The notochord of hagfish Myxine glutinosa: visco-elastic properties and mechanical functions during steady swimming
1 Mount Desert Island Biological Laboratory, Salsbury Cove, Maine, 04672,
USA
2 Department of Biology, Vassar College, Poughkeepsie, New York, 12604,
USA
3 Division of Skeletal Biology, Shriners Hospital for Children, Tampa,
Florida, 33612, USA
* Author for correspondence (e-mail: jolong{at}vassar.edu)
Accepted 9 September 2002
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Summary |
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Key words: notochord, backbone, axial skeleton, hagfish, swimming, bending, damping, stiffness, flexural stiffness, kinematics, visco-elastic properties, Myxine glutinosa
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Introduction |
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We chose to study Atlantic hagfish Myxine glutinosa because (1) as
adults they retain a notochord without vertebral elements
(Cole, 1905), (2) they are
active undulatory swimmers (Adams,
1960
; Martini et al.,
1997
), bending their notochord dynamically, and (3) they belong to
the living sister taxon (Myxiniformes) to vertebrates and thus provide
information to assist in phylogenetic reconstruction of the ancestral craniate
notochord. The notochord of hagfish has also attracted considerable attention
because of its unusual biochemical, molecular and biomechanical properties
(Koob et al., 1994
;
Kielstein et al., 1996
;
Long et al., 1998
;
Welsch et al., 1998
). In a
preliminary biomechanical study (Long et
al., 1998
), the visco-elastic properties of the Atlantic hagfish's
notochord were compared to those of the whole body, and more than half of the
whole body's passive flexural stiffness and damping were attributed to the
notochord alone.
Flexural stiffness and damping are keys to understanding the dynamic
functions of notochords. When vibrations, such as a fish's undulatory body
waves, are driven at a bending frequency (rad s-1) lower than
the structure's resonant or natural frequency, bending motion and internal
stress are dominated by the structure's flexural stiffness, EI (in N
m2). As
approaches and then equals the natural frequency,
flexural stiffness and the mass moment of inertia Im (kg
m2), balance, causing bending motion and internal stress to
increase rapidly and catastrophically unless sufficient damping forces are
present (Den Hartog, 1956
;
Denny, 1988
). This
amplification is expressed as the ratio of the maximal stiffness moment
EI
0, to the maximal applied moment
M0 (N m) driving the vibrations (modified from
Denny, 1988
):
![]() | (1) |
To examine their relative importance in swimming and bending hagfish, we
measured EI and C in situ at physiologically relevant
frequencies and curvatures using a custom-built, dynamic bending machine
(Long, 1998). We determined
the natural physiological parameters for the bending tests from kinematic
analysis of steadily swimming hagfish. To understand the contributions of
specific structures to the overall visco-elastic properties of the body, we
compared, in dead hagfish, the properties of the whole body with those of the
skinless body, the intact notochord, and the core of the notochord.
Specifically, we asked two questions. (1) Over the range of lateral bending
motions seen during swimming, what are the body's passive visco-elastic
properties? (2) Compared to skin and muscle, does the notochord contribute
substantially during swimming to the body's passive mechanical functions?
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Materials and methods |
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![]() | (2) |
From the motion of the reconstructed midline, a number of other kinematic
variables were measured (Fig.
1). For the purpose of designing physiologically relevant bending
experiments, of primary importance were the amplitude of the midline curvature
0, and the tailbeat or undulatory frequency
(rad
s-1). Tailbeat frequency is measured as the quotient of 2
(rad)
and the tailbeat period (s), which is the time required by body point 30, the
tail tip, to twice achieve values of zero lateral displacement following an
initial zero displacement (Fig.
1D). Other kinematic variables measured included lateral amplitude
(`heave') of body point 11,
0,11, lateral amplitude of the
tail tip (point 30),
0,30, and amplitude of the pitch angle
0 of the body segment between midline points 10 and 11
(Fig. 1D). The instantaneous
pitch angle
(rad) is defined as the orientation of the segment
relative to the direction of the freestream flow
(Vogel, 1994
), which is
approximately the x-axis or the axis of progression. The relative
timings of maximal heave, flexion and pitch at midline point 11 were measured
as the phase lag (fraction of a tailbeat period T) between the time
of maximal heave and flexion
y
, and between
the time of maximal heave and pitch
ya. For positive
values of phase lag the maximal heave occurs later in the tailbeat cycle than
either the flexion or pitch maxima.
Using a technique modified from McHenry
(2001), we defined the
curvature half-wave length,
/2, as the distance (as
a proportion of normalized body length L) along the body axis from
zero to zero
, i.e. from inflection point to inflection point;
/2 was measured on the body when the half-wave
included body point 11, the position that corresponds to the portion of the
body bent during dynamic tests (0.37 L from rostrum). Thus, the
reported value of
/2 for each trial is the average
of all the instantaneous values of
/2
(Fig. 1C). The curvature
half-wave is roughly analogous to the so-called propulsive wave or half-wave,
measured as the distance along the axis of progression (roughly the
x-axis in Fig. 1)
between midline nodes, as determined by a variety of methods (for caveats and
a review, see Long and Nipper,
1996
). The
/2 value has the advantage
over the standard propulsive wavelength of being independent of (i) the
estimation of the axis of progression, (ii) the requirement of left-right
symmetry in body bending, (iii) the determination and subtraction of average
body velocity from the midline points and (iv) the lateral position of the
tail.
To determine which variables predicted length-specific swimming speed,
UL (in L s-1), we ran the following
general linear model using the statistical software JMP (SAS Institute, Inc.,
version 3.0). Because the 23 swimming trials were from eight individuals, we
included individual, IND, as a randomized-block effect:
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Dynamic bending tests
In order to determine the flexural stiffness EI and the flexural
damping C of the hagfish body axis, we used the methodology and
machinery of Long (1998). To
produce sinusoidal cantilever bending, the caudal side of a small portion of
the intact body was clamped on a stationary strain gauge and the cranial side
was clamped onto a motor-driven linkage that applied a bending couple of
varying curvature amplitude
0 and frequency
. The
strain gauge measured the bending moment M transmitted through the
body section. Dynamic M signals produced by the strain gauge (two 120
foil gauges; Omega Engineering) were excited and amplified by a
high-frequency (40 kHz response time) bridge amplifier (Omega Engineering
model DMD-520) and were digitally recorded at 1000 Hz (National Instruments
model NB-MIO16E analog-to-digital converter). Simultaneously and in the same
file, we digitally sampled the
, which was determined using a rotary
variable differential transducer (Schaevitz model R30D) mounted colinear with
the bending couple input linkage; it measured
that we converted to
using Equation 2.
We determined EI and C for a given experimental trial
(fixed and
) using the equation of motion for a
single-degree-of-freedom system (see Den
Hartog, 1956
):
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![]() | (5) |
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For a trial at a given 0 and
, the measured
C values were used to calculate the net flexural work W (J),
used over a complete cycle to bend the sample. The W, also known as
the work to overcome damping C, was calculated as follows (see
Den Hartog, 1956
):
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![]() | (8) |
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To determine the relative influence of axial structures on EI, E, C,
W and R, we sequentially removed, by dissection, the skin, the
lateral musculature, and the notochord's outer fibrous sheath
(Fig. 2). The removal of the
lateral musculature (change from the `without skin' to `notochord', structural
treatment categories 2 and 3, respectively), significantly decreased mean
width, area and I of the test sections
(Fig. 2). Removal of the outer
fibrous sheath (change from `notochord' to `core', structural treatment
categories 3 and 4, respectively) significantly decreased the mean width of
the test sections (Fig. 2A).
While changes in morphometrics were not statistically detectable (although a
consistent decreasing trend exists) between the `intact' and `skin-removed'
categories (structural treatment categories 1 and 2, respectively), this
change removes the subcutaneous blood sinus
(Forster, 1997). The skin and
the lateral musculature were removed while the hagfish body was clamped in the
bending machine. In order to remove the outer fibrous sheath without damaging
the core, we dissected the intact axial skeleton from the body, and then
carefully removed the outer fibrous sheath with a scalpel. Because the core
was occasionally damaged in this process, we used the cores from different
individuals than those used for the first three structural treatment
categories.
The body and notochord were tested using section lengths, l, of 0.004-0.010 m long (Table 1), depending on the experimental treatment. The test section was centered at 0.37 L, the same position for which we measured kinematic features of swimming (see previous section). For structural treatment categories 1-3, only the test section was altered; the rest of the body was intact, attached, and unaltered. For structural treatment category 4, the entire notochordal core was kept intact. Prior to each test, the section was conditioned by undergoing at least 20 cycles of bending. Potential order effects (the potential for sample degradation over time to be correlated with serially arranged tests) for structural treatments were unavoidable, given the need for sequential dissection and to minimize the number of animals used. Order effects were also possible for the tests within a structural treatment category, since tests proceeded from lowest frequency and amplitude to highest frequency and amplitude. To test the magnitude of the order effects, we repeated the first tests following the last; in no case did identical test conditions, separated by time and testing, reveal changes in E or C greater than 3%.
The notochordal core treatment confounded our statistical analysis in two
ways. (1) In order to produce a detectable M signal with the
small-diameter of the core, we increased the applied by reducing the
length, l, of the test section. Since the range of
values
was, thus, higher than that used for structural treatment categories 1-3, we
nested the factor
within the factor `structural treatment' TRT, which
produced a single nested factor equivalent to the factor
and the
by TRT interaction term (Zar,
1996
). (2) Cores were from individuals different from those in
structural treatments categories 1-3, therefore we nested the factor
individual, IND, within the
and TRT factors. This produced a compound
nested factor equivalent to the factor IND and the IND by
by TRT
interaction (Zar, 1996
). The
result was a mixed-model nested analysis of variance (ANOVA) with
and
TRT as main effects:
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To test for differences between adjacent levels of the factors and
TRT, we ran a priori contrasts. Analysis was conducted in JMP (SAS
Institute, Inc., version 3.0).
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Results |
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Visco-elastic properties of the notochord and body
Overall, dynamic bending tests at 0.37 L revealed that the
visco-elastic properties of the body varied significantly as functions of the
bending frequency , structural treatment TRT, the interaction of
and TRT, and individual IND, nested within TRT and curvature,
;
nested within TRT was not a significant factor for any response
variable. Specifically, with changes in
, flexural stiffness
EI, flexural damping C and flexural work W, all
varied significantly (Table 4). A priori contrasts between
categories revealed that
EI increased from 2
to 4
and from 4
to 6
rad
s-1, C decreased at each level, and W increased
from 2
to 4
rad s-1 (Fig.
4). With changes in TRT, all five visco-elastic properties varied
significantly; a priori contrasts
(Table 5) between TRT
categories revealed that EI decreased when the notochordal core was
isolated, the apparent Young's modulus E, increased when the axial
muscles were removed from the notochord, C decreased when the skin
was removed and when the notochordal core was isolated, and the resilience
R decreased when the notochordal core was isolated. The interaction
of
and TRT was significant for EI, E, C and W;
inspection of the graphical patterns (see
Fig. 4) reveals that there is
little effect of
on core EI, E shows little or no
effect with the body intact and the skin removed, and C and
W show little or no
effect with the core of the notochord.
Finally, the compound factor of IND nested within TRT and
was
significant for all five variables; since this factor sequestered variance
caused by differences between individual hagfish, it eliminates those effects,
which are not of interest for the purposes of this study, from the responses
of the other factors.
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EI of the intact body did not decrease significantly with the
removal of the skin or the axial muscle. Even though the differences were not
significant, the mean EI of the notochord whole body decreased by 25%
compared to the mean EI of the whole body (values pooled across ,
and IND). Thus, conservatively, the notochord provides the body with
75% of its total EI. When the lateral musculature was removed,
E increased significantly, by an order of magnitude from a mean of
0.38 to 4.97 MPa (values pooled across
,
and IND). The
notochord has a high EI, in spite of a significantly reduced
I (see Fig. 2D),
because of an increased E relative to that of the whole body.
Flexural damping C decreased significantly when the skin was
removed, implicating the subcutaneous sinus as a source of flexural damping;
the mean (values pooled across ,
and IND) decreased 20% from
6.33 to 5.07 kg m3 s-1. Since no significant decreases
in C were detected with the removal of the lateral musculature, the
notochord must be the primary source of the body's flexural damping, providing
80% of the total C. The damping function of the notochord is also
supported by the lack of change in flexural work W, which is a
function of C (see Equation 8) and resilience R, with the
removal of the lateral musculature.
Flexural damping and stiffness were correlated. With increasing C,
EI decreased rapidly (Fig.
5A) for all structural treatments. In contrast, with increasing
damping moments, the stiffness moments increased linearly
(Table 6;
Fig. 5B) for all structural
treatments. As increased, the ratio of stiffness moments to damping
moments increased linearly for the skin-off treatment but not for the others
(Fig. 5C); the ratio derived by
assuming frequency-independent values of EI and C decreased
with increasing
.
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Discussion |
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The mechanical dominance of the notochord has four important implications,
each of which is addressed in the following sections. (1) The
frequency-dependent visco-elastic properties of the notochord interact
dynamically to give it the capacity to amplify and stabilize undulatory
swimming motions of the whole body (Figs
4,
5). This dynamic capacity has
been omitted from computational models of swimmers with notochords, such as
lamprey (Ekeberg, 1993;
Carling et al., 1998
; Isjpeert
et al., 1998
,
1999a
,
1999b
) and amphibian tadpoles
(Lui et al., 1996
,
1997
;
Hoff and Wassersug, 2000
), and
its inclusion may enhance locomotor performance in next-generation simulations
that couple internal and external forces. (2) The similarity in mechanical
properties in notochords and intervertebral joints
(Fig. 6) suggests a common
structural and physicochemical basis. (3) Scenarios of the evolution of
vertebral columns are informed by an understanding of the mechanical
capacities of notochords during swimming
(Koob and Long, 2000
;
McHenry, 2001
). (4) Some of
the unique kinematic features of hagfish swimming may be caused by the
mechanical properties of the notochord.
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Mechanical capacities of the notochord during steady swimming
The visco-elastic properties of the notochord give it the capacity to
function as both a motion stabilizer and a mechanical power amplifier. While
springs acting as force transmitters in oscillating propulsive systems reduce
energy costs by reducing negative work, they may cause unstable dynamics
(Harper et al., 1998). Dynamic
instability may be controlled actively, by sensory-muscular control, or
passively, using the system's constitutive mechanical properties
(Harper et al., 1998
). Passive
stability is preferable in animal locomotion in certain situations, because it
provides faster and simpler mechanical feedback and response
(Dickinson et al., 2000
).
In hagfish, evidence for passive dynamic stability comes from the ratio of
stiffness to damping moments, a special case of Equation 1 that assumes that
the system is always bending at its resonance frequency (see Introduction).
While operating at resonance provides energy savings to swimming animals
(DeMont, 1990;
Oxner et al., 1993
; for
reviews, see Long and Nipper,
1996
; Pabst,
1998
), the amplification of bending or stresses that produces
those savings may lead to catastrophic failure if they are undamped
(Denny, 1988
). Thus, we
examined the ratio of stiffness moments and damping moments at all observed
undulatory frequencies to determine if there might be any situation in which
resonant swimming might lead to break-away bending. Unexpectedly, EI
and C of the hagfish notochord adjust with changes in
(Fig. 5A) to produce a nearly
constant amplification ratio of seven (Fig.
5C). Thus, the notochord would amplify force or curvature and, at
the same time, limit that amplification if hagfish were swimming with an
at or near their resonance frequency. In other words, the
visco-elastic properties of the notochord provide the body with a nearly
constant dynamic stability over a wide range of swimming speeds.
Hagfish could take advantage of their dynamic passive stability by using
their muscles to adjust the stiffness of their body to match the resonance
frequency to any at which they are swimming. Such active tuning was
first suggested by Blight
(1977
), and adjustable body
stiffness has been shown to affect swimming performance in physical sunfish
models (McHenry et al., 1995
)
and computational simulations of sunfish
(Long et al., 2002
) and
lamprey (Ijspeert et al.,
1998
). Myomeric muscles of eel Anguilla rostrata have the
capacity to increase the body's EI and C by a factor of
three and seven, respectively (Long,
1998
). Taken in combination, these results suggest that hagfish
may have the capacity to engage in variable resonance swimming.
Structural and physicochemical basis of mechanical properties
It is surprising that the notochord, a hydrostatic, fiber-wound cylinder,
could have the apparent material stiffness E and size-independent
damping C/A comparable to the intervertebral joints in a segmented,
bony vertebral column in a fish and cetacean
(Fig. 6). For comparison, a
motion segment of the intact human vertebral column, including the invertebral
disc and articular processes of the neural spine, has a storage modulus
G' (roughly equivalent to E, depending on loading
conditions) of 45 MPa (for a review, see
Iatridis et al., 1996). The
G' of the isolated nucleus pulposus and anulus fibrosus are
much lower, ranging from 0.01 MPa
(Iatridis et al., 1996
) to
0.20 MPa (Iatridis et al.,
1999
) in the former and 0.007-0.020 MPa
(Iatridis et al., 1997
) in the
latter. The basis for the correspondence in aquatic species may derive in part
from the similarity between physicochemical properties of the notochord core
and those of intervertebral discs and joints: the mechanical properties of
both systems appear to be based on hydrostatic mechanisms (for reviews, see
Wainwright, 1988
;
Hukins and Meakin, 2000
;
Koob and Long, 2000
).
Physicochemical measurements coupled with mechanical tests provide a
preliminary basis for understanding the mechanical properties of the
notochord. Bulk free-swelling tests on isolated segments have established that
the notochord's core is osmotically active: it swells in solutions with ionic
strength below that in vivo, and shrinks in higher ionic strength
solutions (Koob et al., 1994).
The flexural stiffness EI and apparent material stiffness E
of the notochord are inversely proportional to the osmolarity of the bathing
solution (Sinwell et al.,
1999
). These observations suggest that the core of the notochord
exerts a swelling pressure on the constraining fibrous sheath, imparting the
high, hydrostatically controlled material stiffness.
Unfortunately, far too little is known about intervertebral discs and
joints in non-mammalian species for a satisfactory comparison. Nevertheless,
several common features should be mentioned. Unconstricted notochords in
sturgeon and lungfish are organized essentially the same as the hagfish
notochord, and display similarities with respect to cell morphology in having
large vacuoles bounded by intermediate filaments
(Schmitz, 1998). The yellow
perch's intervertebral joint, which is derived from the notochord, is also
made up in large part of interconnected cells with large vacuoles
(Schmitz, 1995
). The situation
is different in mammalian intervertebral joints. The core of these joints, the
nucleus pulposus, is predominantly extracellular matrix (for a review, see
Urban et al., 2000
).
Evolution of the function of the notochord and vertebrae
Adults in all the extant chordate clades can be found that retain an
unsegmented notochord associated with axial musculature driving undulatory
flexures. Within Urochordata, the appendicularians retain a notochord in a
muscular tail used for locomotion or feeding
(Nishino and Satoh, 2001). In
Cephalochordata (lancelets) and Myxiniformes (hagfishes), free-swimming and
burrowing adults retain a notochord (Gee,
1996
). In vertebrates, the unsegmented notochord has been retained
with the addition of neural and hemal arches of cartilage or bone in lamprey,
some shark, lungfish, sturgeon and paddlefish
(Goodrich, 1930
). Since many
other metazoans without notochords also engage in undulatory swimming
(Clark, 1964
), the evolution of
the notochord was not a prerequisite.
Given this comparative information, the unsegmented notochord of hagfishes
is not degenerate; the unsegmented notochord engaged in undulatory motion is
the ancestral character state of the axial skeleton in Craniata (hagfishes and
vertebrates). This evolutionary direction has received additional support from
the phylogenetic placement of the Lower Cambrian hagfish-like fossil
Haikouella, which possesses an unsegmented notochord, as a clade
between cephalochordates and myxiniformes
(Holland and Chen, 2001). We
argue that by examining the mechanics of the notochord of the marine hagfish,
we investigate an axial skeletal system that is likely to have retained
functional features common to the stem lineage of Chordata. By contrast, we
would not argue the same if we had chosen to examine Branchiostoma, a
lancelet whose notochord appears to have derived features such as intrinsic
paramyosin that actively alter flexural stiffness
(Webb, 1973
).
Within Craniata (hagfishes and vertebrates), the notochord evolved
segmentation in the form of vertebral centra
(Gee, 1996). The number of
vertebrae is inversely proportional to the magnitude of body curvature in
fast-starting fish (Brainerd and Patek,
1998
) and steadily swimming undulatory vertebrates
(Long and Nipper, 1996
). Thus
vertebrae may stiffen the body. An alternative hypothesis, and one that we can
test with our data on hagfish notochords, is that the apparent increased body
stiffness comes not from the vertebrae per se but from increases in
the material stiffness and damping of the connective tissues of the
invertebral joints. Our data do not support this alternative
(Fig. 6). Instead, the apparent
material stiffness, E, and the damping per cross-sectional area,
C/A, are nearly equal in notochords and intervertebral joints.
Vertebrae, being much stiffer, mineralized elements, restrict bending to the
joints. Thus for a given amount of body curvature, the internal strain and
stress on any section of notochordal tissue would be much less than that on
the tissues of an intervertebral joint.
Midline kinematics of swimming hagfish
To our knowledge (see also Vogel and
Gemballa, 2000), this is the first quantitative analysis of the
undulatory swimming motions of hagfish (for qualitative observations, see
Adams, 1960
). While our
original intent in videotaping steadily swimming hagfish was to determine
physiologically relevant ranges for
and
in the bending
experiments, the kinematic results are of interest in their own right
(Fig. 3). At the swimming
speeds measured here, ranging from a speed UL of 0.4 to
1.0 Ls-1, we find that while undulatory frequency
increases linearly with respect to UL, tailbeat amplitude
y0,30 decreases linearly with respect to
(Fig. 3A,B). This is an
unexpected result. From undulatory teleosts, we expected that
and
y0,30 would increase in concert at speeds below 5
Ls-1 (Bainbridge,
1958
,
1963
), or that y0,30
would remain constant (Webb et al.,
1984
; for a review, see
Videler, 1993
). This decrease
in y0,30 may be due, in part, to the increasing stiffness EI
of the notochord with increasing
(Fig. 5). Furthermore, since
the squares of
and y0,30 are proportional to hydromechanical
power output (Wu, 1977
;
Webb et al., 1984
), in order
to swim faster a decrease in y0,30 must be compensated for by either
a disproportionately large increase in
or an increase in propulsive
efficiency. Since
nearly doubles over the observed ranged of speeds
and y0,30 decreases by nearly half
(Fig. 3), the changes in
hydromechanical power should cancel out while notochord and body stiffness
increase. Clearly, more work is required to fully understand the mechanics of
steady swimming in hagfish.
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Acknowledgments |
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