CHAPTER 3
Modeling of Lengthening Muscle:
The Role of Inter-Sarcomere Dynamics
David Morgan
3.1 Inter-Sarcomere Dynamics
Every muscle fiber is made up of a large number of sarcomeres connected in series, so the
tension must be the same in all of them. If the sarcomeres are identical, then velocity will also be
the same in all, so any externally applied movements will be equally distributed among all the
sarcomeres of the fiber. This leads to the common
procedure of modeling a muscle fiber, or even a
whole muscle, as a scaled sarcomere. Most crossbridge modeling of muscle makes this assumption
implicitly, bearing in mind that many experimenters have gone to great lengths, by use of
segment length clamps or diffraction measurements, to make it at least approximately true in
their experiments. Note that throughout this chapter the sarcomere is taken to be the fundamental
unit of contraction. In many ways, the half sarcomere would be a more appropriate unit, and all
the comments and calculations referring to sarcomeres could equally be taken to refer to half
sarcomeres.
However, it is most unreasonable to expect exact uniformity in the isometric tension-generating
capability (herein referred to as strength) of the
sarcomeres. There will always be some variation
in cross-sectional area and in myo-filament overlap. The question is whether and when the nonuniformity is significant. Huxley and Peachey
(1961) showed that fibers tetanized at long lengths
exhibited internal motion, with small regions at
the ends shortening and the rest of the fiber slowly
lengthening. They postulated that this internal
motion, together with the discontinuity of the
slope of the force-velocity curve, gave rise to the
slowly rising phase (tension creep) of "fiber
isometric" tension records at long lengths. A long
length here refers to lengths such that at least
some sarcomeres are on the descending limb of
the sarcomere length-tension diagram (Gordon et
al., 1966b; Julian and Morgan, 1979a, Figure 1).
The explanation runs as follows. Series connected elements with a length-tension diagram that
shows tension decreasing with increasing length
must be unstable. That is, inequalities in length
will be accompanied by inequalities in tension
generating capability that will lead to greater inequalities in length. The strong sarcomeres will
shorten and get stronger, while the weak will be
stretched and get even weaker.
The
force-velocity relation, however, shows an increasing force for increasing lengthening velocity. This
will provide dynamic stability to the sarcomere
length distribution. In fact for any achievable
duration of tetanus, most of the sarcomeres will
not have changed in length by more than a few
tenths of a micron. The internal motion would not
affect the tension (while all sarcomeres were on
the linear descending limb) were it not for the
change in slope of the force-velocity curve about
zero velocity. This can be seen by considering
two sarcomeres of unequal isometric tensions but
a zero sum of velocities - that is, their total length
is kept constant (Morgan, 1985, Figure 1). Their
tensions must be equal and the velocities of equal
magnitude and opposite direction.
If the
force-velocity curve were of constant slope, the
tension that satisfied this condition would be the
average of the two isometric tensions, and there-
Multiple Muscle Systems: Biomechanics and Movement Organization
1.M. Winters and S.L-Y. Woo (eds.), © 1990 Springer-Verlag, New York
3. Morgan; Modeling of Lengthening Musck
fore independent of the amount of difference
between the two sarcomere strengths, provided
that both are on the descending limb. However,
with a force-velocity curve that is steeper for
lengthening than for shortening, the tension must
be nearer to the isometric value of the shortening
(stronger) sarcomere than to that of the lengthening (weaker) sarcomere in order to make the
velocities of equal magnitude. This means that increasing the difference between the lengths and
hence strengths of the two sarcomeres will increase the tension. This same argument can be
generalized for many sarcomeres to show that the
tension in an isometric fiber is not the average of
the isometric tensions of the sarcomeres, but is
closer to the isometric capability of the stronger
sarcomeres.
The problem can be formalized as one of distributions and transformations. The first distribution is in sarcomere lengths. This is transformed into the distribution of sarcomere strengths
by the length-tension diagram. If all sarcomeres
are on the descending limb, then this is a linear
transformation and the mean of the transformed
distribution is equal to the transform of the mean
of the original distribution. That is, the mean
strength is the strength of the mean sarcomere
length.
The second transformation is from
strength to velocity at a particular tension. This is
essentially the force-velocity curve. Therefore
the mean velocity at a given tension will only be
the velocity appropriate to the mean sarcomere
strength at that tension if the force-velocity transformation is linear. In particular the mean
velocity will not in general be zero when the tension is equal to the mean strength if the
force-velocity transformation is non-linear, as is
found experimentally.
Gordon et al. (1966a, 1966b) showed that keeping the length of a more uniform central segment
of a fiber constant, rather than the length of the
whole fiber reduced the rate of rise of the "creep"
tension, in agreement with the explanation offered.
A smaller initial distribution of lengths and
strengths is expected to lead to a lower tension
early in the tetanus and less initial internal motion.
This in turn means that the degree of nonuniformity and the amount of "extra" tension
increase more slowly. Reducing the spread of sarcomere lengths has not been shown to reduce the
final peak tension, nor does modeling predict that
47
it should (Morgan et aI., 1982). If the tetanus is
continued long enough, the non-uniformities will
become large, and so will the tension.
Inter-sarcomere dynamics are also thought to
be significant in slow shortening of muscle fibers
(Julian and Morgan, 1979b, Figures 1 and 2) and
whole muscles (Abbott and Aubert, 1952, Figure
5). Experimentally, the tension in a fiber held
isometric after a slow active shortening from a
long length is less than the tension generated when
stimulation is commenced at the shorter final
length. Rapid shortening, however, produces no
tension deficit. In other words, a fiber is not able
to shorten slowly up its length-tension curve, as
would a crossbridge model of a sarcomere.
Interrupting stimulation of the shortened fiber or
whole muscle removes the deficit. Julian and
Morgan showed that the slow shortening was
largely absorbed by the end regions seen by
Huxley and Peachey (1961). This means that
most of the sarcomeres have not shortened, and so
have not increased their tension generating
capabilities. The initially shorter and stronger end
sarcomeres have shortened out of the descending
limb, and have become weaker again as they
moved onto the ascending limb of the
length-tension relation. However, there is always
a transition zone of sarcomeres with lengths between long and short, and hence strengths greater
than the tension. These are the ones that take up
most of the imposed shortening. When the shortening is more rapid, the tension is less and the
range of velocities for a given range of strengths is
smaller. This leads to a more uniform shortening
and so a final isometric tension nearer to that seen
if stimulation is commenced at the final length.
The model described here is able to simulate this
experiment (Morgan, 1990). The depression of
stiffness, or more accurately the lack of rise of stiffness, which is reported to accompany the lack of
rise of tension (fsuchiya and Sugi, 1988), is also
compatible with this explanation. Most of the sarcomeres have not changed length, and therefore
not stiffness, so that the fiber stiffness remains
near to that at the original long length.
3.2 Sarcomere Behavior During Stretch
Despite the extensive attention given to the behavior of muscles during stretch in recent years
(e.g., Cavagna et al., 1968; Huxley, 1971; Edman
et al., 1978; Flitney and Hirst, 1978; Morgan et aI.,
48
Multiple Muscle Systems. Part I: Muscle Modeling
1978; Julian and Morgan, 1979a; Edman et al.,
1982; Cavagna et al., 1986; Umazume et al., 1986;
Colomo et al., 1988; Sugi and Tsuchiya, 1988;
Tsuchiya and Sugi, 1988; Bottinelli et al., 1989;
Harry et al., 1990 [see also Chapter 2 (Hatze)],
relatively little consideration has been given to
inter-sarcomere dynamics under these conditions.
Lengthening muscle produces a number of unexpected results (see Sections 3.4 and 3.5) and also a
remarkable inconsistency of results. This difficulty in reproducing results in even successive
stretches on the same fiber could be seen to suggest that inter-sarcomere dynamics may be
important, with the initial distribution of sarcomeres being the fundamental inconsistent factor.
Woledge et al. (1985, p.71) highlighted this
variability in respect to the yield ratio (the yield
tension expressed in terms of the isometric
tension) and change in slope of the force-velocity
curve, but it is apparent in many other features of
tension records during lengthening (see also
Chapter 6 (Winters».
The force-velocity curve of a fiber or whole
muscle for lengthening is usually found to have a
yield tension, meaning that increasing the velocity
beyond a certain point does not increase the tension any further (Katz, 1939). This of course
corresponds to zero incremental damping or viscosity.
The statement above, that the
force-velocity curve will stabilize the inherent instability of the sarcomere length distribution, is no
longer true if the sarcomeres are stretching in this
yielding condition. The instabilities will be
catastrophic in the sense of producing very rapid
changes in the sarcomere length distribution.
This becomes clear in a simple thought experiment. If a fiber is stretched at other than a very
small velocity, the tension will rise as elastic structures are stretched, until it reaches the yield point
of the weakest sarcomere. There will always be a
weakest sarcomere, no matter how small the differences between the sarcomeres may be. At this
point the weakest sarcomere will yield, that is,
begin to stretch more rapidly than the others,
without increasing tension. Elastic elements will
also cease lengthening as the tension levels out. If
the weakest sarcomere is on the descending limb
of the length-tension relation, this increased
lengthening will reduce its tension-generating
capability. It will then be unable to support the existing tension at any velocity and so will lengthen
very rapidly indeed, limited only by inertial and
passive viscous forces. Of course this lengthening
will allow shortening of the rest of the fiber which
will cause some reduction in tension. If the number of sarcomeres in the fiber is large, however,
this reduction of the tension is likely to be less
than the reduction in tension-generating capability
of the weakest sarcomere, and so will not stop the
rapid elongation. Eventually the passive structures within the sarcomere will produce a passive
tension equal to the tension in the fiber, and extension of the sarcomere will stop. If the imposed
stretch is continued, the tension will again rise until it reaches the slightly greater yield point of the
next weakest sarcomere, which will then extend
rapidly. This process will be repeated until the
motion stops.
Our thought experiment suggests then that
lengthening of a fiber will not be at all uniform,
but will take place essentially by "popping" of sarcomeres, one at a time, in order from the weakest
towards the strongest. This has far-reaching consequences, which will be discussed in Section 3.4.
Where are the weakest sarcomeres likely to be?
The shortest and strongest have been shown to be
concentrated near the ends. It has been reported
(Colomo et al., 1988), in accord with my own experience, that some fibers do have weak patches that is, the weakest sarcomeres all in one part of
the fiber. In these fibers the yield ratio is low, the
continued rise during stretch is large, and nonuniform lengthening can be seen. Colomo et al.
(1988, Figure 5) also reported a smaller change in
slope of the force-velocity curve at zero velocity.
In a more uniform fiber the weakest are likely to
be randomly distributed throughout most of the
fiber. Of course a fiber consists of many parallel
myofibrils, able to move somewhat independently
of each other. The popping of the randomly distributed weakest sarcomeres in a relatively
unifonn fiber is more likely to be a myofibrillar
phenomenon than a fiber one. The weakest sarcomere in one myofibril may be at a different
point along the fiber to the weakest sarcomere in
the neighboring myofibril. This widespread distribution of elongated sarcomeres in myofibrils
will make them difficult to detect by direct observation. No sarcomere will be extended all the way
across the fiber, but scattered long sarcomeres extending only across one or a few myofibrils will
be scattered in three dimensions throughout the
3: Morgan; Modeling of Lengthening Muscle
fiber. This should be visible as increased disorder
and skewing of sarcomeres. To the best of my
knowledge, no quantitative measurements have
been made of this. The mechanical consequences
of the myofibrillar distribution of elongated sarcomeres will be small, unless significant forces are
generated between myofibrils by the elongation of
a sarcomere in one myofibril. Consequently the
model assumes that each sarcomere has a unique
sarcomere length applicable all across the fiber.
In this sense it could be considered as a model of a
myofibril.
3.3 The Computer Model
The model was closely based on that of
Morgan et al. (1982), ran on a Macintosh computer (Apple Computer Inc., Cupertino, CA), ~d
was written in Lightspeed Pascal (Thmk
Technologies, Bedford, MA, now a division of
Symmantec) using the Programmers Extender
(Invention Software Corp, Ann Arbor, MI). One
half of the muscle fiber was modeled as either 100
or 500 sarcomeres (or contractile units) connected
in series. In order to accommodate different
muscles and temperatures, the unit of time was
defined as the time for all unloaded sarcomeres to
shorten 111m. Thus the unloaded shortening
velocity was 1 micron per sarcomere per time unit
For the usual frog single fibers, this means that
one time unit corresponds to about 500 ms at O°C,
and about 50 ms at 20°C. Each sarcomere was
represented by a Hill type model, consisting of a
contractile component characterized by a
force-velocity curve, a linear series elastic component (default stiffness required 0.024 11m per
half sarcomere to drop the tension from isometric
to zero), and an exponential parallel elastic component. The force-velocity curve was taken as the
classic Hill-Katz curve as quantified by Morgan et
al. (1982). Constants could be entered for a/Fo
(where a is the Hill parameter and Fo is the Hill
contractile element force intercept [see also
Chapter 5 (Winters)]; default here: 0.25), the
change of slope between slow lengthening and
slow shortening (default: 6 times higher for
lengthening), the asymptote for lengthening
(default: 1.8 Fo)' and a curvature coefficient for ~he
lengthening region. The unloaded shortemng
velocities of all sarcomeres were the same. For
each sarcomere the isometric tension (strength)
was taken as the product of the length-tension
49
the
curve of Gordon et al. (1966b) and
specified isometric capability at optimum length
F
of that sarcomere. The distribution of FOG,.
wis specified as an exponential distribution with a
random variation added. Values for the end sarcomere, the central sarcomere, the length constant
of the exponential distribution, the random component amplitude, and the random number
generator seed were all specified by the user. The
random component was generated by smoothing a
series of pseudo-random numbers generated by the
computer, giving an approximately Gaussian distribution. The initial length distribution was very
similarly specified. The basic passive tension
curve was an offset exponential, specified by the
slack length, a length constant of the exponential
and the tension at some specified sarcomere
length, and applied to the sarcomere at the center
of the fiber end. The passive tension curves for
the other sarcomeres were scaled from this so that
the specified sarcomere length distribution
produced the same passive tension in all the sarcomeres.
No provision has been made for
sarcomere lengths less than the slack length, so no
simulations involving slack fibers can yet be run.
The length changes to be applied were
specified by the times of beginning and ending the
ramp, and the fmal average sarcomere length. The
initial average sarcomere length was calculated
from the initial sarcomere length distribution. The
program included facilities to display, save, and
print
the
length-tension
relation,
the
force-velocity relation, the passive curve for any
sarcomere, the movement being applied, and the
distributions of F ,strength and of sarcomere
length, as well asOop'
the tension-time recordb·
emg
produced. Sarcomere length, FOG,., and strength
could also be displayed as histograms. In addition
a "segment length" record was obtained by adding
the sarcomere lengths of the "central" half of the
fiber. The simulation could be stopped at any time
to examine these curves and then resumed.
The solution proceeded iteratively as before
(Morgan et al., 1982). The time intervals for the
calculation were not equal, but varied automatically to accommodate the rate of change of
tension. No interval greater than 0.0001 time units
(less during very rapid ramps) was accepted if tension changed more than 2% of isometric tension or
if the length of any sarcomere changed more than
0.2 11m within that time interval. An option al-
50
Multiple Muscle Systems. Part I: Muscle Modeling
lowed the fiber to be replaced by a single
sarcomere, with parameters equal to the average of
those for the fiber. Tension was plotted on the
screen as simulation progressed.
Experience using the model justified several of
the assumptions made in the thought experiment
above. It was found that for realistic passive tension curves and initial sarcomere lengths, a
popped sarcomere was extended well beyond zero
overlap. Of particular importance was the confIrmation that the sarcomere lengths are instantaneously unstable for any reasonable assignment
of parameters and number of sarcomeres, even as
low as 100. The only way to actually fmd a solution during the rapid elongation phase was to add
a small amount of damping to the series elastic
component of the sarcomere model. This meant
that the tension fell to the minimum of the curve
of total sarcomere tension against sarcomere
length as each sarcomere popped. In practice this
procedure led to a large number of very short time
intervals and excessive calculation time.
Consequently a "quick pop" option was provided,
whereby no attempt was made to track a sarcomere through popping.
If the "quick pop" option was enabled, then in
each time interval, after solving for the tension but
before updating the sarcomere lengths, a check of
the sarcomere nearest to its yield point was made
to see if its updated yield point would be less than
the existing tension, that is, whether the tension
would need to be reduced in the next time interval.
If so, the present time step was repeated with half
the duration. When a tension decrease would be
required and the time interval was at the minimum
allowed (0.0001 time unit), then the sarcomere
would be popped by setting its isometric
capability (F00 ) to zero. This ensured that only its
passive tensi6n would be used from then on, and
that during the next time interval, it would be
stretched appropriately. This led to an instantaneous fall in tension, as the other sarcomeres
shortened. This option reduced the calculations
considerably, but was shown not to affect the
results perceptibly other than during the actual
"popping" of the sarcomere.
easily observed, but produced unrealistic tension
traces. Using 500 sarcomeres produced much
more realistic records, though individual pops are
still discernible. Most real muscle fibers have
many more than 500 sarcomeres, so that an even
smoother trace would be expected. Figure 3.1a
shows the simulated tension during a stretch, and
while isometric at the initial and final lengths.
This figure shows that the model simulates a large
number of the peculiar features of muscle being
stretched.
3.4.1 Continued Tension Rise During Stretch
Most experimenters agree that stretching a
muscle at constant velocity produces a tension that
continues to rise throughout the stretch, whether
the sarcomere lengths are on the plateau or descending limb of the length-tension relation (see
in particular Harry et al., 1990). Shortening at
constant velocity within the plateau of the
length-tension diagram, on the other hand,
produces a tension that is much more nearly constant. The lengthening behavior is inconsistent
with "normal" cross-bridge models, as the distribution of cross-bridge extensions, and hence the
tension, should reach a steady state after stretching
more than a few cross-bridge strokes. On the descending limb, of course, tension should fall as the
overlap of thick and thin filaments is reduced.
This was seen when the model was set to single
sarcomere, as shown in Figure 3.2.
If inter-sarcomere dynamics dominates, then
the continued rise represents popping ever
stronger sarcomeres. The continued rise is inherent in the model. The tension trace is just a
series of yield points, each one greater than the
last. On the descending limb of course there will
be a countervailing effect of the slow reduction of
strength of all the sarcomeres that are slowly
lengthening below their yield point. This may account for occasional observations of tension
falling during stretch, such as in Edman et al.
(1978, Figure 2b). Variations in the pattem of sarcomere non-uniformity will account for the
variable amount of rise seen between experimental
records. A large spread of sarcomere lengths in
the model can lead to a rounding of the yield
comer as sarcomeres are popped from early in the
3.4 Modeling Results and Discussion
The model with 100 sarcomeres showed a large stretch, as shown in Figure 3.2. During shortening
abrupt drop in tension each time a sarcomere the slope of the force-velocity curve ensures that
popped. This enabled the pattern of popping to be the sarcomeres shorten much more uniformly, and
51
3. Morgan; Modeling of Lengthening Muscle
the single sarcomere cross-bridge models are more continued. The final tension is near that for an
nearly correct.
isometric contraction at the original length, and almost independent of the amplitude or speed of the
3.4.2 Permanent Extra Tension
stretch. These are exactly the results found in
The model records also show the phenomenon
muscle fibers. Looking at the final sarcomere
of permanent extra tension [Abbott and Aubert,
length distribution (Figure 3.1b) shows what has
1952; Julian and Morgan, 1979b; Edman et al.,
happened. Most of the lengthening has been taken
1982; Woledge et al., 1985; Sugi and Tsuchiya,
up by the few popped sarcomeres, and the rest
1988; Tsuchiya and Sugi, 1988; see also Chapter 2
have only stretched a little. Furthermore the
(Hatze»). After a stretch at long length, the tenweakest sarcomeres have been "removed" from
sion does not fall to the level appropriate to the
final length, regardless of how long the tetanus is
b)
a)
27
j
24
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oi"-. I ~
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00
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so
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=,11 at'ICI 11iU" ..... unO. me ~ .. 0 .7'020
Figure 3.1: The model behavior during stretch. a)
Tension traces during fixed end activation at initial and
final lengths and during an active stretch of 0.2 11m between these lengths. Note the following points. The
longer length produces less isometric tension, owing to
the length-tension curve. The isometric traces show a
"tension creep" or slow rise phase followed by a slow
decline. The tension continues to rise throughout the
stretch.
The popping of individual sarcomeres
produces the ripple during stretch. The tension after
,.
t ,S J,..-----~r;;:=
"""'"'
=.:-_
=:::, ----....,
S~O
the stretch does not fall to the tension generated while
isometric at the same length. The details in the figure
relate to the stretch record. b) The distribution of sarcomere lengths at the beginning of contraction, at the
end of the stretch, and at the end of the simulation.
Note the "popped" sarcomeres, scattered throughout
the fiber, and extended beyond the length of filament
overlap. After such a long tetanus, the sarcomeres are
quite non-uniform.
52
Multiple Muscle Systems. Part I: Muscle Modeling
the distribution of active sarcomeres, giving a
slight tendency for the tension to rise above that at
the original length. This is counteracted by the
slight elongation of the other sarcomeres.
Combined with the older evidence suggesting sarcomere
non-uniformity
(interrupting
the
stimulation long enough for the tension to fall, and
then resuming produces the tension appropriate to
the final length (Julian and Morgan, 1979b, Figure
6)), these records make the inter-sarcomere
dynamics explanation of permanent extra tension
very attractive. Note that a decrease in stiffness
during the stretch is also predicted by the model,
as observed by Tsuchiya et al. (1988) and Sugi et
aI. (1988). The decrease in this model, however,
is not due to a decreased overlap and hence number of crossbridges as they postulated, but to the increasing number of popped sarcomeres. The stiffness of the passive tension curve is less than that
of an active sarcomere, so popping more sarcomeres decreases the stiffness, even though most
of the sarcomeres have not lengthened.
b)
0.0
1 .0
2.0 Tim.
3.0
•.0
5.0
I'::~~~---------------
I.
O.
0.0
1.0
2.0
T,me
3 .0
•. 0
5.0
Figure 3.2: A fiber with a wider distribution of F..".
a) The upper traces are for a fiber of 500 sarcomeres
with a 10% random variation of isometric capabilities
and a 0.05 11m random variation in initial sarcomere
lengths. The lower traces are for a single sarcomere
with the average properties of those in the upper traces.
The middle trace shows the movement, calibrated in
mm for the fiber. Note the following points. The fiber
has a smaller yield ratio than the sarcomere, and the
yield point is much less distinct. The tension continues
to rise throughout the fiber stretch but falls for the
Inilal
oat"""""
Final
UICOINI. length
length
,
•.C¥n
single sarcomere, owing to the sarcomere
lengtb-tension curve. Isometric contractions can also
involve popping of sarcomeres. For the single sarcomere the tension decays towards that appropriate to
the final length, but stays above for the fiber. b) The
initial and final sarcomere length distributions shown
as histograms. Initial sarcomere lengths 2.35 11m plus
0.05 m random variation. Slack length is 2.1 11m.
Passive tension at 3.1 IJ.ffi is 0.2 F.. Stretch amplitude
is 0.15 11m. Default force-velocity curve is utilized.
3. Morgan; Modeling of Lengthening Muscle
3.4.3 Force-Velocity Cune at High Speeds
Stretching the simulated fiber produces a tension that is almost independent of the stretch
velocity except for very small velocities. This is
to be expected, since the tension during stretch is
determined mainly by the distribution of yield
points for the various sarcomeres. The velocity
has a minor effect through the lengthening of the
other sarcomeres below their yield points.
Experiments also show that the tension of a
muscle or fiber during constant velocity stretch is
essentially independent of velocity beyond the
yield point. A cross-bridge model, however, can
only show this under specific, rather unlikely assumptions. As a sarcomere is stretched at higher
and higher speeds, the opportunity for crossbridges to form becomes less and less, and in most
models the number of attached sarcomeres
decreases. The only way to avoid a fall in tension
is to increase the average tension per cross-bridge,
and hence the average extension of the crossbridges. It has been shown (Harry et al., 1990)
that imposing a limit on cross-bridge extension
even as large as six times the maximum extension
in an isometric muscle produces predictions that
depart significantly from experiment.
Inter-sarcomere dynamics can easily accommodate a sarcomere force-velocity curve that falls
at large velocities. Once a sarcomere yields and
becomes unable to support the existing tension at
any speed, the tension that it can support becomes
essentially irrelevant. This is shown by the fact
that using the "quick-pop" option in the model,
equivalent to a sarcomere force-velocity curve
that falls to zero immediately past the yield point,
does not affect the overall tension trace. In this
way the sarcomere force-velocity curve can fall at
high speed, but the fiber force-velocity curve does
not.
3.5 Explanations of Other Phenomena
The ideas behind the model can also be seen to
provide explanations for other puzzles about
lengthening muscle that have not yet been
modeled.
3.5.1 Isotonic Experiments
Simulations of isotonic experiments are not appropriate to the present model, as they rely heavily
on the moderately fast transient behavior of sarcomeres, which is not realistically modeled by a
53
Hill model. The behavior to be expected from a
consideration of inter-sarcomere dynamics can
however be predicted, and compared to experiment. If a model fiber is subjected to a load
greater than the yield point of some of its sarcomeres, all those sarcomeres unable to support
the imposed tension will quickly pop, giving a
rapid lengthening of the fiber. When that has happened, the rate of stretch of the fiber will
drastically slow to that due to the sub-yield
lengthening of the sarcomeres with yield points
greater than the imposed tension. Such slowing of
the rate of lengthening is the behavior that has
been seen in real fibers, and has proved so difficult
to explain by cross-bridge or any other models
(Huxley, 1971, Figure 6; Huxley, 1980, p. 84).
(How can enough cross-bridges form in a rapidly
extending fiber to resist the imposed load when
they were unable to do so in the isometric fiber?)
This would also explain the difficulty experienced
by Pollack's group (e.g., Granzier et al., 1989) in
plotting the force-velocity curve using isotonic
stretches. They found that increasing the isotonic
load increased the amplitude of the immediate
lengthening, but not the steady lengthening
velocity that followed (Granzier et al., 1989).
Sustained rapid isotonic lengthening does not occur experimentally.
3.5.2 Damage from Eccentric Contractions
Another peculiarity of muscle being lengthened
while active is its propensity to damage. Step tests
(Friden et aI., 1983; Newham et al., 1983) and arm
curls (Clarkson and Tremblay, 1988) in humans,
and downhill running in rats (Armstrong et al.,
1983), all produce damage with the following
characteristics. Immediately after the exercise, the
only changes seen are small areas of elongated sarcomeres, sometimes as small as one half sarcomere in one myofibril. In other cases a group of
elongated sarcomeres, extending part or all of the
way across the fiber, are seen. The next day, the
muscles involved are painful, and histology shows
damaged muscle fibers being replaced. The degree of damage is not related to the general fitness
of the subject. Everyday experience also shows
that sports involving eccentric contractions such as
horse riding and mountain climbing often produce
such "delayed-onset muscle soreness," while concentric exercise sports such as swimming and
bicycle riding usually do not. The proposal of non-
54
Multiple Muscle Systems. Part I: Muscle Modeling
uniform lengthening provides the mechanism for
the initial local damage, which can then lead to
destruction of the fiber (Armstrong, 1984, esp.
p.. ·35).
When a sarcomere "pops", it is extended, probably to the point where there is no overlap of thick
and thin fIlament lUTays. When the muscle
relaxes, it is likely that the inter-digitating pattern
of the fIlaments is not fully resumed immediately.
(The question of the extent and time course of the
recovery of a popped sarcomere is an area that
needs more experimental investigation.) This
provides a weak point during the next stretch, and
increases the stress on the neighboring myofibrils
at that sarcomere. In this way, repeated stretches
can be envisaged to produce a microscopic tear in
the fiber. At some point this tear damages sarcoplasmic reticulum or sarcolemma, allowing
uncontrolled release of calcium, and "clot" formation.
In single fibers, stretching will sometimes kill
fibers that have withstood many isometric and/or
shortening contractions (personal observations).
My own recent observations suggest that these
fibers often have a low fiber yield ratio, consistent
with a wide spread of sarcomere strengths. (See
below and Colomo et al., 1988.)
3.6 Modeling ConclusionslPredictions
predicted by the model. No specific experiments
have yet been undertaken to test this point, but a
brief examination of the literature provides some
support. Certainly the rise during stretch tends to
be greater at longer lengths, where non-uniformities may be expected to be greater.
Similarly, inter-sarcomere dynamics suggests
that the change in slope of the force-velocity
curve about zero for a fiber will always be less
than the change for its sarcomeres, since the tension for a slow stretch will be measured later in
the contraction, when the non-uniformity will be
greater, and hence the yield point of the weakest
sarcomere less. This similarly means that the sarcomere value for the slope change must be at least
as high as the highest ever seen for a fiber. My
measurements of Figure 5 of Colomo et al. (1988)
suggest a value nearer 9 than the classically assumed 6. The isotonic measurements of Granzier
et al. (1989) produced values even higher than
that. If the difficulties found in measuring the
force-velocity curve (continued rise of tension) by
stretching at low velocities are due to non-uniformities, as seems likely, then the isotonic experiments, which quicldy pop all the very weak sarcomeres before non-uniformities become worse,
may be the best method of measurement. Note
that with more sarcomeres in the fiber, the deviation of strengths required to ensure that at least
some are below a given threshold will be less.
This modeling has led to several conclusions
beyond the general principle of the non-uniform
lengthening of muscle. If the yield point of the 3.7 Future Directions
fiber is the yield point of the weakest sarcomere, 3.7.1 Experimental Confirmation
but the isometric tension of the fiber is biased
Perhaps the greatest objection to the suggestion
towards the isometric capability of the stronger
that inter-sarcomere dynamics dominate the
sarcomeres, then the yield ratio of the fiber (yield
response of muscle to stretch is the absence of
tension divided by isometric tension) must be less
direct evidence of such large sarcomere non-uniforthan for its sarcomeres. This means that the yield
mities. Clearly if the popped sarcomeres are few
ratio for a sarcomere must be greater than the
and widely scattered in individual myofibrils, then
highest value ever seen in a fiber of that type.
seeing them will not be easy. In particular their efThat is, the true value for frog single-fiber sarfect on a diffraction pattern is likely to be
comeres should not be taken as the mean of the
complex, and measurements of segment length are
fiber observations (approx. 1.8), but as the largest,
not appropriate.
at least 2.1. In addition, it is concluded that a low
Other indirect evidence can be sought. If the
yield ratio can be taken as indicative of a wide dis- myofibrils of popped sarcomeres do not fully
persion of sarcomere strengths.
retum to their inter-digitating pattern on relaxaIn the model, a steep rate of rise of tension
tion, particularly after long and/or fast stretches,
during a stretch is also indicative of a wide sarthen cumulative effects should be observable.
comere strength distribution. Although the two
Preliminary experiments with Drs. Julian and
parameters are measuring slightly different aspects
Claflin have shown increased apparent series comof the distribution, a general correlation is still
3: Morgan; Modeling of Lengthening Muscle
pliance and shifts in the fiber length for optimum
tension. Both of these were permanent and
cumulative, in that repeating the stretches increased the changes. Further experiments to
explore the various parameters are under way.
Other experiments that could provide useful information include quantitative measurements of
the disorder of sarcomeres, and looking for a correlation between the yield ratio and the rate of
continued tension rise during stretch.
3.7.2 Relevance to Whole Mammalian Muscles
Nearly all the experiments described here have
been for single frog muscle fibers fully tetanized
at near-freezing temperatures (the exception being
damage from eccentric contraction). How directly
relevant are these ideas to whole mammalian
muscle sub-maximally activated at normal body
temperature?
Whole muscles have much more passive tension than single fibers, often to the point of not
having a descending limb in a plot of total tension
against muscle length. However, much of the additional elasticity is probably not effectively in
parallel with the individual (half) sarcomeres, so
that the sarcomeres probably do have a region of
decreasing tension with increasing length.
Permanent extra tension has certainly been shown
in whole toad sartorius (Abbott and Aubert, 1952).
Whole muscle does show a continuing rise during
stretch, although the absence of a clear plateau of
isometric tension without significant passive tension complicates the experiment.
The yield ratio of tetanized mammalian muscle
is commonly less than for frog fibers. One possible interpretation is that mammalian muscles
have a greater range of sarcomere strengths.
However, differences in species and temperature
make this suggestion rather speculative. The effects of submaximal activation are even more
difficult to evaluate, partly because of the difficulty in doing experiments on sub-maximally
activated single fibers. If the effect of motion on a
submaximally activated sarcomere is similar to the
tension traces observed in whole muscle, that is, a
collapse of tension as the stretch is continued, then
instability is very probable and inter-sarcomere
dynamics are likely to be very important
However, inferring the effect of lengthening on
sarcomere tension from observations of the effect
of lengthening on fiber tension in this situation is
55
far from trivial.
3.7.3 Refinement of the Model
Future work in this area is currently concentrated on improving the sarcomere model
along the lines discussed by Zahalak (Chapter 1)
so that isotonic experiments can be simulated.
Consideration of sub-maximal activation and a
study of the energetics of eccentric contractions
should also be facilitated by this development.
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