## SUMMARY

Scaling effects on the kinematics of suction feeding in fish remain poorly
understood, at least partly because of the inconsistency of the results of the
existing experimental studies. Suction feeding is mechanically distinct from
most other type of movements in that negative pressure inside the buccal
cavity is thought to be the most important speed-limiting factor during
suction. However, how buccal pressure changes with size and how this
influences the speed of buccal expansion is unknown. In this paper, the
effects of changes in body size on kinematics of suction feeding are studied
in the catfish *Clarias gariepinus*. Video recordings of prey-capturing
*C. gariepinus* ranging in total length from 111 to 923 mm were made,
from which maximal displacements, velocities and accelerations of several
elements of the cranial system were determined. By modelling the observed
expanding head of *C. gariepinus* as a series of expanding hollow
elliptical cylinders, buccal pressure and power requirement for the expansive
phase of prey capture were calculated for an ontogenetic sequence of catfish.
We found that angular velocities decrease approximately proportional with
increasing cranial size, while linear velocities remain more or less constant.
Although a decreasing (angular) speed of buccal expansion with increasing size
could be predicted (based on calculations of power requirement and the
expected mass-proportional scaling of available muscular power in *C.
gariepinus*), the observed drop in (angular) speed during growth exceeds
these predictions. The calculated muscle-mass-specific power output decreases
significantly with size, suggesting a relatively lower suction effort in the
larger catfish compared with the smaller catfish.

## Introduction

Changes in body size have important consequences for the mechanics of musculo-skeletal systems (Hill, 1950; Schmidt-Nielsen, 1984). One of these consequences is that larger animals need more time to carry out the same movement (e.g. downstroke of the wings during flight or leg extension during jumping) compared with smaller animals (Askew et al., 2001; Bullen and McKenzie, 2002; Schilder and Marden, 2004; Toro et al., 2003). Because of the importance of scaling relationships for the ecology, behaviour, performance and evolution of animals (e.g. Carrier et al., 2001; Walter and Carrier, 2002; Hutchinson and Garcia, 2002; Davenport, 2003), theoretical models have been proposed (e.g. Hill, 1950; Richard and Wainwright, 1995). These models provide quantitative predictions of scaling of kinematics in geometrically similar animals.

Although most scaling studies have addressed animal locomotion, several
experimental studies have focussed on scaling of prey capture kinematics in
aquatic suction feeding vertebrates
(Richard and Wainwright, 1995;
Reilly, 1995;
Cook, 1996;
Hernandez, 2000;
Wainwright and Shaw, 1999;
Robinson and Motta, 2002).
Surprisingly, the results of these studies are largely inconsistent. For
example, the time to open the mouth increases with body length by
*L*^{0.592} in *Micropterus salmoides*
(Wainwright and Shaw, 1999),
*L*^{0.333} in *Gynglymostoma cirratum*
(Robinson and Motta, 2002) and
*L*^{0.314} in *Danio rerio*
(Hernandez, 2000) and is
independent of body size in *Salamandra salamandra*
(Reilly, 1995). Furthermore,
none of the existing geometrical-similarity models are able to explain the
observed results in most cases. Based on the intrinsic dynamics and energetics
of contracting muscle, the model of Hill
(1950) predicts that similar
movements (e.g. a limb rotating a certain angle) should be carried out in
times directly proportional to the linear dimensions
(∼*L*^{1}). On the other hand, the model of Richard and
Wainwright (1995) predicts
that durations of kinematic events are independent of body size
(∼*L*^{0}) by assuming that the shortening velocity of a
muscle is directly proportional to muscle length (or the number of sarcomeres
in series). Consequently, the influence of size on the speed of movements of
the feeding system during suction remains a poorly understood phenomenon.

The maximal speed of a given movement is determined by the equilibrium of forces in the equation of motion for this specific movement. Most of these forces will be subject to size effects. The magnitude of drag force, for example, depends on the surface area of the structures moving through a given fluid and will therefore increase with size. How these external forces scale with animal size, and how this balances with the available muscular power, energy or stress-resistance of bones, will often determine the performance of a given movement (e.g. Hill, 1950; Wakeling et al., 1999).

During the expansive phase of suction feeding, a negative (sub-ambient) pressure is created inside the buccal cavity (see, for example, Van Leeuwen and Muller, 1983; Lauder, 1985). As estimated by Aerts et al. (1987), the force exerted by this negative buccal pressure on the expanding lower jaw is the most important factor to be overcome by the contraction of the mouth-opening muscles. Additionally, a recent study has demonstrated a correlation between the force available from the epaxial musculature and buccal pressure magnitudes in centrarchid fishes (Carroll et al., 2004). This also suggests that intra-buccal pressure is the most constraining factor for the maximal performance of the cranial system of suction feeders. So, if the maximal speed of movements by the cranial musculo-skeletal system during suction is mainly limited by buccal pressure, then scaling effects are probably the results of the size dependency of buccal pressure.

In their theoretical model of suction feeding in fish, Muller et al.
(1982) have shown that peak
sub-ambient buccal pressure increases drastically (approximately∼
*L*^{4.5}) if an expanding fish head (modelled as an
expanding cone) is artificially lengthened without changing the speed of
expansion. Therefore, due to scaling effects on buccal pressure, suction
feeders may be forced to decrease the speed of buccal expansion when they
become larger (resulting in pressure magnitudes that their cranial muscles are
capable of generating). Moreover, the existing scaling models
(Hill, 1950;
Richard and Wainwright, 1995)
may not be able to explain scaling of suction-feeding kinematics because the
more important effects of changes in pressure with size are not taken into
account in these models.

In the present paper, the scaling of prey capture kinematics in the African
catfish *Clarias gariepinus* is investigated. Next, the relationships
between speed of buccal expansion, cranial size and muscular power requirement
are explored by hydrodynamic suction modelling combined with inverse dynamic
modelling in *C. gariepinus*. Finally, we evaluate the hypothesis that
scaling of suction-feeding kinematics may be determined by the size effects
imposed on buccal pressure magnitudes.

## Materials and methods

### Study animals

*Clarias gariepinus* (Burchell 1822) is an air-breathing catfish
(Fam. Clariidae) with an almost Pan-African distribution that is also found in
rivers and lakes of the Middle East and Turkey
(Teugels, 1996). It has a
broad diet that includes mostly fish, shrimps, crabs, insect nymphs, beetles
and snails (Bruton, 1979).
While this species shows different kinds of foraging behaviours, including
bottom feeding, surface feeding or group hunting, prey are generally captured
by a combination of suction feeding and biting
(Bruton, 1979;
Van Wassenbergh et al., 2004).
Juvenile *C. gariepinus* specimens already have a fully ossified
cranial system that appears to be generally similar in shape to the adult
configuration at the ontogenetic stage of 127 mm standard length
(Adriaens and Verraes, 1998).
Adults can grow up to 1.5 m total length
(Teugels, 1986), making this
species particularly suitable for studying scaling effects.

In the present study, we used 17 individuals between 110.8 and 923.0 mm in total length. As the cranial length (defined as the distance between the rostal tip of the premaxilla and the caudal tip of the occipital process) can be measured more precisely and excludes variability in the length of body and tail, we use this metric to quantify size. The individuals used were either aquarium-raised specimens obtained from the Laboratory for Ecology and Aquaculture (Catholic University of Leuven, Belgium) or specimens obtained from aquacultural facilities (Fleuren & Nooijen BV, Someren, The Netherlands). However, catfish from both origins did not show different growth patterns of the feeding apparatus (see Herrel et al., 2005). All animals were kept in a separate aquarium during the course of the training and recording period. In general, it took about two weeks to train the catfish to feed in a restricted part of the aquarium.

### Video recordings of prey captures

Video sequences were recorded of *C. gariepinus* capturing pieces of
cod (*Gadus morhua*) that were pinned onto a plastic coated steel wire
(Fig. 1). In order to obtain a
similar feeding situation for both the small and large individuals, the size
of the prey was scaled according to the size of the catfish (diameter between
25% and 35% of cranial length). The recordings were made using a Redlake
Imaging Motionscope digital high-speed video camera at 250 frames
s^{-1} (for individuals with cranial lengths between 28.01 and 71.00
mm), a JVC GR-DVL9800 camera (PAL recording system) at 100 frames
s^{-1} (for individuals with cranial lengths of 94.13-130.0 mm) or a
Panasonic F15 at 50 frames s^{-1} (for the 210.2 mm cranial length
individual). The feeding sequences were recorded simultaneously in lateral and
ventral view, using a mirror placed at 45°. Two floodlights (600 W)
provided the necessary illumination. Only those prey capture sequences that
were approximately perpendicular to the camera lens were selected and retained
for further analysis.

### Kinematic analysis

Ten recordings were analysed for each individual. Specific anatomical
landmarks were digitised using Didge (version 2.2.0, Alistair Cullum,
Creighton University, Omaha, NE, USA), from which kinematic variables
describing the position of the lower jaw, hyoid, branchiostegal membrane and
neurocranium were calculated (see Fig.
2). From kinematic plots, timings of kinematic events (maximum and
end of the analysed head part movements) were determined, with time 0 being
the start of lower jaw depression. After data filtering (4th order Butterworth
zero phase-shift low-pass filter) and differentiation *versus* time,
velocities and accelerations were calculated. As we are mainly interested in
maximal performance, the maximal values per individual (i.e. largest
excursions, highest peak velocities and accelerations) were used in the
regression analysis. Only for the timing variables were the averages from the
10 analysed sequences for each individual used in the regressions to enable
comparison with previous scaling studies
(Richard and Wainwight, 1995;
Robinson and Motta, 2002).

As growth is an exponential phenomenon, all data were
log_{10}-transformed values (one data point for each individual) and
were plotted against the log_{10} of cranial length. Next, least
squares linear regressions were performed on these data. As the kinematic
variables or the model output (see below; dependent data) probably have a much
greater error than measurements of cranial length (independent data), least
squares regressions are appropriate in this case
(Sokal and Rohlf, 1995). The
slopes of these linear regressions (with 95% confidence limits) were
determined in order to evaluate changes in prey capture kinematics in relation
to changes in body size. A slope of 0 indicates that the variable is
independent of cranial length. Slopes of 1 and -1 denote that the variables
increase or decrease, respectively, proportional to cranial length, while
slopes different from these values stand for a variable changing more than
proportionally with cranial length.

Each regression was tested for statistical significance by an analysis of
variance (ANOVA). *t*-tests were performed to compare the observed
regression slopes against expected values
(Zar, 1996). The basic
assumptions of normality and linearity were met in the presented data. The
significance level of *P*=0.05 was used throughout the analysis.

### Suction modelling and calculation of buccal pressure

First, spatio-temporal patterns of water velocities inside the mouth cavity
were calculated with the ellipse model of Drost and van den Boogaart
(1986). This model has been
shown to give accurate predictions of flow velocities in suction-feeding
larval carp (*Cyprinus carpio*)
(Drost and van Den Boogaart,
1986) and in the snake-necked turtle (*Chelodina
longicollis*) (Aerts et al.,
2001). This model also gives good predictions of the actual flow
velocity for suction feeding of *C. gariepinus*. Results of a
high-speed X-ray video analysis of *C. gariepinus* capturing small,
spherical pieces of shrimp (6 mm diameter) charged with a small metal marker
(0.5 mm diameter) show maximal prey velocities of 1.2 m s^{-1}. After
applying the suction model (see below for details) to the same individual, the
two analysed sequences gave maximal flow velocities of 1.13 and 1.60 m
s^{-1} (Van Wassenbergh et al., in
press). Assuming that small prey behave approximately as a part of
the fluid, these findings suggest that the model output is also realistic for
*C. gariepinus*.

In our suction model, the head of the catfish, from mouth aperture to
pectoral fin, is approximated by a series of hollow elliptical cylinders
(Fig. 3). Each cross-section of
this structure consists of an external ellipse and an internal ellipse. The
length of the major and minor axis of the external ellipses correspond,
respectively, to the height and width of the head at any given position, while
the internal ellipse axes approximate the width and height of the
bucco-pharyngeal cavity (further referred to as buccal cavity or mouth
cavity). Changes in the length of the external ellipse axes were deduced from
the recorded videos. To do so, upper and lower contours of the catfish's head
were digitised frame by frame (50 points each) in the lateral and ventral
view. At the same time, the coordinates of a longitudinal axis connecting the
upper jaw tip to a point equidistant between the base of the right and left
pectoral fin were digitised (Fig.
2). Next, the contour coordinates were recalculated in a new frame
of reference moving with the fish, with the upper jaw tip as origin and the
longitudinal axis above as the *X*-axis. The coordinates of each
contour curve (upper, lower, left and right contours of the head) were then
fitted with 10^{th}-order polynomial functions, using the XlXtrFun
add-in for Microsoft Excel (Advanced System Design and Development, Red Lion,
PA, USA). Next, the distance between the corresponding coordinates of the
upper and lower contours, and between the left and right contours, were
calculated at 201 equally spaced intervals along the longitudinal axis. For
each external ellipse, the profiles of length and width *versus* time
were filtered with a 4^{th}-order Butterworth zero phase-shift
low-pass filter in order to reduce digitisation noise (cut-off frequency of
30, 12 and 6 Hz for videos recorded at 250 Hz, 100 Hz and 50 Hz, respectively;
see above).

The internal dimensions of the mouth cavity of *C. gariepinus* at
rest are approximated using X-ray images from lateral and ventral view X-ray
videos of a preserved specimen with closed mouth (94.13 mm cranial length;
302.0 mm total length). During recording of these X-ray videos, the specimen
was held vertically while a saturated barium solution was poured in the mouth
(for illustration, see Van Wassenbergh et
al., in press). Using this radio-opaque fluid, the boundaries of
the mouth cavity could accurately be distinguished, and the internal area of
the mouth cavity determined, for all positions along the longitudinal axis. To
account for the presence of the gill apparatus, the length of the major and
minor axes of ellipses in the gill region were (arbitrarily) reduced by 10%
(see Van Wassenbergh et al., in
press). It was assumed that this situation (i.e. the internal
volume of the mouth cavity of the preserved specimen at rest) reflects the
moment prior to the start of the suction event. Subsequently, changes in the
height and the width of the head over time (external ellipses) will cause
changes in the width and height of the internal mouth volume ellipses. As
internal volume data were collected for one individual only, we are forced to
assume that the dimensions of the buccal cavity are proportional to the
measured external dimensions of the head in *C. gariepinus*. This means
that allometry in the external dimensions of the head is assumed to be
reflected in a similar allometry of the bucco-pharyngeal cavity. The X-ray
videos were made with a Philips Optimus X-ray generator coupled to a Redlake
Imaging MotionPro digital high-speed camera.

According to the continuity principle, any change in volume must be filled
instantaneously with water and thus generate a flow relative to the fish's
head. So, at each cross-section of the mouth cavity, the total water volume
passing through this cross-section in a given amount of time depends on the
total volume increase posterior to this cross-section. In this way, the mean
flow velocity during a given time increment can be calculated at each of the
modelled ellipse-shaped cross-sections of the mouth cavity by dividing the
volume increase posterior to this ellipse by the area of the ellipse (average
for that time increment). This holds as long as the opercular and
branchiostegal valves are closed. If not, the modelled system becomes
undetermined (Muller et al.,
1982; Muller and Osse,
1984; Drost and van den
Boogaart, 1986). In general, branchiostegal valve opening can be
detected shortly after *C. gariepinus* reaches maximal oral gape.
However, for several of the recorded prey capture sequences, it was
problematic to pinpoint precisely the frames in which the transition from
closed to opened valves occurred. Therefore, we only used the model output
from the start of mouth opening until the time of maximal gape.

Next, pressure inside the expanding modelled profile with closed valves was calculated according to Muller et al. (1982): (1)

where Δ*p* is the pressure (difference from hydrostatic
pressure), ρ is the density of water (1000 kg m^{-3}), *x*
is the position along the longitudinal axis (*x*=0 at the pectoral fin
base; *x*=l at the mouth aperture), *u* is the instantaneous
flow velocity of water in x direction, *u*_{l} is the
instantaneous flow velocity of water in the mouth aperture in a frame of
reference moving forward with the head (see
Fig. 2), *a*_{r}
is the ratio between *u*_{m} (instantaneous flow velocity of
water in the mouth aperture in the earth-bound frame) and
(*u*_{m}-*u*_{l}), and *h*_{l} is
the instantaneous radius of the mouth aperture (assumed to correspond to the
average between the half width and the half height of the ellipse at this
position). The prime sign denotes that the first derivate against time is
taken for this function. A representative example of pressures calculated for
*C. gariepinus* is shown in Fig.
4.

### Inverse dynamic calculation of required muscular power

The power required for expanding the series of hollow elliptical cylinders
was calculated by inverse dynamics. First, for each hollow cylinder, a new
frame of reference is defined with the top of the external ellipse
cross-section as the origin, *X* as the width, *Y* as the height
and *Z* as the depth (Fig.
3). By doing so, the mid-sagittal part of the skull is entirely
motionless in the model, while movement during expansion of the elliptical
cylinders will predominantly occur in the ventral and lateral parts of the
head. The external ellipses and internal ellipses are now respectively given
by the following equations:
(2)
(3)

where *h*_{ext} and *w*_{ext} are,
respectively, the half height and half width of the external ellipse,
*h*_{int} and *w*_{int} are, respectively, the
half height and half width of the internal ellipse, and *c* is the
distance between the centres of the external and internal ellipses. *c*
was determined from the lateral-view X-ray picture of a 94.13 mm cranial
length *C. gariepinus* specimen with a barium-filled buccal cavity and
will logically be constant throughout the expansion. The distance from the
internal to the external ellipse following the horizontal axis of the internal
ellipse (*r*_{hor}) can be calculated by:
(4)

Next, the volume between the internal cylinders and the external cylinders
was divided into 40 segments (Fig.
3). To do so, at 20 equally spaced intervals along the horizontal
axis of the internal ellipses (*X-*coordinates from
-*w*_{int} to +*w*_{int}) the corresponding
*Y*-coordinates were calculated using equation 3. The same was done for
the external ellipses (*X-*coordinates from -*w*_{ext}
to +*w*_{ext}) using equation 2. The area between the external
and internal ellipse at a given cross-section is now approximated by a series
of quadrangles connecting every two adjacent points on the external ellipse
with the two corresponding points on the internal ellipse (see
Fig. 3). After defining depth,
*d*, as the distance between the cross-sections used in the model, we
can calculate the volume, the surface area bordering the buccal cavity
(*A*_{int}) and the surface area bordering the outside of the
fish's head (*A*_{ext}) for each segment
(Fig. 3). We assumed that the
mass of this volume has the uniform density of 1000 kg m^{-3}, which
also implies neutral buoyancy. The *xyz*-coordinates of the centre of
mass (COM) of any given segment were approximated by taking the average *x,
y* and *z* of the eight corners of the segment.

Thirdly, the linear speed (*v*), acceleration (*a*) and
direction of motion of the COM, as well as the inclination of
*A*_{ext} and *A*_{int}, were calculated. As the
expansion is symmetrical for the left and right sides of the head, these and
the following calculations were performed for a single side. Consequently,
final output values (required muscular force; see below) were doubled.
Additionally, as the dorsal part of the head (almost static in the model) is
taken in by the bony neurocranium, no calculations were performed for this
region (see colour scheme in Fig.
3).

To calculate the force required from muscular activity,
*F*_{muscle}, the following equation of motion was used:
(5)

where *m* is the mass, *a* is the linear acceleration,
*c*_{added} is the added mass coefficient,
*F*_{pressure} is the force resulting from pressure differences
between the inside and the outside of the buccal cavity and
*F*_{drag} is the force from hydrodynamic drag. Note that
forces resulting from friction between head parts and from deformation of
tissues (e.g. stretching of the jaw adductor muscles during mouth opening) are
not included. As these forces will probably become important only at the end
of cranial expansion, it is assumed that these forces are small compared with
the other forces during expansion. The orientation of
*F*_{pressure} is perpendicular to the average inclination of
the planes bordering the buccal cavity (*A*_{int}; see
Fig. 3) and the outside of the
head (*A*_{ext}; see Fig.
3). Its magnitude depends on Δ*p* (at a given time
and position along the medio-sagittal axis) and surface area. The average
surface area of *A*_{int} and *A*_{ext} was used
for calculating the magnitude of *F*_{pressure}. We used the
value of 1 for *c*_{added} (added mass coefficient for
cylinders according to Daniel,
1984), although the importance of changes in this value on the
model output will be discussed (see Discussion). *F*_{drag} is
parallel to the direction of motion, and its magnitude was calculated by:
(6)

where *c*_{d} is the drag coefficient, ρ is the density
of water (1000 kg m^{-3}), *A*_{ext-p} is the area of
the external surface of the bar projected onto a plane perpendicular to the
direction of motion, and *v* is the velocity of the COM. A value of 1
was used for the shape-dependent *c*_{d}, corresponding
approximately to drag on an infinite flat plate
(Hughes and Brighton, 1999).
By using the projected area *A*_{ext-p},
*F*_{drag} decreases sinusoidal with the angle of attack of the
surface moving through the water, which is in accordance with experimental
measurements (Munshi et al.,
1999; Bixler and Riewald,
2002).

Finally, total power required from cranial and post-cranial muscles
(*P*_{req}**)** was calculated by:
(7)

Required power will be expressed as a muscle-mass-specific power by
dividing *P*_{req} (calculated as above) by the mass of a
subset of the muscles contributing to the expansions of the cranial system.
From all individual catfish used in the present study, the single-sided mass
of the protractor hyoidei, sternohyoideus, levator arcus palatini, levator
operculi and the part of the hypaxials that is anterior to the pectoral fin
basis was measured and summed (Herrel et
al., 2005). As this measure of muscle mass is only a fraction of
the total mass of the muscles active during the expansive phase of suction
feeding (for example, the epaxial muscles are not included), the presented
muscle-mass-specific power and mechanical work will overestimate the actual
values. They should therefore be regarded as relative, rather than real
values. However, we can use these values for analysing the scaling
relationships of these variables because it can safely be assumed that the
mass of the used sample of muscles will show a similar scaling with cranial
size as the total muscle mass contributing to suction.

## Results

### Linear kinematic variables

Maximum linear displacements of the lower jaw, hyoid and branchiostegal
membrane during prey capture of *C. gariepinus* increase significantly
with increasing cranial size (Table
1; Fig. 5A,B). This
increase does not differ significantly from a size-proportional increase
(slope=1; *P*>0.25). Therefore, no departure from isometry can be
distinguished for these variables.

Maximum peak linear velocities of mouth opening, mouth closing, hyoid
depression and branchiostegal movements do not change significantly with
cranial size (Table 1;
Fig. 5C,D). Maximum peak linear
accelerations, however, decrease significantly with increasing cranial length
(Table 1;
Fig. 5E,F). This decrease is
almost (and statistically not different from) a decrease proportional to
cranial size (*P*>0.07).

### Angular kinematic variables

Maximum angular displacements of the neurocranium (elevation) and the hyoid (lateral abduction) do not change significantly with cranial size (Table 1; Fig. 6A,B). This also means that no departure from isometry can be distinguished for these variables.

Maximum peak angular velocities of hyoid abduction and neurocranial
elevation decrease significantly with increasing cranial length
(Table 1;
Fig. 6C,D). In general, this
decrease approximates (and is not statistically different from) a decrease
proportional to cranial size (*P*>0.13).

### Timings

The scaling coefficients (slopes) of timings of the analysed kinematic
events are very similar to each other
(Table 1). In general, the time
from the start of mouth opening until the start, maximum excursion or end of
all analysed movements increases approximately proportional to increasing
cranial length (*P*>0.29).

### Buccal pressure

The magnitudes of peak sub-ambient pressure (i.e. maximal instantaneous
pressures averaged over the entire buccal cavity, calculated by the present
hydrodynamic modelling) do not change significantly with size (slope=-0.25,
*N*=17, *R*^{2}=0.06, 95% confidence interval between
-0.77 and 0.28, *P*=0.337) (Fig.
7A). A similar scaling relationship is found for the magnitudes of
the highest (per individual) average buccal pressure (averaged over position
and time from start to maximal gape): the linear-regression slope is negative
(slope=-0.46, *N*=17, *R*^{2}=0.21, 95% confidence
interval between -0.99 and 0.03), although not statistically different from 0
(*P=*0.066) (Fig.
7B).

### Required power

The maximal peak power required from the muscular system (calculated by the
present hydrodynamic modelling and inverse dynamics) increases significantly
(*P*=0.048) with increasing cranial length (slope=0.97*, N*=17,
*R*^{2}=0.24, 95% confidence interval between 0.01 and 1.92).
This is also the case for the maximal average power (average from start to
maximal gape), which shows a significant (*P*=0.021) increase with
increasing size (slope=1.08*, N*=17, *R*^{2}=0.31, 95%
confidence interval between 0.19 and 1.97).

However, when expressed as muscle-mass-specific power (expressed in W
kg^{-1} muscle), it also decreases highly significantly
(*P*<0.0001) with increasing cranial length
(Fig. 8). This is true for the
highest (per individual) instantaneous muscle-mass-specific required power
(slope=-2.49*, N*=17, *R*^{2}=0.65, 95% confidence
interval between -3.48 and -1.49), as well as for the highest (per individual)
average (from start to maximal gape) muscle-mass-specific power
(slope=-2.37*, N*=17, *R*^{2}=0.64, 95% confidence
interval between -3.36 and -1.39).

## Discussion

During ontogeny, when *C. gariepinus* becomes larger, important
changes in the speed of movements of the cranial structures during suction
feeding occur (Figs 5,
6;
Table 1). In general, angular
velocities decrease approximately proportional with increasing cranial size
while linear velocities remain more or less constant
(Table 1). These results are
not consistent with the previous studies on scaling of suction feeding in
vertebrates (Richard and Wainwright,
1995; Reilly,
1995; Hernandez,
2000; Wainwright and Shaw,
1999; Robinson and Motta,
2002). Angular velocity does decrease in these studies but at a
lower rate than proportional to body length. Therefore, the results of the
present study add to the relatively large interspecific variability in scaling
coefficients of aquatic feeding kinematics, which have already been observed
among previous studies.

By applying biomechanical modelling to the experimentally observed prey
capture kinematics in *C. gariepinus*, a more detailed insight into the
mechanics of suction feeding in relation to body size can be achieved. In this
way, we may be able to explain the observed scaling relations in prey capture
kinematics for this species. In particular, we focus on the potential
importance of buccal pressures in limiting the maximal speed of buccal
expansions during suction and how this is influenced by size. Results from the
model presented in this paper show that the pressure gradient induced by
expanding the buccal cavity is theoretically responsible for the largest
fraction (>80%) of the total force required from muscular contractions
during this expansive phase of suction feeding
(Fig. 9). This is in accordance
with the findings of Aerts et al.
(1987) and Carroll et al.
(2004). This also implies that
the output of our model (required muscular power) is not very sensitive to
changes in the less important factors: inertia (<20% of total required
muscular force) and hydrodynamic drag (<2% of total required muscular
force). Consequently, the assumptions and approximations that were made in
modelling these factors are not critical. For example, tripling the added mass
coefficient (*c*_{added}) in the model of a representative prey
capture event increases the required power, in general, by less than 3%.

### Scaling relationships predicted by the model

Assuming that the maximal power output during suction feeding is
proportional to the mass of the muscles involved in this process, the
presented model can be used to generate specific predictions concerning the
scaling of kinematics in *C. gariepinus*. If the linear dimensions of
the model are increased isometrically without changing the rate of buccal
expansion (i.e. constant angular velocities), the calculated negative buccal
pressure magnitudes increase approximately proportional to the square of
cranial length (∼*CL*^{2}). As we have shown that
sub-ambient buccal pressure is the most important factor in resisting cranial
expansion in our model (see above), and given the fact that the surface area
of the modelled cranial apparatus to which these pressures apply
(∼*CL*^{2}) and linear velocity of expansion (e.g. velocity
of the hyoid tip) will also increase in this situation (∼*CL*), it
was not surprising that our model also shows that power requirement (i.e.
required force ∼*CL*^{4} multiplied by linear velocity∼
*CL*) increases approximately by *CL*^{5}. However,
the power available from the muscles (i.e. force ∼ muscle cross-sectional
area multiplied by linear velocity ∼ fibre length) only increases by
*CL*^{3} in the case of isometric growth or by
*CL*^{3.4} if accounting for the positive allometry observed in
*C. gariepinus* (Herrel et al.,
2005). For this situation, which corresponds to the predictions of
the theoretical scaling model of Richard and Wainwright
(1995), the required power
will exceed the available power during growth. This `deficit' in power will
increase proportional with *CL*^{2} (isometry) or with
*CL*^{1.6} (*C. gariepinus* allometry) during growth,
forcing *C. gariepinus* to decrease its speed of buccal expansion.

By increasing the time to carry out a given buccal expansion (abbreviated
*T*), buccal pressure and power requirement decreases. In order to
balance the available and required power, *T* would have to increase
during growth of *C. gariepinus* in a way that the calculated power
`deficit' because of growth (∼*CL*^{2}; see above) can be
compensated for. By adjusting the expansion time *T* in our model, we
found the buccal pressure changing approximately ∼*T*^{-2}
and power requirement ∼*T*^{-3}. Consequently, in case of
isometric growth, equilibrium between the available and required power for
buccal expansion is reached when *T* is increased by
*CL*^{2/3}. This corresponds, for example, to angular
velocities scaling proportional to *CL*^{-2/3}, and linear
velocities scaling with *CL*^{1/3}. Accounting for the positive
allometry of muscle mass in *C. gariepinus*, this equilibrium will be
reached when the (angular) speed of buccal expansion changes in proportion
with *CL*^{-1.6/3}.

However, these theoretical predictions are not confirmed by the
experimental data. The observed decrease in the speed of buccal expansion with
growth is considerably larger (∼*CL*^{-1};
Table 1) than the expected
scaling relationship (∼*CL*^{-1.6/3}). In other words, the
larger *C. gariepinus* are slower than predicted, or *vice
versa*. As a consequence, the model output of muscle-mass-specific power
requirement decreases significantly with increasing size
(Fig. 8).

### The model of A. V. Hill

Scaling relationships found for prey capture kinematics in *C.
gariepinus* apparently match the prediction by the scaling model of Hill
(1950), in which it was stated
that geometrically similar animals should carry out similar movements in times
directly proportional to their linear dimensions. In this model, inertial
forces (and not hydrodynamic pressures) are assumed to be dominant. If the
time to fulfil a given movement does not change with size, then the kinetic
energy
()
required to accelerate a specific mass (for example a limb) scales as length
*L*^{5} (*m*∼*L*^{3} and
*v*^{2}∼*L*^{2}). By contrast, the work a
muscle can do is expected to increase merely by *L*^{3},
leaving a `deficit' in available work increasing by *L*^{2}.
This deficit can be overcome by increasing the time to carry out this movement
in proportion to the increasing length, by which the required energy for this
movement is in proportion to the total muscle mass.

The forces resulting from buccal pressures calculated for suction feeding
in *C. gariepinus* apparently scale similar to the inertial forces
outlined by Hill: these forces both increase approximately by
*L*^{4} in the case of constant speed and by the square of
speed in the case of constant size. Consequently, the size and speed
dependence of the energetic demands for suction feeding in *C.
gariepinus* are expected to be identical to the acceleration of a limb.
Nevertheless, there are still two reasons why this model
(Hill, 1950) does not explain
the results for *C. gariepinus*. First, one of the assumptions of Hill
(1950) is that the muscles are
optimized to work at a `reasonable' energetic efficiency. This assumption does
not necessarily apply to occasional, explosive movements such as suction
feeding. Furthermore, the mechanical energy or work required for a single
suction feeding cycle (which only takes fractions of a second to be completed)
is very low compared with the energetic content of prey
(Aerts, 1990). For example, if
we assume that the total work for buccal expansion in *C. gariepinus*
equals twice the work calculated by our model from start to maximum gape,
total work is about 0.0034 J for the smallest and 0.3 J for the largest
individuals used in this study. On the other hand, the energy content of a
small fish prey (1 g wet mass) is approximately 6000 J
(Cummins and Wuycheck, 1971).
Consequently, as the energetic efficiency of buccal expansion is probably an
unimportant aspect of the biology of fishes, we believe that the assumption of
the model of Hill (1950), i.e.
the size independency of muscle-mass-specific work, is not a good starting
point for predicting scaling patterns of an extremely short and anaerobic
action such as suction feeding in fish. Note, in this respect, that the
maximal power of muscle, *P*, has a different speed dependency than
does mechanical work, *E* (*E*=∫*P* d*t*). As
a result, different scaling relationships must be predicted if the
muscle-mass-specific power instead of muscle-mass-specific work is assumed to
be size independent. In the case of suction feeding, maximal power seems a
more appropriate speed-limiting factor than muscular energetics (see also
Carroll, 2004).

Second, *C. gariepinus* does not fulfil the assumption of geometric
similarity. Herrel et al.
(2005) measured, for example,
a significant positive allometry in the mass (in general approximately∼
*L*^{3.4}) and a negative allometry in the fibre length of
the cranial muscles (∼*L*^{0.7}). As a result, whereas
scaling of prey capture kinematics is well predicted by the model of Hill
(1950), the size independency
of muscle-mass-specific work (an intrinsic assumption of his kinematic
predictions) is not observed (Fig.
10).

### Why are large catfish slower than predicted?

If we assume that the muscle-mass-specific power output capacity of the
muscles involved in buccal expansion does not change with size, the results
indicate that larger fish perform sub-maximally compared with smaller fish, in
a way that a smaller proportion of the available power is used
(Fig. 8). This still leaves us
with the question of *why* larger *C. gariepinus* show this
apparently reduced suction effort, despite being presented with similar prey
of which the size was adjusted according to the size of the catfish. Possibly,
faster buccal expansions are not needed for large catfish in order to perform
successful prey captures on the experimental prey types. Indeed, it was
recently demonstrated for *C. gariepinus* that the actual suction
performance (maximal prey distance and size) increases substantially with
size, despite the observed decrease in the speed of buccal expansion
(Van Wassenbergh et al., in
press). For example, it was estimated that the maximal size of a
prey that can still be successfully drawn into the mouth by suction increases
faster than proportional to cranial size if these prey are sucked from the
same absolute distance from the mouth (Van
Wassenbergh et al., in press). The observation by Bruton
(1979) that larger *C.
gariepinus* include a relatively larger amount of evasive prey (i.e. fish)
in their natural diet also indicates this increase in suction performance with
size. Consequently, the immobile prey of which the size is scaled according to
predator size may not have been the ideal situation to induce a comparable
suction effort in both small and large *C. gariepinus*. If, as appears
to be the case for *C. gariepinus*, some taxa of aquatic suction
feeders tend to perform increasingly sub-maximally when becoming larger, this
may potentially also explain the large variability observed in the literature
on scaling relationships of prey capture kinematics in this group of
animals.

### Conclusions

Model calculations have shown that negative buccal pressures are
responsible for the largest part of the power (more than 80%) required for
expanding the buccal cavity during prey capture in *C. gariepinus*. The
size dependency of buccal pressures and the forces required for suction
feeding will force *C. gariepinus* to become relatively slower during
growth; if not, the required power would exceed the expected available power
from its muscles (proportional to muscle mass). In this way, we expected
*C. gariepinus* to decrease the (angular) speed of movement of its
cranial structures during suction proportion to cranial length
*CL*^{0.53}. However, the experimental data show that (angular)
speed decreases more rapidly with size than predicted: approximately
proportional to *CL*^{1}
(Table 1). According to our
model, this would imply a significant decrease in the muscle-mass-specific
power output. Our data therefore suggest that suction effort employed by the
fish to capture similar prey decreases with size. Suction performance,
however, does not (Van Wassenbergh et al.,
in press), leaving the possibility for larger *C.
gariepinus* not to use their full muscular capacity while still performing
successful prey captures on the experimental prey types we used in this
study.

- List of symbols
*a*- linear acceleration (m s
^{-2}) *a*_{r}- ratio between
*u*_{m}and (*u*_{m}-*u*_{i}) *A*_{ext}- surface area bordering the outside of the head of a hollow elliptic
cylinder segment (m
^{2}) *A*_{ext-p}- projected area of
*A*_{ext}onto a plane perpendicular to the direction of motion (m^{2}) *A*_{int}- surface area bordering the buccal cavity of a hollow elliptic cylinder
segment (m
^{2}) *c*- distance between the centre of the external and internal ellipses defining the modelled catfish head (m), function of position along the medio-sagittal axis
*c*_{added}- added mass coefficient
*c*_{d}- drag coefficient
*CL*- cranial length (mm)
- COM
- centre of mass
*d*- distance between two successive points along the medio-sagittal axis for which the cross-section is modelled by ellipses (m)
*E*- mechanical work (J)
*E*_{k}- kinetic energy
*F*_{drag}- drag force (N)
*F*_{muscle}- muscular force (N)
*F*_{pressure}- force resulting from pressure differences (N)
*h*_{int}- height radius of the internal ellipse (m)
*h*_{ext}- height radius of the external ellipse (m)
*h*_{l}- radius of the mouth aperture (m)
*L*- body length (mm)
*m*- mass (kg)
*P*- power (W)
*P*_{req}- power required for buccal expansion (W)
- Δ
*p* - pressure difference from hydrostatic pressure (Pa)
*r*_{hor}- horizontal distance (left or right) between the internal and external ellipse (m)
*t*- time (s)
*T*- duration of the buccal expansion phase (s)
*u*- flow velocity (moving frame of reference) in the direction of the
longitudinal axis (m s
^{-1}) *u*_{l}- flow velocity (moving frame of reference) at the mouth aperture (m
s
^{-1}) *u*_{m}- flow velocity at the mouth aperture in earth-bound frame of reference
(m s
^{-1}) *v*- linear velocity (m s
^{-1}) *x*- position along the longitudinal axis
*w*_{ext}- width radius of the external ellipse (m)
*w*_{int}- width radius of the internal ellipse (m)
- ρ
- density (kg m
^{-3}) - ∼
- proportional to

## ACKNOWLEDGEMENTS

We thank S. Devaere, D. Adriaens, F. Huysentruyt and N. De Schepper for
taking care of some of the catfish used in this study. W. Fleuren is
acknowledged for supplying *C. gariepinus* specimens. Thanks to F.
Ollevier, F. Volckaert and E. Holsters for providing us with *C.
gariepinus* larvae and also the largest individual used in this study. The
authors gratefully acknowledge support of the Special Research Fund of the
University of Antwerp. A.H. is a postdoctoral fellow of the fund for
scientific research - Flanders (FWO-Vl).

- © The Company of Biologists Limited 2005