high frequency characteristics of elasticity of skeletal muscle fibres kept in relaxed and rigor...

]oumal of Muscle Research and Cell Motility 15, 130-144 (1994)

High frequency characteristics of elasticity of skeletal muscle fibres kept in relaxed and rigor state

M. E. M. DE WINKEL,” T. BLANG6 and B. W. TREIJTEL

Department of Physiology, Uniuersily of Amsterdam, Meibergdreef IS, 1105 AZ Amsterdam, The Netherlands

Received 5th June 1993; revised I November 1993; accepted I November 1993

Summary

The viscoelastic properties of crossbridges in rigor state are studied by means of application of small length changes, completed within 30 ps, to isometric skinned fibre segments of the iliofibularis muscle of the frog in relaxed and rigor state and measurement of the tension response. Results are expressed as a complex Young’s modulus, the real part of which denotes normalized stiffness, while the imaginary part denotes normalized viscous mechanical impedance. Young’s modulus was examined over a wide frequency range varying from 5 Hz up to 50 kHz. Young’s modulus can be interpreted in terms of stiffness and viscous friction of the half-sarcomere or in terms of elastic changes in tension and recovery upon a step length change.

The viscoelastic properties of half-sarcomeres of muscle fibre segments in rigor state showed strong resemblance to those of activated fibres in that shortening a muscle fibre in rigor state resulted in an immediate drop in tension, after which half of the drop in tension was recovered. The following slower phases of tension recovery-a subsequent drop in tension and slow completion of tension recovery-as seen in the activated state, do not occur in rigor state. The magnitude of Young’s moduli of fibres in rigor state generally decreased from a value of 3.12 x IO7 N m-’ at 40 kHz to 1.61 x IO7 N mm2 at about 100 Hz.

Effects of increased viscosity of the incubation medium, decreased interfilament distance in the relaxed state and variation of rigor tension upon frequency dependence of complex Young’s modulus have been investigated. Variation of tension of crossbridges in rigor state influenced to some extent the frequency dependence of the Young’s modulus. Recovery in relaxed state is not dependent on the viscosity of the medium. Recovery in rigor is slowed down at raised viscosity of the incubation medium, but less than half the amount expected if viscosity of the medium would be the cause of internal friction of the half-sarcomere. Internal friction of the half-sarcomere in the relaxed fibre at the same interfilament distance as in rigor is different from internal friction in rigor. It will be concluded that time necessary for recovery in rigor cannot be explained by friction due to the incubation medium. Instead, recovery in rigor expressed by the frequency dependence of the Young’s modulus has to be due to intrinsic properties of crossbridges. These intrinsic properties can be explained by the occurrence of state transitions of crossbridges in rigor. Similarity of Young’s modulus of fibre segments in the activated state and in rigor in the frequency range above 5 kHz strongly suggests that the same state transitions occur in force generating crossbridges in the activated fibre.

Introduction

Tension transients induced by small, rapid length changes of isometrically mounted muscle fibres contain infor- mation about elastic properties of muscle fibres, which can be related to elastic properties of crossbridges. Cross- bridges are known to be the force generators of muscle (Huxley, 1957) and are assumed to act as elastic structures (Huxley & Simmons, 1971; Ford et al., 1977). When a contracting muscle fibre is rapidly released over a short distance, tension first decreases, then partially recovers

*To whom correspondence should be addressed.

0142-4319 0 1994 Chapman & Hall

over the next few milliseconds. The decrease in tension concomitant with the shortening reflects this crossbridge elasticity. The following tension recovery is thought to result from changes of state of crossbridges, including attachment and detachment.

One of these states of crossbridges involved in the force generating mechanism is the rigor state, in which the crossbridge is strongly bound to actin. In this study viscoelastic properties of crossbridges in rigor state were examined by determination of the complex Young’s modulus of skeletal muscle fibre segments as a function of frequency. The real part of the Young’s modulus at a certain frequency denotes pure stiffness, normalized to

Stiffness of muscle fibres in rigor state

length and cross-section, and the imaginary part denotes normalized internal viscous friction multiplied by frequency in rad S-I. From the Young’s modulus, if specified as a function of frequency, the tension response of a half-sarcomere on a small step length change can be calculated. The tension change concomitant with the rapid length change corresponds to the complex Young’s modulus at high frequency. The subsequent tension recovery is reflected by a decrease in magnitude of the complex Young’s modulus with decreasing frequency.

We investigated elastic properties of the crossbridges in rigor state for two reasons: (I) in rigor state there is no ATP available and thus the system of possible crossbridge states is simplified, (2) the stable rigor state (AM-state) can be considered as an intermediate state in crossbridge cycle. If occurring, state transitions of crossbridges can cause a change in crossbridge force without a change in overall length of the crossbridge. This leads to recovery in the tension response after a rapid length change. Recovery of tension after a length change, i.e. Young’s modulus increasing with frequency as observed with the fibre in rigor (Jung et al., 1988; de Winkel et al., 1992) could therefore indicate that crossbridges in rigor are not in a single state and that state transitions are possible. It could also be possible that after the length change of the fibre crossbridges in combination with filaments or other structures proceed to change length. Due to their stiffness this could also lead to recovery after the length change.

Therefore the frequency dependence of the complex Young’s modulus of muscle fibres in rigor state (i.e. Young’s modulus which depends on frequency) was examined over a wide frequency range varying from 5 Hz up to SO kHz. It was investigated if the frequency dependence of the complex Young’s modulus could be affected by changing viscosity of the incubation medium, by changing interfilament distance of the fibre in relaxed state and by variation of tension of crossbridges in rigor state. As an approximation of the mean, purely elastic extension of crossbridges, the amount of instantaneous sliding to reduce tension of the half-sarcomere to zero is estimated from the Young’s modulus of the fibre in rigor at 40 kHz.

131

Methods and materials

Preparation Experiments were performed with segments of single muscle fibres of the frog (Rana esculentu), skinned by freeze-drying. Preparation of these freeze-dried muscle fibres has been described by Stienen and colleagues (1983) and by Stienen and Blange (1985). Stienen and Blange (1985) measured T,/T, and T,/T, curves of muscle fibres skinned by freeze-drying and with similar duration of length change, the results were similar to those of Ford and colleagues (1977), performed on intact muscle fibres. Jung and colleagues (1988) compared the results of stiffness measurements of chemically skinned (Triton X-100)

muscle fibres with those of freeze-dried muscle fibres. Jung and colleagues (1988) stated that chemically skinned muscle fibres in the relaxed state were to some extent less stiff and no difference was found for fibres in activated and rigor state. These results indicate that freeze-dried muscle fibres are suitable for taking accurate stiffness measurements with microsecond time resolution. In a skinned muscle fibre rigor can be induced by removal of ATP from the intracellular medium so that the dissociation step in the cycle of actin-myosin ATP-ase &not take place. This process is reversible and addition of ATT’ will cause the fibre to relax. Lengths of the single muscle fibres ranged from 3 to 7 mm and the diameters varied between 70 and 200 Wm.

Soltlfiofls Two types of incubation media were used: a relaxing and rigor solution. A computer program based on that of Fabiato and Fabiato (1979) was used to calculate the concentration of important ions, pH, ionic strength and the amount of KOH/HCl and KC1 necessary to adjust the pH and ionic strength of the solutions to the values desired at a chosen temperature (Jung et al., 1988). Several stability constants mentioned by Godt and Lindley (1982) were used. The composition of the solutions was in mu: (I) relaxing solution: imidazole 60; EGTA 20; creatine phosphate 10; MgATP 5; Ca2+ 0; Mg2+ 1; Na+ 31; K+ 45; Cl- 51 and 50 U ml-’ creatine kinase; (2) rigor solution: imidazole 60; EGTA 20; EDTA 5; Ca’+ 0; Mg’+ 0; Na+ 30; K+ 52; Cl- 72. In some experiments the relaxed fibre was compressed through addition of 3% (w/v) Dextran (MW 200-270 kDa; BDH, Poole; Dorset, UK) to the relaxing solution. In all solutions pH was set to 7.00 ( f 0.05) by addition of KOH and the chosen ionic strength was 160 (f 1) mM. The temperature of the solutions was kept at 4.0’ (f0.5) c.

Apparatus System used was based on that described by van den Hooff and colleagues (1982) and by Stienen and BlangC (1985). The mechanical moving part of the system is based on the principle of a loudspeaker, i.e. a coil moving in the field of a permanent magnet. For a detailed description, see van den Hooff and colleagues (1982).

Tension changes following the length change of the muscle fibre are measured by a homemade force transducer with a resonance frequency of 70 kHz in air, a damping time constant of 500 ps and a noise of 2 PV (de Winkel et al., 1993). When the fibre is attached to the transducer the resonance frequency decreases by 5%. The bandwidth of the force transducer amplifier is 150 kHz.

Experimental protocol The freeze-dried fibre segments were mounted in dry state between the displacement generator and the force transducer with cr-cyanoacrylate glue (Cyanolit 202, 3M Center, St. Paul, MN, USA). Then the fibres were incubated for at least 10 min in the relaxing solution. Sarcomere length was checked in the relaxing solution by means of laser diffraction pattern evoked by a 5 mW Helium-Neon laser (Uniphase, model 105-l: h = 632.8 nm) and adjusted to 2.15 pm.

The standard procedure in the experiments consisted of incubation in the relaxing solution (at least 10 min) before

MRCM I%--D

132 DE WINKEL, BLANGE and TREIJTEL

tension responses to quick length changes were recorded. This was followed by incubation in the rigor solution for at least 5 min. Recording of tension responses to length changes occurred only after development of a steady rigor tension level.

After recording of tension transients in control conditions (fibre in relaxed and rigor state), the fibre in rigor state was stretched or released (less than I% of the original fibre length) in order to obtain another isometric tension level. Tension transients were recorded at various isometric rigor tension levels before the fibre was set to its original length and incubated in the relaxing solution.

Tension responses as well as length changes were recorded by two digital oscilloscopes (Nicolet Explorer IIIA, Madison, WI, USA) at frequencies of 500 kHz and 5 kHz respectively. Recordings at 500 kHz covered a period of 2 ms duration available for analysis, which corresponds to a range from 250 Hz upwards in the frequency domain. Recordings at 5 kHz covered a period of 200 ms. Before sampling at 5 kHz, signals were filtered with a first order low-pass filter at I kHz to suppress aliasing effects in the analysis. Recordings of tension responses and length changes were stored on floppy discs. At each value of length change eight recordings were summed to reduce variations in tension due to noise.

During the procedure, fibre length and diameter were measured by means of a microscope (Zeiss Opmi-1: f = 50 mm (diameter) and f = 175 mm (length), oculars 20x), without finding difference in fibre diameter in the two solutions. Uniformity of the sarcomere length was checked after each incubation in rigor by use of laser diffraction technique and if non-uniformity in sarcomere length developed the fibre was discarded.

A different laser diffraction technique was used to register local movements inside the sarcomere of a muscle fibre segment in rigor state. The method used has been described by Stienen and Blange (1985). The first order diffraction line of the muscle fibre segment in rigor state was projected on a position sensitive photodiode at a distance of 0.94 m. The position sensitive photodiode (Silicon Detector Corp. Newbury Park, California) had a bandwidth of 25 kHz.

Calculations Tension transients were simulated by an automated procedure, based on minimization of the square mean difference between the first 2 ms of measured tension response and the simulation. The digitally recorded tension responses and length changes (the input signal for simulation procedure) were digitally filtered with a linear phase (finite impulse response) filter with pass-band up to 40 kHz and 4 times attenuation in the stop-band above 60 kHz. Use of this filtering procedure prevented the fitting from being dominated by the natural frequency of the force transducer or by noise at high frequencies. The following formulas in Laplace transform notation were used for simulation (Blange & Stienen, 1985):

-Us) -2kl f(s) Z(s) - E 6) Ie”l - e-kI 1

kZ = prs2 + (4p, r~R~‘)~.~s’.~ + rlR-‘s

Z(s)

Y

(1)

(2)

In these equations s = jo, with w the frequency in rad so’ and j”= -1; pm is density and TI viscosity of the surrounding fluid; pr fibre density; R radius of cross-section and 1 length of the fibre; f(s) represents corrections for frequency characteristics of the force transducer, transducer amplifier (cut-off frequency 150 kHz), and of the displacement detector; E(S) = the applied length change, which is the input signal originating from the displacement generator; Z(s) = the elastic impedance (complex Young’s modulus) and T(s)= force signal from transducer divided by the cross-sectional area of the fibre. Note the correction for the surrounding fluid (the second and third term on the right of Equation (2)). See BlangC and Stienen (1985) for a detailed explanation of these equations.

Equation (3) yields a good description of the experimentally observed complex Young’s modulus as a function of frequency, as reported earlier (de Winkel et al., 1992). This empirical formula describes the complex Young’s modulus of a muscle fibre with three parameters: an elastic constant E, (N m-l), a reciprocal time constant a (rad s-r), and an exponent v. These parameters are determined by optimization of similarity of calculated and measured tension transient over 2 ms (de Winkel et al., 1993). At low frequencies (O << ~1) the complex Young’s modulus tends to reduce to the elastic constant (E,), while the exponent (v) represents a linear increase with frequency in the log-log plot for frequencies o >> ~1. The reciprocal time constant (m) is determined by the intersection of the line representing the linear increase with the level of the elastic constant E, in a plot of magnitude of the Young’s modulus versus frequency. According to the formula, phase shift is almost zero at frequencies (a<< a) and rises in a transition range around o = ~1 to v n 2-l rad. The empirical formula yields a better description of Young’s moduli of the fibre in rigor or in the relaxed state than the five parameter model (Jung et al., 1988) and, in these cases, is almost as good as the seven parameter model used by the same authors to describe the Young’s modulus of activated fibres (de Winkel et al., 1993).

Prior to actual simulation, optimization of simulation of the tension was first performed with a scaling factor, inserted in Equation (I), as an additional variable parameter. Deviation of this scaling factor from 1 can be interpreted as a mismatch because of faulty measurements of fibre diameter or length or because of in homogeneity. Since tension is force divided by cross-sectional area, deviation can be expressed as mismatch in (circular) diameter. Diameters found in this way were within the accuracy of 5% of the measured fibre diameters.

Due to phase ambiguity the complex Young’s modulus (Z) at a certain frequency cannot be solved uniquely from the transfer function of the tension response to a step length change as can be seen in Equation (1). However, if the tension transient has been simulated, k(s) and the complex Young’s modulus (Z) are approximately known for all frequencies. Substitution of k by means of Equations (2) in (1) leads to an expression for the transfer function in terms of frequency and elastic impedance. By an iterative approximation, in which the parameters of the Young’s modulus, found by simulation, are used as starting-points, the elastic impedance (complex Young’s modulus) can be determined separately for each frequency from the measured recordings by insertion of T and E in left part of Equation (1) (de Winkel et al., 1993). The performance of the experimental apparatus was regularly checked by similar determination of the Young’s modulus of a silk fibre, which is almost purely elastic (see de Winkel et al., 1993). This way of

Stiffness of muscle fibres in rigor state

determination of the Young’s modulus as a function of frequency is used throughout this study to check the evaluation of the Young’s modulus by parameters according to Equation (3) sufficiently reflects actual changes in Young’s modulus induced by the different experimental conditions.

Results

Local changes in sarcomere length Prior to stiffness measurements it was checked if the applied length change was absorbed by the fibre in rigor state. Therefore the change in position of the first order laser diffraction line after application of a step length change to the muscle fibre in rigor state was detected by a position sensitive photodiode. From this signal the change in sarcomere length at midpoint position of the fibre was calculated and compared to the ratio of length change of mover signal to total fibre length (see Fig. I).

The first trace (A) in Fig. I represents the mechanically applied length change of 0.13%, completed within 30 JJS. The second trace (B) shows the time course of the length change at sarcomere level in the middle of the fibre, and the third trace (C) the resulting tension change. As shown in Fig. 1 trace (B), the length change at sarcomere level is almost immediately completed. The difference in resonance behaviour between the quantities length change at sarcomere level and the tension change is primarily due to their different positions of detection.

From the relative length change applied at the end of the muscle fibre in rigor state 95% is detected in the middle of the fibre at sarcomere level. Therefore it was

133

concluded that the relative length change of the fibre segment was equal within experimental accuracy to the relative length change found at sarcomere level in the middle of the fibre.

Stiffness measurements An example of first 2 ms of measured and simulated tension responses of muscle fibres in rigor state to rapid length changes are shown in Fig. 2B. General features of the first 2 ms of tension responses after a fast length change are: a delay, an abrupt tension change, an

oscillation and a fast recovery. In Fig. 28 the tension response is recorded at a sampling frequency of 500 kHz, while in Fig. 2A the same tension response is shown recorded at a sampling frequency of 5 kHz. The extreme in tension transient in Fig. 2A is heavily truncated because of the low sampling frequency, as can be seen by comparison with Fig. 2B.

From tension transients evoked by small length changes, the viscoelastic properties of the fibre expressed by a complex Young’s modulus can be determined as described in the methods section. To this end the tension response, recorded at a sampling frequency of 500 kHz,

was simulated by a model description which contains the Young’s modulus according to the empirically derived Equation (3). The simulated response is shown too in Fig. 2B.

The difference left between the measured and simulated tension response is evaluated further in the way described in the last paragraph of ‘calculations’ (see Materials and Methods), which leads to determination of

20

Fig. 1. Local changes in sarcomere length. The top trace (A) shows the mechanically applied length change of 0.131% completed within 30 ps, the second trace (B) the length change observed locally at the middle of the fibre (0.128%), third trace (C) the resulting change in tension recorded at sampling rate of 500 klutz. Fibre length: 3.55 mm, diameter: 70 pm.

DE WINKEL, BLANGE and TREIJTEL

A

0 0.5 1.0 1.5 2.0

Tie (ms)

Fig. 2. Examples of the same tension transient of a muscle fibre in rigor state recorded at two different sampling frequencies, in Fig. 2A at 5 kHz and in 2B at 500 kHz. The applied length change was 0.54 nm per half-sarcomere (0.05% of muscle length). The dotted line represents the simulated tension response. Fibre length: 4.22 mm, diameter: 90 pm, rigor tension: 0.21 x IO5 N mm’.

the Young’s modulus at separate frequencies. The result is shown in dots in Fig. 3A and B together with the parametric approximation (drawn line found by simulation with Z(s) = E,( 1 + s/a)” (Equation (3)).

The approximation of the complex Young’s modulus is determined by simulation of the first 2 ms of the tension transient recorded at a sampling frequency of 500 kHz. This same approximation, extended to lower frequencies is drawn in Fig. 3A (also the solid line). The sharp peak in magnitude of the Young’s modulus at 18 kHz, not present in the approximation, was due to a

mechanical resonance, which caused deviation between the recorded and actual length change and which was known from the control procedure with the silk fibre. Over the frequency range from 5 Hz up to 40 kHz the Young’s modulus, represented by the dots in Fig. 3A and B, can be approximated well by the empirical Equation (3).

In the double logarithmic plot of magnitude of the complex Young’s modulus versus frequency, found from recordings with a sampling rate of 500 kHz, the magnitude increased almost linearly with frequency from 250 Hz up to 40 kHz (de Winkel et al., 1992). From the recordings taken at 5 kHz sampling rate, which permit determination down to 2.5 Hz, it was found that at low frequencies the magnitude of the Young’s modulus of a muscle fibre in rigor state was constant and equal to the value of the elastic constant E, in Equation (3). It can be seen in Fig. 3 that the amplitude of the Young’s modulus is independent on frequency in the range below approximately 100 Hz and that above this frequency the Young’s modulus increases steadily. Phase shift of the complex Young’s modulus (lower panels in Fig. 3) was zero up till frequencies of 100 Hz, above which it increased to an almost constant value of 0.3 radians.

Values of parameters of the Young’s modulus of this muscle fibre, acquired by simulation, together with those of three other muscle fibres in rigor state are given in Fig. 4 as a function of the applied step length change (see Fig. 4).

Figure 4 shows that there is almost no difference in magnitude of the elastic constant (E,) resulting from lengthening and shortening steps. The same holds true for the exponent, while the reciprocal time constant (a) varies somewhat. For a small displacement the system can be considered linear. Values of the model parameters obtained from simulations of tension responses to length changes outside the range of f 1.3 nm per half-sarcomere, especially if resulting from shortenings larger than 1.3 nm per half-sarcomere, appeared to be less reliable as judged from a poorer fit of the tension response at optimal simulation.

To achieve estimates of parameters of Young’s modulus which are independent of the applied displacement, the intercept at zero length change of the least mean square regression line was taken. This regression line was calculated from values of model parameters acquired from simulation of tension responses to length changes smaller than 1.3 run per half-sarcomere. The average values of the model parameters taken at zero length change for muscle fibres in rigor state as well as their standard deviations are given in Table 1.

For muscle fibres in rigor state the average amplitude of the complex Young’s modulus at low frequencies was 1.6 x lo7 N mm2 and increased as a function of frequency with an exponent of 0.16. Parameters of Young’s moduli at zero length change had no correlation with fibre segment length. The large variation in reciprocal time

Stiffness of muscle fibres in rigor state 135

, 2

10-z 8

IO -’ 1 1 I

Frequency W-k)

‘.:Y .* : .

. T . ..y . ;

: . .

: 1 1 I I

Frequency &a) I02

Fig. 3. The magnitude of the complex Young’s modulus of the half-sarcomere and its phase shift as a function of frequency for a fibre in rigor state, obtained from the tension transients of Fig. 2, recorded at 500 kHz (Fig. 3B) and 5 kHz (Fig. 3A). The dots represent the measured elastic impedance (Z), the solid line describes the simulation of the complex Young’s modulus by the function: Z&O) = E, (I+ (j~/~r))“. Experimental conditions as in Fig. 2.

constant as well as the variations in other parameters must be due to differences between individual muscle fibres.

Viscous drag

It was investigated if the frequency dependence of the elastic impedance was affected by interaction between the crossbridges and incubation solution. Therefore a rigor solution was used to which 20% (w/v) glucose was added. This raises viscosity of the incubation medium by a factor 2.0 (Weast & Astle, 1981). Increased viscosity of the medium within the half-sarcomere should lead to an increase in viscous impedance of the muscle fibre seg-

ment. Under the assumption that viscous behaviour would be completely due to frictional drag by incubation fluid the reciprocal time constant (a) should be halved at doubled viscosity of the incubation medium.

Tension responses of muscle fibre incubated in rigor solution with and without addition of 20% (w/v) glucose to comparable step length changes, taken after complete rigor tension development, are shown in Fig. 5.

The solid line represents the tension response of a muscle fibre in rigor state to a rapid length change without addition of glucose. The dashed line is the tension response to an almost equal length change obtained from the same fibre incubated in a rigor solution

136 DE WINKEL, BLANGB and TREIJTEL

3. wN/m’) T

0 on q o 0

20 ‘.

0 0 0 o

0 . l

. . .Q . I

. 10 . , . 02

t : : : :c . : : : : , [1

. . . . n

.3

04

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Length change ( q m per hdf-urwmcn )

IOS

T

Alpim (radh) . 0 IO”

0.40

I Expooent

. 0.3ot

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Herb change ( nm per Wf---~ )

I : : : :lMl+4 : : : : ,

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Lcagth chaqe ( nm per batf-awcomere )

Fig. 4. Model parameters E,, ~1, V, as a function of the amplitude of displacement for four different muscle fibres.

with increased viscosity. The slight difference that can be seen in the tension responses is due to changed external and internal friction. The influence of external friction is eliminated by simulation with the relevant value of viscosity inserted in Equation (2). The means of the parameters of Young’s modulus at zero length change determined from measurements with fibres incubated in rigor solution and subsequently transferred to the 20%

(w/v) rigor-glucose solution, are shown in Table 2. Comparison of the difference in means by a paired

Student’s f test revealed that the mean difference in elastic constant (E,) was not significant at 5% level, but the mean difference of reciprocal time constant (a) and exponent (v)

were. However the increase in viscosity of the incubation medium did not lead to a difference of a factor 2 in internal friction of muscle fibres in rigor state. The

Table 1. The model parameters of fibres in rigor state

Paramefers Rigor

EO (1.61 f 0.66) X 10’

cl (4.07 2 4.50) x 10’ V (0.16 + 0.06)

The mean values and standard deviations of the model parameters at zero displacement of muscle fibres in rigor state. E, is given in N m-‘, CL in rad s-l. The number of fibres used is 36.

experimentally observed decrease in reciprocal time constant (a) was only a factor of 0.8 instead of 0.5, as could be expected from frictional drag. Moreover, the effect of the factor of 0.8 in 01 on Young’s modulus is completely offset by the decreased exponent (v) at the high frequency limit of measurement of 40 kHz. The result indicates that recovery of tension in rigor is governed by reaction rates, which are only to a minor extent diffusion controlled (Chang, 19~‘).

The same measurements were repeated for muscle fibres in relaxed state and incubated in a relax-glucose solution. Results, shown in Table 3, indicate that for muscle fibre in relaxed state the three model parameters are even less affected by increase in viscosity of the incubation medium than in the case of rigor. No significant differences in parameters were found from measurements of muscle fibres in relaxing and in relax-glucose solution.

Interfilament distance

Comparison of the frequency dependence of the complex Young’s modulus at high frequencies of muscle fibres in relaxed and rigor state resulted in a remarkable difference in exponent, i.e. the exponent of muscle fibres in rigor state was half the value of muscle fibres in relaxed state. In both states, as shown in Tables 2 and 3, this frequency dependence can hardly or not at all be influenced by variation of the viscosity of the incubation medium.


70

60

g 50

3 40

g .I 2

g 30

“M

8 20

10

0 0

I

0.5 I I I

1.0 1.5 2.0

Time ( ms )

Fig. 5. Tension responses to similar length changes of muscle fibres incubated in rigor solution with or without 20% (w/v) glucose

added. Length changes applied were about I.13 nm half-sarcomere (0.11% of muscle length). Riglu stands for the rigor solution with increased viscosity. In both cases developed rigor tension was 0.29 x 10’ N m-l. Fibre length, 7.10 mm; diameter, 83 pm.

Table 2. Effects of viscosity of incubation fluid for fibres in rigor state

Parameters Rigor (6) Riglu (6) F-value

EO (1.40 + 0.56) X lo7 (1.39 + 0.49) X IO’ 0.87 (-)

a (8.81 + 5.66) X lo3 (7.11 + 5.32) X lo3 0.01 (+)

V 0.20 f 0.05 0.19 + 0.05 0.01 ( + )

The mean values and standard deviations of the model parameters at zero displacement of muscle fibres incubated in rigor or rigor solution with doubled viscosity. Riglu stands for a rigor solution with 20% (w/v) glucose added. Ionic strength 160 mM. E, is given in N mm2 and a in rad s-‘. Behind the solutions the number of fibres used is given between brackets. ( + ) stands for a significant difference in means at 5% level, ( -) means no significant difference.

Table 3. Effects of viscosity of incubation fluid for fibres in relaxed state

Parameters Relax (9) Reglu (9) P-value

EO (0.31 + 0.21) X lo7 (0.30 f 0.21) X lo7 0.48 (-) a (1.88 + 0.90) X 10’ (1.76 f 0.84) X lo3 0.46 (-)

V 0.33 f 0.06 0.33 If: 0.09 0.72 ( - )

The mean values and standard deviations of the model parameters at zero displacement of muscle fibres incubated in a relaxing solution with/or without doubled viscosity. Reglu stands for a relaxing solution with 20% (w/v) glucose added. E, is given in N mm’ and a in rad s-‘. Behind the solutions the number of fibres used is given between brackets (-) means no significant difference in means at 5% level.


I-

Time ( ms 1

Fig. 6. Tension responses to similar length changes of muscle fibre in relaxed state with or without decreased interfilament distance. Rede3 stands for a relaxed fibre with decreased interfilament distance. Length changes applied to the relaxed fibre were 0.09 nm per half-sarcomere (0.08% of muscle length). Fibre length, 5.3 mm; diameter, 167 pm.

The difference in interfilament distance of muscle fibres in relaxed and rigor state, as demonstrated by X-ray measurements (Jung et al., 1989), could affect the difference in amplitude of the Young’s mod& as a function of frequency. To investigate the effect of interfilament distance, measurements were performed on muscle fibres incubated in a relaxing solution with 3% (w/v) Dextran added. Addition of 3% (w/v) Dextran to the relaxing solution compresses the filament spacing comparable to a value observed in rigor state (Jung et al., 1992).

Figure 6 shows tension responses of a relaxed muscle fibre with (Rede3) and without decreased interfilament distance (Relax) to similar step length changes. At a similar length change the tension transient is different for

a relaxed fibre with and without decreased interfilament distance.

Average parameters of the Young’s modulus at zero length change, obtained from stiffness measurements performed on muscle fibres incubated in a relaxing-dextran solution, are summed in Table 4.

Decreasing the interfilament spacing resulted in a significant 17% increase in mean of the elastic constant while the means of the exponent and the reciprocal time constant did not change significantly.

It could be supposed that the Young’s modulus of the fibre in rigor is the sum of the Young’s modulus in the relaxed state and stiffness due to crossbridges. This hypothesis was tested by inspection of differences in

Table 4. Effects of the interfilament distance for fibres in relaxed state

Parameters Relax (7) Rede3 (7) P-value Rigor (7)

EO (0.21 + 0.18)X lo7 (0.25 f 0.19)X 10' 0.03 (+) (1.38 + 0.47) X 10' cf (1.06 + 0.72) X lo3 (0.91 + 0.71) X 10' 0.51(-) (1.47 + 2.37) X lo3 V (0.36 + 0.08) (0.36 f 0.08) 0.98 (-) (0.16 + 0.04)

The mean values and standard deviations of the model parameters at zero displacement of muscle fibres incubated in a relaxing solution with/or without decreased interfilament distance. Rede3 stands for a relaxing solution with 3% (w/v) Dextran added. Ionic strength 160 mM. E, is given in N m-’ and CL in rad SKI. Behind the solutions the number of fibres used is given between brackets. (+ ) stands for a significant difference in means of relax and Rede3 at 5% level. (-) means no significant difference.


Young’s moduli of the fibre in rigor and in the relaxed state with decreased inter-filament distance. Young’s moduli were calculated as a function of frequency by insertion of the parameters at zero length change in Equation (3).

lower rigor tension level and thus lower tension of the crossbridges.

Figure 7 shows the differences in Young’s mod& of rigor and relaxed state with decreased interfilament distance for the different muscle fibres. Frequency dependence of the curves marking the difference between Young’s moduli of muscle fibre in rigor and in relaxed state with decreased interfilament spacing is still clearly present. For some fibres the differences rises with frequency over the whole range, for others it is rising with frequency ,at lower frequencies and decreasing subsequently. Therefore it is unlikely that frequency dependence of Young’s modulus of the fibre in rigor and thus tension recovery after a length change in rigor, can be explained by subtraction of the Young’s modulus of the relaxed fibre with decreased filament distance.

The average values obtained for the parameters of the Young’s modulus at zero length change are shown in Fig. 8 as a function of the ratio of isometric tension before the length change and the isometric tension developed directly after transfer from the relaxing to the rigor solution.

It was found that the elastic constant (E,) increases and the exponent (v) decreases with increasing rigor tension, while the reciprocal time constant (01) is hardly influenced.

It can be seen from the data shown in Table I that at frequencies higher than 1 kHz the Young’s modulus in rigor is rising with frequency. The same is true for the Young’s modulus of fully activated fibres, which also is reported to be of about the same size (Jung et al., 1988). Comparison was performed by taking the ratio of the Young’s modulus of the fully activated fibre over the modulus in rigor. In both cases the moduli determined separately for each frequency were used, because the representation of the modulus of the activated fibre by Equation (3) might conceal details. In Fig. 9 an example of this ratio is shown,

Tension applied to the crossbridges

To investigate the effect of tension in crossbridges on the complex Young’s modulus of muscle fibres in rigor, isometric tension was adjusted to various values by changing length of the fibre, after it had reached a steady level, but prior to the application of rapid length changes. The tension level was increased less than two times normal rigor tension to avoid slippage or detachment of crossbridges. Release of the muscle fibre resulted in a

T I Z( Ri ) - Z( Rede 3 ) I

In accordance with reported results (Jung ef al., 1988) the magnitude of the ratio is constant and close to 1 at frequencies above 2 kHz up till detection limit of 40 kHz and phase shift is zero. Below I kHz the ratio decreases and phase shift increases with decreasing frequency.

0 !

10-2

1 I

10-l

I I

1

Frequency &Hz)

I I I

10 lo2

Fig. 7. Differences in Young’s moduli as a function of frequency for muscle fibres in rigor and relaxed state with decreased interfilament distance. The curves were calculated from the estimated parameters found by simulation of the tension responses. Different symbols represent results of different fibres.


EO (N/m’)

2.5 IO’ -

5.0 106 -

O-

Alpha @ad/s)

S.OlO-

2.5 IO3 -

2.0 I03 -

I.5 Id -

I.0 105 -

5.0 Id -

O-

0 0.5 I I.5 Relative rigor tension

Exponent

0.20-

0.16-

O.l6-

O.l4-

O.lZ-

O.lO- 0 0.5 I I.5 0 0.5 I I.5

Relative rigor tension Relative rigor tension

Fig. 8. The average model parameters E,, ~1, v, for zero displacement versus the relative rigor tension (i.e. normalized to isometrically developed rigor tension). The error bar indicates the standard error of the mean (SEM).

Discussion

The normalized mechanical impedance of a muscle fibre is, apart from its mass density, determined by elasticity and internal viscous friction. The elastic and viscous impedance can conveniently be represented by a complex Young’s modulus. The real part stands for the normalized elastic and ‘the imaginary part for the normalized viscous impedance. Because of the homogeneous structure of the fibre down to the dimensions of the half-sarcomere, the elastic and viscous impedance of the half-sarcomere can be found from the Young’s modulus by accounting for these dimensions. For the fibre in relaxed state or in rigor, the magnitude of the complex Young’s modulus generally increases as a function of frequency above 100 Hz. At lower frequencies the amplitude of the complex Young’s modulus is constant. Halpem and Moss (1976) reported a comparable frequency dependence of the Young’s modulus (in frequency range of lo-600 Hz for whole muscle in relaxed state. In the range 50-100 Hz their dynamic Young’s modulus started to increase with frequency which agrees with our findings as well as their value of the elastic constant (2.8 x lo6 N m-“) at a muscle length of 1.26 L,. Kawai and Brandt (1980) measured

stiffness of rabbit psoas muscle fibre in relaxed (1.0 x lo6 N m-“) and high rigor state (2.0 X lo7 N m-‘) by sinusoidal length changes in low frequency range (O-167 Hz). High rigor state was achieved by addition of Ca”-ions to the rigor solution, leading to a higher rigor tension. Their Young’s moduli agree very well with ours, although we measured muscle fibres in low rigor state.

As shown in this paper the complex Young’s modulus depends on the frequency used and at high frequencies it can be satisfactorily approximated by the empirical equation: Z(s) = E,(l + s/a)“, Equation (3) from the calculations section. In a log-log plot at high frequencies the Young’s modulus increases linearly with frequency.

It is generally supposed that a mean, purely elastic extension of crossbridges can be determined by taking the ratio of isometric force developed by the activated half-sarcomere over ‘instantaneous stiffness’. In our experiments Young’s modulus at 40 kHz is still increasing with frequency. This means that only a higher limit for this elastic extension of crossbridges can be assessed. Based on a Young’s modulus of 3.1 x lo7 N me2 at 40 kHz, an isometric tension of 1.0 x lo5 N m-2 in Ca’+-activated, freeze-dried, muscle fibres and a half sarcomere length of 1.07 pm, this upper limit is 3.4 nm per half-sarcomere.


.

12 1 Activated

1 Z 1 Rigor

I I c

0.2 Frequency &Hz)

10 50

Fig. 9. The ratio of the Young’s moduli of a fibre in activated and rigor state as a function of frequency. For activation solution used, see Jung and colleagues (1988). Fibre dimensions: length, 5.3 mm; diameter, 166 pm; maximally developed isometric tension: T,(Ac) = 1.13 x IO5 N m-‘; rigor tension: T(Ri) = 0.21 x lo5 N m-‘.

Data of Jung and colleagues (1988), determined for the same kind of preparation, but based on another procedure of estimation of the Young’s modulus, indicated a value for the purely elastic extension of crossbridges of 2.5 nm per half-sarcomere. The same authors showed in later work (Jung et al., 1992) that, in the frequency range under consideration and again for the same preparation, Young’s modulus increases linearly with isometric tension, adjusted by activation at different Ca’+-concen- trations. Although they found that above 40 kHz and at (almost) full activation effects of filament elasticity be- came noticeable, it must be concluded that below 40 kHz Young’s modulus is determined by stiffness of the crossbridges. The similarity of Young’s modulus of the fibre in rigor and in activated state at frequencies above 2 kHz (Fig. 9) strongly suggests that the number of crossbridges in the fully activated state and in rigor is about the same and that recovery as observed in rigor makes up part of the recovery in the activated state. At frequencies below 100 Hz the magnitude of the Young’s modulus of the fibre in rigor given by E, (Table 1) is 1.6 x lo7 N mm’. This means that crossbridges in rigor in the fully activated fibre have a mean working stroke, given by isometric force of the fully activated fibre over

E, in rigor, which comes out at 6.6 nm. This is 2-3 times the mean, purely elastic extension of crossbridges. The occurrence of recovery in rigor, corresponding with the difference in working stroke and mean elastic extension of crossbridges in the fibre in rigor state, indicates that even for these crossbridges transitions between states can take place and that rigor is not a single state of the crossbridge. It must be assumed that crossbridges in rigor occur in more than one state and that transitions occur with time constants down to l/a or 0.5 ms.

The increase of complex Young’s modulus with frequency, related to a fast process of tension recovery, can be compared with results of the experiments of Irving and colleagues (1992) who applied repeated length changes to tetanized intact muscle fibres. Application of repeated step length changes (releases) before complete tension recovery had taken place reduced the developed tension, while the tension change simultaneous with each length change remained the same. From the experiments shown in this paper the same tension course in time should be expected after application of repeated length changes to a muscle fibre in rigor state. Application of a second length change before complete tension recovery has taken place will also reduce the final rigor tension, while


Another possible cause for the difference in frequency dependence of the complex Young’s moduli of muscle fibres in relaxed and rigor states is a difference in filament spacing. To be able to compare the values of the model parameters obtained from fibres in relaxed and rigor state the interfilament spacing needed adjustment. Rigor filament spacing can be obtained in a relaxed musde fibre by addition of 3% (w/v) Dextran-500 to the relaxation solution (Jung et al., 1989). For a muscle fibre in relaxed state this basically resulted in a scaling towards higher Young’s modulus (17% increase in elastic constant). The model parameters (apart from E,) were not or hardly affected by the decrease in filament distance, as shown in Table 4. Goldman and Simmons (1986) too found an increase in stiffness when decreasing the lattice spacing. With use of PVP (polyvinyl pyrrolidone) they reduced the spacing of skinned fibres osmotically to that of a relaxed intact fibre and observed that stiffness increased about 1.6 times.

As shown, up to 40 kHz differences in complex Young’s moduli of muscle fibres in relaxed and rigor state cannot be attributed to viscous drag by the surrounding fluid or to differences in interfilament spacing (internal friction).

It could be supposed that Young’s modulus in the relaxed state acts in parallel with the Young’s modulus of crossbridges in rigor and that the two have to be added to give the modulus of the fibre in rigor. The remaining frequency dependence of the difference of moduli of the fibre in rigor and in the relaxed state (Fig. 7) gives no support to this idea. Arguments about the origin of Young’s modulus in the relaxed state are ambiguous. Weakly bound crossbridges could contribute to Young’s modulus of relaxed fibres. For rabbit psoas muscle fibres in relaxed state 80% of stiffness has been attributed to weakly attached crosdbridges (Chalovich et al., 1991). Based on extrapolation of stiffness measurements at low ionic strength a value of less than 5% of the number of crossbridges in rigor is reported for frog muscle (Jung et al., 1989) at the ionic strength used in this study, but it is not clear if crossbridges at low ionic strength are comparable to weakly bound crossbridges under more physiological conditions (Huxley & Kress, 1985). The result that change of viscosity of the incubation fluid had no effect on Young’s modulus in the relaxed state points, even stronger than in the case of rigor, to another mechanism than just elastic structures in combination with fluid friction.

The Young’s modulus changed only by a constant factor over the whole frequency range upon decrease of interfilament distance in the relaxed fibre, which points to a combined increase of stiffness and friction. This could be explained by an increase of the number of weakly bound crossbridges. Weakly bound crossbridges will probably have other viscoelastic properties than crossbridges of the fibre in rigor state. In that case, in accordance with the result shown in Fig. 7, the difference

the tension change simultaneous with each length change and the recovery, following this change, will remain about the same. This can be concluded from the fact that the complex Young’s moduli of muscle fibres in rigor state do not depend strongly on tension (Fig. 8). Of course, in deviation of the activated state, final tension level in rigor will decrease after each additional shortening till it is eventually reduced to zero. In the activated fibre, due to loss of force exerting crossbridges and due to implementation of new ones, pulling force can be maintained at values above zero. These last processes could partly be fast too as evidenced by a growing difference between Young’s modulus below 2 kHz of the fibre in activated state and in rigor.

Comparison of the complex Young’s modulus as a function of frequency for muscle fibres in relaxed and rigor state resulted in a clear difference in linear increase with frequency in the log-log plot. This difference in Young’s moduli of muscle fibres in relaxed and rigor state will be connected with a difference in number of crossbridges and/or different states. We investigated if this difference in frequency dependence in complex Young’s moduli, described by the exponent and the reciprocal time constant, could be due to frictional drag by the incubation medium or to internal friction. First the viscosity of the incubation medium was doubled by addition of 20% (w/v) glucose to the normally used rigor and relaxation solution. The same stiffness measurements were performed and analysed. Comparison of magnitude and phase of the complex Young’s moduli of muscle fibres incubated in a solution with or without addition of 20% (w/v) glucose resulted in the conclusion that for muscle fibres in relaxed state addition of glucose did not change the frequency dependence of the complex Young’s modulus significantly. For a muscle fibre in rigor state doubling the viscosity of the incubation solution resulted in a somewhat different frequency dependence of the complex Young’s modulus. The modulus started to increase with frequency at a lower frequency. In terms of model parameters the elastic constant remained the same while the reciprocal time constant together with the exponent decreased significantly (at 5% level). However, the reciprocal time constant was not halved, but decreased only by a factor of 0.8 in case of rigor. At high frequencies (40 kHz) the effects of decrease of exponent and reciprocal time constant cancel and Young’s modulus does not change at all. Moreover in case of the relaxed fibre no difference at all was found at doubled viscosity. Therefore the difference in frequency dependence of the Young‘s moduli of muscle fibres in relaxed and rigor states can hardly be induced by viscous drag of the surrounding fluid. If it is assumed that recovery in rigor is due to state transitions of crossbridges, this can be explained as partial diffusion control of the transition rate (Chang, 1977). The result agrees with calculations of Tawada and Sekimoto (1991) who assumed only a small contribution to frictional drag by fluid.

DE WINKEL, BLANGE and TREIJTEL 143

of Young’s moduli of the fibre in rigor and relaxed state would at the same time tends to be less than the modulus of rigor itself and shows a more complicated frequency dependence. Taken the arguments together, it can be concluded that evidence is against explanation of the Young’s modulus in relaxed state by purely passive structures and that subtraction of Young’s modulus in relaxed state from that in rigor is not indicated for determination of the viscoelastic properties of crossbridges in rigor.

We investigated the magnitude of the complex Young’s modulus of muscle fibres in rigor state as a function of frequency after the fibres in rigor state were stretched or released to different isometric tension levels. To avoid slippage of crossbridges the muscle fibres in rigor state were not stretched more than 1.5 times normal rigor tension, which is less than Ca”-activated tension. These small stretches did not cause irreversible changes in the fibre segment because control rigor tension did not deviate from the control values before stretching. Suzuki and Sugi (1983) reported that irreversible changes in muscle fibres in rigor state take place after stretches beyond S-S% of their original length (a 8% stretch corresponds with 4.2 T,, with T,= maximal developed tension during activation). Because our displacements were small (-1.3 nm Per half-sarcomere < displacement < 1.3 run per half-sarcomere) it is unlikely that the results shown are due to irreversible changes.

Increase of tension applied to the crossbridges resulted in an increase of the elastic constant and the normalized rigor tension, while the exponent showed a decrease and the reciprocal time constant remained almost the same. The decrease in exponent partly counteracts the effect of increase of E, at high frequencies, but the net effect at 40 kHz is still an increase in Young’s modulus with isometric tension. If the view is accepted that crossbridges can make transitions from one rigor state to another, then at increasing isometric tension, a growing number of crossbridges might not be able any more to go into a subsequent state. This could lead to a growing average mechanical impedance of crossbridges and fibre, and to less recovery of tension, in line with the larger increase of Young’s modulus at low frequencies (E,) and the smaller increase at high frequencies.

In summary stiffness of muscle fibres in relaxed and rigor states depends on frequency in the range above 100 Hz. The difference in frequency dependence of the complex Young’s moduli of muscle fibres in relaxed and rigor state, especially at high frequencies, is hardly affected by changes in viscosity of incubation medium or interfilament distance. This leads to the conclusion that the complex Young’s modulus of muscle fibre in rigor state, as a function of frequency, reflects intrinsic properties of the crossbridges. Variation of tension exerted by the crossbridges does affect frequency dependence of the complex Young’s modulus in rigor state and can be

explained as tension dependence of the viscoelastic properties of crossbridges.

The variation in complex Young’s modulus as a function of frequency, must be assumed to reflect transitions of the crossbridges between different rigor states, which can be associated with tension recovery in tension responses on length changes. In view of the similarity of Young’s modulus of the fibre in rigor and in the activated state at frequencies above 2 kHz, this applies not only to the fibre in rigor, but probably too to force exerting crossbridges in the activated fibre.

References

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high frequency characteristics of elasticity of skeletal muscle fibres kept in relaxed and rigor...

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