Sarcomeric visco-elasticity of chemically skinned skeletal muscle fibres of the rabbit at rest

Sarcomeric visco-elasticity of chemically skinned skeletal muscle fibres of the rabbit

at rest

K. W. RANATUNGADepartment of Physiology, University of Bristol, Bristol BS8 1TD, UK. E-mail: k.w.ranatunga@bristol.ac.uk

Received 19 March 2001; accepeted in revised form 24 July 2001

Abstract

The giant muscle protein titin (connectin), contained in the gap filament that connect a thick filament to the Z-line ina sarcomere, is generally considered to be responsible for the passive force (tension) and visco-elasticity in restingstriated muscle. However, whether it can account for all the features of the resting tension response remains unclear.In this paper, we examine the basic features of the ‘sarcomeric visco-elasticity’ in a single resting mammalian musclefibre and attempt to account for various tension components on the basis of known structural features of asarcomere. At sarcomere length of �2.6 lm, the force response to a ramp stretch of 2–5% is complex but can beresolved into four functionally different components. The behaviour displayed by the components ranges from pureviscous type (directly proportional to stretch velocity, ranging from 0.1 to 30 lengths s)1) to predominantly elastictype (insensitive to stretch velocity at 1–2 s time scale); simulations show two components of visco-elasticity withcharacteristically different relaxation times. The velocity-sensitive components (only) are enhanced by filamentlattice compression (dextran – 500 kD) and by increased medium viscosity (dextran – 12 kD); also, the relaxationtime of visco-elasticity is longer with increased medium viscosity. Amplitude of all the components and therelaxation time of visco-elasticity are increased at longer sarcomere length (range �2.5 – 3.0 lm). The study, andquantitative analyses, extend our previous work on intact muscle fibres and suggest that the velocity-sensitivetension components in intact sarcomere arise from interactions between sarcomeric filaments, filament segments andinter-filamentary medium; the two components of visco-elasticity arise from distinct regions of titin (connectin)molecules.

Introduction

It has been known for over a century that a relaxed(resting) striated muscle possesses long-range elasticityand certain other characteristic visco-elastic properties(see references in Ranatunga, 1994). Pioneering studiesby Maruyama and co-workers (1976, 1977) and byWang and co-workers (1979, 1984) have led to a newunderstanding of the muscle resting tension at the levelof a sarcomere. This is largely due to the identificationof a filamentous protein, titin ( connectin), that isassociated with thick filaments in sarcomeres (seereferences in Keller III, 1995). Briefly, experimentalstudies have shown that the resting muscle elasticity isresident within myofibrils (Magid and Law, 1985) andthat the resting elasticity derives principally from thegap (titin or connectin) filament that extends from thethick filaments to the Z-line in sarcomeres (Horowitsand Podolsky, 1987). Indeed, it has been shown that theelastic behaviour of gap filament can account for thesteady state, passive, force–extension characteristics inskinned muscle fibres (Wang et al., 1991; Horowits,1992; Granzier et al., 1996). A gap filament is formed bythe I-band region of, probably, six molecules of titin (seeWang et al., 1991). Skeletal muscle titin within I-band

consists of 70–90 tandem domains of immuno globulin(Ig regions) flanking an unfolded length, so-calledPEVK region, containing 1000–2200 residues (Labeitand Kolmerer, 1995). Molecular mechanism(s) of titinelasticity and extensibility, that may be considered asphysiologically relevant, remain unclear, althoughmechanisms involving folding–unfolding of Ig domains(Soteriou et al., 1993; Erickson, 1994), configurationaland other entropic mechanisms (see Politou et al., 1995)have been proposed. It appears that titin may have twophysiologically relevant elastic mechanisms; as a sarco-mere is stretched from slack length, the titin straighten-ing in the Ig regions leads to force rise with minimalchange in secondary/tertiary structure. Force rises moresteeply with further stretch due to extension in thePEVK region (Labeit and Kolmerer, 1995; Tskhovreb-ova and Trinick, 1997; Trombitas et al., 1998). Single-molecule biomechanics experiments on titin have alsoshown unfolding of individual Ig domains that form themodular titin, resulting in slow stress relaxation (seeKellermayer et al., 1997; Tskhovrebova et al., 1997).More recently, it has been found that transitionalextension of about 15% can occur due to reversibleand faster (�25 s–1) intermediate unfolding of Ig mod-ules (Marszalek et al., 1999).

Journal of Muscle Research and Cell Motility 22: 399–414, 2001. 399� 2002 Kluwer Academic Publishers. Printed in the Netherlands.

Observations on the molecular mechanisms of titinelasticity, summarised above, can account for somemechanical properties of resting muscle (for e.g. thesteady state resting force–extension relation in fibre types(see above). However, to what extent the gap (titin)filament contributes to determining more dynamic me-chanical properties of resting muscle remains unclear.The resting tension response to a ramp (and hold) stretchin intact muscle fibres (from frog and rat) can be resolvedinto, at least, three components, a viscous (P1), a visco-elastic (P2) and an elastic (P3) component. The identi-fication is made on the basis of their different sensitivitiesto stretch velocity (Bagni et al., 1992, 1995; Mutungi andRanatunga, 1996a, b). However, the underlying struc-tural bases of the components within a sarcomere are notexactly known. It was proposed that the viscous com-ponent arises from resistance to stretch between thickand thin filaments, whereas the visco-elasticity andelasticity results from extension in the gap filament aswell as from other cytoskeletal parts. Present study onshort segments of single skinned, rabbit psoas, musclefibres provides experimental evidence and detailed ana-lyses that basically support the above thesis: the resultsshow that the velocity-sensitive components only (P1 andP2) are enhanced by filament lattice compression andincreased medium viscosity. However, careful examina-tion suggests that the resting tension development maybe more complex, consisting of at least four components;a semi-quantitative interpretation of the velocity-sensi-tive forces in resting muscle fibres is attempted.

Materials and methods

The trough system used in this study was fully describedpreviously (Ranatunga, 1994, 1996). The front experi-mental trough had glass windows in front and bottom;its back wall was lined with aluminium foil, behindwhich were assembled small Peltier modules for use inclamping the trough temperature. The temperature ofthe entire trough system was kept at <10�C by passing acold antifreeze–water mixture through a jacket in theassembly and the temperature of the front trough wasclamped at 10�C by the Peltier module system. Thesolution temperature in the front trough was monitoredby the thermistor (for feedback) as well as by a separatesmall thermocouple (20–100 lm diameter, in differentexperiments) placed very close to the muscle fibre.

The design of the tension transducer was similar tothat described previously (Ranatunga, 1999). It consist-ed of two AE 801 elements (Akers, Norway) housedwithin a small brass box: one element was connected toa fibre preparation for tension recording and the otheracted as a dummy, forming a full bridge in order toreduce the temperature sensitivity. The tension record-ing silicon beam was cut to half its length so as toimprove its dynamic characteristics: the natural reso-nant frequency was 14 kHz. The motor was built using asmall permanent magnet (25 mm outer diameter) taken

from a loudspeaker; the moving coil was wound roundan aluminium-foil cylinder (‘former’); it was held byplastic hinges and its axial movement was monitoredphoto-electrically. The motor was capable of producingramp stretches of up to 60 lm in 200 ls. The transducerhooks were made of 50–100 lm diameter Invar wires(an alloy of steel and nickel; gift from Goodfellow,Cambridge, UK), used because of their low thermalcoefficient of expansion (see Ranatunga, 1996). Thecompositions of the buffer solutions used were the sameas those used in our previous studies (i = 200 mM andpH = 7.1, Goldman et al., 1987; Ranatunga, 1994).Briefly, the standard relaxing solution contained (inmM) 5-MgATP, 20-creatine phosphate, 10-glutathione,15-b-glycerol phosphate (pH buffer) and 20-EGTA and1 mg ml–1 creatine phosphokinase. However, the con-trol solutions also contained 4% dextran of highmolecular weight (500 kDa, D500) for compression offilament lattice spacing (in skinned fibres) to normal,intact fibre dimensions (see Matsubara et al., 1985; Xuet al., 1993).

Experimental procedures

A rabbit was killed by an intravenous injection of anoverdose of sodium pentobarbitone and muscle fibrebundles from psoas muscle were prepared, chemicallyskinned using 0.5% Brij Aldrich Chemical Co. andglycerinated as described previously: the skinning solu-tion contained 1 mM Na-azide (Fortune et al., 1989).Fibres from five animals were used in the study. In anexperiment, a segment of a single fibre (1–3 mm inlength) was mounted (using nitro-cellulose glue) be-tween two hooks, one attached to the tension transducerand the other to the motor. The fibre width and length(L0) were determined by optical microscopy. In addi-tion, the sarcomere length change in a 0.5 mm region ofthe fibre near the tension transducer was monitoredusing He–Ne laser diffraction; the position of the firstorder diffraction was monitored by a diffractometer,constructed essentially as described previously (seeMutungi and Ranatunga, 1996a). In a typical experi-ment, a series of 10–20 ramp (and hold) stretches ofdifferent slopes (range of stretch speed 0.1–40 L0 s–1)were applied when the fibre was immersed in relaxingsolution (pCa 8) in order to determine the stretchvelocity dependence of various tension components. Theamplitude of stretch was kept constant in a series (rangein different experiments, 2–5%L0) and the procedurewas repeated for different conditions on the same fibre(see Results). The steady tension developed undermaximal Ca-activation (pCa 4) was also recorded froma fibre. Typically, the initial sarcomere length wasbetween 2.6–2.7 lm.

Data recording, analyses and presentation

The outputs from the transducer (tension), the motor(fibre length), and the diffractometer (sarcomere length)

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were examined on two digital cathode ray oscilloscopesand, using CED 1401 (plus) laboratory interface andSignal Averager software (Cambridge Electronic DesignLtd, Cambridge, UK), stored in a PC based computer(486, CENCE Ltd, UK). In some experiments, up to fiveresponses were averaged at low stretch speeds, in orderto increase the signal to noise ratio. Measurements ofvarious tension components and their analyses weredone as described previously (Mutungi and Ranatunga,1996a; see also Figure 1). Measurement of the peaktension (Pk), the steady tension, at stretched length,after relaxation (P3) and the tension at break point(inflection) on the rising phase (P1) was made using theSignal Averager software; the steady resting tension, i.e.baseline tension before the stretch, was subtracted fromthese measurements. P2 tension was calculated asPk ) (P3 + P1). Tension values will be normalised tothe fibre cross-sectional area and given in kN m)2 in the

presentations; thus, these normalised tensions represent‘stress’ values. Further analyses of the data, involvingcurve fitting to P2 data (see Mutungi and Ranatunga,1996b), were done using Fig. P software (Biosoft,Cambridge, UK). The curve fitted to the P2 tension vs.stretch velocity was:

P2 ¼ lkðv=lÞrð1� expð�l=vÞ=rÞ;

where l is stretch amplitude (as ol/L0), k is stiffness, r isrelaxation time and v is stretch velocity (as (ol/L0)/s).This is the same basic equation used previously (Schoen-berg, 1988; Bagni et al., 1995), but since the stretchamplitude was constant, the analysis could be madeagainst stretch velocity (see Mutungi and Ranatunga,1996b). Such curve fitting provided the relaxation time(time constant) and the elastic modulus (k) of the visco-elastic (P2) component. Simulation of the tension

Fig. 1. General features of the tension responses. (a) Superimposed tension (top traces) and sarcomere length (middle traces) responses to a series

of constant amplitude ramp stretches at different velocities (bottom traces; 0.5–30 L0s)1) from one fibre (fibre length, L0, 1.59 mm); the fibre was

bathed in relaxing solution containing 4% dextran (500 kD). Note that the sarcomere length follow the induced change in fibre length. The

tension rises during a stretch, reaches a peak at the end of the ramp and, thereafter, decays to the same plateau level (P3) at the stretched length.

(b) Four of the traces displayed at a faster time scale. The tension traces show complexity on the rising and the falling phases. (c) Tension records

at five moderately fast speeds (>6 L0s)1). Note the ‘break’ or ‘inflection’ (P1) on the rising phase (arrows) and on the initial falling phase of the

records; at these moderately fast speeds, increase of peak tension is due to increase of break-point tension (P1), since P3 is constant and P2 has

reached steady level (see Figure 2b). (d) Two tension records along with the differentiated sarcomere length records. As shown by Bagni et al.

(1992) in frog fibres, the P1 inflection on the rising phase corresponds to the end of the sarcomere length acceleration and that on the falling phase

corresponds to the end of deceleration (the interrupted lines). Amplitude of P2 was calculated as Peak � (P3 + P1) [note that Peak, P3 and P1

tensions refer to the tension levels in excess of the initial steady resting tension (12 kN m)2 in this fibre, RT shown in c), which was subtracted

from all the measurements].

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transient using the sarcomere length response and othercalculations were done using Mathcad 7.0 software(MathSoft Inc., MA, USA).

Some general considerations of the experimentaltechniques and analyses

1. Identification of the break/inflection on the risingphase of the tension transient and measurement of theP1 tension level were difficult at low stretching velocitiesand required examination of the response at differenttime scales. The difficulty was partly due to the smalltension levels involved. Additionally, since the ampli-tude of a ramp stretch in a series was constant, the lowvelocity ramps last longer allowing the curving P2

tension response to be relatively more prominent (seeMutungi and Ranatunga, 1996a). Thus, the linearregression fitted to the P1 vs. stretch velocity data, insome cases, had a positive intercept of small, butsignificant, amplitude (see Figure 2b). Whether thisrepresents the slow cycling crossbridges (‘short rangeelasticity’ of Hill, 1968) as suggested by Bagni et al.(1995) remains uncertain. In our analyses, the linearregression was constrained to pass through the origin, inestimating the viscosity coefficient from P1 analyses,since measurement at higher velocities was readilyachieved.2. Analyses of the tension transient induced by a ramp–hold stretch, as described above, assumes that restingmuscle fibre sarcomere can be represented by a me-chanical model consisting of three components arrangedin parallel. Firstly, an elastic component (a spring, P3)develops tension (force) in direct proportion to sarco-meric length so that its force rises during the ramp,

reaches the steady level at ramp-end and remains so atthe extended length. Secondly, a viscous component (adash-pot, P1) develops a steady tension during the rampwhen the sarcomeric stretching velocity is constant; itstension disappears soon after the ramp-end. Thirdly avisco-elastic component (P2, a spring in series with adash-pot) develops tension during the ramp in a timedependent (delayed) manner; tension reaches a peak atramp-end and decays after the ramp at the extendedlength also in a time dependent manner. Thus, the peakof the tension transient is the sum of three componentvalues. The validity of this analysis has been discussedbefore (see Bagni et al., 1992; Mutungi and Ranatunga,1996a, b) and is also borne out by the fact that the basictension transient can be constructed from the sarcomericlength response (see Figure 6).

Results

Analyses of the tension responses

Sample experimental traces from a single muscle fibrethat was immersed in control relaxing solution areshown in Figure 1. Tension response (top traces) andthe sarcomere length change (middle traces) induced byapplication of constant amplitude ramp (and hold)stretches at six different speeds (range 0.5–30 L0 s)1;bottom traces) are shown in Figure 1a; the appliedstretches lead to corresponding changes in sarcomerelength. In each tension response, the tension rises duringthe ramp and the rise is higher at faster stretch speeds;thereafter, the tension decays to the same steady level atthe stretched sarcomere length. The steady tension in

Fig. 2. Stretch velocity dependence of the tension components. (Analysis of data from the same fibre, shown in Figure 1) (a) Peak tension (Pk, �)

and the steady tension after relaxation (P3, �) measured for a wide range of stretch velocities (L0 s)1); the stretch amplitude was constant at

0.037 L0. (Note that Pk has a complex dependence.) P3 tension was not significantly correlated with stretch velocity (P>0.1) and is insensitive to

stretch velocity (elastic). (b) Stretch velocity dependence of break point tension on the rising phase (P1, �) and the amplitude of P2 calculated as

Pk ) (P3 + P1) (�). The solid line through P1 data is the calculated linear regression (P<0.001); intercept is 1.24 kN m)2 and the slope (viscosity

coefficient, g) is 0.54 kN m)2/L0 s)1. Such a significant intercept was not obtained in all cases, and was ignored from further analyses. The dotted

line is the linear regression constrained to pass through the origin, and its slope gives g = 0.6 kN m)2/L0 s)1. The curve fitted to P2 data is that

used in previous studies for analysing visco-elasticity (see Methods); the relaxation time of P2 from the curve is 7.1 ms and the elastic modulus is

713 kN m)2. The dotted curve through P2 data is that based on two sub-components of visco-elasticity.

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this fibre is about 5 kN m)2 above the initial (pre-stretch) resting tension and will be referred to as P3. Oncareful examination of a tension response initiated byfast stretches (Figure 1b,c), the rising phase and thedecaying phase are seen to contain a break/inflection.Figure 1d shows two tension transients and the differ-entiated sarcomere length records (rate of change ofsarcomere length). As shown by Bagni et al. (1992), theoccurrence of the breaks in the tension transientcorresponds to the time (a delay of �0.2 ms) at whicha new steady sarcomere velocity is reached; the delay isprobably due to mechanical propagation time along thefibre. Thus, the tension measured at break-point (P1),represents a component that persists between the break-point and the peak when sarcomere velocity is constant.The tension at the break on the rising phase (P1), thepeak tension (Pk) and the steady tension (P3) weremeasured from each trace and the initial steady restingtension (RT in Figure 1c) removed; P2 tension wascalculated as Pk�(P1 + P3).

The tension data for a range of stretch velocitiescollected from the same preparation as in Figure 1 areshown in Figure 2, where tensions are plotted againstramp stretch velocity as L0 s�1 (fibre length per second).Figure 2a shows that P3 tension (open circles) isrelatively insensitive to stretch velocity (elastic) whilePk tension (filled circles) increases with stretch speed.Figure 1b shows P1 tension (open circles) plotted againststretch velocity and it is seen that P1 tension increases indirect proportion with stretch velocity (viscous). Theslope of the linear regression (dotted line) corresponds toa viscosity coefficient of 0:6� 103 Pa s L�1

0 (small inter-cept obtained in some cases, as here, was ignored – seeMethods). The P2 tension (filled circles) increases withvelocity up to a plateau (visco-elastic; i.e. a viscouselement in series with an elastic element). Analysis of theP2 tension data by curve fitting (see Methods) gives arelaxation time for the visco-elasticity of 7 ms (see figurelegend). Pooled data from seven fibres are given inTable 1.

Effect of dextran (D500 and D12)

Sample records in Figure 3a illustrate the basic effects ofhaving 4% high molecular weight dextran (D500) in therelaxing solution. Figure 3a shows the tension response

induced by the same standard ramp stretch in thepresence (indicated by arrow) and in the absence of 4%D500 in the relaxing solution; two responses in theabsence of D500 were recorded before and after exposureto D500. It is seen that osmotic compression has minimaleffect on P3 (elastic) but it markedly increases Pk, andthe effects are reversible. The same responses aredisplayed at a faster time scale in Figure 3b. They showthat, despite the small amplitude in the absence of D500,the general features of the tension response are similar tothat in the presence of D500; thus a break is seen on therising phase of the response. However, the amplitudes ofboth P1 and P2 components are smaller in the absence ofD500 resulting in a reduced Pk tension. The observationsare basically similar to those reported from skinned frogfibres after osmotic shrinking with other agents (Gold-man and Simmons, 1986). Figure 3c shows tension andsarcomere length traces from another fibre that wasexposed to 4% of dextran of small molecular weight(12 kD, D12) that would enter the filament lattice. Theeffects are similar to the above; the peak tension, but notthe steady elastic P3, is considerably increased in thepresence of D12. The definition of the break on the risingphase, however, was not clear in D12 containing solu-tions. The effect of dextran was examined in nine fibres.In each fibre, the tension responses to three to sevendifferent standard velocities were recorded with andwithout D500 or D12 and the average ‘with dextran/without dextran’ ratio determined for peak and P3

tensions. In every case the peak tension with dextran wasclearly larger than without dextran, but there weredifferences between fibres in the extent of increase,particularly with D12. The mean (s.e.m.) tension ratiosfrom different fibres are shown as histograms inFigure 3d. Thus, the amplitudes of velocity-sensitivecomponents (only) are increased by about 1.5–2-foldwith D500 and about 3-fold with D12; change in P3 is notsignificant (P > 0:1). The active tension was found to belittle affected by 4% D12; the active tension ratio, with/without D12, was 1.005 (±0.025, n ¼ 4), not significantlydifferent from 1 (P > 0:1). The active force has beenshown to be relatively insensitive to decrease of latticespacing (i.e. compression by D500), over a wide range, inprevious studies (see Gulati and Babu, 1985).

Complete analyses as shown in Figure 2a were notfeasible without 4% D500. Therefore, changes in the

Table 1. Summary data from psoas muscle fibres at 10�C (with 4% D500)

n S.L. (lm) g(kN.m)2/Lo.s)1) P2 , s (ms) P2 (kN.m)2) P3 (kN.m)2) Po (kN.m)2)

7 2.57 (0.03) 0.59 (0.09) 8.0 (1.1) 677 (99) 167 (24) 220 (24)

5 2.92 (0.05) 1.0 (0.1) 16.0 (3.2) 1364(193) 826 (158) 201(7)*

*n = 3.

Data are from 8 fibres in four of which data were collected at the two sarcomere lengths. Each value gives the mean (± s.e.m.); n = number of

fibres. The data given are the initial sarcomere length (S.L., in lm), the viscosity coefficient (g, in kN.m–2/L0.s)1) from P1 analyses, the time

constant (s, in ms) and elastic modulus (in kN.m)2) of visco-elasticity from P2 analyses, the elastic modulus for P3 (in kN.m)2) and the maximum

Ca-activated force (P0, in kN.m)2). The difference in the data between longer and shorter sarcomere lengths is significant (P < 0.05, t-test),

except for P0.

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tension components produced by 4%D12 were examinedin four fibres that were already compressed with 4%D500.Peak and P3 tension data at a range of stretch velocitiesfrom one fibre are shown in Figure 4a. It is seen that,compared to the data in control condition (i.e. with D500

only – open symbols and include data before and afterexposure to D12), D12 produces an increase of peaktension, but not ofP3. Figure 4b shows the analyses ofP1

and P2 as in Figure 2b; despite the scatter, an increase ofboth the viscosity coefficient (slope of P1 vs. velocity) andof the plateau P2 tension is seen. Pooled data from fourfibres are shown in Figure 4c where mean (s.e.m.) tensionratios are shown for the different tension components(open columns); the viscosity coefficient increased by�30%, plateau P2 tension by 20%: the apparent smallincrease (6%) in P3 is not significant (P > 0:1). Ofparticular interest is the finding that the relaxation timeof P2 is also increased (�30%) in the presence of D12.

Dependence on sarcomere length

Figure 5a and b illustrate experimental data from onefibre in which data similar to Figure 2 were collected attwo different sarcomere lengths. The data clearly showsthat the tension values of all the components (squares,dotted lines) are reversibly increased at the longersarcomere length. Similar experiments at short and longsarcomere lengths were done on four fibres, and thepooled data are shown in Figure 5c (illustration similarto Figure 4d). The data show that, compared to valuesobtained at the short sarcomere length of �2.6 lm, thetension values at �3.0 lm (open columns) are approx-imately two to three times larger for viscosity coefficient(P1), and for plateau P2 tension; P2 relaxation time(hatched column) is also increased (�2-fold). Addition-ally, and unlike with dextran, the steady P3 tension ismarkedly increased (6–8-fold). The maximal Ca-activated

Fig. 3. Effect of 4% dextran (a) Superimposed tension responses (upper traces) and sarcomere length responses (bottom traces) to a standard

stretch from a muscle fibre, when it was bathed in relaxing solution with or without 4% dextran 500 kD (D500). The tension response with dextran

is indicated by arrow and the other two were recorded 5–10 min before and after dextran exposure. [Note that in the presence of 4% D500, the

peak tension is increased but P3 is little affected.] (b) Same traces as in (a) but displayed at an expanded time scale. Tension responses with and

without D500 have the same features and, since P3 is not changed, the increased peak tension is due to increased P1 (arrows) and increased P2. (c)

Superimposed tension responses and sarcomere length records to a standard stretch from another fibre, with and without 4% dextran 12 kD

(D12). An increased peak tension with little change of P3 is seen; the definition of the break on the rising phase, however, was not clear in D12. (d)

Pooled data from nine fibres in each of which peak and P3 tensions were measured at three to seven different standard stretch velocities, with and

without 4% D500 (four fibres) or D12 (five fibres) added to the relaxing solution. Four percent dextran/no dextran tension ratio (mean ± s.e.m.)

is shown as histograms. Results indicate that filament lattice compression (D500) and increased medium viscosity (D12), greatly enhance velocity-

sensitive tension components.

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tension was not increased but was reduced (see Table 1)at the longer sarcomere length. Figure 5d show pooleddata from 12 fibres and they illustrate the general trendswith respect to sarcomere length dependence of the threecomponents; the amplitude of all three components andthe relaxation time of P2 are increased at longersarcomere length. The data at short and long sarcomerelengths are also given in Table 1.

Further analyses: simulation of tension transients

The experiments presented above, and the mean valuesfor various characteristics given in Table 1, clearly showthat the data from single skinned rabbit psoas fibres arebasically similar to those previously obtained fromintact fast (extensor digitorum longus, EDL) musclefibres at the same temperature (see Mutungi andRanatunga, 1996b). The differences in absolute values(e.g. larger P2 and P3 amplitudes in psoas) are probably

expected because of shorter titin isoform in psoas thanin EDL muscles (Labeit and Kolmerer, 1995).

To determine whether the above analysis is satisfac-tory to fully describe a given tension response, a threecomponent tension response was generated from asarcomere length record using the indices determinedfrom the analysis (as in Figure 2). Thus using thesarcomere length response, P3 component was generatedby direct scaling, P1 component was made by differen-tiating (see Figure 1d) and P2 component obtainedusing instantaneous velocity and the general formula forvisco-elasticity given in Methods with appropriate(single) time constant. The data from one fibre shownin Figure 6 illustrate the main findings. It was foundthat the rising phase of the tension response (includingthe break-point) was satisfactorily simulated, but dis-crepancies were found on the relaxation phase. Eachframe in Figure 6 shows the experimentally recordedsarcomere length and tension responses (continuous

Fig. 4. Effect of 4% dextran (12 kD), in the presence of 4% dextran (500 kD). (a) Peak (d, s) and P3 (m, n) tension data from a fibre, plotted

against stretch velocity. Open symbols and dotted curves are in control condition (with 4% D500, before and after exposure to 4% D12) and filled

symbols and solid curves are with 4% D12 and 4% D500. (Note increased peak tension with little change in P3.) (b) P2 and P1 tension data from

the same experiment (presentation similar to Figure 2b). The data show increased P2 and P1 in the presence of D12 (¤, d). (c) Similar data

analyses made in four fibres and the pooled data. Open columns show 4% D12/control ratios (mean ± s.e.m.) for viscosity coefficient (labelled

P1), elastic modulus for P2 and elastic modulus for P3. Note that P1 and P2 levels are increased 20–30%, whereas P3 increase is <10% (not

significant). Hatched column shows the ratio for the relaxation-time (r), from P2-analyses; the relaxation time is increased by �30% with D12.

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lines) and the generated tension trace (crosses). Fig-ure 6a–c show responses for three different stretchvelocities and frame d shows traces in c displayed atan expanded time scale. It is seen that the simulatedtension trace (crosses) deviates from the experimentalone in the tension relaxation phase; this is pronouncedat higher stretch velocities (compare c and d with a).

The discrepancy on the tension decay phase could belargely eliminated if the simulation is done by assumingthat the tension response contains two (rather than one)visco-elastic components. This is illustrated in Figure 7,where the visco-elastic (P2) component was assumed toconsist of two sub-components – each characterised by adifferent time constant and relative stiffness (see figurelegend). Same traces as in Figure 6c and d are shown inFigure 7a and b; they illustrate that the entire tensionresponse is modelled with such a four-component model.Similar analyses of tension traces from three fibres

(n ¼ 10) showed that the two visco-elastic sub-compo-nents would be characterised by average time constants(and range) of 4.7 ms (3–9.5 ms) and 56 ms (30–90 ms)and average relative amplitudes (and range) of 0.71(0.65–0.75) and 0.29 (0.25–0.35), respectively. The mainfinding from these analyses is that previous analysis(based on measurement of a ‘breakpoint’ on the risingphase, Figure 2) is too simplistic and that the P2 visco-elasticity consists of a predominant ‘fast’ component(�70%, 5 ms) and a small ‘slow’ component (�30%,50 ms). The dotted curve drawn through the P2 data inFigure 2b is that calculated for two sub-components.

Discussion

In general, the tension (force) responses to rampstretches presented here from skinned psoas muscle

Fig. 5. Effect of increased sarcomere length. (a) and (b) An experiment on one muscle fibre, in which data were collected at sarcomere lengths of

2.65lm (d, s), 3.0 lm (j, h) and finally at 2.65 lm (r, ); data presentation is similar to Figure 2 (a) Shows Pk tensions (d,j,¤) and P3

tensions (s,(,) and (b) Shows the P1 (s,(,) and P2 (d,j,¤). P3 tension at either sarcomere length was not correlated with stretch velocity

(P>0.1). Note that all tension components have, reversibly, increased in amplitude at the longer sarcomere length. Maximum Ca-activated

tension was decreased (16%) at the longer sarcomere length. (c) Similar paired data from four fibres. Open columns show the (mean ± s.e.m.)

tensions at the longer sarcomere length (�3lm) as a ratio of that from the same fibre at the shorter sarcomere length (�2.6lm). All the tension

components are increased at extended length and it is particularly pronounced in P3 (elastic tension). Hatched column shows the increased

relaxation time (r) for visco-elasticity (P2 component). (d) Pooled data for three different sarcomere length ranges (abscissa) from a total of 12

fibres (each data point is a mean ± s.e.m. from 4 to 7 fibres). Data shown are viscosity coefficient (P1 – s, lower left ordinate), elastic modulus of

P2 (() and of P3 – (d, right ordinate) and the relaxation time of P2 (j, upper left ordinate). Between long (�3lm) and short (�2.6lm)

sarcomere lengths, the differences for all the components are significant (P<0.02, t-test).

406

fibres in relaxed state are basically similar to thosepreviously reported from intact rat and frog fibres (seeIntroduction), and similar analyses can qualitativelydescribe them. Thus, their analyses indicate the occur-rence of three components (in parallel). One tensioncomponent is directly proportional to stretch velocityand persists steady during a constant velocity ramp(viscous, P1). A second component is insensitive tovelocity and remains steady from the end of the ramp, atthe extended length (elastic, P3). A third component isvisco-elastic (a viscous element in series with an elasticelement, P2) that develops with a characteristic relax-ation time (�10 ms) during the ramp and declinesafterwards at the stretched length. As shown in Fig-ure 3, present results show that the velocity-sensitivecomponents (P1 and P2) are enhanced by 4% dextran,whereas the elastic component (P3) is little affected. D500

(large molecular weight) is known to compress filamentlattice in skinned fibres (Xu et al., 1993), whereas D12

(small enough to enter filament lattice) would increasemedium viscosity around filaments within the lattice.Results in Figure 4 show that, when examined in fibresalready compressed with D500, D12 increased the ampli-tude of both P1 and P2 and also the relaxation time ofP2:P3 was again not affected. Thus, the findings identifytwo factors that influence the P1 and P2 components(only), namely the filament lattice spacing and themedium viscosity. At least qualitatively, these findingswould be consistent with the basic thesis proposed inprevious studies (Bagni et al., 1995; Mutungi andRanatunga, 1996b; see Figure 8). A viscous-like force,P1, may arise from resistance to sliding between mod-erately stiff sarcomeric filaments (thick and thin fila-ments) and their interaction with medium viscosity;increase of medium viscosity and/or decrease of inter-filamentary distance by filament lattice compressionwould enhance it. The P2 visco-elasticity may resultfrom lengthening in a more compliant extensible sarco-

Fig. 6. Construction of tension responses. Tension responses to ramp-sarcomere lengthening (lower-most trace) at three different speeds (a, b and

c) are shown superimposed on those (·) modelled using the analysed data; frame d shows the traces in c at a faster time scale. Using the analysed

data and the sarcomere length record, a simulated transient (·) was constructed as the sum of three components, an elastic (P3), a viscous (P1) and

a visco-elastic (P2) component – with a single time constant. In the simulation, an amplitude adjusted sarcomere length record was P3 and its first

differential, the P1. The rising phase of P2, and its relaxation from end of the ramp, was calculated using the general formula for visco-elasticity

given in Methods. Note that simulation satisfactorily produces the general features of the tension transient, including the break on the rising

phase. This indicates the general validity of the analysis employed here. Thus, the peak tension is the sum of three components: one that persists

during the ramp and is an analogue of sarcomere velocity (viscous, see Figure 1d); a second scales with sarcomere length (elastic) and a third that,

in a time dependent manner, increases during ramp and decreases afterwards (visco-elastic). The simulation is unsatisfactory on the tension

relaxation phase and the discrepancy is particularly pronounced at moderate to high ramp speeds (see b, c and d).

407

meric filament (titin filament) and its ‘polymer-like’interaction (see Rouse, 1953) with the medium. En-hancement of the P2 amplitude and its increasedrelaxation time (increased g, see below) in the presenceof D12 is therefore expected. Why lattice compressionwith D500 has a pronounced effect (Figure 3a and b)remains unclear, although it may come about fromappropriate re-alignment of filaments due to compres-sion. Additionally, D500 may contain some smaller-sizeddextran molecules that may indeed enter the filamentlattice and increase the viscosity. The P3 componentmay represent tensile elastic force in titin filament andother cyto-skeletal parts, and it would not be expected

to be sensitive to changes in medium viscosity orfilament lattice spacing, as indeed found in this study.

The amplitude of the three tension componentsincreased with sarcomere length (Figure 5) as found inintact fibre experiments (Bagni et al., 1995; Mutungiand Ranatunga, 1996b). Extension of sarcomere lengthwill lengthen resting force bearing elements (titin fila-ment), as well as decrease the inter-filamentary spacingin sarcomeres. Thus, extended sarcomere length willincrease the amplitude of the velocity-sensitive compo-nents in much the same way as D500 increases them, aswell as by titin extension. Additionally, present resultsshow that the relaxation time of P2 visco-elasticity is

Fig. 7. Simulation assuming that visco-elasticity (P2) consists of two sub-components. Simulation was done as in Fig. 6, but taking the visco-

elastic (P2) component as consisting of two sub-components – each characterised by a different time constant and a relative stiffness. The time

constant and the relative stiffness (amplitude) used here were 3.5 ms and 0.7 and 50 ms and 0.3 for the two sub-components. Same traces are

shown at two different time scales in a and b; note that both the rising and the relaxation phases of a tension response are satisfactorily simulated

by this model.

Fig. 8. (a) General features of the fractional extension of Ig region (·) and the PEVK region () of titin molecule in I-band region, when

sarcomere length is increased. Computation is done basically as described by Granzier et al. (1997) and Trombitas et al. (1998); note that Ig region

is extended by �70% at sarcomere length of 2.6lm, whereas PEVK is extended by only �30%. (b) and (c) Schematic arrangement of filaments

and their segments in a sarcomere at rest and at extended positions, as used in the Discussion. Thus, visco-elasticity arises largely from the PEVK

region due to extension of its end-to-end length and viscous tension arises from others that do not get significantly extended but get displaced –

thin filaments, thick filaments and Ig segments.

408

lengthened at the extended sarcomere length. Therelaxation time of a visco-elastic component is giveng � E, where g is viscosity coefficient of viscous elementand E is elastic modulus of elastic element that arein series; increased E (titin filament stiffness) at theextended sarcomere length would result in longerrelaxation time of P2. Increase of P3 could be linkedto extension in titin filament (and other cyto-skeletalelements) that bears the resting tension.

On the basis of the above considerations, it shouldbe possible to construct a tension transient from thesarcomere stretch-record. Indeed, the basic features ofthe tension transient were evident in such simulations(see Figure 6). However, the analyses presented inFigures 6 and 7 also show that the visco-elasticity (P2)consists of two sub-components; a predominant fastcomponent with a time constant (relaxation time) of�5 ms, and a smaller slow one with a time constant of�50 ms. Titin filament is thought to consist of twomechanically different regions (Ig and PEVK regions;see Introduction) and, it may be argued that, twocomponent visco-elasticity is a manifestation of thisarrangement.

In an Appendix below, the velocity-sensitive tensioncomponents are considered in more detail in an attemptto account for their origin, at least qualitatively, fromknown features of the sarcomeric filaments and theirinteractions in a typical sarcomere of a vertebrateskeletal muscle, but not taking cycling crossbridges asa contributory factor (see below). The aim is to extendthe calculations made previously that indicated thatvelocity-sensitive tensions are rather small in restingsarcomeres. Calculations given in Appendix lead tosignificantly larger values as upper estimates for thesetension components and the considerations can accountfor a number of basic observations. They show that amajor contribution to viscous force (P1) in restingmuscle fibres may arise from thin and thick filaments,particularly at rest length (SL� 2.6 lm). A two com-ponent visco-elasticity (P2) with different relaxationtimes and amplitudes can arise from gap (titin) filamentsegments (Ig and PEVK). The increase of the tensionamplitudes at longer sarcomere length (Figure 5) is,generally, due to increased participation of stiffened gapfilaments in determining these components; in the caseof velocity-sensitive components, it is also due tofilament lattice change.

The marked velocity sensitivity of tension develop-ment during stretch implies that, under more dynamicconditions, the force–sarcomere length relation of rest-ing sarcomeres would be shifted to a significantly lowersarcomere length range than under steady-state condi-tions. This may be of physiological significance since itmay operate as a mechanism that enhances sarcomerelength uniformity (and A-band stability) and mechanicaltransmission along muscle fibres. When physiological,nerve-induced low-frequency activation propagatesalong fibres, it will result in shortening of somesarcomeres and stretching of others; the results indicate

that, within limits, inactive sarcomeres will resiststretching in a speed dependent manner. Since, theviscous tensions decrease less with rise of temperature(Mutungi and Ranatunga, 1998), where as the speedof active contraction and power output increase muchmore prominently (see Ranatunga, 1984, 1998), therelative magnitude of these effects would be greater atphysiological body temperatures.

Uncertainties

The experiments reported here are from chemicallyskinned fibres that may contain, around them, structuralelements other than myofibrils e.g. collagen, intermedi-ate filaments etc. Being elastic structures their partici-pation during stretch would contribute to elastic tension(P3). The particular emphasis in this study is with regardto velocity-sensitive tension components; that non-myofibrillar filaments would contribute to such tensioncomponents that correlate with sarcomeric stretchingvelocity seems unlikely, but remains uncertain. It wouldbe of particular interest to repeat these experiments inmechanically skinned fibres and/or isolated myofibrils.

The question as to whether any of the tensioncomponents arises from slow or fast cycling crossbridges(Hill, 1968; Brenner et al., 1982) was not addressed inthe above discussion. A number of studies have foundno supporting evidence of significant involvement ofcycling crossbridges (see Bagni et al., 1992, 1995 andMutungi and Ranatunga, 2000: see also, Campbell andLakie, 1998). One main observation that does notsupport cycling crossbridge involvement is the findingthat the tension components are increased at the longersarcomere length (Figure 5). Additionally, in a study onrelaxed myofibrils, Bartoo et al. (1997) reported thatcycling crossbridges might contribute to tension only atlow temperature (5�C) and low ionic strength (20 mM).However, the upper estimates for tension values deter-mined here by calculation (see Appendix) are stillapproximately five to ten times smaller than thoseexperimentally measured. Thus, the significance of theuncertainties referred to above cannot be under-esti-mated. An aspect that may need further consideration isthat inter-filamentary interactions, that may not neces-sarily be cycling crossbridges, (e.g. titin–titin, titin–actininteractions) may contribute to tension response tostretch in resting sarcomeres (Li et al., 1995; Linkeet al., 1997; Trombitas and Granzier, 1997).

Appendix

The main purpose of doing the following calculationswas to extend those done by previous workers so as toobtain an upper estimate for the viscous-like resistanceto filament sliding and visco-elastic relaxation onstretch, in a resting sarcomere of a vertebrate skeletalmuscle. Thus, various parameter values taken are those

409

taken in previous estimations and that would apply to atypical vertebrate sarcomere.

Some quantitative aspects

The schematic diagrams illustrated in Figure 8 areconstructed as an aid to the following discussion. Whereapplicable, the following symbols and values are used inthe calculations presented below. Tension calculationswill be made for half-sarcomere (see Huxley, 1980).v ¼ 1 lm s)1, unit velocity for half sarcomere; gm ¼2� 10)3 Pa s)1, fluid (myoplasmic) viscosity (about2� g of water, calculated for a soluble protein concen-tration of 0.2 g ml)1, with intrinsic viscosity of 3 ml g)1

and a Huggin’s constant of 1 for determining specificviscosity, taken from Huxley, 1980).

Thick filamentN ¼ 6� 1014, number of thick filaments per squaremetre cross-section of muscle (Huxley, 1980): a1 ¼7:5 nm, radius of thick filament backbone. M ¼ 300,number of myosin heads in half a thick filament (i.e. perhalf sarcomere): a myosin head thought of as anellipsoid with a major width (W) of 15 nm and a minorwidth (w) of 5 nm. C ¼ 21, number of C-proteinmolecules per half the thick filament, five Ig domains(width 5 nm) exposed and floating (G. Offer, personalcommunication).

Thin filamenta2 ¼ 24:7 nm, radius of a uniform cylinder of thinfilaments around one thick filament (i.e. taking filamentlattice spacing with 4% dextran as 37 nm (Xu et al.,1993; Millman, 1998).A ¼ 400, number of actin monomers (350) + tropo-

nins (50); each spherical and with a radius, b ¼ 2:75 nm.The number of thin filaments per square metre is takenas 2� N . (a fibrous protein known as nebulin is closelyassociated with thin filaments (see Keller III, 1995), butits contribution is not considered here; see below).

Gap filamentn ¼ 6, number of titin (connectin) molecules per gapfilament (i.e. per thick filament per half sarcomere).Ig = 80–90 – number of Ig domains per titin moleculein I-band, Ig domain width 5 nm, lig is the random linklength, 15 nm. P ¼ 1600 – number of residues in PEVKregion, each 0.35 nm in width (Tskhovrebova et al.,1997), lP is the random link length 2 nm.

Viscous tension (P1)

Although the occurrence of an apparent viscous tensionduring a ramp stretch in resting muscle and muscle fibreshas been well documented (see Ford et al., 1977), theorigin of viscous tension has not been fully accountedfor. The viscosity coefficient, or the viscous tension atunit velocity (or at 1 lm s)1 for half sarcomere),obtained in the present study is �0.5 kN m)2 at 10�C

(see Table 1). This compares with the values obtained at10�C for intact mammalian (rat) fast muscle fibres(0.8 kN m)2; the coefficient for slow fibres was higher –2 kN m)2, see Mutungi and Ranatunga, 1996b). Forintact frog fibres at 0–3�C, Ford et al. (1977) give arange 1.5–4�108 N s m)3 for half-sarcomere; they cor-respond to viscous tensions of 0.15–0.4 kN m)2 at theunit velocity. More recently Bagni et al. (1997) reporteda value of 0.27 kN m)2 for intact frog fibres at 15�C.The temperature coefficient for the muscle viscoustension is similar to that for viscosity of water(Q10 ¼ 1:3, Mutungi and Ranatunga, 1998), probablyindicating that its origin is from some interaction withthe aqueous medium.1. Huxley (1980) made a calculation by assuming a thickfilament to be a (uniform) rod of radius a1, slidingwithin a (uniform) cylinder of radius a2 formed by thinfilaments. The viscous resistance to movement wouldthen be:

F1 ¼ Nð2pgmlvÞ= lnða2=a1Þ;

where l ¼ 0:8 lm (length of thick/thin filament overlap)at resting sarcomere length. The calculation gives atension at unit velocity of 5 N m)2. The value is about100 times too small, but the calculation does show thatthe viscous tension would increase with compression offilament lattice (decreasing a2). However, for a filamentlattice compression from �45 nm (with no dextran) to�37 nm (with 4% D500, see Xu et al., 1993), thecalculation gives a tension increase of �20% which ismuch less than the �75% increase obtained in the data(see Results).

The presence of myosin heads on the thick filamentwould increase the effective radius of thick filament. Asimple calculation may be made by lumping all myosinheads together to cover uniformly a certain length, lm,of the thick filament, so that a thick filament (l) consistof two regions, lm and a myosin head free (l� lm). Theradius of (l� lm) would be a1, but increasing theeffective radius of lm by 1.5–2.5 times – to take accountof the myosin heads in ‘sleeping’ (major width parallel tofilament axis) or ‘standing’ (major width perpendicularto filament axis) positions – increased F1 to 6–7 N m)2.Thus the estimate (F1) from this type of model is<10 N m)2 which is <1/50th of the measured value.2. An alternative calculation would be to determine theresistance to axial displacement arising from the differ-ent filaments in a sarcomere. A thin filament may bethought of as two arrays spherical beads in which eachbead interacts with fluid viscosity; since the filament issufficiently stiff, all beads along its length will be movingat the same velocity. An estimate of the total frictionalforce arising from thin filaments can then be made fromindividual bead contributions and using Stoke’sformula:

F2 ¼ 2NAð6pvgmÞð3pðb3=2ÞÞ

410

where 3pðb3=2Þ represents the radius of the reduced (to

50%) effective bead volume to take account of the lossof interactive volume and freedom due to overlapamong beads in the filament. The calculation yields avalue of 40 N m)2; the value is 30 N m)2, if the effectivebead volume is reduced further to 25% (and it is50 N m)2 if no reduction to the volume is made). It isdifficult to estimate the extent to which the presence ofthree to four nebulin molecules, that run parallel with athin filament (see Keller III, 1995), increases thefrictional force of a moving thin filament. The elasticmodulus of nebulin filament is two to three orders ofmagnitude smaller than that of the actin filament(Yasuda et al., 1995), but the structural arrangementbetween the two is unclear.3. A gap (titin) filament that connects a thick filament tothe Z-disc in sarcomere consists of the I-band regions oftitin molecules (n ¼ 6), each molecule spanning the fullI-band. It is known that a titin molecule has two distinctregions arranged in-series (Ig and PEVK regions – seeIntroduction) which have different mechanical charac-teristics (Labeit and Kolemerer, 1995). Thus, the Igregion has been shown to extend at lower range ofsarcomere length, whereas PEVK region determinesmuch of the force–extension relation at longer sarco-mere lengths (Granzier et al., 1997; Trombitas et al.,1998; see Figure 8A). At a sarcomere length of�2.6 lm, the gap-length (half I-band) in a sarcomerewould be �500 nm ((SL�1.6 lm)/2). Calculations showthat the Ig region of titin molecules (contour length400–450 nm) would be extended to about �70%,whereas the PEVK region (contour length �560 nm)only partially straightened (�30%) (see above). Eachregion of the titin molecule in the I-band may be likenedto a flexible polymer consisting of a series of randomlinks (of a standard length) that interact with themedium viscosity. In the following calculations, therandom link length used were 15 nm for Ig region (lig)and 2.5 nm for PEVK region (lP) and they are similar tothe persistence lengths used previously (Granzier et al.,1996, 1997).

A calculation may be made for Ig region in gapfilaments as,

F3 ¼ NnRigð6pvgfrPfÞc

where Rig is the number of random links, r is the radiusof a sphere equivalent in volume to the link, i.e. =3pð2:5 nm3 (lig=5 nm)) and Pf (Perrin’s factor) = 1.25(Tanford, 1961). Since force transmission to Z-disc willbe reduced due to flexibility, an approximate correctionfactor (c ¼ 0:7, extension ratio) is included. F3 is13 N m)2. A corresponding calculation for PEVKregion (F4), with a flexibility correction factor of 0.2(i.e. 0.7 � extension ratio of PEVK =0.3), yields3 N m)2.

4. Since two gap filaments suspend each thick filamentacross a sarcomere, the thick filament also will getdisplaced during sarcomere lengthening. If they are

moving at unit velocity, the force resulting from myosinheads in half-sarcomere, may also be estimated fromStoke’s formula, taking r as radius of a sphere that has avolume equal to myosin head (ellipsoid), calculated asr ¼ 3

pðW � w� w=8Þ. The frictional force due to myosinheads in thick filament is 30 N m)2. Similarly, thecontribution from freely projecting C-protein modulesin thick filaments would be about 6 N m)2, so that thetotal for a thick filament (F5) is 36 N m)2.

Assuming linear summation of individual compo-nents, the total force (as F1 + F2 + F3 + F4 + F5) atunit velocity would be �0.1 kN m)2 which is about 1/5th of that measured experimentally. Contributionsfrom at least two factors (F1 and F3) would increase,whereas others may remain unchanged at longer sarco-mere length. It is possible that a discrepancy factor (D)of �5 represents a resistance arising from other inter-actions not considered here (see below).

Visco-elastic (P2) and Elastic (P3) tensions

As had been suggested previously, the origins of visco-elasticity (P2) and, at least, part of elasticity (P3) arelikely to be in the titin (connectin) containing gapfilament, since the compliance of a resting sarcomereresides in the gap filament. Moreover, the fibre typedifferences in visco-elasticity could then be attributed totitin isoforms (Wang et al., 1991; Mutungi and Rana-tunga, 1996b).

The increase of steady (or elastic) tension (P3) inducedby a length step of 4% is 3 kN m)2 [calculated as EM �0.04/SL, where EM is elastic modulus for P3

(�200 kN m)2 – Table 1)]. This corresponds to atension increase of about 5 pN per gap (and per thick)filament, or �1 pN per titin molecule. A similarcalculation for visco-elastic tension (P2) yields �3 pNper titin molecule. These values are considerably smallerthan the tensions recorded in isolated single titinmolecules (�50–100 pN), during extension-induced un-folding (Kellermayer et al., 1997; Tskhovrebova et al.,1997) of their Ig domains. A tension of 1–3 pN for 4%extension is in the range expected from a titin moleculewithout unfolding (see Trombitas et al., 1998). Indeed acorresponding calculation using dimensions of psoasmuscle titin molecule (Labeit and Kolmerer, 1995;Tskhovrebova et al., 1997) predicts a steady entropictension increase of �1.0 pN for 4% extension atsarcomere length of 2.6 lm.

As mentioned above, at a sarcomere length of�2.6 lm, the PEVK region (contour length �560 nm)only partially straightened (�30%) (see above). Thissuggests that the dominant visco-elastic component (P2-fast) may arise from the PEVK-region, since much ofthe stretching will occur in that region (see Figure 8C).This is further indicated by the fact that the ratio ofcontour length/root mean square length is much higherfor each PEVK region (�40, calculated as 560 nm/[p1600� 0:35 nm]) than for an Ig region (about 10,

450 nm/[p90� 5 nm]). Since, the titin molecules in the

411

A-band region are firmly bound to the thick filament,the entire gap–thick filament complex becomes per-turbed by sarcomere lengthening, the gap filamentdetermining the centering of each thick filament. Thus,it seems appropriate to consider the gap–thick filamentcomplex as the relaxation unit. Since the thick filamentis stiff, it would be the two gap filaments on either endsof a thick filament that will undergo relaxation. Ap-proximate calculations can be made assuming thatPEVK region of titin molecule behaves as a flexiblepolymer in a viscous medium, so that the force inducedby its displacement has two components, an entropicone and another, arising from frictional interaction withthe medium (Rouse, 1953). Taking two PEVK regionsin sarcomere to consist of six to eight flexible segmentsof equal length that can undergo relaxation (Rouse’sapproximation), the relaxation time sðpÞ at differentmodes (p ¼ 1; 2 . . .) can be calculated using Rouse’s(1953) formula (see Ward and Hadley, 1993);

sðpÞ ¼ 3Dðls2fcÞ=½24kT sin2ðpp=2ðnþ 1ÞÞ�;

where k is the Boltzmann constant, T is absolutetemperature, n is number of flexible segments, ls ismean square length of segment – calculated from linklength and number, fc is frictional coefficient persegment length – calculated from friction in a link (asabove). The relaxation time for the largest component(p ¼ 1) is �2 ms, when an arbitrary discrepancy factorof 3� D is included (see below). This may well accountfor the major (P2-fast) component of visco-elasticity.The smaller amplitude P2-slow component probablyarises from the Ig region. A similar calculation for Igregions taking number of segments as 2 (since it isextended) and the same discrepancy factor (3� D), givesa relaxation time of (for p ¼ 1) of �20 ms. It may bethat the reversible intermediate unfolding of some Igmodules as reported by Marszalek et al., 1999 (at�25 s)1 or relaxation time 40 ms) also contribute toP2-slow visco-elastic component; indeed differences instability have been found between Ig-modules of titin(Politou et al., 1994). Since a given stretch producesdifferential relative stretching in the two regions (prob-ably about 6� more relative to its rest length in PEVK),the amplitude of the component arising from PEVK isexpected to be larger. These relative differences betweenPEVK and Ig components are in the appropriatedirection to explain the amplitudes and time constantsof the two sub-components of visco-elasticity (P2).However, the absolute values are an order of magnitudetoo low.

The above analyses raise two fundamental questions.Firstly, although higher than in previous determina-tions, the calculated velocity-sensitive tension valuesare about five to ten times too small. It is necessary tosee whether the myoplasmic viscosity is underestimat-ed. The value taken for the concentration of proteins inmyoplasm (200 mg ml)1) is probably not low (seeLuby-Phelps, 1994). If one takes values (from Tanford,

1961) of 3.6 ml g)1 as intrinsic viscosity for globularproteins and a Huggin’s constant of 2 (as for un-charged spheres), the myoplasmic viscosity will increaseby about 30%; the correction falls far short of therequirement. The study of Kao et al. (1993) on tissueculture cells indicates that the fluid phase of cytoplasmmay be as crowded as a 13% solution of dextran.Taking conservative estimates of 40 ml g)1 for intrinsicviscosity of dextran (i.e. as for cellulose acetate,molecular weight 12 kD, Tanford, 1961) and a Hug-gin’s constant of 0.5, this translates to a cytoplasmicviscosity of 0.02 Pa s, which is indeed about 10� largerthan that taken for our calculations. The presence ofany ‘structured water’ around myofilaments could raisethe apparent viscosity of water in the inter-filamentaryspace. Secondly, it seems likely that additional factorsare involved such as electrostatic, ionic and hydropho-bic interactions that are not taken account of in thesecalculations. Indeed forces in resting muscle fibres aregreatly augmented by decreased ionic strength in themedium (Bagni et al., 1995). In particular, inter-fila-mentary interaction may be an important contributorto these components. Even if thick and thin filamentinteraction through cycling crossbridges may be min-imal (Figure 5), there may be titin–titin interactionwithin gap filaments and/or actin–titin interactionbetween filaments. For instance, the exact arrangementof six titin polypeptides in a gap filament is not known,but the possibility of interaction between them cannotbe excluded (see Wang et al., 1984). Evidence ofinteraction between titin and actin has been obtainedin some studies (Li et al., 1995; Linke et al., 1997;Trombitas and Granzier, 1997). Also not considered inthe above analyses is a contribution from structures inZ-lines and M-lines in the sarcomere.

Acknowledgements

I thank the Wellcome Trust for its support of myresearch. I am grateful to Dr Gerald Offer (Bristol) withwhom I discussed some aspects of the work, and toProfessor G. Cecchi (University of Florence, Italy) andProf. R. M. Simmons (Kings College, London) forreading and making useful comments on an earlierversion of the manuscript.

References

Bagni MA, Cecchi G, Colomo F and Garzella P (1992) Are weakly

binding bridges present in intact muscle fibers? Biophys J 63: 1412–

1415.

Bagni MA, Cecchi G, Colomo F and Garzella P (1995) Absence of

mechanical evidence for attached weakly binding cross-bridges in

frog relaxed muscle fibres. J Physiol 482: 391–400.

Bagni MA, Cecchi G, Cecchini E, Colombini B and Colomo F (1997)

Force responses to fast ramp stretches in stimulated frog skeletal

muscle fibres. J Muscle Res and Cell Motil 18: 1–10.

412

Bartoo ML, Linke WA and Pollack G (1997) Basis of passive tension

and stiffness in isolated rabbit myofibrils. Am J Physiol 273: C266–

C276.

Brenner B, Schoenberg M, Chalovich JM, Greene LE and Eisenberg E

(1982) Evidence for crossbridge attachment in relaxed muscle at

low ionic strength. Proc Natl Acad Sci 79: 7288–7391.

Campbell KS and Lakie M (1998) A crossbridge mechanism can

explain the thixotropic short-range elastic component of relaxed

frog skeletal muscle. J Physiol 510: 941–962.

Erickson HP (1994) Reversible unfolding of fibronectin type II and

immunoglobulin domains provides the structural basis for stretch

and elasticity of titin and fibronectin. Proc Natl Acad Sci 91:

10,114–10,118.

Ford LE, Huxley AF and Simmons RM (1977) Tension responses to

sudden length change in stimulated muscle fibres at near slack

length. J Physiol 269: 441–515.

Fortune NS, Geeves MA and Ranatunga KW (1989) Pressure

sensitivity of active tension in glycerinated rabbit psoas muscle

fibres: effects of ADP and phosphate. J Muscle Res Cell Motil 10:

113–123.

Goldman YE and Simmons RM (1986) The stiffness of frog skinned

muscle fibres at altered lateral filament spacing. J Physiol 378: 175–

194.

Goldman YE, McCray JA and Ranatunga KW (1987) Transient

tension changes initiated by laser temperature jumps in rabbit

psoas muscle fibres. J Physiol 392: 71–95.

Granzier H, Helmes M and Trombitas K (1996) Nonuniform elasticity

of titin in cardiac myocytes: a study using immunoelectron

microscopy and cellular mechanics. Biophys J 70: 430–442.

Granzier H, Kellermayer M, Helmes M and Trombitas K (1997) Titin

elasticity and mechanism of passive force development in rat

cardiac myocytes probed by thin-filament extraction. Biophys J 73:

2043–2053.

Gulati J and Babu A (1985) Critical dependence of calcium-activated

force on width in highly compressed skinned fibers of frog. Biophys

J 48: 781–787.

Hill DK (1968) Tension due to interaction between the sliding

filaments in resting striated muscle. The effect of stimulation. J

Physiol 199: 637–684.

Horowits R (1992) Passive force generation and titin isoforms in

mammalian skeletal muscle. Biophys J 61: 392–398.

Horowits R and Podolsky RJ (1987) The positional stability of thick

filaments in activated skeletal muscle depends on sarcomere length:

evidence for the role of titin filaments. J Cell Biol 105: 2217–2223.

Huxley A (1980) Reflections on Muscle. The Sherrington Lectures XIV.

(pp. 66, 67) Liverpool University Press, Liverpool.

Kao HP, Abney JR and Verkman AS (1993) Determinants of the

translational mobility of a small solute in cell cytoplasm. J Cell

Biol 120: 175–184.

Keller III TCS (1995) Structure and function of titin and nebulin. Curr

Opin Cell Biol 7: 32–38.

Kellermayer MSZ, Smith SB, Granzier HL and Bustamante C (1997)

Folding–unfolding transitions in single titin molecules character-

ized with laser tweezers. Science 276: 1112–1116.

Labeit S and Kolmerer B (1995) Titins: giant proteins in charge of

muscle ultra-structure and elasticity. Science 270: 293–296.

Li Q, Jin J-P and Granzier HL (1995) The effect of genetically

expressed cardiac titin fragments on in vitro actin motility. Biophys

J 69: 1508–1518.

Linke WA, Ivemeyer M, Labeit S, Hinssen H, Ruegg JC and Gautel M

(1997) Actin–titin interaction in cardiac myofibrils: probing a

physiological role. Biophys J 73: 905–919.

Luby-Phelps K (1994) Physical properties of cytoplasm. Curr Opin Cell

Biol 6: 3–9.

Magid A and Law DJ (1985) Myofibrils bear most of the resting

tension in frog skeletal muscle. Science 230: 1280–1282.

Marszalek PE, Lu H, Li H, Carrion-Vazquez M, Oberhauser AF,

Schulten K and Fernandez JM (1999) Mechanical unfolding

intermediates in titin molecules. Nature 402: 100–103.

Maruyama K (1976) Connectin, an elastic protein from myofibrils. J

Biochem 80: 405–407.

Maruyama K, Matsubara S, Natori R, Nonomura Y, Kimura S,

Ohashi K, Murakami F, Handa S and Eguchi G (1977) Connectin,

an elastic protein of muscle: characterization and function. J

Biochem 82: 317–337.

Matsubara I, Umazume Y and Yagi N (1985) Lateral filamentary

spacing in chemically skinned murine muscles during contraction.

J Physiol 360: 135–148.

Millman BM (1998) The filament lattice of striated muscle. Physiol Rev

78: 359–391.

Mutungi G and Ranatunga KW (1996a) The tension relaxation after

stretch in resting mammalian muscle fibers: Stretch activation at

physiological temperatures. Biophys J 70: 1432–1438.

Mutungi G and Ranatunga KW (1996b) The viscous, visco-elastic and

elastic characteristics of the resting fast and slow mammalian (rat)

muscle fibres. J Physiol 496: 827–836.

Mutungi G and Ranatunga KW (1998) Temperature-dependent

changes in the viscoelasticity of intact resting mammalian (rat)

fast- and slow-muscle fibres. J Physiol 508: 253–268.

Mutungi G and Ranatunga KW (2000) Do crossbridges contribute to

the tension during stretch of passive muscle? A response. J Muscle

Res Cell Motil 21: 301–302.

Politou AS, Gautel M, Pfuhl M, Labeit S and Pastore A (1994)

Immunoglobin-type domains of titin: same fold, different stability.

Biochemistry 33: 4730–4737.

Politou AS, Thomas DJ and Pastore A (1995) The folding and stability

of titin immunoglobulin-like modules, with implications for the

mechanism of elasticity. Biophys J 69: 2601–2610.

Ranatunga KW (1984) The force–velocity relation of rat fast- and

slow-twitch muscles examined at different temperatures. J Physiol

351: 517–529.

Ranatunga KW (1994) Thermal stress and Ca-independent contractile

activation in mammalian skeletal muscle fibers at high tempera-

tures. Biophys J 66: 1531–1541.

Ranatunga KW (1996) Endothermic force generation in fast and slow

mammalian (rabbit) muscle fibers. Biophys J 71: 1905–1913.

Ranatunga KW (1998) Temperature dependence of mechanical power

output in mammalian (rat) skeletal muscle. Exp Physiol 83: 371–

376.

Ranatunga KW (1999) Effects of inorganic phosphate on endothermic

force generation in muscle. Proc Soc B 266: 1381–1385.

Rouse PE (1953) A theory of linear viscoelastic properties of dilute

solutions of coiling polymers. J Chem Phys 21: 1272–1280.

Schoenberg M (1988) Characterization of the myosin adenosine

triphosphate (M.ATP) crossbridge in rabbit and frog skeletal

muscle fibers. Biophys J 54: 135–148.

Soteriou A, Clarke A, Martin S and Trinick J (1993) Titin folding

energy and elasticity. Proc R Soc B 254: 83–86.

Tanford C (1961) Physical Chemistry of Macromolecules. (pp. 324–

328) Wiley, New York.

Trombitas K and Granzier H (1997) Actin removal from cardiac

myocytes shows that near the Z-line titin attaches to actin while

under tension. Am J Physiol 42: C662–C670.

Trombitas K, Greaser M, Labeit S, Jin J-P, Kellermayer M, Helmes M

and Granzier H (1998) Titin extensibility in situ: entropic elasticity

of permanently folded and permanently unfolded molecular

segments. J Cell Biol 140: 853–859.

Tskhovrebova L and Trinick J (1997) Direct visualization of exten-

sibility in isolated titin molecules. J Mol Biol 265: 100–106.

Tskhovrebova L, Trinick J, Sleep J and Simmons RM (1997) Elasticity

and unfolding of single molecules of the giant muscle protein titin.

Nature 387: 308–312.

Wang K, McCarter R, Wright J, Beverly J and Ramirez-Mitchell R

(1991) The regulation of skeletal muscle stiffness and elasticity by

titin isoforms: a test of the segmental extension model of resting

tension. Proc Natl Acad Sci 88: 7101–7105.

Wang K, McClure J and Tu A (1979) Titin: major myofibrillar

components of striated muscle. Proc Natl Acad Sci 76: 3698–3702.

413

Wang K, Ramirez-Mitchell R and Palter D (1984) Titin is an

extraordinarily long, flexible, and slender myofibrillar protein.

Proc Natl Acad Sci 81: 3685–3689.

Ward IM and Hadley DW (1993) An Introduction to Mechanical

Properties of Solid Polymers. John Wiley UK.

Xu S, Brenner B and Yu LC (1993) State-dependent radial elasticity of

attached cross-bridges in single skinned fibres of rabbit psoas

muscle. J Physiol 461: 283–299.

Yasuda K, Anazawa T and Ishiwata S (1995) Microscopic analysis of

the elastic properties of nebulin in skeletal myofibrils. Biophys J 68:

598–608.\scale120%

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