To determine the possible locomotor functions of the hagfish notochord, we measured its flexural stiffness EI (N m-2) and flexural damping C (kg m3 s-1), under in vitroconditions that mimicked the body curvature and bending frequency measured during steady undulatory swimming. To assess the notochord's contribution to the mechanical behavior of the whole body, we also measured EI and C of the whole body, the body with skin removed, and the notochord with the outer fibrous sheath removed. When subjected to dynamic bending at angular frequencies from π to 6π rad s-1 and midline curvatures from 11 to 40 m-1, 1 cm in situ body segments(N=4), located at an axial position of 37% of the body length, showed significant changes in EI, C, the Young's modulus or material stiffness (E, MPa), the net work to bend the body over a cycle(W, J) and resilience (R, % energy return). When skin,muscles and the outer fibrous sheath of the notochord were removed sequentially, each structural reduction yielded significant changes in mechanical properties: C decreased when the skin was removed, E increased when the muscles were removed, and EI and R decreased when the outer fibrous sheath was removed. Although occupying only a small portion of the cross-sectional area, the notochord provides the body with 75% of its total EI and 80% of total C, by virtue of its high E, ranging from 4 to 8 MPa, which is an order of magnitude greater than that of the whole body. Thus, as the body's primary source of EI and C, the notochord determines the passive (i.e. internal, non-muscular) mechanical behavior of the swimming hagfish. EI and C covary inversely and non-linearly such that as C increases, EI decreases. However, the bending moments M (Nm) produced by each property increase proportionally, and the ratio of stiffness to damping moments, also known as the amplification ratio at resonance, is nearly invariant (approximately 7) with changes in driving frequency. If the body operates in life at or near resonance, the variables EI and C interact over a range of swimming speeds to produce passive mechanical stability.

Chordates have evolved an unique hydrostatic axial skeleton, the notochord,that is present in all taxa of that phylum early in development; it is retained in the adults of some taxa and modified by vertebral elements in others (Gee, 1996). Notochords are hypothesized to have evolved to stiffen the body(Goodrich, 1930) and to prevent body compression during muscle activation(Clark, 1964). In addition,notochords may adjust function by means of dynamically variable mechanical properties (Long et al.,1998). Our goal was to determine the notochord's locomotor functions by measuring its mechanical properties and behavior, relative to those of the intact body, under conditions that mimic those of steady undulatory swimming.

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 ω (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 ω 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 EIκ0, to the maximal applied moment M0 (N m) driving the vibrations (modified from Denny, 1988):
$\ \frac{EI{\kappa}_{0}}{M_{0}}=\frac{EI}{\sqrt{(EI-I_{\mathrm{m}}l{\omega}^{2})^{2}+(C{\omega})^{2}}},$
1
where κ0 is the amplitude of curvature (m-1), C is the flexural damping (kg m3 s-1) and l is the length of the structure (m). This ratio is a relative measure of the amplification of the dynamic moments experienced by the oscillating structure, with values above 1 indicating amplification and those below indicating attenuation (Denny,1988).

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?

### Animals

Hagfish Myxine glutinosa L. were purchased from a commercial supplier (Huntsman Marine Laboratory, New Brunswick, Canada), who caught the specimens in baited traps in the Bay of Fundy. At Mount Desert Island Biological Laboratory, Salsbury Cove, Maine, hagfish were kept in circular 400 l tanks with running seawater at 10-12°C. 16 post-absorptive hagfish were used, 8 in swimming trials and 8 in bending experiments (for lengths and masses, see Table 1). Following swimming trials and prior to bending experiments, animals were killed with an overdose of propylene phenoxetol. During testing, tissues were continually bathed in chilled hagfish Ringer solution(Riegel, 1978).

Table 1.

Morphometrics of hagfish and test sections used in each experimental group

HagfishBody mass (kg)Body length Lb (m)Test section length l (m)
Group 1: Swimming trials
0.034 0.284 0.009
0.036 0.329 0.011
0.038 0.298 0.010
0.039 0.314 0.010
0.040 0.315 0.011
0.042 0.298 0.010
0.046 0.330 0.011
0.049 0.324 0.011
Mean±S.D. 0.041±0.0050 0.311±0.0167 0.010±0.0006
t test*
Group 2: Structural treatments 1-3
0.039 0.377 0.012
10 0.041 0.380 0.010
11 0.045 0.375 0.010
12 0.058 0.385 0.011
Mean±S.D. 0.046±0.0083 0.379±0.0044 0.011±0.0010
t-test A, B
Group 3: Structural treatment 4
13 0.070 0.415 0.006
14 0.069 0.427 0.003
15 0.044 0.415 0.004
16 0.065 0.425 0.004
Mean±S.D. 0.062±0.0122 0.0421±0.0064 0.004±0.0013
t-test
HagfishBody mass (kg)Body length Lb (m)Test section length l (m)
Group 1: Swimming trials
0.034 0.284 0.009
0.036 0.329 0.011
0.038 0.298 0.010
0.039 0.314 0.010
0.040 0.315 0.011
0.042 0.298 0.010
0.046 0.330 0.011
0.049 0.324 0.011
Mean±S.D. 0.041±0.0050 0.311±0.0167 0.010±0.0006
t test*
Group 2: Structural treatments 1-3
0.039 0.377 0.012
10 0.041 0.380 0.010
11 0.045 0.375 0.010
12 0.058 0.385 0.011
Mean±S.D. 0.046±0.0083 0.379±0.0044 0.011±0.0010
t-test A, B
Group 3: Structural treatment 4
13 0.070 0.415 0.006
14 0.069 0.427 0.003
15 0.044 0.415 0.004
16 0.065 0.425 0.004
Mean±S.D. 0.062±0.0122 0.0421±0.0064 0.004±0.0013
t-test

t-tests were for two-tailed distributions with heteroscedastic variances conducted on the three pairwise comparisons of the three group samples. The same letter indicates that there was no detectable difference in the means of the groups (P>0.05).

See Materials and methods and Fig. 2 for an explanation of the structural treatment categories.

### Kinematic analysis of steady swimming

From a distance of 3 m, we simultaneously videotaped (SVHS format, 60 images s-1, shutter speed 0.001 s; Hitachi Model KP-M1U) lateral and dorsal views of swimming hagfish through transparent walls and by the use of a front-surface mirror (Edmund Scientific) mounted at 45° above a still-water circuit (2 m straight sections connected by turns of 1 m radius). After gentle manual prodding, hagfish swam readily at speeds of their own volition. We used only those trials in which fish swam (i) without touching the walls and (ii) steadily, as defined by linear regression(r2>0.98 for one tailbeat cycle) of axial displacement against time of body point 11 (Fig. 1D; see below). By these criteria, 23 trials were selected (3 trials from 7 fish and 2 trials from 1 fish). Body point 11, the position that corresponds to the portion of the body bent during dynamic tests (0.37 L from rostrum, where L is body length), was chosen because,as the propulsive wave of bending propagates from head to tail, it was the first point to contain a nearly complete half-wave of curvature (see Fig. 1B; Table 3). With a complete half-wave on both sides of this position, anterior and posterior bending moments are roughly balanced and can thus be imposed as such on experimental sections (see below). Since we did not measure kinematics or mechanical properties at other axial positions, extrapolations should be made cautiously.

Fig. 1.

Swimming motions and kinematic variables. (A) A swimming hagfish(TL 31.5 cm), as represented by overlaid, reconstructed midlines (31 points each) from one tailbeat cycle. (B) Midlines displaced vertically (time descends) to show curvature and forward progression (17.5 cm s-1). Inset: Midlines at times 0.20 s and 0.27 s are magnified to show placement of points 0-30 evenly along the midline. (C) The curvature half-wave length,λ κ/2 (L), is the axial distance along the midline from the points of 0 to 0 midline curvature, κ (m-1).(D) Other kinematic variables include amplitude of the midline curvature,κ 0, amplitude of the lateral displacement of body points 11,γ 0,11, and 30, γ0,30, amplitude of the pitch angle, α0. The phase lags δy—κand δy—α, are the differences in time, as a fraction of the normalized tailbeat period (T), between maximalγ and maximal κ and α, respectively.

Fig. 1.

Swimming motions and kinematic variables. (A) A swimming hagfish(TL 31.5 cm), as represented by overlaid, reconstructed midlines (31 points each) from one tailbeat cycle. (B) Midlines displaced vertically (time descends) to show curvature and forward progression (17.5 cm s-1). Inset: Midlines at times 0.20 s and 0.27 s are magnified to show placement of points 0-30 evenly along the midline. (C) The curvature half-wave length,λ κ/2 (L), is the axial distance along the midline from the points of 0 to 0 midline curvature, κ (m-1).(D) Other kinematic variables include amplitude of the midline curvature,κ 0, amplitude of the lateral displacement of body points 11,γ 0,11, and 30, γ0,30, amplitude of the pitch angle, α0. The phase lags δy—κand δy—α, are the differences in time, as a fraction of the normalized tailbeat period (T), between maximalγ and maximal κ and α, respectively.

Table 3.

Kinematic variables correlated with undulatory frequency, ω,and/or amplitude of midline curvature κ0 as determined by linear regression*

Lateral tail amplitude y0,30=0.2272-0.0086ω 0.255 0.220 0.014
Pitch angle amplitude α0=0.198+0.0189κ0 0.332 0.300 0.004
Phase lag, curvature δy-κ=-0.0522+0.0118ω 0.242 0.206 0.017
Phase lag, curvature δy-κ=-0.0736+0.0074κ0 0.190 0.151 0.038
Lateral tail amplitude y0,30=0.2272-0.0086ω 0.255 0.220 0.014
Pitch angle amplitude α0=0.198+0.0189κ0 0.332 0.300 0.004
Phase lag, curvature δy-κ=-0.0522+0.0118ω 0.242 0.206 0.017
Phase lag, curvature δy-κ=-0.0736+0.0074κ0 0.190 0.151 0.038
MeanS.D.
Phase lag, pitch angle δy-α=-0.25T  ±0.067
Curvature half-wavelength λκ/2=0.427L  ±0.0422
Lateral body amplitude y0,11=0.065L  ±0.0232
MeanS.D.
Phase lag, pitch angle δy-α=-0.25T  ±0.067
Curvature half-wavelength λκ/2=0.427L  ±0.0422
Lateral body amplitude y0,11=0.065L  ±0.0232

T=tailbeat period, L=body length.

P<0.05 are significant and indicated as such in bold, N=23.

Means are given when no significant linear regression was found.

*

From the videotape of each steady swimming trial, we manually digitized the dorsal midline of the hagfish for a complete tailbeat cycle by overlaying(Digital Vision, Inc. model TelevEyes Pro) a computer-generated x—y coordinate grid (National Institutes of Health version 1.46 on an Apple Corp. model Macintosh G3) onto a digitally paused video image(Sony Model MD 9500 MDR). 20 points along the midline, starting at the tip of the rostrum and ending at the tip of the tail, were selected sequentially at points equidistant from the lateral edges of the body. Each set of 20 digitized points was fitted with a smooth function using a custom compound-spline software package (Jayne and Lauder, 1993). The spline function was used to generate a segmented midline of 31 points (numbered 0-30), 30 segments of equal length,and 29 joints (Fig. 1A,B). From this reconstructed midline we measured the x and y position of each point and each joint's flexion, the angular deviation θ (in rad)of the joint (Fig. 1D) from a straight line (Jayne and Lauder,1993). Because the entire section was bending not as a hinge, as implied by θ, but as a small cantilevered beam, we converted θ to midline curvature κ (in m-1), using the following formula,which was derived from trigonometry assuming uniform curvature along the small length l, of the body segment:
$\ {\kappa}=\frac{2\mathrm{sin}\left(\frac{{\theta}_{0}}{2}\right)}{l}.$
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 curvatureκ 0, and the tailbeat or undulatory frequency ω (rad s-1). Tailbeat frequency is measured as the quotient of 2π (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, γ0,11, lateral amplitude of the tail tip (point 30), γ0,30, and amplitude of the pitch angleα 0 of the body segment between midline points 10 and 11(Fig. 1D). The instantaneous pitch angle α (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 δy—κ, and between the time of maximal heave and pitch δy—a. 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, λκ/2, as the distance (as a proportion of normalized body length L) along the body axis from zero to zero κ, i.e. from inflection point to inflection point;λ κ/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 λκ/2 for each trial is the average of all the instantaneous values of λκ/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 λκ/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:
$\ U_{\mathrm{L}}=c+\mathrm{IND}+{\kappa}_{0}+{\omega}+y_{0,11}+y_{0,30}+{\alpha}_{0}+{\delta}_{\mathrm{y}-{\alpha}}+{\delta}_{\mathrm{y}-{\kappa}}+\frac{{\lambda}_{{\kappa}}}{2},$
3
where c is a constant. In addition, we used univariate linear regressions to test if κ0 and ω were, individually,significant predictors of the remaining kinematic variables.

### 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 κ0 and frequency ω. The strain gauge measured the bending moment M transmitted through the body section. Dynamic M signals produced by the strain gauge (two 120Ω 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 κ, which was determined using a rotary variable differential transducer (Schaevitz model R30D) mounted colinear with the bending couple input linkage; it measured θ that we converted toκ using Equation 2.

We determined EI and C for a given experimental trial(fixed κ and ω) using the equation of motion for a single-degree-of-freedom system (see Den Hartog, 1956):
$\ M_{0}\mathrm{sin}({\omega}t)=EI{\kappa}_{0}\mathrm{sin}({\omega}t-{\delta})+C{\omega}{\kappa}_{0}\mathrm{cos}({\omega}t-{\delta}),$
4
where the term κ0sin(ωt—δ) is the curvature as a function of time t,ωκ 0cos(ωt—δ) is the rate of change of curvature with respect to time, and δ is the phase lag (rad)between the maxima of κ and M. Note that inertia in the particular experimental apparatus is negligible and thus, for measuring EI and C, is ignored(Long, 1998). Because the EI and C terms are 90° out of phase, when the EI term is maximal the C term is zero. With the help of trigonomic identities, Equation 4 simplifies and can be rewritten:
$\ EI=\frac{M_{0}\mathrm{cos}{\delta}}{{\kappa}_{0}}.$
5
Likewise, when EI is zero, C is maximal and thus Equation 4 simplifies and can be rewritten:
$\ C=\frac{M_{0}\mathrm{sin}{\delta}}{{\omega}{\kappa}_{0}}.$
6
For a trial at a given κ0 and ω, 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):
$\ W={\pi}Cl{\omega}{\kappa}_{0}^{2}.$
7
In addition, EI can be divided by I, the area moment of inertia (m4), in order to yield the apparent material stiffness E (in MPa), or Young's modulus. Since hagfish have a nearly circular cross-section (see Fig. 2), I is given as follows (Denny,1988):
$\ I=\frac{{\pi}}{4}r^{4},$
8
where r is the radius of the hagfish's cross-section. The resilience R (% energy return) of the test section was calculated as follows(Wainwright et al., 1976):
$\ R=\frac{100}{e^{{\pi}\mathrm{tan}{\delta}}}.$
9
Finally, to determine if these mechanical properties varied with changes inκ and ω, we varied those parameters over a range (11-40 m-1; π-6πrad s-1, respectively) that included physiological values of midline curvature κ0 and undulatory frequency ω (see Results).
Fig. 2.

Structural treatments 1-4 used in bending experiments. (A) Cross-sectional view of body axis at 0.37 L from rostrum. The body segment (1) was serially dissected (depicted from left to right) with the skin removed (2),then the muscle removed (leaving the notochord) (3), and finally the outer fibrous sheath removed (4). (B) Width of the body. (C) Cross-sectional area of the body. (D) Second moment of area I of the body. Asterisks denote significant differences (P<0.05) between adjacent means as determined using planned constrasts in a repeated-measures ANOVA (N=8 individual hagfish).

Fig. 2.

Structural treatments 1-4 used in bending experiments. (A) Cross-sectional view of body axis at 0.37 L from rostrum. The body segment (1) was serially dissected (depicted from left to right) with the skin removed (2),then the muscle removed (leaving the notochord) (3), and finally the outer fibrous sheath removed (4). (B) Width of the body. (C) Cross-sectional area of the body. (D) Second moment of area I of the body. Asterisks denote significant differences (P<0.05) between adjacent means as determined using planned constrasts in a repeated-measures ANOVA (N=8 individual hagfish).

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 κ by reducing the length, l, of the test section. Since the range of κ values was, thus, higher than that used for structural treatment categories 1-3, we nested the factor κ within the factor structural treatment' TRT, which produced a single nested factor equivalent to the factor κ and theκ 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 κ and TRT factors. This produced a compound nested factor equivalent to the factor IND and the IND by κ by TRT interaction (Zar, 1996). The result was a mixed-model nested analysis of variance (ANOVA) with ω and TRT as main effects:
$\ (El,E,C,W,\mathrm{or}R)={\omega}+\mathrm{TRT}+{\omega}{\times}\mathrm{TRT}+{\kappa}[\mathrm{TRT}]+\mathrm{IND}[\mathrm{TRT},{\kappa}].$
10

To test for differences between adjacent levels of the factors ω and TRT, we ran a priori contrasts. Analysis was conducted in JMP (SAS Institute, Inc., version 3.0).

### Swimming kinematics

In a still-water tank, hagfish swam steadily using undulatory body waves. The length-specific swimming speed UL, was predicted by a single variable, the undulatory frequency ω, which increased linearly with increasing UL(Table 2, Fig. 3). In turn, ω was inversely proportional to the tailbeat amplitude y0,30, and was directly proportional to the phase lag between lateral displacement and curvature δy-κ(Table 3, Fig. 3). The other kinematic variables were independent of UL and ω. In addition,while the amplitude of midline curvature, κ0, was not a significant predictor of UL, it was directly proportional to the amplitude of the pitch angle, α0 andδ y-κ.

Table 2.

Swimming speed ULof hagfish as a function of the midline kinematic variables, determined by multiple regression

Independent variable (d.f.)Sum of squares (s.s.)F ratioP
Individual (7) 0.0791 3.17 0.078
κ0 (1) 0.0101 2.83 0.137
ω (1) 0.0387 10.82 0.013
λκ/2 (1) 0.0012 0.34 0.455
y0,11 (1) 0.0022 0.62 0.455
y0,30 (1) 0.0082 2.30 0.173
α0 (1) 0.0031 0.86 0.384
δy-α (1) 0.0040 1.13 0.324
δy-κ (1) 0.0003 0.11 0.755
Independent variable (d.f.)Sum of squares (s.s.)F ratioP
Individual (7) 0.0791 3.17 0.078
κ0 (1) 0.0101 2.83 0.137
ω (1) 0.0387 10.82 0.013
λκ/2 (1) 0.0012 0.34 0.455
y0,11 (1) 0.0022 0.62 0.455
y0,30 (1) 0.0082 2.30 0.173
α0 (1) 0.0031 0.86 0.384
δy-α (1) 0.0040 1.13 0.324
δy-κ (1) 0.0003 0.11 0.755
Summary of fitANOVAF ratioP
N=23 Model s.s.=0.3985 7.44
r2=0.941, adj. r2=0.814 Error s.s.=0.0250  0.006
Summary of fitANOVAF ratioP
N=23 Model s.s.=0.3985 7.44
r2=0.941, adj. r2=0.814 Error s.s.=0.0250  0.006

P<0.05 is significant and indicated as such in bold, N=23.

In a post-hoc univariate regression, UL=0.3178+0.0419ω (r2=0.460,adj. r2=0.434; P=0.0004).

See List of symbols for definitions of variables and units.

Fig. 3.

Kinematic analysis of swimming. (A) Undulatory frequency ω is the only significant (P<0.05) predictor of swimming speed, UL, in a multivariate regression with individual hagfish treated as a random effect (see Table 2). Also, ω is significantly correlated, as determined by univariate regression (see Table 3), with tailbeat amplitude y0,30 (B) and phase lag between heave and curvature at body point 11, δy-κ (C). (D) The amplitude of the midline curvature at point 11 (0.37 L),κ 0, is not a significant predictor of UL. It is, however, a significant predictor of pitch angle α0 (E)and phase lag, curvature δy-κ (F). Lines are significant regressions (see Tables 2, 3). Symbols in A and D indicate eight different individuals (total N=23).

Fig. 3.

Kinematic analysis of swimming. (A) Undulatory frequency ω is the only significant (P<0.05) predictor of swimming speed, UL, in a multivariate regression with individual hagfish treated as a random effect (see Table 2). Also, ω is significantly correlated, as determined by univariate regression (see Table 3), with tailbeat amplitude y0,30 (B) and phase lag between heave and curvature at body point 11, δy-κ (C). (D) The amplitude of the midline curvature at point 11 (0.37 L),κ 0, is not a significant predictor of UL. It is, however, a significant predictor of pitch angle α0 (E)and phase lag, curvature δy-κ (F). Lines are significant regressions (see Tables 2, 3). Symbols in A and D indicate eight different individuals (total N=23).

### 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 ω, structural treatment TRT, the interaction ofω and TRT, and individual IND, nested within TRT and curvature, κ;κ nested within TRT was not a significant factor for any response variable. Specifically, with changes in ω, flexural stiffness EI, flexural damping C and flexural work W, all varied significantly (Table 4). A priori contrasts between ω categories revealed that EI increased from 2π to 4π and from 4π to 6πrad s-1, C decreased at each level, and W increased from 2π to 4πrad 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 ω 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 ω on core EI, E shows little or no ωeffect with the body intact and the skin removed, and C and W show little or no ω effect with the core of the notochord. Finally, the compound factor of IND nested within TRT and κ 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.

Table 4.

Summary of F-values (Type III sums of squares) from nested ANOVAs performed separately on mechanical variables from bending hagfish bodies

Variableω (3,180)TRT (3,60)ω × TRT (9,180)κ[TRT] (12,180)IND[TRT,κ] (48,180)
EI 34.26 (<0.00016.43 (0.00110.77 (<0.00010.06 (1.00) 31.57 (<0.0001
E 0.19 (0.902) 11.98 (<0.000137.92 (<0.00010.13 (0.999) 42.52 (<0.0001
C 239.70 (<0.000134.03 (<0.000125.30 (<0.00010.29 (0.989) 6.71 (<0.0001
W 8.04 (<0.00013.54 (0.0195.18 (0.00011.76 (0.082) 8.27 (<0.0001
R 2.17 (0.093) 7.74 (0.00021.39 (0.195) 0.37 (0.967) 12.48 (<0.0001
Variableω (3,180)TRT (3,60)ω × TRT (9,180)κ[TRT] (12,180)IND[TRT,κ] (48,180)
EI 34.26 (<0.00016.43 (0.00110.77 (<0.00010.06 (1.00) 31.57 (<0.0001
E 0.19 (0.902) 11.98 (<0.000137.92 (<0.00010.13 (0.999) 42.52 (<0.0001
C 239.70 (<0.000134.03 (<0.000125.30 (<0.00010.29 (0.989) 6.71 (<0.0001
W 8.04 (<0.00013.54 (0.0195.18 (0.00011.76 (0.082) 8.27 (<0.0001
R 2.17 (0.093) 7.74 (0.00021.39 (0.195) 0.37 (0.967) 12.48 (<0.0001

P-values are indicated in parentheses; P<0.05 is significant and indicated as such in bold, N=256.

Degrees of freedom (model, error) are given in parentheses below each factor.

κ[TRT] is κ nested within TRT', and represents the combined effects of κ and the κ×TRT interaction (see Zar, 1996); analogous reasoning is used for the compound nested term, this nested term IND[TRT,κ]. For experimental justification, see Materials and methods.

See List of symbols for definitions of variables and units.

Fig. 4.

Visco-elastic properties of the body axis (at 0.37 L) of hagfish during dynamic bending. The diameter of the plotted circle is proportional to the experimental angular frequency ω at π, 2π, 4π, 6π rad s-1. Statistically significant (P<0.05) overall differences for each property are listed in the far right-hand column (see Table 4); differences between treatments TRT (planned contrasts) are indicated by the labeled double-headed arrows between graphs (see Table 5). As indicated by the label for fish number, the notochord properties (penultimate right-hand column) were measured on different individuals, a condition necessitated by the isolation procedure and that accounted for use of a nested design (indicated as κ[TRT], or curvature nested within treatment', when significance is indicated). Values of circles are means; error bars are ± one standard error from pooling of individuals (N=4).

Fig. 4.

Visco-elastic properties of the body axis (at 0.37 L) of hagfish during dynamic bending. The diameter of the plotted circle is proportional to the experimental angular frequency ω at π, 2π, 4π, 6π rad s-1. Statistically significant (P<0.05) overall differences for each property are listed in the far right-hand column (see Table 4); differences between treatments TRT (planned contrasts) are indicated by the labeled double-headed arrows between graphs (see Table 5). As indicated by the label for fish number, the notochord properties (penultimate right-hand column) were measured on different individuals, a condition necessitated by the isolation procedure and that accounted for use of a nested design (indicated as κ[TRT], or curvature nested within treatment', when significance is indicated). Values of circles are means; error bars are ± one standard error from pooling of individuals (N=4).

Table 5.

Planned a priori contrasts within the factors of angular frequency ω, and structural treatment

Angular frequency ω
Structural treatment
Variableπ versusversusversusWhole versus skin removedSkin off versus notochordNotochord versus core
EI -1.30 (0.196) -3.35 (0.001-4.62 (<0.00010.42 (0.679) 0.73 (0.466) 2.83 (0.007
E -0.09 (0.925) -0.25 (0.804) -0.35 (0.727) -0.07 (0.944) 4.31 (<0.00010.20 (0.839)
C 14.78 (<0.00012.70 (<0.00012.96 (0.0042.35 (0.0221.52 (0.519) 5.82 (<0.0001
W -0.49 (0.622) -3.07 (0.003-2.71 (0.787) 0.41 (0.681) 0.65 (0.519) 1.95 (0.056)
R -0.39 (0.701) 0.50 (0.621) -2.24 (0.0260.22 (0.828) -0.19 (0.850) 3.98 (0.0002
Angular frequency ω
Structural treatment
Variableπ versusversusversusWhole versus skin removedSkin off versus notochordNotochord versus core
EI -1.30 (0.196) -3.35 (0.001-4.62 (<0.00010.42 (0.679) 0.73 (0.466) 2.83 (0.007
E -0.09 (0.925) -0.25 (0.804) -0.35 (0.727) -0.07 (0.944) 4.31 (<0.00010.20 (0.839)
C 14.78 (<0.00012.70 (<0.00012.96 (0.0042.35 (0.0221.52 (0.519) 5.82 (<0.0001
W -0.49 (0.622) -3.07 (0.003-2.71 (0.787) 0.41 (0.681) 0.65 (0.519) 1.95 (0.056)
R -0.39 (0.701) 0.50 (0.621) -2.24 (0.0260.22 (0.828) -0.19 (0.850) 3.98 (0.0002

P-values are indicated in parentheses for each value of t;P<0.05 are significant and indicated as such in bold, N=256.

See List of symbols for definitions of variables and units.

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 ω,κ 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 ω, κ 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 ω, κ 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 ω 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 ω.

Fig. 5.

Coupling of stiffness and damping in the body and notochord of hagfish during dynamic bending. (A) As flexural damping C increases, flexural stiffness EI decreases. Parameterized within the power curves (see Table 6) is driving frequencyω. (B) The maximal bending moments produced by stiffness and damping are directly and linearly proportional (see Table 6 for equations). (C)Experimentally derived values for the ratio of stiffness moments to damping moments change minimally, compared to theoretical ratios assuming constant EI and C (dotted lines), as ω changes (see Table 6). If ω is equal to the resonance frequency, the ratio is a special case of Equation 1, and estimates the maximum possible amplification of internal stresses. Filled squares, the intact' structural treatment category; open squares, the skin removed' category; open circles, the notochord' category; filled circles, thecore' category.

Fig. 5.

Coupling of stiffness and damping in the body and notochord of hagfish during dynamic bending. (A) As flexural damping C increases, flexural stiffness EI decreases. Parameterized within the power curves (see Table 6) is driving frequencyω. (B) The maximal bending moments produced by stiffness and damping are directly and linearly proportional (see Table 6 for equations). (C)Experimentally derived values for the ratio of stiffness moments to damping moments change minimally, compared to theoretical ratios assuming constant EI and C (dotted lines), as ω changes (see Table 6). If ω is equal to the resonance frequency, the ratio is a special case of Equation 1, and estimates the maximum possible amplification of internal stresses. Filled squares, the intact' structural treatment category; open squares, the skin removed' category; open circles, the notochord' category; filled circles, thecore' category.

Table 6.

Regression equations relating stiffness to damping

TreatmentEI versus CEIκ0versus Cκ0ωEI/Cω versus ω, empiricalEI/Cω versus ω, theoretical
Intact EI=357.0C-3.86 EIκ0=6.51 Cκ0ω+0.27 EI/Cω=0.042ω+6.37 EI/Cω=20.8ω-1.0
r2=0.80 r2=0.80; P<0.0001 r2=0.55; P=0.257 r2=1.00
Skin off EI=29.3C-1.77 EIκ0=6.11Cκ0ω+0.40 EI/Cω=-0.049ω+8.29 EI/Cω=25.8ω-1.0
r2=0.56 r2=0.68; P<0.0001 r2=0.96; P=0.020 r2=1.00
Notochord EI=14.18.0C-1.69 EIκ0=6.15Cκ0ω+0.78 EI/Cω=0.007ω+7.13 EI/Cω=22.9ω-1.0
r2=0.59 r2=0.77; P<0.0001 r2=0.07; P=0.729 r2=1.00
Core EI=0.03C-3.03 EIκ0=2.58Cκ0ω+0.39 EI/Cω=0.066ω+3.26 EI/Cω=10.7ω-1.0
r2=0.80 r2=0.11; P=0.207 r2=0.87; P=0.059 r2=1.00
TreatmentEI versus CEIκ0versus Cκ0ωEI/Cω versus ω, empiricalEI/Cω versus ω, theoretical
Intact EI=357.0C-3.86 EIκ0=6.51 Cκ0ω+0.27 EI/Cω=0.042ω+6.37 EI/Cω=20.8ω-1.0
r2=0.80 r2=0.80; P<0.0001 r2=0.55; P=0.257 r2=1.00
Skin off EI=29.3C-1.77 EIκ0=6.11Cκ0ω+0.40 EI/Cω=-0.049ω+8.29 EI/Cω=25.8ω-1.0
r2=0.56 r2=0.68; P<0.0001 r2=0.96; P=0.020 r2=1.00
Notochord EI=14.18.0C-1.69 EIκ0=6.15Cκ0ω+0.78 EI/Cω=0.007ω+7.13 EI/Cω=22.9ω-1.0
r2=0.59 r2=0.77; P<0.0001 r2=0.07; P=0.729 r2=1.00
Core EI=0.03C-3.03 EIκ0=2.58Cκ0ω+0.39 EI/Cω=0.066ω+3.26 EI/Cω=10.7ω-1.0
r2=0.80 r2=0.11; P=0.207 r2=0.87; P=0.059 r2=1.00

P<0.05 is significant and indicated as such in bold, N=16 for each regression.

Note that P-values for regressions using power functions are not normally given.

During dynamic bending that mimics the motion of steady swimming, the visco-elastic properties of the hagfish's intact body are dominated by those of the notochord (Fig. 4),which provides 75% of the body's flexural stiffness EI, and 80% of its flexural damping C. This conclusion is surprising for two reasons. (1) We expected that the passive visco-elastic properties of the lateral musculature would contribute substantially to both EI and C because the musculature is responsible for the majority of the body's cross-sectional area (Fig. 2B). (2) The notochord occupies a position at or near the presumed neutral axis of bending and, hence, should have a disproportionately small contribution to EI, as predicted by the notochord's reduced second moment of area I (Fig. 2C). These expectations are violated(Fig. 5): the notochord compensates for its small I by having, relative to the whole body, a disproportionately large apparent Young's modulus E(Fig. 4).

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.

Fig. 6.

The range of visco-elastic properties of different axial skeletons subjected to sinusodial bending. The notochords of hagfish (N=4) have apparent material stiffnesses (or Young's modulus) E, greater than that of the notochord (N=1) of white sturgeon Acipenser transmontanus (Long,1995), the intervertebral joints (N=6) of blue marlin Makaira nigricans (Long,1992) and the intervertebral joints (N=3) of saddleback dolphin Delphinus delphis (Long et al., 1997). Flexural damping C is normalized per cross-sectional area A of the notochord or intervertebral joint (note units of momentum); the notochord of hagfish has a range of values overlapping the range for sturgeon and marlin. For each species, the range of mean values are given (black bars) from dynamic cantilevered bending tests conducted at physiologically relevant curvatures and driving frequencies. The ranges include regional variation, when measured.

Fig. 6.

The range of visco-elastic properties of different axial skeletons subjected to sinusodial bending. The notochords of hagfish (N=4) have apparent material stiffnesses (or Young's modulus) E, greater than that of the notochord (N=1) of white sturgeon Acipenser transmontanus (Long,1995), the intervertebral joints (N=6) of blue marlin Makaira nigricans (Long,1992) and the intervertebral joints (N=3) of saddleback dolphin Delphinus delphis (Long et al., 1997). Flexural damping C is normalized per cross-sectional area A of the notochord or intervertebral joint (note units of momentum); the notochord of hagfish has a range of values overlapping the range for sturgeon and marlin. For each species, the range of mean values are given (black bars) from dynamic cantilevered bending tests conducted at physiologically relevant curvatures and driving frequencies. The ranges include regional variation, when measured.

### 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, EIand C of the hagfish notochord adjust with changes in ω(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 anω 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 ω 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 Eof 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 ω and κ 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 ωincreases linearly with respect to UL, tailbeat amplitude y0,30 decreases linearly with respect to ω(Fig. 3A,B). This is an unexpected result. From undulatory teleosts, we expected that ω 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 EIof the notochord with increasing ω(Fig. 5). Furthermore, since the squares of ω 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 ω or an increase in propulsive efficiency. Since ω 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.

List of symbols used

• A

cross-sectional area (m2)

•
• C

flexural damping (kg m3 s-1)

•
• E

Young's modulus, apparent material stiffness (MPa)

•
• EI

flexural stiffness (N m2)

•
• G′

storage modulus (MPa)

•
• I

moment of inertia (m4)

•
• Ia,

area moment of inertia (m4)

•
• Im

mass moment of inertia (kg m2)

•
• L

normalized body length (%)

•
• Lb

body length (m)

•
• l

section length (m)

•
• M

bending moment (N m)

•
• M0

amplitude of bending moment (N m)

•
• r

•
• R

resilience (elastic energy return) (%)

•
• t

time (s)

•
• T

normalized tailbeat period

•
• U

swimming speed (m s-1)

•
• UL

length-specific swimming speed (L s-1)

•
• W

flexural work, net for one cycle (J)

•
• x

position along dimension of forward body motion (m)

•
• y

position along dimension of lateral body motion (m)

•
• y0,11

amplitude of lateral displacement at midline point 11 L

•
• y0,30

amplitude of lateral displacement at midline point 30 L

•
• α

•
• α0

•
• θ

•
• θ0

•
• δ

•
• δy-α

phase lag from y0,11 to α0(T)

•
• δy-β

phase lag from y0,11 to β0(T)

•
• κ

midline curvature (m-1)

•
• κ0

amplitude of midline curvature (m-1)

•
• λκ/2

curvature half-wave length along axis (L)

•
• ω

This work was supported by grants from the Office of Naval Research to J.H.L. (ONR no. N00014-997-1-0292) and Shriners of North America to T.J.K.(no. 8610). B.J.S. was supported by a Fellowship from the Undergraduate Research SummerInstitute of Vassar College. The authors thank Matt McHenry,Wyatt Korff, Robert Suter, Ben Johns, and two anonymous reviewers for their constructive comments on the manuscript. We also thank the staff of the Mount Desert Island Biological Laboratory for their patience and support.

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