Muscle fiber angle, segment bulging and architectural gear ratio in segmented musculature
Department of Biology and Program in Organismic and Evolutionary Biology, University of Massachusetts, Amherst, MA 01003, USA
Author for correspondence (e-mail:
brainerd{at}brown.edu)
Accepted 29 June 2005
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Summary |
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Key words: biomechanics, muscle architecture, segmentation, myomere, myosepta, swimming, sonomicrometry, fish, salamander, Siren, Urodela
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Introduction |
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Despite the structural complexity of myomeres and myosepta, sonomicrometry
studies have demonstrated that the bodies of most fishes bend like a simple,
homogeneous beam during swimming (Coughlin
et al., 1996; Shadwick et al.,
1998
; Katz et al.,
1999
; with the exceptions being tuna and mako sharks;
Shadwick et al., 1999
;
Donley et al., 2004
).
Beam-like behavior means that, for longitudinally oriented red muscle fibers,
muscle fiber strain (
f) is equal to longitudinal strain
(
f=
x), when
x is defined as
the local longitudinal strain at the same mediolateral position as the muscle
fiber. Therefore,
f for red fibers can be calculated from
video images of fish curvature and measurement of the distance of the fibers
from the vertebral axis, or
f can be measured directly with
longitudinally arranged pairs of sonomicrometry crystals (reviewed in
Long et al., 2002
).
The oblique orientations of fibers in the white musculature, however, cause
white muscle fiber strains to differ from local longitudinal strains
(f
x). Alexander
(1969
) proposed a model in
which obliquely oriented muscle fibers within the myomeres participate in
helical trajectories that cross myoseptal boundaries. Alexander showed that
when fibers are oriented obliquely, fiber strain is less than longitudinal
strain (
f<
x). Therefore, a smaller
f is required to produce a given
x in
obliquely oriented fibers than in longitudinally oriented fibers, and this
amplification of
f results from fiber rotation (i.e. increase
in fiber angle) during contraction (Azizi
et al., 2002
).
To measure f in the white musculature with sonomicrometry,
the crystal pairs must be aligned along the
and
fiber angles. A
few studies have used sonomicrometry to measure
f in
superficial areas of the dorsal white musculature, within 2 mm of the skin
surface (Franklin and Johnston,
1997
; James and Johnston,
1998
; Wakeling and Johnston,
1998
). Proper crystal alignment is more difficult to achieve in
the deep white musculature (Wakeling and
Johnston, 1999
), but one study measured both deep and superficial
f, with crystal alignment precision within ±10° of
the muscle fiber angles (Ellerby and
Altringham, 2001
).
Other techniques have also been used to measure or estimate white muscle
fiber strain. Rome and Sosnicki
(1991) measured white
f by bending freshly killed carp to various curvatures and
letting them set in rigor mortis. Sarcomere lengths were then measured from
frozen sections and compared with resting sarcomere lengths to calculate
f. Other studies have measured
x with video
or sonomicrometry and used Alexander's helical trajectory model
(Alexander, 1969
), van
Leeuwen's mediolateral bulging model (van
Leeuwen, 1990
) or Wakeling and Johnston's centroid technique
(Wakeling and Johnston, 1999
)
to calculate
f for the white fibers
(Rome et al., 1988
;
Lieber et al., 1992
;
Johnston et al., 1995
;
Spierts and van Leeuwen, 1999
;
Wakeling and Johnston,
1999
).
Measures of red and white muscle fiber strain can be combined to calculate
the red-to-white gearing ratio, defined as red f divided by
white
f (Rome and
Sosnicki, 1991
; Wakeling and
Johnston, 1999
). When defined in this way, the gearing ratio
combines the effect of muscle fiber angulation in the white musculature with
the effect of greater distance from the vertebral axis in the red musculature.
To separate these two effects, we define here an `architectural gear ratio',
AGR=
x/
f, in which
x and
f are longitudinal strain and fiber strain at the same
mediolateral position. In longitudinally oriented fibers,
f=
x and AGR=1. In obliquely oriented fibers,
f<
x and AGR>1
(Azizi et al., 2002
).
The primary goal of the present study is to explore the effect of muscle
fiber angle on AGR in segmented musculature. The lateral hypaxial musculature
(LHM) of salamanders is a good model system for this work because is
approximately constant within each layer of the LHM, and
is
approximately zero (Fig. 1A).
It is also a good model system because the hypaxial myomeres of salamanders
are approximately planar, rather than forming nested cones, and the myosepta
run vertically rather than obliquely as in fishes. This relatively simple
geometry makes it possible to model the LHM segments as planar rectangles in
the xy plane with some thickness along an orthogonal
z-axis (Fig. 1B;
Azizi et al., 2002
).
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A key component of our model is that the volume of each muscle segment is
assumed to remain constant during contraction
(Azizi et al., 2002). This is a
reasonable assumption because precise volumetric measurements have shown that
muscle volume decreases only slightly during isometric contraction, for
example 0.007% in the frog sartorius and 0.003% in the frog gastrocnemius
(Baskin and Paolini, 1967
). To
maintain this approximately constant volume during contraction, muscles must
bulge out in one or both of the dimensions orthogonal to shortening
(Otten, 1988
). This
isovolumetric constraint is central to muscular hydrostat models and leads to
the conclusion that, for cylindrical muscular hydrostats, the contraction of
muscle fibers with angles lower than 54.44° causes the cylinder to shorten
and contraction of muscle fibers with angles higher than 54.44° causes the
cylinder to lengthen (Kier and Smith,
1985
).
A previous model of fish musculature used the isovolumetric constraint to
correct for the effect of mediolateral bulging on the distance of fibers from
the neutral axis (van Leeuwen,
1990). If a good measure of mediolateral bulging is available, and
x-ray studies have disagreed on how much mediolateral bulging actually occurs
in fishes (van Leeuwen, 1990
;
Wakeling and Johnston, 1998
),
then this model provides a useful correction to simple beam theory for
calculating longitudinal strain from body curvature. Models of the nested-cone
geometry of myomeres have also used the isovolumetric constraint to model the
shape changes of myomeric cones during muscle contraction in fish axial
musculature and lizard tails (Westneat et
al., 1998
; Zippel et al.,
1999
).
Our segmented musculature model is similar to a model proposed by Alexander
(1969) in which a block of
muscle is assumed to bulge equally in height and depth when it shortens in
length. Our model differs in that segment height and depth are allowed to vary
semi-independently within the isovolumetric constraint
(Azizi et al., 2002
). This
semi-independence allows us to explore the effects of dorsoventral
versus mediolateral bulging on fiber strain, longitudinal segment
strain and AGR.
In a previous study (Azizi et al.,
2002), we used a preliminary version of our model to interpret the
morphology and function of hypaxial myosepta in an aquatic salamander,
Siren lacertina (Fig.
1A). Preliminary model results showed that, in muscle segments
with angled fibers, dorsoventral bulging increases longitudinal segment strain
for a given amount of muscle fiber strain (i.e. bulging increases the AGR of
the segment; Azizi et al.,
2002
). Myomeres are fundamentally muscular hydrostats, surrounded
by collagenous myosepta and skin
(Wainwright, 1983
;
Westneat et al., 1998
). The
stiffness of these connective tissues may constrain segment bulging in one
dimension and permit bulging in another, thereby modulating the bulging
condition and AGR of the segments. In S. lacertina, the collagen
fibers in the hypaxial myosepta are oriented mediolaterally, indicating that
these myosepta constrain mediolateral bulging and permit dorsoventral bulging,
thereby increasing the AGR of the hypaxial segments
(Azizi et al., 2002
).
Here, we derive a generalized equation for the relationship between muscle fiber strain, segment bulging, muscle fiber rotation and longitudinal segment strain. We simulate four specific bulging conditions to explore the effects of initial muscle fiber angle and segment bulging on AGR, test our model assumptions and predictions with sonomicrometry and electromyography of the hypaxial myomeres in Siren lacertina and then use our validated model to explore the effects of muscle fiber angulation and bulging on force and work in segmented musculature.
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Materials and methods |
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Although the model includes segment strain in three dimensions, the muscle fibers lie in just one plane, making the model an `isovolumetric planar' model (Fig. 1B). In the future, it may be possible to expand the model to include fiber shortening and rotation in three dimensions, but our planar model is appropriate for the approximately planar structure of the salamander LHM (Fig. 1).
The purpose of the model is to calculate the effects of initial muscle
fiber angle and segment bulging on the magnitude of segment strain for a given
muscle fiber strain (i.e. the architectural gear ratio, AGR). The final
geometry of the segment after shortening
(Fig. 1B, contracted) can be
expressed in terms of initial length of the muscle fiber (f),
extension ratio of the muscle fiber (f), extension ratio of
the segment (
x), initial muscle fiber angle (
) and
final muscle fiber angle (ß) in radians):
![]() | (1) |
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![]() | (5) |
This generalized model (Eqns 25) may be further simplified by
setting constraints on how the segment bulges during contraction. We define
four `bulging conditions' that represent points along a continuum of possible
shape changes: (1) segment depth remains constant (z=1) and
segment height increases to maintain the constant volume of the segment; (2)
the segment bulges equally (
y=
z) in
height and depth; (3) the segment height remains constant
(
y=1), and segment depth increases to maintain the constant
volume of the segment; (4) segment height decreases in proportion to segment
shortening (
y=
x), and segment depth
increases to accommodate shortening of both x and y.
Equations for calculating segment bulging and longitudinal segment strain are
derived for each of the four conditions in Appendices 1 and 2. We used
Microsoft Excel X for Mac to run simulations of the four bulging conditions
with a range of input values for
and
f.
Assumptions
Three primary assumptions of our model are: (1) the segment does not shear
into a non-rectangular parallelepiped during contraction; (2) the LHM segments
are planar at rest and the muscle fibers remain in the same plane as they
contract; and (3) the muscle fibers are active and generating force during
segment shortening. We tested our model and these assumptions with
sonomicrometry and electromyography (EMG) of the external oblique (EO) and
internal oblique (IO) in Siren lacertina L.
(Fig. 1).
Segment shearing and violation of assumption 1 would occur if the two
myosepta bordering each segment were to translate differentially in the
vertical direction or deform differentially during segment shortening. We
assume that shear does not occur, and therefore all muscle fiber shortening is
converted into fiber rotation. This is a reasonable assumption in salamanders
because at least two muscle layers with opposite muscle fiber directions are
always present in each segment (EO and IO;
Fig. 1), and their positive and
negative vertical force components balance each other to prevent shear.
Previous studies of swimming in two salamanders, Dicamptodon and
Ambystoma, support the absence of substantial shear during steady
swimming in salamanders, because shear would be associated with long axis
torsion, and substantial long axis torsion was not observed
(Carrier, 1993;
Bennett et al., 2001
).
Assumption 2 is violated by the curvature of the segments around the
circumference of the animal at rest (Fig.
1) and by the likelihood that muscle fiber curvature increases as
segments bulge out between the myosepta during segment contraction (as
observed in S. lacertina by Azizi
et al., 2002). Testing the model predictions with sonomicrometry
will determine how severely the violations of assumption 2 affect the validity
of the model.
We tested assumption 3 with simultaneous EMG of the EO and IO and
sonomicrometry of the EO during steady swimming in S. lacertina.
Assumption 3 is supported by previous studies in which all layers of the LHM
were found to be active during steady swimming in two salamanders,
Dicamptodon and Ambystoma
(Carrier, 1993;
Bennett et al., 2001
).
Testing the model with sonomicrometry and electromyography
We used sonomicrometry to test whether segment shear (assumption 1),
deviations from planar (assumption 2) or perhaps violation of some
unrecognized assumption causes the model predictions to differ substantially
from measured segment strains. Initial muscle fiber angles () and fiber
lengths (f) measured during the surgeries (see below) were combined
with changes in
f and
y from
sonomicrometry and substituted into equations 5 and 2 to calculate a predicted
value for the magnitude of longitudinal segment strain
(
x=
x1). The mean predicted
x was calculated for each of the swimming sequences analyzed
and compared with empirically measured mean
x from
sonomicrometry measurements of the same sequence.
Three adult Siren lacertina, ranging in total length from 38 cm to 43 cm, were purchased from a licensed herpetological vendor. The salamanders were housed in individual glass aquaria, which were maintained at approximately 22°C. The salamanders were fed a diet of four earthworms per week but were not fed three days prior to the surgery. The University of Massachusetts Institutional Animal Care and Use Committee approved all experimental and animal care protocols.
Salamanders were anesthetized by immersion in a buffered solution of
tricaine methanesulfonate (1 g l1). Intraspecific variation
in the muscle fiber angles of the LHM of salamanders is substantial
(Simons and Brainerd, 1999)
and therefore it was necessary to measure muscle fiber angles in each
individual before implanting the sonomicrometry crystals. We made an incision
in the skin to expose the lateral aspect of a myomere located at 70% of the
total body length (0.7 TL) from head to tail, and we measured the EO
muscle fiber angle to within ±0.5°. The longitudinal position (0.7
TL) was selected because the myomeres and myosepta of the LHM in this
region are roughly planar, and substantial axial bending occurs in this region
during steady swimming in Siren
(Gillis, 1997
). We found that
the EO muscle fiber angles of the three individuals examined were 36.5°,
40.5° and 43.0°.
After measuring fiber angle, we closed the incision with 6-0 silk suture
and implanted the crystals on the other side of the body in the same
longitudinal position (0.7 TL). To minimize the surgical trauma to
the myomere of interest, we used small incisions (5 mm) in the two
adjacent myomeres to gain access to the EO muscle layer. The tips of the
crystals were then pushed into the myomere of interest through small incisions
(1 mm) in the myosepta, and the leads were sutured tightly to both the
myosepta and the skin (the crystals themselves were embedded in the EO
musculature). Three crystals were arranged into a right-angled triangle
(Fig. 1): two in a vertical
series along one myoseptum and the third on an adjacent myoseptum at the
appropriate dorsoventral position to be aligned with muscle fiber angle
(measured previously as above). Skin incisions were closed with 6-0 silk
suture. The right-angled-triangle crystal configuration allowed us to measure
muscle fiber strain, longitudinal segment strain and dorsoventral segment
strain.
Prior to data collection, the salamanders were allowed 1 h to recover from
the anesthetic. Longitudinal segment strain and muscle fiber strain were
collected at 200 samples s1 with a Sonometrics TRX-6 sonic
micrometer (Sonometrics Corporation, London, Ontario, Canada) as the
salamanders swam in a 2.5 m-long aquatic trackway. Analysis of the
sonomicrometry data was conducted with Sonometrics Sonoview 3.1.4 and limited
to swimming bouts that contained a minimum of four complete tailbeat cycles.
Instantaneous length measurements were converted to extension ratios
(=length/rest length) prior to a quantitative comparison of empirical
and model results.
Trackway swimming included a range of swimming speeds and maximum
longitudinal segment strains for each animal, and we used this variation to
test the model under these varying conditions. We plotted the measured
x versus the predicted
x from the
model to test assumptions 1 and 2 and to look for bias in the model.
To test assumption 3, EMGs from the EO and IO of two S. lacertina were recorded during steady swimming, and axial bending was measured with sonomicrometry. Fine-wire, hooked electrodes were constructed from 0.05 mm-diameter, insulated nichrome wire (California Fine Wire Co., Grover Beach, CA, USA). Electrodes were implanted percutaneously with 25 gauge hypodermic needles at a depth predetermined to be appropriate for the EO or IO. Signals were amplified 1000x with A-M Systems AC amplifiers (model number 1700; Everett, WA, USA) with the low and high filters set to 100 Hz and 1000 Hz, respectively, and the 60 Hz notch filter on. Each signal was digitized at 4000 samples s1 with an instruNet analog-to-digital converter with Superscope II software (GW Instruments, Somerville, MA, USA).
Modeling force and work
Our model can also be used to explore the effects of muscle fiber angle and
segment bulging on the force and work produced by the segment. The vector
component of muscle fiber force (Ff) in the direction of
longitudinal segment shortening (Fx) depends on the
instantaneous muscle fiber angle (from
Alexander, 1968
):
![]() | (6) |
Eqn 6 is often used to calculate the force produced by pennate muscles
(e.g. Calow and Alexander, 1973
and many subsequent studies of pennate muscle force production), but it makes
the simplifying assumption that muscle fiber angle does not change during the
contraction cycle, which is approximately true only for small changes in
muscle fiber length. When angled muscle fibers in pennate or segmented muscles
contract, the fibers rotate and
changes from
to ß. In
such cases, the mean relative force
(
R) can be calculated by
integrating relative force from
to ß and averaging over the
change in fiber angle:
![]() | (7) |
![]() | (8) |
From equations for force (Eqn 7) and displacement (Eqn 2), expressions for segment work in the x and y dimensions can be derived (Appendix 3).
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Results |
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In the second bulging condition, y=
z,
muscle volume is conserved by bulging equally in the dorsoventral and
mediolateral dimensions (the same assumption as in
Alexander, 1969
). By assuming
that
y=
z, Eqn 1 becomes
x3cos2
(
f2
x)+sin2
=0
(Appendix 2; Eqn 16). Substituting a range of values for
at a fixed
muscle fiber strain yields values for AGR over a range of initial muscle fiber
angles (Fig. 2). In this
bulging condition, if
=25° and the muscle fibers shorten by 10%,
then the muscle segment will shorten by 14.7% (
x=0.835),
corresponding to an AGR of 1.47.
In the third bulging condition, y=1, all of the segment
bulging occurs in the z (mediolateral) dimension. By assuming that
y=1, Eqn 1 becomes
x=[(
f2sin2
)/cos2
]1/2
(Appendix 2; Eqn 17). In this bulging condition, if
=25° and the
muscle fibers shorten by 10%, then the muscle segment will shorten by 12.4%
(
x=0.876), corresponding to an AGR of 1.24
(Fig. 2).
In the fourth bulging condition, x=
y,
we assume that as the segment shortens longitudinally, y also
shortens by an equal proportion. It is plausible that y might
decrease because obliquely oriented muscle fibers generate a vertical force
component that will tend to decrease y
(Fig. 1A). When
x=
y, the decrease in segment height
prevents rotation of the muscle fiber during contraction (ß=
), and
Eqn 1 simplifies to
x=
f (Appendix 2; Eqn
18). This result demonstrates that, without muscle fiber rotation, muscle
fiber strain and longitudinal segment strain are equal, corresponding to an
AGR of 1.00 at all initial muscle fiber angles
(Fig. 2).
To visualize the effect of different segment bulging conditions and AGRs on
overall body bending, we combined seven segments to create a hypothetical
aquatic vertebrate and assumed that all of the segments contract
simultaneously (Fig. 3). We
created drawings to scale by setting the combined length of the seven segments
to 4 cm and the diameter of the hypothetical animal to 0.6 cm [radius
(r)=0.3 cm]. We set the muscle fiber strain to 10%
(f=0.1), calculated the segment strain
(
x) for each bulging condition and calculated the radius of
curvature (R) for a given segment strain using beam theory
(R=r/
x). Figs
2 and
3 demonstrate the importance of
segment bulging on AGR in this model. The more the segment bulges in the
dorsoventral (y) dimension, the greater the AGR and the greater the
axial bending for a given amount of muscle fiber shortening.
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Empirical tests of the model and its assumptions
Sonomicrometry data were collected from the EO muscle layer of three adult
Siren lacertina (Fig.
4). During steady swimming, longitudinal segment strain is greater
than muscle fiber strain, indicating that AGR is greater than one
(Fig. 4A). We observe a
consistent pattern of increasing segment height (y>1)
during longitudinal segment shortening (
x<1), with
changes in segment height falling between the
y=1 and
y=
z bulging conditions
(Fig. 4B).
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To test the assumption that the LHM is active and generating force during segment shortening, we recorded segment length and EMGs from the EO and IO. During steady swimming, both the EO and the IO become active shortly before peak contralateral bending, and activity ceases shortly before peak ipsilateral bending (Fig. 6).
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Increasing the initial muscle fiber angle, and therefore AGR, comes at the cost of force production (Fig. 7B). These results show that relative segment force in the longitudinal direction decreases with increasing muscle fiber angle and that this decrease is accelerated by rotation of muscle fibers due to bulging in segment height.
The observed trade-off between segment shortening and force can also be
examined through calculations of segment work (Appendix 3). Previous equations
for work in pennate muscles have shown that muscle work is independent of
muscle fiber angle and depends only on muscle fiber length and muscle fiber
force (Otten, 1988). In this
model, we expand previous calculations of work to incorporate the orthogonal
shape changes associated with segment bulging (Appendix 3). For these
calculations, directional (x and y) force components are
integrated over the displacement of the fiber in each direction. The work
components are then summed to calculate total segment work. Similar to
previous results, we find that work depends only on muscle fiber length and
fiber force and is independent of muscle fiber angle or bulging condition.
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Discussion |
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The effect of bulging on AGR is mediated by muscle fiber rotation. In our
model, fiber rotation is expressed as the increase in muscle fiber angle from
to ß (Fig. 1B).
The AGR increases as ß increases (Eqn 4), and ß in turn increases as
the final height of the segment increases (Eqn 5). Therefore, the more a
segment bulges in dorsoventral height, the more the muscle fibers will rotate
and the more the segment will shorten for a given muscle fiber shortening.
To emphasize the relationship between dorsoventral bulging and fiber rotation, segment shortening and segment bulging can be thought of as occurring sequentially rather than simultaneously. If the muscle fiber shortens by a given amount, and y is keep constant, then the fiber will rotate to an initial value for ß and the segment will shorten. Then, if the segment lengthens in the y dimension, and muscle fiber length is kept constant, the fiber will rotate even more and the segment will shorten more, thereby increasing the AGR.
Model results indicate that, in segmented muscles with oblique muscle
fibers, longitudinal segment strain will generally be greater than muscle
fiber strain (AGR>1). The AGR is equal to one only when is zero or
if y shortens in the same proportion as x
(
y=
x;
Fig. 2). In all other cases,
the AGR increases with increasing
and with increasing dorsoventral
bulging, indicating a synergistic relationship between initial muscle fiber
angle and segment bulging (Fig.
2).
Measurements of muscle fiber strain and longitudinal segment strain in the
EO of Siren lacertina indicate that, as expected, the AGR in this
segmented muscle layer is greater than one
(Fig. 4A). The AGR of the EO is
determined by both the high (mean of 40°) and by dorsoventral
bulging of the segments (Fig.
4A;
y increases when
f and
x decrease). From empirically measured
x, we calculated the expected
y if
y=
z, and we also plotted
y=1. The magnitude of the empirical
y
curve falls between the
y=1 and
y=
z bulging conditions
(Fig. 4B;
Azizi et al., 2002
).
Testing the model and its assumptions
The model provides an unbiased prediction of longitudinal segment strain,
with predicted strain varying from the measured values by less than 5% strain
over most of the range (Fig.
5). The deviation between measured and predicted strains for the
individual swimming sequences could result from violation of assumption 1 (no
shear), assumption 2 (segments planar) or both. The lack of bias in the
predictions indicates that violation of assumption 1 is the most likely source
of the variation. The segments do bulge out laterally between the myosepta
during swimming, thereby violating assumption 2, but the associated increase
in muscle fiber curvature would tend to produce a bias in favor of higher gear
ratios. Slight torsion of the body, tending to produce left or right turns
equally, would produce unbiased variation. We attempted to select sequences of
steady, rectilinear swimming, but slight turning in unconstrained trackway
swimming could be associated with body torsion and unbiased deviations between
predicted and measured longitudinal segment strains.
As was expected from previous work on other species of salamanders
(Carrier, 1993;
Bennett et al., 2001
), EMG
confirmed that the EO and IO in Siren lacertina are active during
segment shortening, thereby validating assumption 3. Work loop studies would
be necessary to determine the actual contributions to positive and negative
work, but our finding that EO activity begins shortly before the beginning of
segment shortening and ceases shortly before the beginning of lengthening is
consistent with EO force generation during segment shortening and the
contribution of positive work to body bending.
Force and work
As expected from the conservation of work principle
(Otten, 1988), the model
predicts that changes in AGR affect the longitudinal force produced by
segmented musculature. Segment shortening increases (in three of four bulging
conditions) and longitudinal force production decreases with increasing
initial fiber angle and with increasing dorsoventral bulging
(Fig. 7). Our mean segment
force equation (Eqn 8) is similar to the equation that is commonly used to
calculate muscle force from fiber force in pennate muscles (Eqn 6), but we
include a correction for change in muscle fiber angle from
to ß
during contraction. Our modified equation applies equally well to pennate
muscle as to segmented muscle and may be useful for estimating mean muscle
force in pennate muscle contractions with substantial fiber rotation.
Incorporating changes in y (dorsoventral bulging) into the equations for total segment work demonstrates that total segment work equals muscle fiber work and work is conserved (Appendix 3). If we only consider work done in the x (longitudinal) direction, then the conservation of work principle is violated because changes in segment shortening are not the exact inverse of changes in segment force with increasing initial fiber angle (compare Figs 7A and 7B). However, if work done in the y direction is added to work done in the x direction, then segment work and fiber work are equal (Appendix 3). As with our force equation, our work equations apply equally well to pennate muscle and may be useful for estimating work in pennate muscles during contractions in which the width of the muscle does not remain constant.
Function of connective tissues in segmented musculature
Because our model includes the 3D shape changes of the LHM myomeres, and
because myomeres are wrapped in collagenous myosepta and skin that may
constrain their shape changes, our model provides an explicit, quantitative
link between muscle fiber and connective tissue architecture in a simple
segmented muscle system. The strong effect of bulging on AGR suggests that the
structural and material properties of connective tissues may affect the most
fundamental aspects of segmented muscle mechanics the speed and force
of shortening (Azizi et al.,
2002).
The shapes of our sonomicrometry traces suggest that maximum dorsoventral
bulging may sometimes be limited by connective tissues
(Fig. 4). Traces for fiber
strain (f) and segment strain (
x) are
close to sinusoidal, and the trace for dorsoventral bulging
(
y) is sinusoidal when y is decreasing, but the
peaks are flattened when y increases. These flattened peaks would be
consistent with the J-shaped stressstrain curve that is typical for
soft tissues. Stiffness is low in the toe of the J, but then increases at
higher strains, and could be limiting dorsoventral extension and flattening
the peaks. This might explain why the measured trace for
y
appears to follow the model
y=
z at low
strains but then the
y=1 model at higher strains
(Fig. 4B). This analysis is
highly speculative, however, because we did not observe the same, flattened
shape for the
y curve in all trials analyzed, and the shape
could be caused by other factors, such as adjacent muscle layers limiting the
dorsoventral bulging of the EO.
Previous studies have explored the roles of connective tissues in force
transmission and the modulation of body pressure and stiffness (e.g.
Long and Nipper, 1996;
Long et al., 1996
;
Westneat et al., 1993
), and
recent work on collagen fiber orientations in myosepta has demonstrated a set
of highly conserved myoseptal tendons in cartilaginous and ray-finned fishes
(Gemballa et al., 2003
). We
propose that one function of these tendons may be to constrain myomere
bulging, thereby affecting the speed and force of segment shortening. This
`bulge control hypothesis' is not mutually exclusive of other proposed
functions; the skin and myosepta may well contribute to force transmission,
bulge control and the modulation of body pressure and stiffness
simultaneously.
Limits to initial and final muscle fiber angles
If dorsoventral bulging is less than or equal to 1
(y<1), then the only geometric limit to muscle fiber and
segment shortening occurs when the final muscle fiber angle, ß,
approaches 90° and segment length approaches zero
(Table 1). In the
z=1 condition, the area defined by x and
y must remain constant as x decreases and y
increases. In this case, muscle fiber length will be shortest when
ß=45°; beyond 45°, dorsoventral bulging would cause the muscle
fibers to lengthen. In the
y=
z
condition, we calculated the limit on ß to be 54°. Beyond this angle,
the muscle fibers would have to lengthen for further segment shortening to
occur. This is the same angle (54.44°) at which helically wound
cylindrical muscular hydrostats begin to lengthen rather than shorten with
further muscle contraction (Kier and
Smith, 1985
).
Larger strains cause the muscle fibers to rotate through a larger angle and
therefore decrease the maximum initial fiber angle, , that will allow
the fibers to contract by a given amount
(Table 1). We can compare these
maximum initial muscle fiber angles with the actual muscle fiber angles
observed in the lateral hypaxial musculature of salamanders. In eight
representative species from eight families, the fiber angles in the external
and internal oblique layers are generally in the range of 2040°
(Brainerd and Simons, 2000
;
Simons and Brainerd, 1999
).
This range of initial muscle fiber angles would allow maximum muscle fiber
strains of up to
15% for most of the bulging conditions defined by our
models.
Muscle fiber angles in the transverse abdominis (TA) and the external
oblique superficialis (EOS; present only in some salamanders) are generally
higher, ranging from 60 to 80° in the TA and from 50 to 70° in the EOS
(Simons and Brainerd, 1999;
Brainerd and Simons, 2000
).
These high angles indicate that the fibers in these layers probably undergo
very low strains or even active lengthening during segment shortening (EMG
studies indicate that the EOS and TA are active during swimming;
Carrier, 1993
;
Bennett et al., 2001
). The EOS,
in particular, may undergo substantial lengthening because this layer is
located far from the neutral axis and therefore is subjected to large
longitudinal segment strains. The TA, by contrast, is located closest to the
neutral axis of bending. The TA will experience smaller longitudinal segment
strains, but we still expect that the dorsoventral bulging of the segments
will cause the muscle fibers of the TA to undergo some active lengthening
during swimming. Our calculations predict that the EOS and TA contribute
little to axial bending, indeed they may generate forces that oppose lateral
bending, but they may function to balance torsional moments and modulate body
pressure and connective tissue stiffness during swimming
(Brainerd and Simons, 2000
;
Bennett et al., 2001
).
Comparison with models of pennate muscle architecture
Our model of segmented musculature is similar to the most widely used model
for relating muscle fiber strain and fiber force to tendon excursion and
muscle force in pennate musculature
(Benninghoff and Rollhäuser,
1952; Gans and Bock,
1965
; Alexander,
1968
). More sophisticated pennate muscle models have also been
developed, in which curved muscle fibers and deformation of the aponeuroses
have been modeled (Woittiez et al.,
1984
; Huijing and Woittiez,
1984
; Zuurbier and Huijing,
1992
; Van Leeuwen and Spoor,
1993
). The basic pennate model assumes that the distance between
the tendon sheets, usually drawn as the width of the muscle, does not change
during muscle contraction (Otten,
1988
). With this assumption, the pennate model is mathematically
identical to our
y=1 model of segmented musculature,
rotated 90° such that y becomes the width of the muscle and
x becomes the direction of tendon movement.
When comparing the mechanics of two or more muscles with different resting
fiber angles (), pennate muscle models show that increases in
produce increases in tendon excursion and contraction velocity (higher AGR),
as long as resting muscle fiber length is held constant (e.g.
Muhl, 1982
;
Gans and Gaunt, 1991
;
Zuurbier and Huijing, 1992
).
In actual pennate muscles, however, it is more common to compare muscles with
similar overall length and width. In this case, resting muscle fiber length
decreases as
(pennation angle) increases, so increases in velocity and
excursion with increasing
are generally offset by decreases in fiber
length (Calow and Alexander,
1973
).
This trade-off between muscle fiber angle and fiber length is the source of
some confusion. Many text books emphasize that pennate muscles generate
greater forces over shorter tendon excursion distances than do fusiform
muscles with the same length and width (e.g.
Kardong, 2001;
Liem et al., 2001
). In most
muscles this is true because increasing the pennation angle increases force,
by allowing more muscle fibers to attach to the tendon, and decreases tendon
excursion because the fibers are shorter. Confusion arises because the
increased force and decreased excursion are attributed directly to the
increase in pennation angle, when in fact they result from keeping overall
muscle length and width constant while allowing fiber number to increase and
fiber length to decrease with increases in pennation angle. If the number and
length of fibers is held constant, then a muscle with a higher pennation angle
will generate less force and greater tendon excursion for a given muscle fiber
shortening. Thus, the AGR of a pennate muscle is always higher than the AGR of
a fusiform muscle, but the higher AGR is offset by a decrease in fiber length
if the muscle must be packed in to the same available space
(Calow and Alexander,
1973
).
By contrast, our segmented muscle model differs from pennate muscle models
in that when we explore the effect of increasing , we hold segment
length constant, and therefore fiber length increases rather than decreases
with increases in
. In pennate muscles, this would be equivalent to
keeping muscle length constant while allowing muscle width to increase as
pennation angle (
) increases. The assumption of constant segment length
means that segment volume increases with
; therefore, we use the
assumption of constant fiber length rather than constant segment length for
calculations of segment work (Fig.
7; Appendix 3). However, keeping segment length constant for
exploring the effect of
makes biological sense in segmented
musculature because
varies from medial to lateral within the same
segment of both fishes and salamanders
(Alexander, 1969
;
Simons and Brainerd, 1999
;
Gemballa and Vogel, 2002
).
The relative effects of increasing AGR and increasing muscle fiber length
can be seen by comparing the curves in Fig.
2, in which segment length was held constant and fiber length was
allowed to increase, with the curves in
Fig. 7A, in which fiber length
was held constant. Increases in fiber length do contribute to increasing the
magnitude and speed of shortening, but the effect of increasing fiber length
is small when compared with the effects of changes in and changes in
the magnitude of dorsoventral bulging.
![]() |
Appendix 1. Equations for relative segment bulging |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
![]() | (9) |
This equation can be used to derive expressions for the magnitude of bulging in the y and z dimensions for a given amount of longitudinal (x-dimension) segment shortening for each of our four bulging conditions.
Bulging condition 1: z=1
In this condition, z remains constant, so we can substitute
z1 for z2 and
xx1 for x2 in Eqn 9
and solve for y2:
![]() | (10) |
![]() |
List of symbols |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Bulging condition 2: y=
z
Assuming that y and z have equal extension ratios
(y=
z=
yz), then
y2=
yzy1 and
z2=
yzz1 and
x2=
xx1.
Substituting into Eqn 9 yields
x1y1z1=
xx1
yz2y1z1,
which simplifies to
yz=1/(
x1/2) and:
![]() | (11) |
![]() | (12) |
Bulging condition 3: y=1
Since y remains constant, we can substitute y1
for y2 and xx1 for
x2 in Eqn 9 and solve for z2:
![]() | (13) |
Bulging condition 4: y=
x
Assuming that the height and length of the segment decrease by the same
proportion (y=
x), we can substitute
xx1 for x2 and
xy1 for y2 in Eqn 9
and solve for z2
![]() | (14) |
![]() |
Appendix 2. Equations for segment strain |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Bulging condition 1: z=1
From Eqn 10,
y2=y1/x, and since
y1=fsin
(from
Fig. 1B), then
y2=fsin
/
x. Substituting
y2=fsin
/
x into Eqn 1
yields
f2=
x2cos2
+sin2
/
x2
and:
![]() | (15) |
Bulging condition 2: y=
z
From Fig. 1B,
y2=ffsinß and
y1=fsin
. Substituting into Eqn 11,
ffsinß=fsin
/(
x1/2)
and
fsinß=sin
/(
x1/2).
Substituting sin
/(
x1/2) for
fsinß in Eqn 1 yields:
![]() | (16) |
Bulging condition 3: y=1
In this case, y2=y1 and
fsinß=sin
(from
Fig. 1B). Substituting
sin
for
fsinß in Eqn 1 yields
f2=sin2
+
x2cos2
,
and solving for
x yields:
![]() | (17) |
Bulging condition 4: y=
x
Fiber angle is constant in this shortening condition (the before and after
conditions are similar triangles), so we can substitute for ß in
Eqn 2 such that
x=
f(cos
/cos
) and:
![]() | (18) |
![]() |
Appendix 3. Work in segmented musculature |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
![]() | (19) |
![]() | (20) |
![]() | (21) |
![]() | (22) |
![]() | (23) |
![]() | (24) |
![]() | (25) |
![]() | (26) |
![]() | (27) |
![]() | (28) |
![]() | (29) |
![]() | (30) |
For calculations of Wx, we keep segment height constant
at the initial height (y2=y1). For
calculations of Wy, we keep segment length constant at
final segment length (x1=x2). This
allows us to calculate the two work components independently in two separate
steps:
![]() | (31) |
![]() | (32) |
![]() | (33) |
![]() | (34) |
![]() | (35) |
![]() | (36) |
Eqns 31 and 32 can be used to calculate shortening work and bulging work, and Eqn 36 confirms that the total work done by the segment is equal to the work done by the muscle fibers.
![]() |
Acknowledgments |
---|
![]() |
Footnotes |
---|
![]() |
References |
---|
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---|
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