Muscle fiber capacity in Low conductivity solution

Itrsric~rte of Applied ibfattiernrmrics arrd Sfafisrics attrl Depilrfme!mi cf Phj>sicrlogp', Universify 01' British Coiupatbia, Varacosrvea, British Collrrnbiu VB'P' I FV.5

Received April 28, 1975

Loo, D., and &'ALJC;HAN, P. G. 1976. Muscle fiber capacity in low condracdividy solution. Can. I. Physiol. Pharrnacol. 54, 107-1 12.

A method is described for cornpa~ting the effective capacity of rnalscle fibers, C - Q/V where Q is the charge stored, and V is the mernbrane potential, using a standard two-microelectrode, constant current injection techtaiclue. The method is used to compare physical (or elffecdive) capacity of frog muscle fibers bathed in a Iow conductivity, 2.5 m M K + solution, with circuit- theory derived quantities in the same cells and in control fibers. No difTerences can be discerned and it is concluded that low cornductivity of physiological solutions, per se, does not significantly reduce the lerngtlm constarrt of frog muscle transverse tubules.

Loo, D. et VAUGIIAN, P. C . 1976. Muscle fiber capacity in low conductivity solution. Can. J. EPhysioB. Pharmacol. 54, 107-112,

On dCcrit une mkthode permcttank de calculer la capaciti efljcace (C - Q/ I' oil Q =- claarge accumul&, V = potenticl de membrane des fibres m~;scu%aires) h I'aide d'i~ne technique d'in- jection de courant utilisant deux ~nicroklectrodes. On utilise cette mkthode pour cornparer Ba capacitk physique (011 efiective) des fibres ralusculaires de la grenouille baignant dans une solution norrnale (2.5 171.4f) (K +) avcc les quantites thkoriqraes calcul6es pour ies m h e s fibres et pour les fibres t6moins. 11 n'existe pas de difT6reiace; on en conclut que la basse conductivitO des solutions physiologiques en elk-n9me ne r6duit pas significativemernt la constante de longueur des tubes T n~usculaires chez la grenouille.

[Traduit par le jolarnal]

Introduction the steady state passage of current and the

Recent controversy over the effects of low ionic strength solutions on the measurable capacity of skeletal muscle fibers (Vaughan e l nl. 1972; Adrian and Almers 1973, 1974) culminated in Adrian and Almers (1974) stressing the difrer- ence between physically defined electric capacity QC = Q / V where Q is the charge stored and V is the membrane potential) of muscle fibers and membrane capacitance. The latter is a function only of the mernbsane dielectric, and does not appear to be affected by alterations of bathing solution ionic strength at least over a 'physio- logical' range (Valdiosera et a/. 1974), but re- quires sensitive analysis according to circuit theory precepts for its determination.

Adrian and A h c r s (1974) pointed out that if the accessibility of some element of membrane (for example that of the transverse tubules, deep within thc core of the muscle fiber) to the outside solution is restricted in some way, all the memn- brane would not be uniformly charged during

'Please address correspondence to Dr. P. Vaughan, Department of Physiology, University of British Colum- bia, Vancouver, B.C. V6T 1 W5.

effective &pacity df the fiber would be less than that anticipated from circuit theory analysis.

In the experiments, in which they introduced the method, Adrian and Almers (1974) found that in solutions in which potassium concentra- tion was raised above co11trol values effective capacity was reduced. The fall was rnssumed to be caused by a seduction of transverse tubular length constant brought about by an increase in tubular membrane conductance in elevated [Kt 1. Surprisingly, Adrian and Almers (11974) also found a small decrease in the capacity of fibers bathed in 0.24 Mingcr, a so%ution containing laormal potassium (Vaughan er a%. 1972).

Impedance measurements (Valdiosera et hs[.

1974) in a solution similar to 0.24 Ringer irndi- cated a large increase in the resistance of the tubule lumen. If other resistances remain un- changed this should reduce tubule length con- stant and probably effective capacity; but the latter was not measured by these workers.

In thc present paper we introduce a ~rnethod h r measuring the effective capacity of cells using a two-microelectrock (or in theory, at least, a one-microelectrode and Wheatstone

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I08 CAN. 9. PHMSJOL. PHARMACOL. V8L. 54, 1976

bridge arrangement) current clamp experiment. (Adrian and Almers (1974) used three micro- electrodes and voltagc clamp, a system that is not always practicable.) We report, too, on experiti-aents designed to test for differences between erective capacity and metnbrane (circuit theory) capacity valucs obtained from fibers immersed in 0.24 Ringer and tc- compare these values with those fiom control cells.

Theory

In general the capacity of the cell referred to a unit area of surfkce membrane is given by

where A = effective area of cell surface electroni-

cally exposed to the current pulse = 2ruX for an infinite cable (a is cable

(rz~uscle fiber) radius, and X is dc length constant)

r ( 0 , ~ ) = magnitude of stcady (dc) current injected into a cell (at % = 0, tl-ne geometric reference point)

V(Q,w) = steady-state voltage V(8,t) = voltage response at the site of current

injection at any time. Note that the integral has the dimension of

time. Upon integration, in fkct, we obtain a phenomenological 'time constant'. When multi- plied by the applied current, %((?,a), the numera- tor gives us the total charge stored in the system and on dividing by the steady-state voltage, V(O,a)), we obtain the total capacity. I-Her~ce we have an empirical definition of capacity which agrees with our intuitive iaotion.

For a more rigorous justification, we proceed as follows: we assume that we have a linear one-di~mensional cable (the muscle fiber) of half-length I ; the inside-to-outside admittance 43f the cable is y (mho/cm), and the internal resistance of the cable (sarcoplasm) is ri (fj,/cm). Bur boundary conditions are:

V(X,O) = 0 for all x

s V(0,f) - -- I@, t ) - ri- a~ 2

FIG. I . Tracings of the electrotonis response to the injection of a current pulse into the center of a muscle fiber. .4, shows the decay phase only, om an expanded 5weep. B, shows the response on a slow time base. The method involves measurement of the area under the decay phase (vertically hatched area) after the end of the current pulse. Calibration (refers to curve A) 2 mV, 50 ms.

where Y(x,t) is the deviation of the structure from resting potential. The generalized cable equation in Laplace Transform space is

L 1 sinh - x (s)

where A($) = I,/ J;jI,r>

For an infinite cable (1 >> A), Eq. I becomes

The input admittance for the cable is

The expression [d~(s ) /d~] i ,~=~ has been shown (Adrian and Almers 1974) to be the physical capacity for a cable network, and for a current clamp

Hence, relating this to a unit area of surface we have

En practice, one measures the area under a relaxation curve (as shown in Fig. I), and norm- alizes by dividing by the peak amplitude and dividing by the product of the geornstric terms and the input resistance.

The pertinence of the method is underscored by pointing out that in situatio~ls where capac- ity in series with a resistance is itself shunted by

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LOO AND VAUGHAN 109

another resistance, it will not be fully charged at dc and circuit theory capacity (rr~embrane capacity) will exceed fiber eff'ective capacity.

Methods such as this for determining capacity have their usefulness in their independence fro111 the necessity of the experimenters having a precise description of the voltage (in the present case, current where voltage clamp is used) waveform.

Materials and Methods

Solutboras Control ciata were obtained in Ringer's solution and test

data from '0.24 Ringer', a solution in which sodium chloride was replaced by equiosmotic amounts of sucrose (Vaughan et ul. 1972).

The solutions contained (mEquiv. /l): Ringer, Nai 1 18, Ki 2.5, Ca2+ 1.08, 61- 119.7, HP042- 1, M2P04- 1; 0.24 Ringer, Na 22.8, K ' 2.5, Can 1.08, 61- 24.4, HP04:- 1, H2P04-- 1 , sucrose approximately 190 mM/l. The pH of the solutions was 7.2, and 8.24 Ringer had 0.23 times the ionic strength of the standard solution.

Procedure Standard two-microelectrode methods were used to

obtain input resistance and length constant determina- tions for fibers in small bundles from musculus semi- tendinosus of R N ~ I U pipieras. HyperpoBariaing pulses of about 2 M ms duration were employed with control fibers and of 500 ms with fibers in 0.24 Ringer. With these latter fibers a 2.5-fold increase in membrane resistance (see Results) so lengthened fiber time constants tlnat the longer pulses were required, and where records of timecourse sf decay of electrotonic potentials or of spatial decrement of membrane voltage were being made, a suficiently small current was injected that at 'zero' (i.e., less than 50 pm) electrode separation, the peak ansplitude of the voltage response was less than 7.5 mV. Current was injected using a device similar to that described by New (1972); the voltage follower was that employed by Eisenberg and Gage (1969).

In experiments of this kind one of the major uncertain- ties is the cross-sectional area of the muscle fiber. Hodgkrn and Nakajima (1972) used single fibers in their study of the effect of diameter on menabrane capacity and resis- tance and rotated the fibers (approximately elliptical in cross section) under the microscope so that their major and minor diameters could be measured. Generally, however, internal resistivity is assumed and diameters (assuming circular cross-section) are calculated from measured electrical properties (Adrian and Almers 1974) or circular cross-section is assumed and diameters esti- mated from visual inspection.

In the present experiments, after the measurements necessary to compute input resistance and length constaint, together with relaxation information, were made, visual estimates of fiber diameter were checked with sequential impalements with the current electrode across the bundle, while watching for the appearance of an electrotonic pulse on the voltage record. The distance between the extrema at which an electrotonic pulse was recorded was

a~sumed to be 5 pm less than fiber diameter; fibers were aswmed to have circular cross-section, for simplicity. These measurements probably incorporated an error of about 1OC/n in the measured diameter; the error is much larger when one colasiders that the fibers are elliptical in cross-section. However, since the procedures were the same for all cells examined, these errors are more likely to contribute to large variances than to a bias in the results.

Effective muscle fiber capacity (C,,,) was measured int fibers in test solution by the integration procedure, whereas measurement in controls was made by circuit- theory methods alone. The electrotonic potential decayed with a timecourse indistinguishable from an error func- tion, with a single time constant, and the membrane capacities (C,,), estimated from 0.5 decay and 0.84 decay times, were not significalltly different irn these controls, so that C,,, and C,, were presunxed identical.

Results

Fibers in Normal Ringer Cable measurcnlents were made on a number

of control fibcrs, which were required to have resting potentials of at least 90 mk' with both electrodes inserted. The infinite cable model was used for computation (as pointed out in the Theory section, the current-injecting electrode was inserted in the middle of the fiber: the half- length of the fiber was about 8 A) and fibers were rejected if thcre were not at least three colinear points on the semi-logarithmic plot of steady- state electrotonic voltage against electrode sepa- ration used to measure length ~onstamt. .-

A value of 278 f 26 (SE) 52 - crn was obtained for internal resistivity (Ri) from measurements on 12 acceptable (by the criterion outlined above) control cells. The mean is not signifi- cantly different from that obtained by Wodgkin and Nakajima (1972) (1 69 f 4 (SE) Q cm) or by Vaughan et a/. (1972) (1 62 f 16 (SE) Q cn~). Thus, mean fiber diameter (for an equivalent circular-cross-section fiber) was not overesti- mated by more than about 2.5y0 in the present experiments, compared with the measurements of H~dgk in and Nakajima (1972), assuming real Ri9s are identical.

When the resistance separating tubular mem- brane capacity from surface membrane capacity is small and tubular membrane conductance is low, the equivalent circuit for an element of the fiber network approximates an in-parallel resist- ance-conductance couple and the time constants of voltage relaxation are inseparable. A simple, although not rigorous, test of this condition is to compare the 84YQ decay time of the electro-

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110 CAN. J. PHYSIOL. PHAWMACOL. VOL. 549 1976

TABLE 1. Fiber constants in normal Ringer

Diameter 4, CEC. R1 ~ ~ m ) (kn .cm2) (pF/cm2) (fd .cW

60 0.85 8 .2 69 60 1.54 6.5 150 80 1.92 6.3 171 8 0 3.06 6 - 2 385 8 5 3.55 5 - 6 260

Mean 67 2.33 7 .0 1 78 $ SE 3 . 3 0 . 3 tr .4 2 41

tonic pulse at zero electrode separation (this is the time constant of an error function) with the half-decay time As pointed out by Gage and Eiseraberg (1969), for the error function,

a (time constant)) - 4.398'TIi2

For each control fiber an estimate of Inern- brane capacity was made froan the $4y0 decay time (C0.84) and 4.4T'1,;2 (CO5). The values obtained are listed with other derived constants in Tabli: I and are not significantly difTercnt.

fibers in 0.24 Riager VaIdiosera et a&. (1974)) f ~ u n d that when they

immersed nluscle fibers in solutio~as of low ionic strength there was an increase in radial resistance (n resistance associated with the tubule lumen). Such a change may cause a nc~ticerable diff'eucnce between 84%; decay time and 4.4 X half-decay time. The latter would be reduced if

there were no change in surface conduetarace (Adrian arnd Almers 1973). A decrease iaa tubular length constant would also result if the conduct- ance of tubular nlembrane did not fa%). Hence the polarization sf centrally located tubular membrane would Fdll below that of more peripheral membrane and fiber capacity would be reduced, whereas membrane capacitance would remain unaltered.

Data were gathered from fibers immersed in 0.24 Ringer accepted according to the: same criteria as were controls.

One very noticeable diftercnce between con- trol and test fibers was the large value c~f Ri (303 + 43 (SE) $2-cn~) in fibers equilibrated in 0.24 Ringer. A less extreme increase was ob- served by Vaugkan eb QII . (1972) in fibers inn- mersed in a similar solution ('%$.24 I.S.", isotonic sucrose) which had a higher potassium concen- tration.

Iaa addition to the estimates of membrane capacity made from the decay times of the electrotonic pulse, a measure of fiber capacity was obtained by the integration procedure (Theory). The derived constants are listed in Table 2. Tinere is no significant difference be- tween the estimates s f membrane capacity (C0., and Cfi.,,) or between those and the eff'ecbive fiber capacity, related to area of surhce membrane, despite an ordering of means such that efyective capacity appears to exceed mem- brane capacity. The aneans are made more nearly identical if data from the last fiber in Table 2, for which a very high fiber capacity was esti- mated. are deleted.

We attacl-a no significance to the observation that capacities measured in test fibers are slightly higher than in controls.

TABLE 2. Fiber constants in 0.24 Ringer

Diameter RE, Co.5 C n . h 4 ~ C I P R1 (pm) (kf:. crn2) (pF/s;rn2) (p%;/cm2) (pF/crnq ((n. cm)

Mean 73.3 5.83 8.5 9.2 9.7 303 'SE 8.7 1.82 0.3 - 0.5 0.9 43

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LOO AND VAUGHAN

TABLE 3. Conductance values from the literature

Conductance

Condition Symbol Value Reference

Fibers in Ringer's GL * 5 .45 mmholcm Valdiosera et a/.

Fibers in 0.24 Ringer's 6 ' " 0.878 rnmho/cm Valdiosera et al. All fibers c,? 0.15 m h o / c r n U d r i a n et nl.

*Lumen conductivity. tConductance of tubular membrane, per unit transverse-system volume.

It seems not unreasonable to conclude that the accessibility of elements of membrane capa- city is not diminished by immersion of fibers in 0.24 Ringer.

The increase seen in membrane resistance (about 2.5 times control) is probably a conse- quence of the reduction of external chloride concentratioin. Usirng the simplifying assump- tions that (i) both potassium and chloride ions are in electrochemical equilibrium across the membrane at a membrane potential of - 90 mV, (ii) the permeability coefficients of potassium and chloride are identical and are not functions of concentration or membrane potential, and (iii) near tlne resting potential the current-voltage relation is adequately predicted by the constant field equation

(C, - Ci) exg (FV/WT) Icr V-

1 - exp (FV/RT)

where I is membrane current, V is nlembrane potential, Co is the external concentration of ion, Ci is the intracellular concentration of ion. and F, R, and T have their usual thermodynamic meanings, we find about a twofold incrcase in R,,. The increase seen by Adrian and Almers (1974) was about 1.4-fold.

Discussion This pager uses a simple method for obtaining

the effective capacity of muscle fibers (i-e., the amount of charge that can be stored per unit surface area) from records obtained with a two- electrode current clamp technique. The method allows comparison of this physical capacity with circuit-theory derived capacities ('membrane capacitance') measured from the same records. It has been shown that when the conductivity of Ringer's solution is reduced by isosmotic re- placement of sodium chloride with sucrose there is no substantial change in the capacity of muscle fibers bathed in that solution, whether the

capacity is determined by circuit theory means or from a physical definition of electric capacity.

An incrcase in lumen resistivity (Valdiosera C J ~ ul. 1974) reduces tubule length constant (other resistances remaining constant) and therefore, possibly, effective capacity.

We can make some predictions from values that have appeared in the literature. The fibers used for the experiments in 0.24 Ringer were slightly smaller than thosc used by Valdiosera et al. (1974), or those used in the example given in the Appendix of Adrian et al. (1949), but their assumptions and measurements (Table 3) were used.

Unfortunately the value of C& is uncertain; nor can we be certaiin that low ionic strength of the bathing medium per se does not influelncc it. According to Adrian et al. (1949)

Volume to surface ratio of tubules = cm Network factor, a = 1/2

We then arrive at the values for tubular length constant

AT = 75 pm (fibers in Ringer's solution) AT = 30 pm (fibers in 0.24 Ringer)

If surface capacity = B pF/cm2, and, according to Adrian eb a?. (1949)

cw = 3 X lo3 pFjcm3

whcre C, is the capacity of tubular membrane per unit volume of transverse-system, we obtain, for a 34-pm radius fiber

CPff = 6. I p F / ~ 1 ~ 2 in normal Ringer = 5 pF/cm"n 0.24 Ringer

from Adrian and Almers (1974), Eq. 28. This prediction indicates that the measure-

ments of Valdiosera et a?. (1974) can adequately account for the observation of Adrian and Alnners (1974) that there is a slight fall in in 0.24 Ringer.

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112 CAN. J. PMYSPOE. BWAWMACOE. VOL. 54, 1996

However, Adrian et cml. (1969) proposed that AT in normal Ringer might be as inuch as 170 pm (40-prn radius fiber). If this were true A, would not fall suficiently in 0.24 Ringer for a change in Ceff to be detectable and our present result would be expected.

Consequently, while a somewhat arbitrary choice of values for pertinent variables allows us t s reconcile the results of Adrian and Almers (1974) with those of Valdiosera et al. (19741, our own results cannot lead to the conclusion that there is a functionally significant decrease in AT.

Acknowledgment B.L. is in receipt of a predoctoral fellowship

from the Muscular Dystrophy Association of Canada. This work was supported by grant number MA 4381 from the Medical Research Council sf Canada.

Membrane capacity measurements on frog skele- tal muscle in media of low ion content. J. Physiol. 237, 573-685.

ADRIAN, M. H., CHANDLER, W. K., and W O ~ K I N , A. L. 1969. The kinetics of mechanical activation in frog muscle. J. Physiol. 204, 207-2620.

EBSENBEWG, Ha. S., and GAGE, P. 1969. Ionic conductances of the surface and transverse tubular membranes of frog sartorius fibers. J. Gen. Physiol. 53, 279-297.

GAGE, P., and EISENBEWG, R. S. 19690 Capacitance s f the surface and transverse tubular membrane of frog sartorius fibres. J. I'hysiol. 53, 265-278.

HODGKIN, A. L., and NAKAJIMA, S. 1972. The effect of diameter on the electrical constants of frog skeletal muscle fibres. J. I'hysiol. 221, 105-120.

NEW, W., JR. 1972. Constant current source for micro- electrodes. J. Appl. Physiol. 32, 885-88'5.

&'ALDIBSERA, R., CLAUSEN, C., and EISENBERG, R. S. 1974. Impedance of frog skeletal muscle fibers in various solutislas. J. Gen. Physiol. 63, 468-491.

VAUGHAN, P. C., HOWELL, J. N., and EHSENBI':RC;, K. S. 1972. The capacitance of skeletal muscle fibers in solutions of low ionic strength. J. Gen. Physiol. 59. 337-359.

ADRIAN, R. H., and ALMERS, W. 1973. Measurement of membrane capacity in skeletal muscle fibres. Nature (London), New Biol. 242, 62-64.

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