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The notochord of hagfish Myxine glutinosa: visco-elastic properties and mechanical functions during steady swimming

John H. Long, Jr^1,2,*, Magdalena Koob-Emunds^1,3, Benjamin Sinwell^1,2 and Thomas J. Koob^1,3

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

View larger version (31K): [in a new window]   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, /2 (L), is the axial distance along the midline from the points of 0 to 0 midline curvature, (m-1). (D) Other kinematic variables include amplitude of the midline curvature, 0, amplitude of the lateral displacement of body points 11, 0,11, and 30, 0,30, amplitude of the pitch angle, 0. The phase lags y— and y—, are the differences in time, as a fraction of the normalized tailbeat period (T), between maximal and maximal and , respectively.

View larger version (25K): [in a new window]   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).

View larger version (27K): [in a new window]   Fig. 3. Kinematic analysis of swimming. (A) Undulatory frequency 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, 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, y- (C). (D) The amplitude of the midline curvature at point 11 (0.37 L), 0, is not a significant predictor of UL. It is, however, a significant predictor of pitch angle 0 (E) and phase lag, curvature y- (F). Lines are significant regressions (see Tables 2, 3). Symbols in A and D indicate eight different individuals (total N=23).

View larger version (36K): [in a new window]   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 at , 2, 4, 6 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 [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).

View larger version (23K): [in a new window]   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 frequency . (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 changes (see Table 6). If 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, the `core' category.

View larger version (10K): [in a new window]   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.