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Laboratoire d'Optique Appliquée-Ecole Polytechnique, Institut National de la Santé et de la Recherche Médicale U451, 91125 Palaiseau cedex; and Service d'Explorations Fonctionnelles, Centre Hospitalier et Universitaire de Bicêtre, 94275 Le Kremlin-Bicêtre, France
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
ABSTRACT 
Coirault, Catherine, Denis Chemla, Jean-Claude Pourny,
Francine Lambert, and Yves Lecarpentier. Instantaneous
force-velocity-length relationship in diaphragmatic sarcomere.
J. Appl. Physiol. 82(2): 404-412, 1997.
The simultaneous analysis of muscle force, length, velocity, and
time has been shown to precisely characterize the mechanical
performance of isolated striated muscle. We tested the hypothesis that
the three-dimensional force-velocity-length relationship reflects
mechanical properties of sarcomeres. In hamster diaphragm strips,
instantaneous sarcomere length (SL) and muscle length were simultaneously measured during afterloaded twitches. SL was measured by means of laser diffraction. We
also studied the influence of initial
SL, abrupt changes in total load, and
2 × 10
7 M dantrolene.
Baseline resting SL at the apex of the
length-active tension curve was 2.2 ± 0.1 µm, whereas
SL at peak shortening was 1.6 ± 0.1 µm in the preloaded twitch and 2.1 ± 0.1 µm in the "isometric" twitch. Over the whole load continuum and at any
given level of isotonic load, there was a unique relationship between instantaneous sarcomere velocity and instantaneous
SL. Part of this relationship was time
independent and initial SL independent and was markedly downshifted after dantrolene. When five different muscle regions were considered, there were no significant variations of
SL and sarcomere kinetics along the
muscle. These results indicate that the time- and initial
length-independent part of the instantaneous force-velocity-length
relationship previously described in muscle strips reflects intrinsic
sarcomere mechanical properties.
diaphragm contractility; sarcomere kinetics; laser diffraction; sarcomere length inhomogeneity
INTRODUCTION THE CONTRACTILE PROPERTIES of diaphragm muscle have
been widely assessed by using isolated strips in both physiological and pathological conditions, as muscle length, loading conditions, and
stimulation can be precisely controlled (12, 28). In these latter
studies, muscle contraction was assessed by measuring the peak velocity
of shortening, the peak rate of force rise, or total isometric force.
Other studies have shown that simultaneous analysis of force
development, shortening length, shortening velocity, and time can more
precisely characterize the mechanical performance of isolated striated
muscle (1, 9, 21, 27), including the diaphragm (8). During a specific
part of the muscle contraction phase and for a given isotonic cardiac
muscle tension (IMT), instantaneous muscle shortening velocity
(MV) is a unique function of
instantaneous muscle length (ML),
regardless of time and initial muscle length (1, 9, 21, 27). Intrinsic
muscle contractility has been defined as the time- and initial
length-independent part of the three-dimensional
IMT-MV-ML
surface (1). The time- and initial length-independent part of this
surface is considered to represent a fundamental mechanical property of
active striated muscle.
It remains to be determined whether the three-dimensional
IMT-MV-ML
relationship is also a measure of contractility in sarcomeres, the
basic contractile units of striated muscle. A muscle fiber is composed
of numerous sarcomeres connected in series, whose nonuniformity may
contribute to the observed mechanical properties of the muscle,
together with viscoelastic elements in series. Because of both
sarcomere nonuniformity and the complex nonlinear properties of
viscoelastic elements coupled to the overall mechanical behavior of
muscle, the basic mechanical properties observed in muscle are not
necessarily present at the sarcomere level (10, 11, 15, 24, 25).
The purpose of this study was to determine whether the instantaneous
IMT-MV-ML
relationship of diaphragm muscle strip (8) was also observed in
sarcomere, thus providing a mechanical definition of contractility at
the subcellular level. We used laser diffraction technique to measure
the real-time kinetics of sarcomere contraction at all load levels
(22). To describe the subcellular mechanical contractile processes
accurately, the sarcomere kinetics were recorded simultaneously with
those of isolated hamster diaphragm muscle. Hamster diaphragm muscle
was selected because it yields excellent diffraction patterns (28),
thus allowing precise sarcomere length
(SL) measurements to be made.
SL inhomogeneity has been reported in
some skeletal muscles (2, 16-19), although it was not observed in
all studies (28). As the
IMT-MV-ML
relationship may critically depend on
SL, the first part of our study
evaluated critically whether hamster diaphragm exhibited significant
inhomogeneities of SL and kinetics
along the muscle. To this end, we recorded sarcomere kinetics at
five different locations along the muscle length. Because our results
showed that the midlength portion of diaphragm was a good
estimate of sarcomere measurements obtained in other regions,
subsequent sarcomere analyses were performed on the midlength portion
of diaphragm. Over a large range of external loads, the sarcomere and
muscle tension-velocity-length relationships were simultaneously
analyzed at baseline. We studied the influence of initial
SL, abrupt changes in total load
(i.e., abrupt load clamps), and dantrolene, a negative inotropic
drug (23, 29). The hypothesis was that, for a given load level and for
a given initial SL, there was a unique
relationship between instantaneous SL and velocity, regardless of the
history of loading conditions and characterizing the inotropic status.
METHODS This study was carried out on 20 golden hamsters. Care of the animals
conformed to good laboratory practice, and the study was approved by
our institution (Institut National de la Santé et de la Recherche
Médicale). The animals were anesthetized with ether. After median
laparotomy, a muscle strip from the ventral costal diaphragm was
dissected free from the muscle in situ. Insertions on both ribs and the
central tendon were left intact (5, 6). In this way, diaphragmatic
fibers were aligned in parallel and were approximately of equal length
(28). The muscles were vertically suspended in a bath containing the
following Krebs-Ringer solution (in mM): 118 NaCl, 4.7 KCl, 1.2 MgSO4 · 7 H2O, 1.1 KH2PO4,
24 NaHCO3, 2.5 CaCl2 · 6 H2O, and 4.5 glucose. The solution
was maintained at 22°C and equilibrated with a 95%
O2-5%
CO2, giving a pH of 7.4. The
costal extremity of the muscle preparation was held in a stationary
clip at the bottom of the bath, and the extremity of the central tendon
was held in a spring clip that was attached to an electromagnetic lever
system. Muscle strips were preloaded at the initial muscle length
(Lo)
corresponding to the apex of the length-active tension curve. Preload
is the resting tension that stretches the muscle before stimulation.
Afterload is the external tension developed by the muscle. Total load
is the sum of preload and afterload. The muscles were maximally
stimulated in twitch mode (12/min stimulation frequency with
rectangular pulses of 2-ms duration). Stimuli were provided through two
platinum electrodes placed longitudinally on either side of the muscle. At the end of the study, the cross-sectional area
(mm2) was calculated from the
ratio of fresh muscle weight to muscle length at
Lo, assuming a
muscle density of 1. The characteristics of the diaphragm muscles
studied were as follows:
Lo, 12.4 ± 0.4 mm; cross-sectional area, 0.9 ± 0.1 mm2; resting tension, 9.2 ± 0.4 mN/mm2. The force transducer
and the electromagnetic lever system have been previously described (5,
6).
Optical Technique
Mechanical Parameters
/sin 
, where = 0.6328 µm (wavelength of the laser beam) and is the angular separation
of the first-order diffraction line relative to the zero-order
reference line. From the meridional diffraction pattern (Fig. 1), the
first-order line was selected through an adjustable aperture and
collected through a lens that induced a parallel beam. The beam was
then split (B) into two equal parts. One beam was focused onto the PIN
10 diode (D2) producing an
electrical signal (h), which
represents the intensity variations of the first-order line. The other
beam passed through an optical density wedge (W) whose transmission was
related to the lateral shift of the beam and, consequently, was related
to variations in SL linked to
variations in the diffraction angle (22). This beam was then
collected and focused onto the PIN 10 diode (D1), giving the electrical
signal ( g) that represents the product of
intensity variations of the first-order diffraction line and intensity
variations due to the beam's lateral movement on the densitometric
wedge. o is the angular
separation of the first-order diffraction line at rest relative to the
zero-order reference line.
SLo
was calculated from the relation
o = Arc sin
/SLo, where
SLo
is the resting SL at
Lo.
During contraction, the electronic system computed the ratio
g/h in real time; this ratio is
directly related to instantaneous SL
through the relationship SL = SLo/(1 + 
log g/h), where is a
constant of the optical setup. In our experimental conditions, this
expression can be linearized as SL = SLo
[1 + (1 
g/h)].
As demonstrated in Appendix, potential changes in first-order line shape or width during contraction did not
modify the accuracy of our laser diffraction technique.
Fig. 1.
A: schematic representation of experimental setup; laser
diffractometer. L, laser; m, diaphragm muscle; B, beam splitter; W,
densitometric wedge; D1 and
D2, photodiodes;
g and
h, optical signals converted
electronically by D1 and
D2, respectively; T, electromagnetic transducer; M, tension and shortening curves vs. time
for diaphragm muscle; S, instantaneous sarcomere shortening length
curve vs. time. B: influence of
1st-order line width on densitometric wedge (W) transmission.
x, Position of middle of diffraction
line on the wedge; 2 
F, width of
diffracted line; Io, intensity of
diffracted line before transmission through densitometric wedge;
I(x), transmission of diffracted
line by wedge at location x; see also
Appendix.
[View Larger Version of this Image (19K GIF file)]
L; in
%Lo); maximum
muscle shortening velocity at preload (Vc max; in
Lo/s); isometric
peak tension (Pt; in
mN/mm2), and the positive peak
tension derivative of the isometric twitch (+dP/dtmax; in
mN · mm2 · s1).
During these two contractions, we measured initial
SL
(SLo,
in µm), SL at peak shortening
(SLmin;
in µm), maximum extent of sarcomere shortening length
(SLo SLmin = SL; in µm), time to peak
sarcomere shortening (TESL; in
ms), and peak velocity of sarcomere shortening
(SVmax; in
µm/s).
Experimental Protocols
Comparison of sarcomere behavior along the muscle length (n = 10). Uniformity in SL along the muscle length was evaluated in a subset of 10 diaphragm muscles. Throughout this protocol, diaphragm muscles were preloaded at Lo. In each muscle strip, five different regions of equal length were considered. They were numbered 1-5 from the lower end up to the upper end of the muscle (i.e., 1, lower region, 2, medium-low region; 3, central region; 4, medium-upper region; 5, upper-end region. At each location along the muscle length, sarcomere parameters were recorded in both preloaded and isometric contractions.Force-velocity-length relationships (n = 10). Three different protocols were carried out, the sarcomere dynamics being recorded in the central region of the muscle strip (region 3).
EFFECTS OF AFTERLOAD. Mechanical parameters were recorded at Lo over the complete load continuum. To this end, five to eight contractions with regularly incremented loads from preload up to full isometric contraction were performed on each muscle strip (Fig. 2). At each load level, the SL was recorded simultaneously with ML and with muscle tension (MT) (Fig. 3A). A clockwise plot of changes in instantaneous SL vs. instantaneous shortening velocity (SV) was recorded (Fig. 3B). This made it possible to analyze the entire pattern of sarcomere kinetics during shortening independently of time. The instantaneous SV-SL phase planes of various afterloaded contractions were plotted as a function of IMT. This resulted in an instantaneous three-dimensional IMT-SV-SL relationship over the whole load continuum (Fig. 3C). In this way, tension, SL, sarcomere velocity, and time were analyzed simultaneously.
7 M dantrolene (dashed
traces). Three-dimensional
IMT-SV-SL
diagrams were obtained by plotting instantaneous
SL-SV
phase planes as a function of isotonic muscle tension (IMT). Only the
contraction phase of
IMT-SV-SL
phase planes is represented. After dantrolene addition, isometric peak
tension was markedly lower than that of control twitch. Moreover, for
each load studied, time-independent part of the
SV-SL
phase plane was globally shifted downward after dantrolene treatment,
compared with controls.
EFFECTS OF PRELOAD. Preload was modified so that the initial resting muscle length was 95% Lo. Then, instantaneous SL, muscle shortening length, and corresponding velocity-length phase planes were recorded at various afterload levels. INFLUENCE OF DANTROLENE. Experiments were repeated after addition of 2 × 10
7 M dantrolene (Sigma
Chemical, lot 18F0673). Dantrolene exerts a negative inotropic effect
on skeletal muscle by inhibiting
Ca2+ release from the sarcoplasmic
reticulum (23, 29). It has been demonstrated that 2 × 107 M dantrolene markedly
depresses the isometric Pt of
isolated diaphragm muscle (23). Dilutions were made in dimethyl
sulfoxide (Sigma Chemical, lot 30H0608). The volume of drug added never exceeded 1 per 1,000 of the bath volume. This concentration was selected on the basis of a preliminary study in which 1/1,000 (vol/vol)
dimethyl sulfoxide did not alter the isometric
Pt of isolated diaphragm muscle.
Experiments were conducted over the whole load continuum, both at
Lo and at 95%
Lo. All the
mechanical parameters were measured 10 min after dantrolene addition.
Statistical Analysis
The data are expressed as means ± SE. Comparisons of sarcomere measurements at five locations along the muscle length were performed by using one-way analysis of variance with repeated-measures on locations. Comparisons of sarcomere mechanics before and after dantrolene were performed by using Student's paired t-test after analysis of variance. A P value <0.05 was required for statistical significance.RESULTS
The classic mechanical parameters of the diaphragm muscle at
Lo were as
follows: (L) = 13 ± 1% Lo;
Vc max = 2.5 ± 0.1 Lo /s;
Pt = 79 ± 4 mN/mm2, and
+dP/dtmax = 1,755 ± 94 mN · mm2 · s1.
Comparison of Sarcomere Behavior Along the Muscle Length
Figure 2 shows SL, ML, and MT as a function of time in a typical muscle contracting at five increasing total loads, from preload only (contraction 1) up to isometric load (contraction 5). In heavy loading conditions, no external shortening of the whole diaphragm strip occurred, whereas a small degree of sarcomere shortening was still observed. To determine whether sarcomere behavior depended on the region examined, SL and SVmax were measured at different positions along the muscle length. The mechanical characteristics of the twitch with preload only and the isometric twitch are given in Table 1. SLo did not significantly differ along the diaphragm length. In the twitch loaded with preload only, there was no significant difference between regions regarding SL at peak shortening (i.e., ESL), TESL, extent of sarcomere shortening (SL), or
SVmax. Moreover,
in isometric contraction, no significant difference was observed
between regions regarding ESL,
TESL,
SL, or
SVmax. Thus, in
hamster diaphragm, sarcomere parameters did not significantly differ
along the muscle length strips. This indicated that sarcomere
measurements obtained from the midlength region of the diaphragm
accurately reflected those obtained in other regions of the muscle,
both at rest and during contraction.
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Force-Velocity-Length Relationships in Diaphragmatic Sarcomere
During the shortening phase, sarcomere kinetics from the midlength of the diaphragm were simultaneously analyzed independently of time by means of phase-planes of instantaneous SV as a function of instantaneous SL (Fig. 3B). When various afterloaded twitches were considered, plotting of each instantaneous SV-SL phase plane as a function of IMT resulted in a three-dimensional IMT-SV-SL relationship (Fig. 3C). Effects of abrupt load clamps during the sarcomere shortening phase. Figure 4 shows typical sarcomere behavior after abrupt modifications in loading conditions (load clamps). Two different afterloaded contractions (twitches a and b) were first recorded. In a third contraction (twitch c), the sarcomere began to shorten with the same total load as that of twitch a. Contraction c was abruptly clamped during the first third of sarcomere shortening and at the same final total load as that of contraction b. Thus the same final total load was obtained in contractions b and c. Just following the load clamp, the traces of instantaneous sarcomere shortening vs. time of contractions b and c were distinct, indicating that a given degree of sarcomere shortening occurred at different times (Fig. 4A). However, instantaneous SV adapted quasi-instantaneously to instantaneous SL, and the sarcomere velocity-length phase planes of contractions b and c presented a common pathway (Fig. 4D). Thus, for a given level of isotonic total load and during a particular part of the SV-SL phase plane, instantaneous SV was a unique function of instantaneous SL, regardless of the time at which this SL occurred. Similar mechanical comportment was observed with the diaphragm muscle strip. After the load clamp, the traces of instantaneous muscle shortening vs. time of twitches b and c were dissociated (Fig. 4B), whereas, after a brief oscillation period, the corresponding muscle velocity-length phase planes merged (Fig. 4E). Moreover, after the load clamp, length and velocity changes occurred synchronously in whole muscle strip and sarcomere (Fig. 4, A and B). However, owing to the brief oscillation period in the whole muscle, there was a slight difference in the time-independent velocity-length phase planes in muscle and sarcomere (Fig. 4, D and E).
Effects of varying initial SL on sarcomere kinetics. In Fig. 5, initial SL was modified from one contraction to the next by varying preload, total load being kept constant. Thus sarcomere shortening occurred at two similar isotonic total loads but with two different initial SLs (SLo and 95% SLo). The traces of sarcomere shortening vs. time were distinct, indicating that a given SL was reached at two different times (Fig. 5A). However, over a large part of the sarcomere shortening phase, the instantaneous SV-SL phase planes merged (Fig. 5D). Thus, during a specific part of the sarcomere shortening phase and for a given isotonic total load, the instantaneous shortening SV was a unique function of the instantaneous shortening SL, regardless of the initial SL and the time at which the sarcomere shortening length occurred. Similar mechanical behavior was observed with the whole diaphragm muscle strip, i.e., during a large part of the contraction phase; the instantaneous MV-ML traces merged regardless of time and of initial muscle length (Fig. 5, B-E). Furthermore, the onset and end of the time-independent part of velocity-length common pathway were similar in whole muscle strip and in sarcomere (Fig. 5, B and E).
Time and initial length independence of the instantaneous IMT-SV-SL relationship. The above experiments indicate that for a given load level and during a specific stage of the IMT-SV-SL three-dimensional relationship, the sarcomere velocity-length phase plane was time and initial SL independent. This mechanical property was observed at all load levels and in all the muscles studied. When the total load was progressively increased from preload up to isometric tension, the whole continuum of the common pathway on the SV-SL phase planes described the specific time- and initial SL-invariant part of the three-dimensional IMT-SL-SV relationship (Fig. 3C). This specific part characterizes the level of contractility of the diaphragm sarcomere. Influence of 2 × 10
7 dantrolene.
In the diaphragm muscle at
Lo, dantrolene
induced a negative inotropic effect, as attested to by the decline in
Vc max (2.5 ± 0.1 Lo/s at baseline vs. 1.9 ± 0.1 Lo/s after dantrolene, P < 0.001), in L (13 ± 1 vs. 9 ± 1% Lo,
P < 0.001), and in
Pt (79 ± 4 vs. 54 ± 3 mN/mm2,
P < 0.001). The effects of
dantrolene on the mechanical parameters of sarcomeres are summarized in
Table 2. At preload, both
SL and
SVmax
decreased significantly. Conversely, sarcomere shortening at
Pt was not altered relative to
control value after dantrolene addition. For a given isotonic total
load, instantaneous SV remained a
unique function of instantaneous SL,
regardless of the time at which this
SL occurred (Fig.
6D).
This mechanical property was observed over the whole load continuum and
in all diaphragm muscles. Figure 3C
shows instantaneous
IMT-SV-SL
relationships at four load levels, both at baseline (thick trace) and
after dantrolene treatment (dashed trace). For a given instantaneous SL, the instantaneous
SV was slower after dantrolene than
before dantrolene, and the time-independent part of the
IMT-SV-SL
three-dimensional diagram was globally lower after dantrolene, compared
with the control twitch.
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DISCUSSION
We found that, for a constant isotonic total load and a given contractile state, whenever a given degree of instantaneous SL shortening occurred, only one instantaneous SV corresponded to it, regardless of initial SL and/or prior loading conditions. This indicates that the instantaneous force-velocity-length relationship previously described in isolated diaphragm muscle (8) reflects sarcomere properties, since both show exactly the same behavior.
Numerous studies have evidenced SL variability along fibers at rest and during contraction, especially at long fiber length (2, 16-19). At fiber length above Lo (i.e., SL > 2.2 µm), relatively shorter sarcomeres at the ends of the muscle have been shown to shorten at the expense of the longer sarcomeres in the central portion of the fibers (16-19). Such longitudinal nonuniformity is believed to greatly influence the descending limb shape of the SL-tension relationship (17-19). In the first part of our study, variability of SL along diaphragm muscle was examined at the resting SL at which the peak of active isometric tension occurs, i.e., 2.2 µm (2-4, 13, 14). At rest, average SL did not significantly differ between regions, as previously reported in diaphragm muscle at greater SLs (28). Moreover, during both isotonic and isometric contractions, we found similar sarcomere behavior at all locations along the muscle (Table 1). Thus sarcomere dynamics obtained by transilluminating the midlength portion of isolated diaphragm muscle were good estimates of sarcomere measurements obtained in other regions, both at rest and during contraction.
As laser light diffraction measures average SL values, we cannot exclude the possibility that there is some degree of nonhomogeneity in SL values within each sampled region. Moreover, it is also conceivable that SL dispersion of a given region increases during contraction in diaphragm muscle strip, as previously reported in frog skeletal muscle (24, 25). As SL dispersion has been related to the width of the first-order line (2, 20, 25), the influence of line broadening on the accuracy of our SL measurements was evaluated: as demonstrated in the Appendix, an upper limit increase of 50% in diffraction line breadth provides a relative error in the lateral shift determination of <1%. Thus the accuracy of our method is not significantly altered by the broadening of the diffraction line that may occur during contraction. Finally, it has been suggested that a population of sarcomeres that is oriented under an angle with respect to the longitudinal axis of the muscle generates asymmetrical diffraction patterns both in intensity and gravity center of the first-order line diffraction (26, 30). These so-called Bragg angle reflections might cause preferential sampling of the sarcomere population oriented under an angle with respect to the longitudinal axis of the muscle (26). The sampled population could potentially change as the muscle shortens, giving rise to erroneous results when SL measurements are based on the gravity-center displacement of the first-order diffraction line (26, 30). However, our measurements of SL were not based on the analysis of the gravity-center displacement of the first-order diffraction line but consisted of a densitometric wedge, whose transmission was related to the lateral shift of the whole first-order line.
Contractile performance depends on complex intracellular mechanisms that regulate both the number and kinetics of active cross bridges. In whole muscle, the time-independent part of the IMT-MV-ML relationship is thought to reflect "a perfect state of equilibrium of the degree of activation due to the interaction of multiple activating and inactivating influences" within the muscle cell (1). Activating influences lead to the formation of cross bridges and include membrane excitation, increased intramyoplasmic calcium, interaction of calcium with myofilaments, and concomitant enzymatic reactions. By contrast, inactivating influences lead to the detachment of cross bridges and the removal of intramyoplasmic calcium by membraneous systems (1). The time- and initial SL-independent part of the IMT-SV-SL relationship thus corresponds to the transition phase in which activation processes exactly counterbalance inactivation processes (1). Modifications in this state of equilibrium have been shown to modulate the level of the IMT-MV-ML relationship (1, 21). Consistently, the negative inotropic agent dantrolene downshifted the IMT-SV-SL three-dimensional surface relative to the control surface.
Except immediately after the load clamp, we found that whole muscle velocity and length changes ran parallel to those recorded in sarcomeres, in both the time and phase plane analyses. This mechanical correlation indicated a high degree of sarcomere synchronization during the whole contraction phase. It also suggested that viscoelastic elements connected in series with sarcomeres had somewhat simple mechanical behavior during this phase of contraction. Interestingly, we observed a brief period during which whole muscle velocity and length changes differed from those of sarcomere immediately after a load clamp. Similar transient high velocity has been observed in diaphragm (8) and other striated muscle strips (1, 9, 21, 27). It has been attributed to the passive elastic recoil of muscle due to elastic elements in series and/or in parallel with sarcomeres (1). As no oscillation occurred on the sarcomere velocity-length phase plane, our results suggested that the transient oscillations observed in the muscle strips were due to passive elastic elements in series with sarcomeres.
In conclusion, in diaphragm muscle contracting over the whole load continuum, and at any given level of load, there was a unique relationship between instantaneous SV and instantaneous SL. Part of this relationship was time and initial SL independent and precisely characterized the contractile performance of sarcomeres. These results indicate that the three-dimensional force-velocity-length relationship observed at the whole muscle level reflects intrinsic sarcomere properties.
ACKNOWLEDGEMENTS
We thank André Antonetti for providing laboratory facilities and Daniel Milly for technical assistance.
FOOTNOTES
Address for reprint requests: C. Coirault, INSERM 451-LOA-Ecole Polytechnique, Batterie de l'Yvette, 91125 Palaiseau cedex, France.
Received 9 May 1996; accepted in final form 1 October 1996.
APPENDIX
Influence of the Diffraction Line Width on the Accuracy of Sarcomere Length Determination (Fig. 1B)
We assumed a square input shape for the diffracted line with a width of 2F on the
densitometric wedge (W). Let xo, the position of
the middle of the diffraction line. The transmission of the wedge at
location x is
I(x) = 10kx = exp
(k
x), where k is a
constant characteristic of the wedge.
As k = 11.2%,
k = 0.112 × 2.3 = 0.258.
At the input of the wedge, the light power
Eo = 2 F
Io, where Io is the
intensity of the diffracted line before encountering the densitometric
wedge. Ec = Eo exp
(kxo).
At the output of the wedge the light power E on the diode D1 was
|
![E = − <FR><NU>I<SUB>o</SUB></NU><DE><IT>k</IT>′</DE></FR> exp (−<IT>k</IT>′<IT>x</IT><SUB>o</SUB>)[exp (−<IT>k</IT>′&dgr;<IT>F</IT> ) − exp (<IT>k</IT>′&dgr;<IT>F</IT> )]](/content/vol82/issue2/fulltext/404/img002.gif)
|
|
Under our experimental conditions, 2 F ~ 1 mm at rest (width of the
first-order diffraction line on the wedge). If we consider a 50%
increase in the width of the diffracted line during contraction (which
was an upper limit), the relative error on the light power (i.e., on
SL measurement) was
|
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