skeletal muscle. We model the irregular system of tubules as a random network of miniature transmission lines, using differential equations to describe the potential between the nodes and difference equations to describe the potential at the nodes. The solution to the equations can be accurately represented in several approximate forms with simple physical and graphical interpretations. All the parameters of the solution are specified by impedance and morphometric measurements. The effect of wide circumferential spacing between T-system openings is analyzed and the resulting restricted mesh model is shown to be approximated by a mesh with an access resistance. The continuous limit of the mesh model is shown to have the same form as the disk model of the T-system, but with a different expression for the tortuosity factor. The physical meaning of the tortuosity factor is examined, and a short derivation of the disk model is presented that gives results identical to the continuous limit of the mesh model. Both the mesh and restricted mesh models are compared with experimental data on the impedance of muscle fibers of the frog sartorius. The derived value for the resistivity of the lumen of the tubules is not too different from that of the bathing solution, the difference probably arising from the sensitivity of this value to errors in the morphometric measurements.     

Electromechanical coupling in tubular muscle fibers. II. Resistance and capacitance of one transverse tubule

In tubular muscle fibers of the yellow scorpion the transverse tubules are arranged in a radial symmetry. This particular morphology, enables one to derive values for electrical components of one transverse tubule (TT) by treating the TT as a core conductor rather than a complex network. The electrical properties of tubular muscle fibers were completely characterized and analyzed by measuring two independent functions of frequency, i.e., the characteristic impedance and the propagation function. The impedance of a single tubular muscle fiber was determined with microelectrodes over the frequency range 1 Hz to 1.5 kHz. The results were fitted to a possible equivalent circuit model which is based on morphological evidence. The average component values for this model are: Ri = 209 omega-cm, Rm, and RT = 980 omega-cm2 (referred to unit area of surface membrane), Cm and CT = 0.9 muF/cm2, and RL = 103 omega-cm. Relating the equivalent circuit to ultrastructure shows that the average component values are consistent with the hypothesis that the TT is open to the extracellular medium, the electrical capacity of surface and TT membranes is about 1 muF/cm2, and the spread of surface depolarization into the TT is attenuated by about 25%.     

An analysis of the electrical properties of a skeletal muscle fiber containing a helicoidal T system

The linear electrical properties of skeletal muscle fibers have been analyzed using lumped circuit analogues of helicoidal T system. The geometry of a helicoid is assumed to produce two electrical effects, modeled separately. One model is motivated by the pitch or tilt of the T system, which forces the current flowing in the lumen of the tubules to have a longitudinal projection. The second model is motivated by the longitudinal continuity of a helicoid, which forms a structure similar to a cable within the fiber. The pitch or tilting of the T system plane modified the longitudinal resistance of the fiber, making it slightly frequency dependent; however, the magnitude of the change was less than 0.1%. The longitudinal connections between T system networks had a more complicated effect; the magnitude of the correction was again less than 0.1%. The conclusion from this analysis is that a helicoidal T system, whose pitch is constrained by the sarcomere spacing, will not affect electrical signals recorded intracellularly in intact fibers.     

An improved vaseline gap voltage clamp for skeletal muscle fibers

A Vaseline gap potentiometric recording and voltage clamp method is developed for frog skeletal muscle fibers. The method is based on the Frankenhaeuser-Dodge voltage clamp for myelinated nerve with modifications to improve the frequency response, to compensate for external series resistance, and to compensate for the complex impedance of the current-passing pathway. Fragments of single muscle fibers are plucked from the semitendinosus muscle and mounted while depolarized by a solution like CsF. After Vaseline seals are formed between fluid pools, the fiber ends are cut once again, the central region is rinsed with Ringer solution, and the feedback amplifiers are turned on. Errors in the potential and current records are assessed by direct measurements with microelectrodes. The passive properties of the preparation are simulated by the "disk" equivalent circuit for the transverse tubular system and the derived parameters are similar to previous measurements with microelectrodes. Action potentials at 5 degrees C are long because of the absence of delayed rectification. Their shape is approximately simulated by solving the disk model with sodium permeability in the surface and tubular membranes. Voltage clamp currents consist primarily of capacity currents and sodium currents. The peak inward sodium current density at 5 degrees C is 3.7 mA/cm2. At 5 degrees C the sodium currents are smoothly graded with increasing depolarization and free of notches suggesting good control of the surface membrane. At higher temperatures a small, late extra inward current appears for small depolarizations that has the properties expected for excitation in the transverse tubular system. Comparison of recorded currents with simulations shows that while the transverse tubular system has regenerative sodium currents, they are too small to make important errors in the total current recorded at the surface under voltage clamp at low temperature. The tubules are definitely not under voltage clamp control.     

SELECTIVE DISRUPTION OF THE SARCOTUBULAR SYSTEM IN FROG SARTORIUS MUSCLE : A Quantitative Study with Exogenous Peroxidase as a Marker

Skeletal muscles which have been soaked for 1 hr in a glycerol-Ringer solution and then returned to normal Ringer solution have a disrupted sarcotubular system. The effect is associated with the return to Ringer's since muscles have normal fine structure while still in glycerol-Ringer's. Karnovsky's peroxidase method was found to be a very reliable marker of extracellular space, filling 98.5% of the tubules in normal muscle. It was interesting to note that only 84.1% of the sarcomeres in normal muscle have transverse tubules. The sarcotubular system was essentially absent from glycerol-treated muscle fibers, only 2 % of the tubular system remaining connected to the extracellular space; the intact remnants were stumps extending only a few micra into the fiber. Thus, glycerol-treated muscle fibers provide a preparation of skeletal muscle with little sarcotubular system. Since the sarcoplasmic reticulum is not destroyed and the sarcolemma and myofilaments are intact in this preparation, of the properties of the sarcolemma may thus be separated from those of the tubular system.     

Electric potential in cylindrical syncytia and muscle fibers

The model developed in an earlier paper using two coupled partial differential equations for calculating the intracellular and extracellular electric potentials in a syncytium is applied here to cylindrical geometry. Eigenfunction expansions are obtained for the potentials resulting from an intracellular point source of current. The required orthogonality relations for the two sets of coupled radial eigenfunctions are derived. The model is applied to the structure composed of the interior and the transverse tubules of a muscle fiber. Asymtotic expansions for ζ and β→0 are obtained, where ζ is the product of the effective intracellular resistivity, the fiber radius and the outer surface membrane admittance per unit area, and β is the ratio of the effective intracellular resistivity to that of the tubular lumen. Earlier results from the distributed circuit model of a muscle fiber are recovered when ζ and β are small, and for a nerve axon when β=0.     

Electrical properties of sheep Purkinje strands. Electrical and chemical potentials in the clefts

The impedence of sheep Purkinje strands, measured to 3-5 kHz, is interpreted with circuit models based on morphology. The strand is described as a one-dimensional electrical cable. Clefts between myocytes of the strand allow radial current to flow in parallel with current across the outer membrane. A lumped model of the clefts, in which all the cleft membrane is in series with 100 omega-cm2, fits only below 20 Hz. Two distributed models, pie and disk, fit at all frequencies with somewhat different (31%) luminal resistivities, but with similar membrane parameters. Series resistance representing the endothelial sheath is small. Simulations of voltage clamp experiments include measured linear parameters and nonlinear membrane channels, as well as radial variation of cleft concentration, membrane flux, voltage, and current. Cleft potential is drastically nonuniform when sodium current flows. Cleft potential is reasonably uniform when calcium and potassium currents flow, but the calcium and potassium concentrations change markedly, enough to turn off the calcium current, even if the calcium channel did not inactivate. We conclude that physiological current flows produce significant nonuniformities in electrochemical potentials in the clefts of this cardiac preparation.     

Action currents, internodal potentials, and extracellular records of myelinated mammalian nerve fibers derived from node potentials.

The potential distribution within the internodal axon of mammalian nerve fibers is derived by applying known node potential waveforms to the ends of an equivalent circuit model of the internode. The complete spatial/temporal profile of action potentials synthesized from the internodal profiles is used to compute the node current waveforn, and the extracellular action potential around fibers captured within a tubular electrode. For amphibia, the results agreed with empirical values. For mammals, the amplitude of the node currents plotted against conduction velocity was fitted by a straight line. The extracellular potential waveform depended on the location of the nodes within the tube. For tubes of length from 2 to 8 internodes, extracellular wave amplitude (mammals) was about one-third of the product of peak node current and tube resistance (center to ends). The extracellular potentials developed by longitudinal and radial currents in an anisotropic medium (fiber bundle) are compared.     

The ultrastructure of normal and glycerol treated muscle in the ghost crab,Ocypode cursor

The ultrastructure of normal and glycerol treated fibers of the closer muscle of the ghost crab, Ocypode cursor, was studiedmthe muscle is composed of presumably phasic (short sarcomeres) and tonic (long sarcomeres) fibers, the latter greatly predominating. Horseradish peroxidase (HRP) was used as an extracellular tracer to delineate the tubular system (TS), and to determine to what extent this system becomes detached from the extracellular space as a result of glycerol treatment. Sarcolemmal clefts invade deeply into the muscle at Z-lines and I-bands; tubules invaginate into the muscle from the clefts and from the surface sarcolemma at the Z-lines, A-I overlaps and A-bands. A tubules are in frequent diadic or tetradic contact with the sarcoplasmic reticulum (SR), whereas Z tubules appear to be randomly associated with SR, terminal cisterns (TC) and Z-line fibrils. When HRP was administered to normal muscle, black reaction product was found adjacent to the outer surface of the sarcolemma, within the clefts and within profiles of the TS throughout the tissue. In glycerol treated muscle peripheral vacuolation frequently occurred; black reaction product penetrated only as far as the vacuoles and into dilated Z-line tubules, but was virtually absent from the rest of the TS. This lack of continuity between the extracellular space and the A tubules indicated disruption or constriction of the A tubules as a result of glycerol treatment, although Z tubule contact with the extracellular space appeared unimpaired. These findings provide ultrastructural correlates of the electrophysiological changes produced by glycerol treatment of the closer muscle of the ghost crab (Papir, 1973), namely, interference with excitation-contraction (e-c) coupling. The random association of the Z tubules with SR and TC, and their resistance to disruption by glycerol treatment, tend to endorse the claims that the Z tubules in crustacean muscle are not directly involved in e-c coupling (Brandt et al., 1965; Peachey, 1967; Selverston, 1967).     

Linear electrical properties of the transverse tubules and surface membrane of skeletal muscle fibers

With the use of two intracellular microelectrodes and a circuit designed to compensate for the effects of stray capacitances around the electrodes, transfer impedance measurements were made at frequencies from 0.5 to 1000 c/s on frog sartorius muscle fibers bathed in 7.5 mM K Ringer solution. Complete AC cable analyses performed at 46, 100, 215, 464, and 1000 c/s showed that the fibers behaved as ideal one-dimensional cables having purely resistive internal impedances (R_i = 102 ± 11 Ω cm). Two circuits were considered for fiber inside-outside impedance, a four lumped parameter circuit and a parallel resistance and capacitance shunted by the input impedance of a lattice model for the T-system. Least squares fits to fiber input impedance phase angles were better with the latter circuit than with the former. With the use of the lattice model the specific capacitance of both the surface and transverse tubule membranes was found to be 1 µF/cm² and the internal resistivity of the tubules to be about 300 Ω cm.     

Occurrence of unusual tubular invaginations of the plasma membrane in smooth muscle cells of the lamprey,Lampetra japonica

Numerous tubular structures were observed in the surface region of smooth muscle cells making up the vascular walls in the lamprey, Lampetra japonica; they were designated as surface tubules. The limiting membrane of the surface tubules was connected to the plasma membrane, allowing communication of the lumen of the tubule with the extracellular space. Tannic acid reacted with osmium, serving as an extracellular marker, penetrated into the tubules but not into the intracellular organelles, such as the endoplasmic reticulum and the Golgi complex. The surface tubules were grouped in longitudinal parallel rows, separated from each other by tubule-free areas where dense plaques were present. Each tubule was fairly cylindrical (approximately 60 nm in diameter) and often ramified into two or three branches with a blind end. Occasionally, these tubules were encircled by the sarcoplasmic reticulum which was located immediately beneath the plasma membrane. Similar tubules were also observed in the surface region of vascular endothelial cells and fibroblasts in the adventitial connective tissue. The possibility that the surface tubules in the present observations are analogous to the smooth muscle caveolae or the striated muscle T-tubule is discussed.     

Predicted Delays in the Activation of the Contractile System

The capacitance C'(e), presumed to be located across the walls of the transverse tubules of twitch fibers, was identified in earlier impedance measurements by virtue of having a resistance in series with it. When the voltage V(m) across the surface membrane is made to vary, the voltage V(c) across C'(e) will be delayed with respect to V(m), the extent of the delay depending on the location of the series resistance. Model 1 assumes that the resistivity of the lumen of the tubules is negligible; model 2 assumes that the series resistance arises entirely in the tubular lumen; model 3 assumes that the resistivity of the tubular lumen is small, but not negligible and that the bulk of the resistance arises in a structure directly in series with C'(e) and having a similar geometric distribution. If V(m) varies sinusoidally, the relative value of V(c(max)) will fall with increasingly higher powers of the frequency at the center of the fiber if model 2 is applicable, whereas models 1 and 3 predict that V(c(max)) will fall at high frequency only in proportion to the frequency everywhere in the cross-section of the fiber. Equations have been derived for the voltage change V(c) in response to a step change of V(m) and during an action potential. On the assumption that contraction is initiated when V(c) reaches mechanical threshold, the delay between the activation of myofibrils on the axis of the fiber and at the surface would amount to 2.6 msec in model 2 and 0.25 msec in model 3 for frog fibers of about 100 mum diameter during a twitch.     

Unit membrane parameters of electrically syncytial tissues.

A change in the holding voltage, exposure to channel-blocking agents, and similar interventions will induce changes in the membrane properties of electrically syncytial tissues. The altered membrane characteristics will produce changes in the input resistance (RIN) and the phase angle (phi) of the complex admittance of the whole preparation. Exact geometry-independent formulas are derived that give the intervention-induced changes in the membrane capacitance and conductance in terms of the measured changes in RIN and phi. The formulas automatically account for the effects of extracellular resistance in tissues such as skeletal muscle fibers, cardiac Purkinje fibers, and small cardiac "aggregates." The size, shape, and resistance of the extracellular space may be arbitrary and need not be measured. The surface (invaginated) membranes, which face the bath (extracellular space), are assumed to be characterized by an RC circuit with specific capacity Cme (Cmi) and specific conductivity gme (gmi). It is assumed that the intracellular voltage gradient between the electrodes and the membranes is negligible or reliably calculable. The intervention is assumed to leave the geometry and resistivity of the extracellular space unchanged. Under these circumstances the intervention-induced changes in Cme, Cmi, gme, and gmi are determined exactly in terms of the corresponding changes in RIN and certain frequency domain integrals over phi. The technique is illustrated by synthetic data for RIN and phi generated by the "disk" model of a skeletal muscle fiber in which Cme and Cmi depend upon holding voltage. The corresponding voltage dependence of RIN and phi is successfully "inverted" to expose the underlying voltage dependence of Cme and Cmi. These computations suggest that the formulas for Cme and Cmi will be useful in realistic situations, since they are not too sensitive to experimental error in the data for RIN and phi. This method makes it possible to detect voltage-dependent capacity changes due to unit membrane processes (e.g., charge movement) as long as the intrinsic time constant of that process is very small (e.g., less than 1/30 ms). As a second example I consider a disk model that is exposed to increasing concentrations of a channel-blocking agent. The drug dependence of RIN and phi is used to calculate the drug dependence of the total membrane conductivity (the sum of gme and gmi, weighted by the areas of surface and invaginated membranes, respectively).     

Calcium dependence of the stimulating action of cadmium on organic acid transport in the frog kidney

The stimulatory effect of cadmium ions on the Na-dependent fluorescein transport into the frog renal proximal tubules ceased with decreasing Ca++ concentration in solution on both the sides of the cell layer down to micromolar level. The decrease in Ca++ concentration per se stimulated fluorescein uptake during short-term incubations. A further diminution of Ca++ concentration in the tubular lumen with the aid of EGTA resulted in a sharp inhibition of the organic acid transport. Amiloride, which prevented the stimulatory effect of cadmium, inhibited the fluorescein transport at both millimolar and micromolar levels of Ca++ concentration, but it failed to affect the transport process after introducing EGTA into the tubular lumen. The results are discussed within the frames of a model regarding extracellular Ca++ as an allosteric inhibitor, and intracellular Ca++ as an allosteric activator of sodium channels in the apical membrane. Cd++ is assumed to compete with Ca++ for binding to centers of the allosteric inhibition, thereby accelerating the sodium ion flux across the cells of the proximal tubules.     

Evidence for Anion-Permselective Membrane in Crayfish Muscle Fibers and Its Possible Role in Excitation-Contraction Coupling

Under certain conditions only, isolated crayfish skeletal muscle fibers change in appearance, becoming grainy, darkening, and seemingly losing their striations. These changes result from development of large vesicles on both sides of the Z-line. The longitudinal sarcoplasmic reticulum remains unaffected. The vesicles are due to swelling of a transverse tubular system (TTS) which is presumably homologous with the T-system tubules of other muscle fibers. The vesiculations occur during efflux of water or on reducing external K or Cl, but only when KCl can leave the fiber. They never result from osmotic, ionic, or electrical changes when KCl cannot leave. Inward currents, applied through a KCl-filled intracellular cathode, also cause the vesiculations. These are not produced when the cathode is filled with K-propionate, nor by outward or longitudinal currents. Thus the transverse tubules swell only when Cl leaves the cell. Accordingly, their membrane is largely or exclusively anion-permselective. These findings also indicate that the TTS forms part of a current loop, connecting with the exterior of the fiber probably through radial tubules (RT) possessing membrane of low conductivity. Thus, part of the current flowing inward across the sarcolemma during activity can return to the exterior through the membrane of the TTS. The structure and properties of the latter offer the possibility for an efficient electrical mechanism to initiate excitation-contraction coupling.     

The capacitance of skeletal muscle fibers in solutions of low ionic strength

The capacitance of skeletal muscle fibers was measured by recording with one microelectrode the voltage produced by a rectangular pulse of current applied with another microelectrode. The ionic strength of the bathing solution was varied by isosmotic replacement of NaCl with sucrose, the [K] [Cl] product being held constant. The capacitance decreased with decreasing ionic strength, reaching a value of some 2 µF/cm² in solutions of 30 mM ionic strength, and not decreasing further in solutions of 15 mM ionic strength. The capacitance of glycerol-treated fibers did not change with ionic strength and was also some 2 µF/cm². It seems likely that lowering the ionic strength reduces the capacitance of the tubular system (defined as the charge stored in the tubular system), and that the 2 µF/cm² which is insensitive to ionic strength is associated with the surface membrane. The tubular system is open to the external solution in low ionic strength solutions since peroxidase is able to diffuse into the lumen of the tubules. Twitches and action potentials were also recorded from fibers in low ionic strength solutions, even though the capacitance of the tubular system was very small in these solutions. This finding can be explained if there is an action potential—like mechanism in the tubular membrane.     

Muscle fiber capacity in low conductivity solution.

A method is described for computing the effective capacity of muscle fibers, C = Q/V where Q is the charge stored, and V is the membrane potential, using a standard two-microelectrode, constant current injection technique. The method is used to compare physical (or effective) capacity of frog muscle fibers bathed in a low conductivity, 2.5 mMK+ solution, with circuittheory derived quantities in the same cells and in control fibers. No differences can be discerned and it is concluded that low conductivity of physiological solutions, per se, does not significantly reduce the length constant of frog muscle transverse tubules.     

Histological and three dimensional organizations of lymphoid tubules in normal lymphoid organ of Penaeus monodon

The normal lymphoid organ of Penaeus monodon (which tested negative for WSSV and YHV) was composed of two parts: lymphoid tubules and interstitial spaces, which were permeated with haemal sinuses filled with large numbers of haemocytes. There were three permanent types of cells present in the wall of lymphoid tubules: endothelial, stromal and capsular cells. Haemocytes penetrated the endothelium of the lymphoid tubule's wall to reside among the fixed cells. The outermost layer of the lymphoid tubule was covered by a network of fibers embedded in a PAS-positive extracellular matrix, which corresponded to a basket-like network that covered all the lymphoid tubules as visualized by a scanning electron microscope (SEM). Argyrophilic reticular fibers surrounded haemal sinuses and lymphoid tubules. Together they formed the scaffold that supported the lymphoid tubule. Using vascular cast and SEM, the three dimensional structure of the subgastric artery that supplies each lobe of the lymphoid organ was reconstructed. This artery branched into highly convoluted and blind-ending terminal capillaries, each forming the lumen of a lymphoid tubule around which haemocytes and other cells aggregated to form a cuff-like wall. Stromal cells which form part of the tubular scaffold were immunostained for vimentin. Examination of the whole-mounted lymphoid organ, immunostained for vimentin, by confocal microscopy exhibited the highly branching and convoluted lymphoid tubules matching the pattern of the vascular cast observed in SEM.