First published online June 6, 2005
Journal of Experimental Biology 208, 2289-2301 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01642
The mechanical scaling of coasting in zebrafish (Danio rerio)
Matthew J. McHenry1,* and
George V. Lauder2
1 Department of Ecology and Evolutionary Biology, University of California,
321 Steinhaus Hall, Irvine, CA 92697, USA
2 Department of Organismic and Evolutionary Biology, Harvard University, 26
Oxford Street, Cambridge, MA 02138, USA
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Fig. 1. Experimental setup. (A) The routine swimming of zebrafish was recorded from
a dorsal perspective with high-speed video. High-contrast video images were
generated by illuminating the translucent floor of the aquarium with a
fluorescent light, which was prevented from heating the floor of the tank by
the use of a computer fan. (B) Drag was measured by tethering dead fish from
the ventral surface and exposing the body to flow (see text for details).
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Fig. 3. The duration of the coasting phase. (A) Mean coasting duration (filled
circles) and distribution (± 1 S.D.) among three coasts for
individuals of different body length (N=21). The horizontal solid
line shows the mean for all individuals, and the broken lines denote the
distribution (± 1 S.D.) about the mean. (B) The distribution
of coasting durations for 70 coasts for fish of all body lengths
(N=21).
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Fig. 4. The scaling of coasting performance in zebrafish. (A) Video frames of a
larva (orange border), juvenile (purple border) and adult (green border)
illustrate the difference in size and shape of three stages. Scale bars, 2 mm.
(B) The position function (Eqn 4;
coloured lines) fit to measurements of body position (filled circles;
down-sampled for clarity) provided measurements of the time constant,  ,
and initial speed, U0. Glide distance, d, was
measured at the mean glide duration for all fish (broken grey line). (C)
Changes in speed are drawn according to measured parameter values entered into
the speed equation for each fish (Eqn
3). (D-F) The mean (filled circles) and distribution (± 1
S.D.) of kinematic parameters plotted against the body length
(L) for each fish. Orange, purple and green points correspond to the
same individuals shown in panels A-C.
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Fig. 5. Drag coefficient measurements and hydrodynamic regimes for adult zebrafish.
Black circles denote dead drag measurements from tethering experiments
(N=6; nine Re values per individual) and red circles are
in vivo drag measurements (N=4; three coasts per
individual). (A) The broken line lies at the mean value for the inertial drag
coefficient (Cinert) at Re>1000. (B) The
broken line denotes the mean value of the viscous drag coefficient
(Cvisc) at Re<300. (C) The viscous regime
occurs where Cvisc remains relatively constant, and the
inertial regime is defined as the range where Cinert does
not vary with Re.
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Fig. 6. Measurement of in vivo drag in a coasting fish at
Remax=1050. The inertial drag coefficient for this fish
(L=25 mm) was calculated from measurements of body mass and surface
area and (A) the inverse of instantaneous speed measurements during coasting
(filled circles). The drag coefficient (Cinert=0.05) was
found from a least-squares linear fit (red line) to the data, which (B)
provided a prediction (red line) for the measured changes in the cranial
position (filled circles).
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Fig. 7. Reynolds numbers (Re) spanned during coasting by zebrafish of
different size. (A) Dorsal and lateral silhouettes illustrate the shape of the
body (grey) and fins (black) at different stages of growth. (B) The range of
Re traversed during coasting in fish of different size. The upper and
lower edges of vertical bars and error flags denote, respectively, the mean
and 1 S.D. of maximum and minimum values (as shown in key) for
three coasts per fish (N=22). Stages of growth are colour coded for
small larvae (orange), large larvae (blue), juveniles (purple) and adults
(green). The hydrodynamic regimes (broken lines) from drag measurements
(Fig. 5) are shown to the
right.
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Fig. 8. The influence of momentum and hydrodynamics on coasting performance. Coasts
were simulated using the mean parameter values for four ontogenetic stages.
Simulations used the momentum, p, of adults
(p=1.04x10-4 kg m s-1; purple), juveniles
(p=8.05x10-7 kg m s-1; green), large
larvae (p=8.28x10-8 kg m s-1; blue) and
small larvae (p=7.01x10-9 kg m s-1;
orange). Model coasting that included (solid line) and excluded (broken line)
the added mass is shown. (A-C) Curves show the predicted change in body
position as a function of time for coasting simulations in the viscous regime
(Eqn 4), with the body lengths of
(A) small larvae (L=4.35 mm), (B) large larvae (L=8.19 mm)
and (C) juveniles (L=13.81 mm) shown. (D) Simulations of coasting in
the inertial regime (Eqn 8) with
the wetted surface area of adults (S=660 mm2) were run at
each level of momentum.
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Fig. 9. The effects of mechanical parameter values on coasting distance in
juveniles and adults. Coasting distances were calculated (Eqns
4,
8) for three glide durations in
five simulations having unique parameter combinations. Simulations were run
for each of the three measured juvenile and adult parameter values. The length
of bars shows the mean (± 1 S.D.) of these predictions.
(A,B) Predictions were generated from the position function for the viscous
regime (Eqn 4) using body length
as the size parameter. (A) The values for juvenile fish were used for all four
parameters in this simulation. (B) Model fish were scaled to the length of
adults in these simulations. (C,D) Predictions of coasting distance were
calculated from the position function for the inertial regime
(Eqn 8) using wetted surface area
as the size parameter. (C) The effect of hydrodynamic regime alone may be
assessed by comparison with B. (D) The model fish in these simulations were
given the mass of adults. (E) All parameters were set at the measured adult
values in these simulations.
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Fig. 10. Schematic diagram of the major changes in coasting dynamics over ontogeny.
Larval (A) and juvenile (B) stages operate in the viscous regime (symbolized
by a blue flow gradient), whereas the adult stage (C) operates in an in
inertial regime (denoted by the blue streamlines). Despite this change in
hydrodynamics, it is primarily the body mass (orange circle) and initial speed
(grey arrow) that cause the observed differences in coasting performance
(Fig. 4).
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© The Company of Biologists Ltd 2005
)
and angular fin position (
) are illustrated in the first frame. (B) Fin
position (