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Mechanisms of helical swimming: asymmetries in the morphology, movement and mechanics of larvae of the ascidian Distaplia occidentalis

Matthew J. McHenry^*

Department of Integrative Biology, University of California, Berkeley, CA 94720, USA

View larger version (17K): [in a new window]   Fig. 1. Typical shape of the resting larval body. (A) Silhouettes of lateral and dorsal views of a larva, traced from video images. The tail fin is shown in white. The concave-left bend in the tail of this individual can be seen from the dorsal view. (B) The midline measured from the dorsal view of the same individual. Trunk angle () is the angle between the trunk’s midline and the first anterior segment of the tail. Curvature for a tail segment () is equal to the angular flexion, , between the neighboring segments, divided by the length of the segment, s (Thomas and Finney, 1980).

View larger version (46K): [in a new window]   Fig. 2. The symmetry of larval tails and trunks. All measurements are relative to the midline axis and are expressed in terms of body length (see Materials and methods). The lower plots in A and B give the mean values for the surface of the trunk and tail (vertical dashes). Dotted lines connect body positions along the longitudinal axis from the lower image with their corresponding position in the box plot data shown above. The upper plots show the median and quartiles of data on opposite sides of the midline (N=18). (A) In the lower graph, the average shape of the larval body from a dorsal view is traced with vertical tick marks. The thickness of the cellular portion of the tail is visible in the tail region. The distance between the midline and the left margin is statistically indistinguishable from the distance between the midline and the right margin at all antero-posterior positions. (B) In the lower graph, the shape of the average larva as viewed from a lateral perspective is traced with vertical tick marks. In the tail region, the profile of the tail fin is traced. No significant differences were found between the distances of the dorsal (filled columns) and ventral (open columns) margins from the midline at any position along the length of the body.

View larger version (25K): [in a new window]   Fig. 3. Typical changes in tail midline shape during a tail beat illustrated by one individual. On the left are coordinates for the midline taken from a dorsal view and the curves that approximate their shape. The numbers at the end of the curves specify the corresponding time (t) in terms of the tail-beat cycle. The curvature profiles corresponding to these curves are shown on the right. (A,B) t=0.03 tail-beat cycles (shown in black), is within the first half of the tail beat (t<0.5) and tail motion is directed towards the right side of the body. At this moment, the section of the tail anterior to the inflection point (s<si) bends concave-right with a curvature equal to CR, and positions posterior to the inflection point (s>si) are bent concave-left with a curvature of CL. t=0.57 tail-beat cycles (shown in red) is within the second half of the tail beat (t>0.5), and the tail is moving towards the left side of the body. Tail curvature is equal to CL anterior to the inflection point (s<si) and to CR posterior to the inflection point (s>si). (C,D) Tail motion shown at 4ms intervals (roughly 0.03 tail-beat cycles) for the first half of a tail beat (t<0.5), when the tail is moving towards the right side of the body. Note that the magnitude of curvature in both directions (CL and CR) changes with time. (E,F) The motion during the left-directed, second half of the tail beat (t>0.5) shown at 4ms intervals. Again, curvature in both directions varies with time.

View larger version (31K): [in a new window]   Fig. 4. Kinematic variables as a function of time. Body movement for 1.2 tail beats is shown for one individual moving with the typical pattern observed during steady swimming. Thick lines show trends found by least-squares curve-fitting and thin lines denote 95% confidence intervals (calculated by Matlab version 5.2, Mathworks). See Results for the functions fitted to these data. Negative values for curvature are directed concave-left and positive values are concave-right. (A) Graphs of curvature as a function of body position at five different instants separated by intervals of 0.2 tail-beat cycles. The values at each instant for concave-right curvature (CR), concave-left curvature (CL) and the position of the inflection point (si) define the shape of this step function. (B,C) Variation in CR (B) and CL (C). Note that the amplitude of changes in concave-right curvature (CR) is smaller than the amplitude of concave-left curvature changes (CL). Oscillations in CR are of opposite sign and are half a tail beat out of phase with the changes in CR. (D) Variation in the inflection point in curvature (si) with time demonstrates a linear propagation of the inflection point down the length of the tail. (E) The angle between the trunk’s midline and the first anterior tail segment (see Fig.1) with time in phase with CR.

View larger version (16K): [in a new window]   Fig. 5. Possible sources of kinematic asymmetry. (A) Differences between concave-left (CL) and concave-right (CR) curvature amplitude, (B) differences in wave speed () between left-directed and right-directed half tail beats, and (C) non-zero baseline trunk angles (ß), N=11. All were hypothesized to generate kinematic asymmetries (see Materials and methods), but asymmetries were measured only in the differences in curvature amplitude (A) and the non-zero values for baseline trunk angle (C). As in Fig.2, the middle line in each box plot represents the median, the top and bottom edges of the box show the first quartile, error bars denote the second quartile and plus signs indicate values outside the second quartile.

View larger version (18K): [in a new window]   Fig. 6. Resting tail curvature compared with tail curvature during swimming. is the mean tail curvature over the length of the tail. (A) Silhouette of a resting individual from a dorsal view with a mean tail curvature () of 0.47radmm-1. (B) A midline for the individual shown in A is drawn with a tail curvature equal to the median value for tail curvature measured during swimming (=0.51radmm-1). The range of tail excursion during swimming is shown in gray. (C) Comparison of the tail curvatures of resting larvae with the median tail curvatures of an independent sample of swimming larvae shows the statistically indistinguishable differences in curvature between samples (N=11). The middle line in the box plot represents the median, the top and bottom edges of the box show the first quartile and error bars denote the second quartile.

View larger version (17K): [in a new window]   Fig. 7. Flexural stiffness EI of the tail in the two lateral directions compared. (A) Representative data for one individual showing the linear changes in tail deflection that result when forces between 0.1x10-7 and 4.5x10-7N are applied laterally. The calculations for EI (see Materials and methods) derived from the least-squares slopes suggest that bending stiffness is greater in the concave-right direction than in the concave-left direction for this individual. (B) Summary results for flexural stiffness in both directions for all individuals tested (N=13). Stiffness measured when bending in the concave-left direction was not significantly different from bending in the concave-right direction. The middle line in the box plot represents the median, the top and bottom edges of the box show the first quartile, error bars denote the second quartile and plus signs indicate values outside the second quartile.

View larger version (26K): [in a new window]   Fig. 8. Theoretical predictions for the turning moments generated by hydrodynamic forces. (A) The midline of a larval body shown from a dorsal view. At this instant, the tail is beating towards the right of the body, so the normal component of the tail’s velocity (Vn) points in that direction. Neither the normal nor the tangential components (Vt) are drawn to scale. The inset detail of the hydrodynamic forces acting on a tail segment shows the acceleration reaction force acting in opposition to the quasi-steady normal force. The total force (shown in red) is the vector sum of these forces and the quasi-steady tangential force. (B,C) Representative forces and moments predicted for a single tail beat. Red arrows represent the total hydrodynamic force on individual tail segments; gray arrows show the direction of tail motion. For clarity, only the forces for the odd-numbered segments are drawn (segment length 0.11mm). (B) In the first half of the tail beat, the tail moves to the right and experiences fluid forces opposing its motion. These forces generate moments in the clockwise direction with a time-averaged mean magnitude of -1.7x10-8Nm. (C) In the second half of the tail beat, the tail moves leftwards, and forces opposing this motion generate a mean counterclockwise moment of 2.3x10-8Nm. The time-averaged moment predicted for the entire tail beat is 0.3x10-8Nm, which will act to rotate the body in the counterclockwise direction. (D) The distribution of time-averaged moments generated by larvae (N=11) shows that all but one larva generated a counterclockwise-directed moment during swimming. The middle line in the box plot represents the median, the top and bottom edges of the box show the first quartile, error bars denote the second quartile and plus signs indicate values outside the second quartile.

View larger version (22K): [in a new window]   Fig. 9. The dynamics of swimming in ascidian larvae. A free-swimming larva is drawn from a dorsal and posterior perspective for an instant of time when the tail is beating towards the left. Some of the forces likely to influence the swimming mechanics are drawn to illustrate how moments acting on the center of mass could be generated in one (A) or two (B) dimensions. (A) Only hydrodynamic forces acting on the frontal plane of the body are drawn. As this larva swims forward, a yawing moment in the counterclockwise direction will tend to rotate the body in a counterclockwise direction around an axis perpendicular to the frontal plane. If only these hydrodynamic forces acted on the body for the entire tail beat, this larva would follow a circular trajectory lying on a plane coincident with the frontal plane of the body. (B) In addition to hydrodynamic forces, this larva has a buoyancy force and the weight of the body acting on it. The buoyancy force acting at the body’s center of volume is posterior to the center of mass and therefore generates a pitching moment. With both pitching and yawing moments, the body would tend to rotate around an axis that is not perpendicular to the frontal plane. As a result, the larva would swim along a helical trajectory.