| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
Biophys J, December 1998, p. 2971-2983, Vol. 75, No. 6
*Department of Biochemistry and Biophysics, and the Cardiovascular Research Institute, University of California, San Francisco, California 94143; #UCSF/UCB Bioengineering Graduate Group, University of California, San Francisco, California 94143, and Berkeley, California 94720; and §Department of Integrative Biology, University of California, Berkeley, California 94720 USA
 |
ABSTRACT |
|---|
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References
|
|---|
Active glycerinated rabbit psoas fibers were stretched at constant velocity (0.1-3.0 lengths/s) under sarcomere length control. As observed by previous investigators, force rose in two phases: an initial rapid increase over a small stretch (phase I), and a slower, more modest rise over the remainder of the stretch (phase II). The transition between the two phases occurred at a critical stretch (LC) of 7.7 ± 0.1 nm/half-sarcomere that is independent of velocity. The force at critical stretch (PC) increased with velocity up to 1 length/s, then was constant at 3.26 ± 0.06 times isometric force. The decay of the force response to a small step stretch was much faster during stretch than in isometric fibers. The addition of 3 mM vanadate reduced isometric tension to 0.08 ± 0.01 times control isometric tension (P0), but only reduced PC to 0.82 ± 0.06 times P0, demonstrating that prepowerstroke states contribute to force rise during stretch. The data can be explained by a model in which actin-attached cross-bridges in a prepowerstroke state are stretched into regions of high force and detach very rapidly when stretched beyond this region. The prepowerstroke state acts as a mechanical rectifier, producing large forces during stretch but small forces during shortening.
| |
INTRODUCTION |
|---|
|
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References
|
|---|
When muscle fibers, either activated or at rest,
are subjected to constant velocity stretch, force rises in two phases.
The first phase is very steep, with force in an active fiber increasing to at least twice the isometric tension (P0)
over an extension of only a few nanometers per half-sarcomere (nm/hs).
The second is much less steep, with force rising less than
1P0 over an extension of tens of nanometers per
half-sarcomere. The transition between the two phases of force rise
occurs after some critical amount of stretch. The critical amount of
stretch has been interpreted as a critical strain of each cross-bridge
(Hill, 1968
; Flitney and Hirst, 1978; Edman et al., 1978; Lombardi and
Piazzesi, 1990; Stienen et al., 1992), beyond which it is obliged to
detach from actin. Mechanically detached cross-bridges must also
reattach rapidly, for force to be maintained in the second phase (Hill, 1968; Flitney and Hirst, 1978; Güth and Kuhn, 1978; Griffiths et
al., 1980). Colomo et al. (1989b) also noted that the detachment rate must be small over some range of cross-bridge strain to keep that
range populated at high velocities of stretch.
When these ideas are implemented in models of cross-bridge mechanics, a
difficulty becomes apparent: the rapid attachment necessary to maintain
force during stretch at high velocities is, in the simplest models,
inconsistent with the decline in cross-bridge number during shortening.
Harry et al. (1990) varied the detachment rate transition function
g(x) of the model put forward by A. F. Huxley in 1957, in an attempt to fit both lengthening and shortening data. They found that force during stretch in this model was maintained by cross-bridges extended to extreme lengths, exceeding the assumed repeat distance between actin sites. The introduction of a maximum allowable cross-bridge strain implied that force would decrease as
stretch velocity increased. Lombardi and Piazzesi (1990); Piazzesi et
al. (1992, 1995) proposed models that solved this problem. Rapid
reattachment of mechanically detached cross-bridges populates force-producing states. They hypothesize a special detached state for
these cross-bridges, which is only populated during stretch.
We present a new model that explains the mechanics of stretched fibers, which is derived from identified biochemical states and is compatible with current structural models of the contractile proteins and with shortening mechanics. In this model most of the force during stretch is produced by cross-bridges in a prepowerstroke state that acts as a mechanical rectifier, producing large forces during stretch but small forces during shortening. Cross-bridges in this state are stretched to produce large forces, detach at a critical strain, then reattach rapidly. The same rapid kinetics apply during shortening. However, the prepowerstroke state produces little force in this case because negatively strained cross-bridges detach at small strains. We show that the properties of this prepowerstroke state are similar to those of prepowerstroke states proposed by others from different data.
The critical stretch and its velocity dependence are important
parameters for defining cross-bridge models. A number of investigators have studied the mechanics and energetics of muscle fibers during stretch, but most of these studies have employed living frog muscle (Hill, 1968; Flitney and Hirst, 1978; Edman et al., 1978, 1981, 1984;
Cavagna et al., 1994; Colomo et al., 1988a, 1989b;
Lombardi and Piazzesi, 1990; Piazzesi et al., 1992; Månsson 1994;
Tsuchiya and Sugi, 1988). More recently, skinned mammalian fibers have also been used (Stienen et al., 1992). Measurement of the critical stretch requires sarcomere length control, so that the compliance in
the ends of the fiber is not added to the measurement. Critical stretch
has been measured with sarcomere length control in living frog fibers
(Lombardi and Piazzesi, 1990), but not in mammalian fibers.
We stretched fully activated glycerinated rabbit psoas fibers over a
wide range of constant velocities (0.1-3.0 lengths/s), under sarcomere
length control, using an apparatus designed to eliminate stray
compliance. We measured the critical stretch, the force at critical
stretch, stiffness during each phase of the force response, and the
velocity dependence of each of these measures. Finally, we measured the
response to stretch of fibers in the presence of high levels of
vanadate, a phosphate analog that inhibits tension by increasing the
fraction of cross-bridges in a prepowerstroke state (Dantzig and
Goldman, 1985; Goodno, 1982).
| |
MATERIALS AND METHODS |
|---|
|
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References
|
|---|
Fibers
Bundles of psoas fibers ~2 mm in diameter were dissected from
New Zealand White rabbits. The bundles were tied to wooden sticks and
chemically skinned by incubation at 4°C in 50% glycerol, 0.12 M KCl,
5 mM MgCl2, 5 mM EGTA, and 20 mM
3-(N-morpholino)propanesulfonic acid (MOPS) (pH 7.0). The
solution was changed after the first 24 h, and the fibers were
stored at 
20°C for 2 weeks to 3 months before use. For
experimentation, single fibers were dissected from a bundle on a cold
glass microscope slide.
Fibers were attached to the apparatus before being lowered into
solution. One end of the fiber was laid across an arm (a glass capillary tube) connected to a silicon beam force transducer, and the
other end was laid across a metal arm connected to the armature of a
motor. The fiber ends that lay on the arms were then fixed with 2.5%
glutaraldehyde (Tousimis Research Corporation, Rockville, MD),
following the methods of Chase and Kushmerick (1988). Excess
glutaraldehyde was removed by vacuum. After fixation, the ends of the
fiber were glued to the arms using fingernail polish diluted with
acetone. Excess glue was removed by vacuum. The fiber was then lowered
into the relaxing bath, and fiber length and thickness were measured
with the reticule of a stereo microscope (SMZ-2T; Nikon, Tokyo, Japan).
Fiber diameter was estimated by assuming a circular cross section. The
length of the unfixed portion of the fiber between the arms varied from
3 to 4 mm, and the fiber diameter ranged from 50 to 100 µm.
Force transducer and motor
The apparatus has been described fully in Pate and Cooke (1988).
Briefly, the force transducer (Aksjelskapet Microelectronic, Horten,
Norway) with the attached arm had a resonant frequency of ~2.5 kHz
when immersed in solution. The motor (model CX-660; General Scanning,
Watertown, MA) with attached arm responded to a step position command
with an exponential time course, with a time constant of 0.4 ms.
Bathing solutions
Fibers were immersed in a 1-ml well maintained at 10°C, containing a relaxing solution consisting of a rigor buffer (5 mM MgCl2, 1.0 mM EGTA, 20 mM MOPS, pH 7.0), an ATP regenerating system (20 mM creatine phosphate and 1 mg/ml creatine phosphokinase), 4 mM ATP, and 3 mM inorganic phosphate (Pi). Addition of small amounts of Pi help to stabilize the diffraction pattern, allowing control of sarcomere length by feedback of the pattern. The activating solution also contained 1.1 mM CaCl2. The ionic strength of the relaxing and activating solutions was 112 mM and 115 mM, respectively.
Stock solutions of sodium metavanadate (NaVO3) (EM Science,
Gibbstown, NJ) were prepared at 100 mM, pH 10.0, and boiled immediately before use to minimize polymerization (Goodno, 1982). A small (3% by
volume) aliquot of the vanadate stock solution was added directly to
the bath containing the activating solution. Changes in the pH of the
activating bath were less than 0.1 pH unit as a result of vanadate addition.
Laser diffraction
Initial sarcomere length (L0) was determined by measuring the distance between the central and first maxima of a diffraction pattern using a 20-mW He-Ne laser (0.632-µm wavelength; Uniphase Corp., Manteca, CA). During the fiber stretch, changes in sarcomere length were tracked by movement of the centroid of the first diffraction maximum across a 15-mm-wide position-sensing photodiode (model SL-15; UDT Sensors, Hawthorne, CA). The photodiode currents were converted to voltages by high-input-impedance operational amplifiers, and the difference was amplified by an instrumentation amplifier with a flat frequency response up to 3 kHz.
Because the intensity of the diffraction pattern varied from fiber to fiber and between trials, the change in photodiode output voltage corresponding to a specified change in position of the diffraction pattern was calibrated immediately before each stretch, whether passive or active. The calibration procedure was as follows. The first maximum of the diffraction pattern was centered on the photodiode by moving the photodiode until the output voltage was zero. The position was also inspected visually, to ensure that the pattern was centered on the photodiode and not offset because of stray light from the central maximum. A baffle was inserted between the bath and the photodiode to block light from the central maximum. The photodiode was then moved by micrometer by 1 or 2 mm from the center, and the voltage from the photodiode was sampled. The calibration varied from 2 to 6 analog-to-digital units/nm of sarcomere length, so that the spatial resolution of the photodiode sarcomere length measurement was 0.2-0.5 nm. If the change in photodiode output with a change in detector position did not exceed an equivalent spatial resolution of 0.5 nm, the fiber was discarded. Diffraction patterns of the discarded fibers were not necessarily more scattered, but instead were weak in intensity. Finally, the photodiode was moved back to center the light pattern (zero voltage) before the fiber was stretched. The fiber was initially calibrated in relaxing solution, activated, and recalibrated in activating solution before stretching.
A major problem in measuring sarcomere length by laser diffraction is that some intensity from the zero-order diffraction maximum may also reach the photodiode, resulting in an erroneous measure of the position of the first diffraction peak. The effect of this contaminating signal was quantified by comparing the strength of the scattered light from the zero-order diffraction pattern to the strength of the first-order diffraction signal. The scatter in the zero-order diffraction pattern was predominantly due to the coverslips that form the walls of the bathing wells and to the bathing solutions through which the laser beam passes, and so was quantified by measuring photodiode output voltages from the laser beam repositioned so that it did not strike the fiber. The scattered light was found to be 2-6% of diffracted light, in both passive and active fibers, so that the error introduced by scattered light does not produce a significant error in the measurement of sarcomere length.
To check that the photodiode measurement accurately reflected sarcomere length, we changed sarcomere length in a relaxed fiber, where end compliance plays a minimal role, and compared the photodiode measurement with that obtained by direct visual observation of the position of the first diffraction maximum projected onto a screen 20 cm from the fiber and with that expected from the imposed fiber length change. In passive fibers, these three measures agreed to within 5%. When this measurement was made in active fibers, the photodiode measurement agreed with direct observation again to within 5%, but because of end compliance the sarcomere length change was only 35-50% of that expected from the fiber length change. These measurements show that the output of the photodiode provides a reasonable measurement of fiber sarcomere length.
To produce sarcomere length control during a stretch (Fig. 1), the position of the centroid of the first-order diffraction maximum was sampled and converted to sarcomere length. The difference between this estimate of sarcomere length and the command sarcomere length, the error in sarcomere length, was then converted to a fiber length error by multiplying by the fiber length divided by the initial sarcomere length. This fiber length error was used to drive the motor. Departures from linearity in the motor input show corrections made to the length command to achieve ramp stretches of sarcomeres and indicate the necessity of sarcomere length control. Typically the motor was required to move faster than the average velocity near the beginning of the stretch to compensate for end compliance, as shown in Fig. 1.
|
The open loop gain was adjusted to drive the feedback system close to
the command sarcomere length, while maintaining stability, and was
optimized for control at the beginning of the ramp. In addition to the
gain we set, the closed loop gain also depended on the compliance in
the fiber-machine connections, more compliant connections requiring a
larger change in motor position to drive the same change in sarcomere
length. The closed loop gain thus varied between fibers, and ranged
from 0.35 to 0.75 in the experiments reported here. The dependence of
gain on fiber compliance explains the slight curvature we often saw at
the long end of our sarcomere length recordings (e.g., upper
trace in Fig. 2): as the fiber stretched, it became stiffer (less compliant), so that the gain that
was adequate to maintain a ramp in sarcomere length at a shorter length
became too small to maintain a straight line at longer length. This
error, however, does not affect our measurements, as they are made in
the first few nanometers of the stretch, during which a constant
velocity is well maintained (a straight line fit to the first 90% of
the ramp in Fig. 2, for example, fits with R2 = 1.0000). We recorded many trials without sarcomere length control. In
these open loop recordings, the rapid rise in force (phase I) and the
slower rise (phase II) are still apparent, but the transition is much
less distinct: the open loop records looked like the force traces of
Stienen et al. (1992).
|
Data collection
During stretches, fiber force and the position of the first diffraction maximum were sampled at 1 or 5 kHz for 0.6 s by a 486 PC interfaced to the force transducer and photodiode by an analog/digital board (HSDAS-12; Analogic, Wakefield, MA). Input to the motor was calculated by the computer at a rate of 1 kHz based on movement of the diffraction maximum so as to produce the desired change in sarcomere length. For each ramp stretch, the command sarcomere length, actual sarcomere length (photodiode signal), motor input (position), and force were saved during the 50 ms of isometric contraction before stretch, the 3-5% stretch, and the isometric contraction for the remainder of the 600-ms sampling time. The computer program was written in Borland C++ (Version 3.1; Borland International, Scotts Valley, CA).
Experimental protocol
Fibers were mounted and immersed in a relaxing solution. The
fiber diameter was measured, and the force reading was set to zero. The
fiber was scanned for the clearest diffraction pattern by moving the
laser beam across it. If a clean pattern could not be found near the
center of the fiber, the fiber was discarded. Initial sarcomere length
(L0) was measured from the diffraction pattern
projected on a calibrated screen. L0 varied from
2.0 to 2.4 µm. The photodiode was then calibrated, and the fiber then stretched at a specified command velocity for 3-5% of
L0 under sarcomere length control, held at the
final length for the remainder of the 600-ms sampling time, and
returned to the initial length by a slow (0.1 L0/s) shortening ramp. Stretch velocities ranged from 0.1 to 3.0 L0/s. The maximum unloaded
shortening velocity for glycerinated psoas at 10°C is 1.6 L0/s (Cooke et al., 1988). Although the
measurements of Cooke et al. were made at an ionic strength greater
than that used in the present study, maximum shortening velocity has
been found to be independent of ionic strength (reviewed in Seow and
Ford, 1993). Commanded sarcomere length, actual sarcomere length,
force, and voltage input to the motor (corresponding to motor position)
were displayed on the computer monitor and used for quality control. If
actual sarcomere length was not proportional to command sarcomere
length, or if the motor input was anomalous, the data were rejected. In
relaxed fibers, the force response was small, typically 2-5% and
never exceeding 15% of the force change in activating solution at full stretch (1.05 lengths). If all measurements were in order, the fiber
was then transferred to activating solution, and photodiode calibration
and ramp stretch of the fiber were repeated. Maximum isometric stress
(P0) was typically 120 kN/m2. Once
force stabilized at P0, calibration and stretch
were performed as quickly as possible (10-60 s), as the diffraction
pattern was not long-lived. For most fibers, only one or two active
stretches were possible.
Stretches were also performed on some fibers in the presence of vanadate (Vi). The fractional decrease in isometric force and isometric stiffness produced by 3 mM Vi was determined on five fibers by the following procedure before the stretching experiments. After mounting, each fiber was immersed in a relaxing bath without Vi, and the force reading was set to zero. Fiber stiffness was measured from the change in force resulting from a rapid (<0.5 ms) 0.5% extension of fiber length. The fiber was then transferred to an activating bath without Vi. Once the force had stabilized and been recorded, active fiber stiffness was measured. Then 3 mM Vi was added directly to the activating bath. Force declined over the course of ~60 s, then stabilized. The ratios of the isometric force and stiffness in the presence of Vi to the isometric force and stiffness without Vi were recorded for each fiber. In the stretching experiments, each fiber was initially immersed in a relaxing bath without Vi. The photodiode was calibrated and the fiber was stretched following the protocol given in the preceding paragraph. The fiber was then transferred to the activating bath containing 3 mM Vi. Force rose initially, then fell. Once the force had stabilized, the photodiode was recalibrated, and the fiber was stretched.
Data analysis
For stretches faster than ~1L0/s, there
was a clear break point in the force record (Fig. 2, upper
traces). Digital differentiation of force (3-point Lancos
differentiating filter; Hamming, 1983) revealed a sharp transition
between rates of force rise at this point, with faster stretches often
showing a zero crossing in d(force)/dt. The transition time
was defined as the time after the beginning of the stretch when the
rate of force development (d(force)/dt in Fig. 2) first
became zero or negative. Critical strain (LC)
and critical force (PC) are strain per
half-sarcomere and force, relative to P0, at the
transition time. For slower stretches, a transition time was usually
much more difficult to define: there was no clear break point, and
d(force)/dt revealed no clear transition time (Fig. 2,
lower traces). We therefore report critical strains only for
records in which there was a clearly definable transition. For model
parameter identification, it was important to obtained an estimate of
PC at low stretch velocities (<0.5
L0/s). For these slow stretches, we estimated the break point by fitting tangents to the two phases of force rise
(Stienen et al., 1992). However, both the force and
d(force)/dt traces in Fig. 2 make it clear that force during
phase I is by no means linear, so this method cannot be expected to
yield as reliable results.
Stiffness of the isometric fiber was calculated by dividing the increase in force (relative to P0) after the small length step by the length change per half-sarcomere, as measured from the diffraction record. Because length steps are difficult to distinguish in the diffraction records during a stretch, stiffness during phase II was measured relative to isometric stiffness, by computing the ratio of the force rise in response to the step during Phase II to the force response to the step during isometric steady state (the command step sizes were the same).
The small length steps used to assess fiber stiffness produced transient changes in force. Decay half-times were measured as the time from peak force (at the time of the step stretch) to half the difference between peak force and the force extrapolated from the force trajectory before the steps.
| |
RESULTS |
|---|
|
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References
|
|---|
Force response to constant velocity stretch
In activated fibers, the force response to a constant-velocity
stretch consisted of two phases: an initial rapid increase occurring
over a very small stretch (phase I), followed by a slower rise lasting
the remainder of the stretch (phase II). The two phases were much
clearer in force responses to stretches faster than
~1L0/s (Fig. 2, upper traces), than
in responses to slower stretches (lower traces). In the
150-500 ms after the stretch, force declined in two phases well fit by
the sum of two exponentials (Colomo et al., 1989a,b; Cavagna,
1993). The faster rate constant was 200 s
1, independent
of stretch velocity, and the slower was about an order of magnitude slower.
Critical stretch and force at critical stretch
The break point in the force trajectory, identified in the 39 records with clear-cut zero crossings of d(force)/dt,
occurred at a stretch of 7.7 ± 0.1 nm/half-sarcomere (mean ± SEM). The critical stretch appears to be larger at the slowest
velocities of stretch (Fig. 3
A), but this difference is not statistically significant
(
< 0.05).
|
Force at critical stretch (PC) increased rapidly with velocity up to ~1L0/s, then leveled off at 3.26 ± 0.06 P0 (N = 34, Fig. 3 B). In relaxed fibers, the force response was typically less than 5% and at most 15% of the force change in activating solution, at full stretch (1.05 lengths). The passive component of force at critical stretch (i.e., at 1.008 lengths) was therefore at most 2.4% of the total.
Response to small stretches
Fiber stiffness was evaluated by superimposing small step stretches (command steps of 2.5 nm/half-sarcomere, producing actual steps of 0.7-1.9 nm/half-sarcomere) before, during, and after the constant-velocity ramp and measuring the corresponding increase in force. Steps were imposed 35 ms before the start of the stretch, at 40% and 80% of the stretch, and 10 and 60 ms after the stretch (Fig. 4).
|
Isometric stiffness (k0), defined as the ratio
of normalized force change to imposed length change, was 0.208 ± 0.014 nm1 (N = 127), comparable to the
value (0.239 nm1) measured by Lombardi and Piazzesi
(1990) in living frog fibers, using a similar length-step protocol.
Stiffness in phase II exceeded isometric stiffness by 15 ± 3.5%
(N = 102), similar to the 10-20% increase found by
Lombardi and Piazzesi (1990). It should be noted that the speed of our
motor (2.5 kHz) and slow sampling rates (1 or 5 kHz) may not be
adequate to measure stiffness accurately, especially during phase II.
If the force response to a step is falling with a time constant of 1.5 ms (as during phase II in Table 1),
sampling at 5 kHz may miss the early response, amounting to 20% of the
force, thus underestimating stiffness by 20%. Stiffness returned to
isometric stiffness within 10 ms after the stretch (1.04 ± 0.03 k0, N = 128) and remained the
same (1.081 ± 0.04 k0, N = 107) 50 ms later.
|
The rate of decay during phase II of the stretch was much faster than during the isometric steady state, or during the decline in force after the stretch (Table 1 and Fig. 4). The decay rate was the same throughout phase II (assessed at 40% and 80% of stretch) and was fast in passive fibers.
Effect of vanadate
To assess the contribution of prepowerstroke states to force rise
in phase I, we stretched fibers in an activating solution containing 3 mM vanadate. Vanadate (Vi), an analog to phosphate, forms a
stable complex with myosin and ADP, thus increasing the fraction of
cross-bridges in a prepowerstroke state (Goodno, 1982; Dantzig and
Goldman, 1985; Chase et al., 1993). The reduction in isometric force
and stiffness after the addition of Vi was monitored in
five fibers before the stretch experiments. The addition of 3 mM
Vi lowered isometeric force to 7.6 ± 0.5% of control
P0 and lowered isometric stiffness to 10.4 ± 0.6% that of control stiffness.
Fibers stretched in activating solution plus 3 mM Vi demonstrated both phases of force rise (Fig. 5). Vi (3 mM) attenuated PC much less than isometric force. At stretch velocities exceeding 1.0L0/s, PC was 0.82 ± 0.06 times control P0 (N = 12), as compared to 3.26 ± 0.06 P0 (N = 34) in fully active fibers. Thus, in 3 mM Vi the ratio of PC to isometric tension was ~10, compared to 3.2 in control fibers. As discussed below, we interpret this result as evidence that the prepowerstroke state contributes substantially to the increase in force observed when muscles are stretched.
|
| |
DISCUSSION |
|---|
|
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References
|
|---|
Comparison with previous data
Critical stretch
An important experimental parameter for constraining models of cross-bridge mechanics is the distance the sarcomere has been stretched when the phase transition occurs. An accurate determination of this distance requires sarcomere control to eliminate the fiber compliance that is external to the sarcomere. Such experiments have been performed with living frog muscle by Lombardi and Piazzesi (1990), who stretched
tibialis anterior fibers at constant velocities and found that force
overshot its steady-state value at the transition between phase I and
phase II. They measured the stretch at peak of force overshoot
(lp). Their Fig. 4 shows that
lp declines as stretch velocity increases and
asymptotes between 9 and 12 nm/half-sarcomere (above 0.56 µm/s per
half-sarcomere). Our studies are the first to apply sarcomere length
control to mammalian fibers during stretch. Our value for
LC (8 nm) is only a little shorter than the
value of lp for living frog fibers (9-12 nm).
Almost all of our data are at velocities
(>1L0/s) at the upper end of the velocity range they tested. In the absence of sarcomere length control, the critical stretches were longer (14-16 nm/half-sarcomere) for frog fibers (Edman
et al., 1981, 1984; Colomo et al., 1988, 1989) and for permeabilized rabbit psoas fibers, probably because of the contribution of compliances in series with the fiber (Stienen et al., 1992). In
fact, if we had stretched without sarcomere length control, we would
have obtained a value for LC similar to that of
Stienen et al. (1992), because in our experiments the motor moved
approximately twice as far as did the sarcomere length. We conclude
that the critical stretch in mammalian fibers is similar to that in
frog fibers, and that it is independent of velocity from
1L0/s to at least 3L0/s.
Force at critical stretch
Generally, investigators find that the steady-state force at high lengthening velocity in living frog fibers is ~2P0, independent of velocity above 0.5-1 µm/s per half-sarcomere (Sugi, 1972; Flitney and Hirst, 1978; Edman
et al., 1978, 1981, 1984; Cavagna, 1993; Cavagna et al., 1994; Colomo
et al., 1988a,b, 1989a,b; Lombardi and Piazzesi, 1990;
Piazzesi et al., 1992; Månsson 1994; Tsuchiya and Sugi, 1988). At
lower velocities, steady-state force is a rapidly increasing function
of velocity, with half-saturation at ~0.5 µm/s per half-sarcomere.
In permeabilized rabbit psoas at 15°C, Stienen et al. (1992) found
maximum forces asymptoting at ~2P0 for
isovelocity stretches up to 1L0/s. We found a
value half again as large. However, much of the difference is probably
due to three factors: temperature, phosphate, and ionic strength.
Cavagna (1993) found a strong temperature dependence of the force rise
during stretches of living frog fibers. Force rose to
2P0 at 4°C and to only
1.4P0 at 14°C. This change was due to a strong
temperature dependence of P0 and little
temperature dependence of the absolute force produced by stretch.
Similar results were found by Edman et al. (1984). In preliminary
experiments, we find a similar temperature dependence in skinned rabbit
psoas fibers.
Our relatively large PC may also be partially
due to the higher phosphate concentration at which we worked (3 mM
added Pi in our experiments, 0 mM in Stienen et al.). Added
Pi reduces P0 to a greater extent
than stretch force (Stienen et al., 1992; Iwamoto, 1995). Stienen et
al. added 15 mM Pi and found an attenuation of isometric
force to 58% of control and an attenuation of stretch force to 71% of
control. Interpolating from Stienen's data and using the logarithmic
relationship between Pi and P0
determined by Pate and Cooke (1989), we estimate that the added 3 mM
Pi in our experiments may have increased the relative force
enhancement during stretch by ~10%, by lowering
P0 more than PC.
The ionic strength of our activating solution, 115 mM, is reasonably
similar to physiological ionic strength (140-180 mM; Gordon et al.,
1973; Godt and Maughan, 1988), but lower than the 200 mM ionic strength
used by Stienen et al. (1992) in the previous study of skinned
mammalian fibers. Ionic strength varies the relative populations of
pre- and postpowerstroke states. In relaxed fibers, low ionic strength
increases the population of a weakly bound attached state (Brenner et
al., 1982; Schoenberg, 1991). In activated fibers, a prepowerstroke
state is populated by high ionic strength. These two prepowerstroke
states are probably not equivalent (Seow and Ford, 1993; Iwamoto,
1995). Experiments on active fibers have shown that fiber stiffness is
less affected by ionic strength than is isometric force (Piazzesi et
al., 1994; Seow and Ford, 1993; Månsson, 1989), suggesting that it is
not the number of attached cross-bridges that is altered by ionic
strength, but the distribution between pre- and postpowerstroke states.
The moderately low ionic strength of our bathing solutions would
therefore be expected to raise isometric force, with little effect on
stretch force (Piazzesi et al., 1994; Månsson 1994) or maximum
shortening velocity (reviewed in Seow and Ford, 1993). We performed
control stretch measurements on fibers bathed in a solution with ionic strength raised to 200 mM by the addition of 85 mM KAc, for comparison with the data of Stienen et al. (1992). Increased ionic strength lowered the isometric force by 30% but had no effect on force during stretch.
Phase II force rise
A continued force rise after the critical stretch was sometimes but not always present in our data. Force tended to continue to rise during slow but not necessarily during fast stretches. Stienen et al. (1992), also using permeabilized rabbit psoas fibers, consistently
measured monotonic force rise. However, stretches of permeabilized
rabbit soleus fibers (Stienen et al., 1992) and living frog fibers
(Lombardi and Piazzesi, 1990) display an overshoot in force, followed
by steady force. Edman et al. (1978, 1996) also using living frog
fibers, sometimes recorded force rise after the break point, and other
times did not. We conclude that the phase II force rise depends on the
preparation, and that it may be due to factors outside the
cross-bridge, such as sarcomere heterogeneity, as proposed by Morgan
(1990) and by Edman and Tsuchiya (1996). This portion of the data is
not considered in the model proposed below, which would predict a
constant force in phase II, due to cross-bridge kinetics alone.
Step responses
Piazzesi et al. (1992) imposed steps of different sizes, both
lengthening and shortening, on stretched living frog fibers. During
slow stretches they recorded four phases of force recovery, similar to
those of Huxley and Simmons (1971). For faster stretches the force
response was dominated by phase IV, reflecting rapid attachment and
detachment. For stretches at speeds comparable to those we imposed
(faster than 1 µm/s per half-sarcomere), the rate of recovery was
comparable to what we measured (0.4-0.7 ms for 63% recovery; their
figure 23).
Recovery of force and stiffness after stretch
Like Cavagna (1993), we found that the force decline in the first
few hundred milliseconds after stretch was well fit by two exponentials. The fast, early component had a rate constant of ~200
s1, and the second was at least an order of magnitude slower.
Although others have measured stiffness during stretch with much
greater time resolution than ours (Piazzesi et al., 1992), we know of
no data comparable to ours investigating the recovery of stiffness
after a stretch. Both the magnitude of the force response to a step
change in length and the rate of decay of that response recover within
tens of milliseconds of the end of the stretch.
Effect of vanadate
In addition to creating a long-lived prepowerstroke state (Goodno, 1982; Dantzig and Goldman, 1985; Chase et al., 1993), vanadate (Vi) also appears to reduce the total number of attached
cross-bridges: both isometric force and stiffness are decreased in the
presence of Vi, although at low temperatures (10°C), the
reduction of stiffness is somewhat less than the reduction of force
(this paper; Chase et al., 1993). Thus stretch force is reduced by
Vi, presumably because of the decrease in number of
attached cross-bridges, but much less than is isometric force, because
Vi holds cross-bridges in a prepowerstroke state that
exerts low isometric force but high force during stretch.
Model
Data to be fit
Any model of fiber mechanics during isovelocity stretch must account for the following data. 1. Force rises rapidly in the first few nanometers per half-sarcomere of stretch. Force may continue to rise after this initial phase but does so relatively slowly, if at all. The transition in force rise occurs after a critical amount of stretch that is independent of stretch velocity. 2. Fibers exhibit mechanical rectification. Both force and stiffness decline with increasing velocity during shortening, but the response to stretch is production of large forces (up to 2-3P0), with only a moderate change (10-20% increase) in stiffness. 3. The kinetics of the force response to small length steps are much faster during the stretch than before or (>10 ms) after the stretch. 4. Addition of Pi or Pi analogs reduces isometric force to a greater extent than the force generated during stretch. 5. After the stretch, in a fiber held isometric, tension relaxes with a fast phase at a rate of ~200 s1, followed by a slower
decline to P0. There is rapid (<10 ms) return
of stiffness to its prestretch value.
Previous models
The simplest cross-bridge model, proposed by A. F. Huxley in 1957, does not accurately describe force during stretch, because it predicts a decrease in steady-state force with increasing stretch velocities. By modifying the off rate, Zahalak (1981) was able to fit a
Huxley two-state model to steady-state lengthening data. However, to
maintain a large force at high stretch velocities, it was necessary to
allow some cross-bridges to extend to extreme lengths, exceeding the
assumed repeat distance between actin sites. Harry et al. (1990)
attempted several variations on the 1957 Huxley model to fit their data
on steady-state force as a function of stretch velocity. They found
that the introduction of a maximum allowable cross-bridge strain
implied that force would decrease as stretch velocity increased.
Otherwise, rapid attachment during stretch would be required to
maintain force at high velocities. Such rapid attachment is, however,
inconsistent with shortening of the Huxley 1957 model.
The necessity of a region of rapid attachment for maintenance of steady
force during stretch has long been recognized (Hill, 1968; Flitney and
Hirst, 1978; Güth and Kuhn, 1978; Griffiths et al., 1980). Colomo
et al. (1989b) also noted that the detachment rate must be small
over some range of cross-bridge strain to keep that range populated at
high velocities of stretch. However, rapid attachment will also
maintain high fiber stiffness and force during shortening.
The major problem in developing a model that incorporates these
concepts is that the rapid attachment into a force-generating state, required to maintain force during rapid stretches, also maintains force and stiffness during rapid shortening. Lombardi, Piazzesi, and colleagues (1990, 1992, 1995) developed models
that solved this problem by proposing a new detached state that is populated only by a rapid detachment of highly strained cross-bridges during stretch. The rapid reattachment of cross-bridges from this state
maintains a high force during stretch. Because this state is not
populated during shortening, force and stiffness decrease with
increasing shortening velocity. Thus this model is consistent with both
lengthening and shortening mechanics. Månsson (1994) put forward a
similar model, also with a special detached state entered only during
stretch. Based on the observation that stretch force is unaffected by
high extracellular tonicity while isometric force is depressed,
Månsson further proposed that stretch force is produced by the "low
force" state that precedes a high force state during isometric cycling.
Proposed model
We propose a new model to account for the mechanics of muscle during steady stretch. The model is, in essence, a synthesis of the biochemically detailed model of Pate and Cooke (1989), with some of the
kinetic concepts proposed by Lombardi, Piazzesi, and colleagues (1990,
1992, 1995) and by Månsson (1994). Our new model involves only
biochemically identified states and hypothesizes no special detached
state. Based on the observation that force rise during stretch is less
affected by Pi and Vi than is the isometric
force, we propose that the force during rapid stretch is produced
primarily by cross-bridges in a prepowerstroke (AMDP) state. We choose
rate constants associated with this state such that the cross-bridges
in it act as mechanical rectifiers, producing large forces during
stretch and small forces during shortening. In addition, the model
reconciles the critical length with recent measurements of the
powerstroke length and with structural models of myosin function
(Molloy et al., 1995; Rayment et al., 1993).
The Pate and Cooke model has three attached states. During shortening
or under isometric conditions, cross-bridges first attach in a
prepowerstroke state (AMDP), at the beginning of a region of positive
force production. Pi release results in transition to a
strongly bound state (AMD). ADP release and detachment due to ATP
binding to the AM state occur at the end of the region of positive
force production. The known energetics of the actin-myosin-nucleotide interaction in solution define the free energy minima of states in the
fiber (as in the model of Eisenberg et al., 1980). Pate and Cooke
assumed that attached cross-bridges act as Hookean springs. The free
energy minima and an assumed elastic force constant of 0.56RT nm2 then determine the free energy
profiles of the model. Pate and Cooke chose transition rate functions
(Table 3) to fit isometric and shortening data. The critical kinetic
assumptions that make shortening mechanics work in this model are MDP
cross-bridges bind very rapidly to actin over a short range, around the
minimum free energy of the AMDP state; the transition from weakly to
strongly bound states is limited by the release of phosphate, at ~50
s1; transition from the AMD state to AM is very slow
until near the end of the power stroke, then is quite fast; detachment
from AM to MT is quite rapid over the entire power stroke, and very rapid for x < 0. Our model retains these kinetic
assumptions, but with simple piecewise constant transition rates (Table
3), as it is intended to demonstrate feasibility, rather than to fit data precisely. The detached states MDP and MD are lumped into the
model, so an attachment rate from MD to AMD (R24 = 60 s1) is included. During stretch, cross-bridges are
assumed to detach from the AMD state to MD with the reverse rate
constant (R42). The Pate and Cooke model does
not include an off rate from AMD to MD, because this detachment is not
important in shortening.
The essential changes in rate constants required to implement our new
lengthening mechanism are in the assumed transitions between the
prepowerstoke state (AMDP) and its detached state (MDP). During
steady-state lengthening, AMDP cross-bridges are assumed to detach at
or before a maximum strain, implemented by a very large detachment rate
constant for distortions beyond 8 nm. Once detached, MDP cross-bridges
are assumed to reattach rapidly with a rate constant
(R23) that is large in a region of little strain
(2.26 nm 
x 4.0 nm). Rapid detachment and
reattachment has been shown to occur for both strongly and weakly bound
cross-bridges (Brenner et al., 1986). A relatively slow detachment rate
in a region of positive force production (4 nm x 8 nm) allows the AMDP bridges to produce large forces during stretch.
With these changes, the AMDP state becomes a mechanical rectifier. The
rapid attachment rate populates the positive force production region of
the AMDP state during lengthening, permitting forces up to 3P0. The very rapid detachment at
x > 8 nm ensures that force reaches a plateau as
lengthening velocity increases. Rapid attachment also occurs during
shortening or isometric contraction, but the AMDP state produces little
force under these conditions because positive- and
negative-force-producing regions are about equally populated, because
cross-bridges are assumed to detach rapidly from small negative strains.
To account for the short critical length we measured, and to keep total
cross-bridge distortion consistent with current structural models of
the contractile proteins, we compressed the x scale (cross-bridge distortion), so that the free energy minimum of the AMDP
state was at 4 nm, instead of the 7.5 nm assumed by Pate and Cooke.
This modification does not affect shortening behavior, because "the
model is invariant with respect to linear transformation of all values
of distortion," as Pate and Cooke point out. Maintenance of the same
free energy scale, but a compressed x scale, requires a
larger elastic force constant, 1.8kT nm2 per
cross-bridge.
The model has a power stroke of 4 nm. The ultimate strain of the
prepowerstroke (AMDP) state is also assumed to be 4 nm (beyond its
zero-force length of 4 nm), so that cross-bridges are assumed to be
distorted over a range of at most 8 nm, from isometric to rapid
stretch. These distances are compatible with current structural models
of the contractile apparatus, which suggest that force is produced by
the rotation of the 8-nm neck region of myosin (Rayment et al., 1993),
and consistent with in vitro motility assays (Molloy et al., 1995).
Simulations
The model with free energy functions as shown in Fig. 6 and listed in Table 2, and with transition rate constants given in Table 3, was simulated in Borland C++ on a 486 PC, using the method of characteristics (Carrier and Pearson, 1976). Fig. 3 B illustrates that the model fits
steady-state force as a function of velocity.
|
|
|
). Thus a stretch of 4 nm per cross-bridge corresponds
to a stretch of 8 nm/half-sarcomere. In the following discussion all
distances described in the model are for the cross-bridges only, and a
comparison with data must take actin compliance into account. After the
transition, cross-bridges broken off at the ultimate length quickly
reattach, with highest probability at low strain (near
x = 4), and the rapid on rate into the AMDP state at
low strain ensures a steady state. The AMD population also undergoes a
transition. AMD cross-bridges break off once their free energy exceeds
that of the detached MD state. At high speeds of stretch
(>1L0/s), reattachment from MD to AMD is not
fast enough to repopulate the AMD state fully, so force from AMD
declines as the isometric distribution breaks off and is not replaced.
|
1, chosen to match the fast component of force decline
at the end of the stretch. Cross-bridges without Pi also
reattach, to reestablish isometric force. The slow decline of force at
the end of stretch is thus not simulated by this model.
The mechanical rectification property of the AMDP state, simulated by
itself, is exhibited in Fig. 7 B. During lengthening, the
rapid on rate at negative distortions of this state populates it, and
the slow off rate from 4 to 8 nm keeps it populated, producing large
force. During shortening, the rapid off rate for sizable negative
distortions ensures that cross-bridges in this state produce little force.
The slow component of force decline after stretch is not explained by
our model, nor is the long-lasting force enhancement by stretch, nor
the force rise sometimes observed in phase II. All of these may occur
because of mechanisms beyond the level of a single cross-bridge or even
a single sarcomere, and all have been attributed to inhomogeneities
between lengths of sarcomeres in series (Morgan, 1994; Edman and
Tsuchiya, 1996).
While our model explains the stretch-induced increase in force by
postulating different kinetics of the prepowerstroke state when
stretched, our data do not exclude the possibility that stretch alters
the actomyosin interaction by changing the thin filament.
Data explained by this model
1. The observed effects of added phosphate or phosphate analogs motivated important aspects of the model, and the model simulates these effects well. Phosphate (Iwamoto, 1995; Stienen et al., 1992; Elzinga
et al., 1989) and analogs (this paper) affect PC much less, because the AMDP state they populate produces most of the
force during rapid stretch. A simulation of the model with sufficient
Pi to drop isometric tension to 50% of control showed that
PC was lowered modestly to two times control
P0.
2. The 10-20% increase in fiber stiffness (Lombardi and Piazzesi,
1990) during rapid stretch is explained in the model by an increase in
cross-bridge number during stretch, as more cross-bridges are added to
the AMDP state. A 20% increase in fiber stiffness is accomplished by a
50% increase in the number of attached cross-bridges, assuming that
half the fiber compliance is in actin, as described above.
3. The model explains the changing kinetics of force responses to the
stiffness-testing steps (Table 1). Under isometric conditions, the
force response is dominated by the slow kinetics of the AMD state.
Stretch populates a prepowerstroke AMDP state, which has very rapid
attachment and detachment rates. The very rapid decay in force during
steady-state rapid stretch occurs because even a small step moves some
of the AMDP cross-bridges past their ultimate length (x = 8 nm). These cross-bridges also reattach rapidly, so that force does
not drop below the steady-state level. During slow stretch, the
kinetics of force responses to stretch may be intermediate between
those for fast stretch and isometric (cf. Piazzesi et al., 1992).
4. The rapid return to initial conditions after stretch, including
return to isometric stiffness, and to a slower rate of force decline
after a step in length, are accounted for by the 200 s1
detachment rate for the strained AMDP cross-bridges (cf. Colomo et al.,
1989b). The rate of detachment from the high-force state also
determines the velocity required to populate this region. At very slow
stretches cross-bridges carried into this region are detached, whereas
at very high velocities they are carried through the region and
detached beyond the ultimate strain. The velocity of half-saturation
occurs at ~0.5L0/s, or at a relative filament
velocity of 600 nm/s. A simple calculation shows that half-saturation
occurs at a velocity approximately equal to the rate constant of
detachment times the length of the region in question. This would
suggest a region ~3 nm long, close to the length of the region
suggested by the critical length. Thus a rate constant, chosen to fit
force decline after stretch, also explains how the increase in force
depends on velocity.
5. Piazzesi et al. (1992) found that a rapid step shortening by 2 nm is
sufficient to bring force to zero. The distribution of cross-bridges in
the force-producing state during isometric contraction (AMD) is
consistent with this measurement: they are assumed to be equally
distributed over 4 nm, so that step shortening of the cross-bridges by
2 nm (shortening of the fiber by 4 nm per half-sarcomere, assuming half
the compliance of the fiber is in actin) would bring their force to
zero. A step of 2 nm would carry the distribution of prepowerstroke
(AMDP) cross-bridges, distributed between 2 and 6 nm during isometric
contraction (Fig. 6), into a region of very rapid detachment (0-4 nm),
so that they would not contribute force.
6. The model accounts for the low rate of hydrolysis of ATP in
stretched fibers (Abbott and Aubert, 1951), because force during stretch is generated by a state that precedes Pi release
(the AMDP state).
7. Finally, the model predicts a small negative force production early
in activation, as the negative force-producing region of the
prepowerstroke state has by far the fastest on rate. This may account
for the observation of latency relaxation, the reduction of force
observed in muscle after stimulation and just before a twitch (Abbott
and Ritchie, 1951).
Thus a simple model based on known biochemical states and structural
features can account for a variety of observations of the mechanics of
active fibers, including the magnitude, time, and velocity dependence
of force and stiffness during both stretch and shortening.
| |
ACKNOWLEDGMENTS |
|---|
We thank Dr. Ed Pate for ongoing critical review, Marc Chinn for help with the mechanical measurements, and one of the reviewers for the idea that changes in the thin filament may explain the data.
This work was supported by National Institutes of Health Grant HL 32145 (RC), a University of California Mentorship award, a Bank of America-Giannini Foundation Fellowship (EBG), and a grant from the Muscular Dystrophy Association.
| |
FOOTNOTES |
|---|
Received for publication 7 July 1997 and in final form 19 May 1998.
Address reprint requests to Dr. Steven L. Lehman, Department of Integrative Biology, University of California, 3060 Valley Life Sciences, MC 4480, Berkeley, CA 94720. Tel.: 510-642-5893; Fax: 510-643-2439; E-mail: slfr{at}socrates.berkeley.edu.
| |
REFERENCES |
|---|
|
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References
|
|---|
