Biomechanics of jumping in the flea

Abstract

It has long been established that fleas jump by storing and releasing energy in a cuticular spring, but it is not known how forces from that spring are transmitted to the ground. One hypothesis is that the recoil of the spring pushes the trochanter onto the ground, thereby generating the jump. A second hypothesis is that the recoil of the spring acts through a lever system to push the tibia and tarsus onto the ground. To decide which of these two hypotheses is correct, we built a kinetic model to simulate the different possible velocities and accelerations produced by each proposed process and compared those simulations with the kinematics measured from high-speed images of natural jumping. The in vivo velocity and acceleration kinematics are consistent with the model that directs ground forces through the tibia and tarsus. Moreover, in some natural jumps there was no contact between the trochanter and the ground. There were also no observable differences between the kinematics of jumps that began with the trochanter on the ground and jumps that did not. Scanning electron microscopy showed that the tibia and tarsus have spines appropriate for applying forces to the ground, whereas no such structures were seen on the trochanter. Based on these observations, we discount the hypothesis that fleas use their trochantera to apply forces to the ground and conclude that fleas jump by applying forces to the ground through the end of the tibiae.

Full text

836
INTRODUCTION
For almost one hundred years the mechanisms of how fleas jump
has continued to be a challenging biological problem (Russel, 1913;
Snodgrass, 1946; Rothschild, 1965). The jumps were known to be
quick and energetic, so that the power requirements were beyond
what a muscle could produce by a direct contraction (Askew and
Marsh, 2002; Roberts and Marsh, 2003; Vogel, 2005). A series of
studies then used anatomical and engineering analyses, together with
high-speed films to show that fleas propel their jumping by the
storage and release of elastic energy. They first lock the thoraco-
coxal joints of the two hindlegs and then contract two large
dorsoventral muscles to compress part of the skeletal structure of
the thorax that contains the elastic protein resilin, so that it acts as
a tensed spring (Bennet-Clark and Lucey, 1967; Rothschild et al.,
1972; Rothschild et al., 1973; Rothschild and Schlein, 1975;
Rothschild et al., 1975). The lock on the hindlegs is then released
and the rapid expansion of the spring releases the stored energy,
which propels the jump (Bennet-Clark and Lucey, 1967; Rothschild
et al., 1972; Rothschild et al., 1973; Rothschild and Schlein, 1975;
Rothschild et al., 1975). These studies were in agreement about how
fleas stored energy for a jump, but provided two different hypotheses
for how the force was transmitted to the ground. The hypothesis
put forward by Rothschild et al. (Rothschild et al., 1972; Rothschild
et al., 1973; Rothschild and Schlein, 1975; Rothschild et al., 1975),
henceforth called the Rothschild hypothesis, argued that the
expansion of the spring pushed the hind trochanter onto the ground
to transmit the force. This argument was based on two observations:
first, in the preparatory phase of the jump, fleas placed their hind
trochantera on the ground; second, amputation of the hind tarsi had
only a small effect on the frequency of jumping. By contrast, the
hypothesis proposed by Bennet-Clark and Lucey (Bennet-Clark and
Lucey, 1967), henceforth called the Bennet-Clark hypothesis,
argued that expansion of the spring applied a torque about the coxo-
trochanteral joint that was carried through the femur and tibia, and
finally resulted in a force applied to the ground by the hind tibia
and tarsus. This argument was based on the observation that the
average acceleration during a jump was consistent with the spring
expansion having a significant mechanical disadvantage; the ground
forces were approximately a factor of ten fewer than the forces
present in the spring.
The ability of a flea to control the direction of its jump is
determined by how the forces from the spring reach the ground. If
forces were channelled directly to the ground through the hind
trochanter, as proposed by the Rothschild hypothesis, then recoil
of the spring would push the trochanter straight down, propelling
the animal vertically. Generating jumps with a large component in
the horizontal direction would thus be difficult. If, however, forces
were directed through the hind tibia and tarsus, as proposed by the
Bennet-Clark hypothesis, then the tibia could be rotated about the
fermoro-tibial joint to direct the forces and generate jumps with a
more horizontal trajectory.
To distinguish between these two hypotheses, we have used
scanning electron microscopy and ultraviolet microscopy to
reveal relevant structures on the hind trochantera and tarsi, high-
speed video to analyse the joint movements of the legs during
natural jumping, and kinetic modelling. Analysis of natural
jumping shows that fleas do not have to place the hind trochantera
on the ground to jump and that even when they do, there is no
reduction of the acceleration when the trochantera leave the
ground before take-off. Jump trajectories also had large horizontal
The Journal of Experimental Biology 214, 836-847
© 2011. Published by The Company of Biologists Ltd
doi:10.1242/jeb.052399
RESEARCH ARTICLE
Biomechanics of jumping in the flea
Gregory P. Sutton* and Malcolm Burrows
Department of Zoology, University of Cambridge, Downing Street, Cambridge CB2 3EJ, UK
*Author for correspondence (RScealai@gmail.com)
Accepted 16 November 2010
SUMMARY
It has long been established that fleas jump by storing and releasing energy in a cuticular spring, but it is not known how forces
from that spring are transmitted to the ground. One hypothesis is that the recoil of the spring pushes the trochanter onto the
ground, thereby generating the jump. A second hypothesis is that the recoil of the spring acts through a lever system to push the
tibia and tarsus onto the ground. To decide which of these two hypotheses is correct, we built a kinetic model to simulate the
different possible velocities and accelerations produced by each proposed process and compared those simulations with the
kinematics measured from high-speed images of natural jumping. The
in vivo
velocity and acceleration kinematics are consistent
with the model that directs ground forces through the tibia and tarsus. Moreover, in some natural jumps there was no contact
between the trochanter and the ground. There were also no observable differences between the kinematics of jumps that began
with the trochanter on the ground and jumps that did not. Scanning electron microscopy showed that the tibia and tarsus have
spines appropriate for applying forces to the ground, whereas no such structures were seen on the trochanter. Based on these
observations, we discount the hypothesis that fleas use their trochantera to apply forces to the ground and conclude that fleas
jump by applying forces to the ground through the end of the tibiae.
Supplementary material available online at http://jeb.biologists.org/cgi/content/full/214/5/836/DC1
Key words: trajectory control, kinematics, Siphonaptera.
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837Jump of a flea
components and were never in the vertical direction. To determine
whether the observed accelerations were consistent with the
Rothschild or Bennet-Clark hypotheses, we developed two
mathematical models to predict the velocities and accelerations
that would result from each hypothesis. We then compared the
model predictions with the data from natural jumping. Modelling
indicates that both hypotheses can generate take-off velocities that
are consistent with those observed in nature. If forces were
transmitted through the trochanter, however, accelerations should
be higher and briefer than those observed in natural jumping. If
forces were transmitted through the end of the tibia and the tarsus,
the model predicted accelerations that were consistent with those
observed in natural jumping. These experimental and theoretical
analyses support the hypothesis that fleas jump by transmitting
forces from a spring in their thorax through a lever system to
their tarsi and thus to the ground.
MATERIALS AND METHODS
Ten adult hedgehog fleas, Archaeopsyllus erinacei (Bouncé 1835),
taken from hedgehogs were kindly supplied by staff at St
Tiggywinkles Wildlife Hospital Trust, Aylesbury, Bucks., UK. To
determine leg movements and jump trajectory, sequential images
of jumps were captured at rates of 5000framess–1 and with an
exposure time of 0.067ms with a single Photron Fastcam 1024 PCI
camera [Photron (Europe) Ltd, West Wycombe, Bucks, UK] that
fed images directly to a computer. 51 jumps by 10 adult fleas were
analysed. Each flea jumped between three and nine times (median
five). Jumps occurred in a chamber of optical quality glass 80mm
wide, 80mm tall and 10mm deep at its base expanding to 25mm
at the top, in which the temperature was 20–25°C. All analyses of
the kinematics were based on the two-dimensional images provided
by the single camera. The flat body plan of the body of a flea and
the orientation of the chamber made it possible determine which
jumps were within 20deg of the sagittal plane. Within this 20deg
arc, errors in the calculation of jump velocity and jump trajectory
will be small (a maximum 6% underestimate of velocity and a
maximum 2deg error in the elevation of the jump).
Jumps were either spontaneous, elicited by a light touch of a
paintbrush, or in reaction to turning on lights. The fleas jumped
from a high-density Styrofoam floor, which was flat and stable, but
allowed them to grip the substrate firmly. The motion of the flea
over the first four frames after take-off was used to calculate the
take-off velocity and elevation of the jump. When analysing the
sequences of images of a jump, a line was drawn through the body
to represent its transverse axis and its orientation with respect to
the ground. If the centre of this line does not represent the true centre
of mass of the flea, there will be an error in the estimation of the
velocity and trajectory of the centre of mass that is proportional to
the flea’s rotation rate multiplied by the distance between the centre
of the measurement line and the real centre of mass. Fleas do not,
however, rotate at sufficiently high rates for this error to be large.
For example, even if it is assumed that the centre of a line drawn
through the flea is as much as half a body length away from the
true centre of mass then the error will be only 8% and thus negligible
for all reasonable estimates of the centre of mass.
Jump trajectory, linear velocity and angular velocity were
measured. For statistical analyses, one jump was taken randomly
from each flea and data were compared with the data on locusts
from Sutton and Burrows (Sutton and Burrows, 2008) using an F-
test (the var.test function) in the software package R (R Foundation
for Statistical Computing). Data are presented as means ± standard
deviation unless otherwise stated.
The morphology of the hindlegs was photographed and drawn
using a Leica stereomicroscope fitted with a drawing tube, or with
a Nikon DXM 1200 camera. The presence of the elastic protein
resilin was revealed by its characteristic fluorescence (Weis-Fogh,
1960; Andersen, 1963) established using an Olympus BX51WI
compound microscope with OlympusMPlan ⫻10/0.25NA and
LUCPlanFLN ⫻20/0.45NA objective lenses, under ultraviolet
(UV) or white epi-illumination. Images were captured with a Q-
imaging Micropublisher 5.0 digital camera (Marlow, Bucks., UK)
as colour (RGB) TIFF files. The UV light was provided by an X-
cite series 120 metal halide light source, conditioned by a Semrock
DAPI-5060B Brightline series UV filter set (Semrock, Rochester,
NY, USA) with a sharp-edged (1% transmission limits) band from
350nm to 407nm. The resulting blue fluorescence emission was
collected in a similarly sharp-edged band at wavelengths from
413nm to 483nm through a dichromatic beam splitter. Images
captured at the same focal planes under UV and visible light were
superimposed in Canvas X (ACD Systems of America, Miami, FL,
USA). To look for structures on the hind tarsus and trochanter that
might be associated with improving traction with the ground, dried
specimens were sputter-coated with gold and images were taken
with a Phillips XL-30 scanning electron microscope.
Two kinetic models were used to calculate the kinematics
predicted by the Rothschild and the Bennet-Clark hypotheses. The
equations of motion for both models were written and implemented
in Mathematica 5.0 (Wolfram Research, Champaign, IL, USA)
(Appendix 1, 2). The parameters for each model are based on the
data presented for Xenopsylla cheopis in (Bennet-Clark and Lucey,
1967; Rothschild and Schlein, 1975; Rothschild et al., 1975).
RESULTS
Anatomy
Hedgehog fleas have a body mass of 0.7±0.16mg (N10) and a
body length of 1.8±0.19mm (N10). The body is flattened laterally
and is dominated by large hindlegs that rotate in the sagittal plane
(Fig.1A,B). A hindleg is 2.8mm long, with the coxa, femur and
tibia all of similar lengths (0.5–0.6mm), the trochanter is smaller
(0.2mm) and the tarsus is longer (1.0mm; Table1). The ratio of leg
lengths is 1:1.3:1.9 (front:middle:hind). The hindlegs have a ratio
of 3.1 relative to the cube root of the body mass and are 154% of
the body length, compared with 83% for the front legs and 110%
for the middle legs.
Table 1. Body form of the flea
Archaeopsyllus erinacei
Parameter Value
Body mass (mg) 0.7±0.16
Body length (mm) 1.8±0.19
Hindleg length (mm)
Coxa 0.5±0.06
Trochanter 0.2±0.04
Femur 0.5±0.03
Tibia 0.6±0.04
Tarsus 1.0±0.06
Ratio of leg lengths
Front 1
Middle 1.3
Hind 1.9
Leg length (% body length)
Front 83
Middle 110
Hind 154
Hindleg length (mm)/mass (mg)0.33 3.1
Values are means ± s.d.,
N
=10.
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838
Consistent with extensive previous findings, the scanning electron
micrographs of the hindlegs showed that the five segments of the
hind tarsus and the distal end of the tibia have a number of prominent
bristles and spines that would normally contact the ground and thus
improve traction (Fig.1C). Moreover, the distal end of the most
distal tarsal segment has two large hook-like cuticular spines, which
could also be used to gain traction on a substrate. The trochanter
has a series of hairs but these are much smaller than the spines and
claws present on the tarsus (Fig.1D).
The hind coxae pivot with the thorax so that the whole hindleg
can be rotated through 15deg in a plane parallel to the long axis of
the body (Fig.2A,B). Each coxa is reinforced by sclerotised struts
and a distal sclerotised rim at the joint with the trochanter. The
trochanter itself can rotate through 100deg in the same plane but
moves little relative to the femur, so that both can be treated as a
single element in modelling the movements of a hindleg. Rotation
of the trochanter about the coxa is key to jumping; levation of this
joint moves the whole hindleg forwards into a cocked position ready
for jumping so that the femur fits into a lateral indentation of the
coxa (Fig.2). The tibia can rotate through 130deg about the femur,
again in the same plane as the more proximal joints. The long tarsus
can bend at the various joints between its five segments.
When the hindlegs were examined under UV light a consistent
patch of blue fluorescence was seen in the thorax (Fig.3A,B). With
the specific wavelengths used this blue fluorescence is a
characteristic signature of the elastic protein resilin (Weis-Fogh,
1960; Andersen, 1963). The fluorescence was associated with
sclerotised elements of the internal thoracic skeleton (Fig.3A,C).
The metathorax has two symmetrically arranged areas of resilin
associated with each hindleg (Fig.3C), but no resilin could be
G. P. Sutton and M. Burrows
detected associated with the front and middle legs at the equivalent
sites. Bennet-Clark and Lucey suggested that this lump of resilin
was the energy store for the flea, based on a calculation that stated
that a lump of resilin this size could, in theory, store the necessary
energy (Bennet-Clark and Lucey, 1967). In froghoppers and locusts,
however, a significant amount of energy is stored not only in
resilinous lumps, but also in the bending of sclerotised cuticle
(Bennet-Clark, 1975; Burrows et al., 2008), thus pointing to the
possibility that the flea may also store energy by bending the
reinforced cuticle surrounding the resilin lump. There is agreement,
however, that the recoil of the energy storage applies a force at the
coxo-trochanteral joint (Bennet-Clark and Lucey, 1967; Rothschild
and Schlein, 1975).
Fig.1. Anatomical features of the flea
Archaeopsyllus erinacei
involved in
jumping. (A)Scanning electron
micrograph of the whole flea with the
hindlegs outstretched. (B)Drawing of the
right hindleg and part of the thorax to
show the joints and the skeletal
reinforcements in the coxa and thorax.
(C)Scanning electron micrograph of the
tibio-tarsal joints and the tarsi of the
hindlegs in another flea to show the
spines and claws that make contact with
the ground when jumping. (D)The hind
trochanter of a third flea showing that
there are few structures that could aid
traction with the ground.
Table 2. Jumping performance of the flea,
Archaeopsyllus erinacei
Parameter Formula Mean (
N
10) Fastest jump
Body mass (
M
; mg) 0.7±0.16 1
Body length (
L
; mm) 1.8±0.19 2
Time to take off (
t
; ms) 1.4±0.25 1.2
Take-off velocity (
v
; m s–1) 1.3±0.21 1.9
Take-off angle (deg) 39±5.7 45
Body angle at take-off (deg) 23±8.7 39
Acceleration (
a
; m s–2)
a

v
/
t
960±233 1600
g force (
g
)
g
f/9.86 98±23.7 160
Energy (
e
; J)
e
0.5
Mv
20.6±0.28 1.8
Power (
p
; mW)
p

e
/
t
0.43±0.19 1.5
Force (F; mN) F
ma
98±23.7 160
Power/muscle mass* (W kg–1)
p
/(0.11
M
) 6000±2100 14,000
Values are means ± s.d.
*This assumes that fleas have the same percentage of their body mass
devoted to jumping (11%) as locusts do (Bennet-Clark, 1975).
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839Jump of a flea
Kinematics of natural jumps
The jump of a flea was propelled by the rapid and simultaneous
movements of the two hindlegs (Table 2). Preparation for a jump
began with an initial phase where the body was rotated largely by
the action of the front and middle legs thus setting the different
attitudes of the body from which the rapid, propulsive movements
of the legs could be delivered (Figs4 and 5). The hind trochantera
were then levated, moving the more distal segments of the hindlegs
forwards so that the femur fitted into an indentation of the coxa.
The legs then depressed at the coxo-trochanteral and extended at
the femoro-tibial joints propelling the flea forwards.
In 45 of the 51 jumps (88%), the first propulsive movements of
the hindlegs in a jump began from a position in which both hind
tarsi were on the ground and both hind trochantera were close to
or touching the ground (Fig.5 and supplementary material Movie
1), confirming the observations made by Rothschild et al.
(Rothschild et al., 1975). The initial depression of the trochantera
thus occurred while they were either directly in contact with the
ground, or very close to it. Despite this initial contact or closeness
to the ground, the trochantera were obviously clear of the ground
approximately 0.6ms after their initial movement. This would
indicate that if the forces were applied to the ground by the
trochantera, the acceleration of the body would drop once they left
the ground. In fact, the acceleration did not decrease when the
trochantera left the ground, but, instead, the body continued to be
accelerated until take-off approximately 0.6ms later.
In six of the 51 jumps (12%) by A. erinacei the two hind
trochantera were both clear of the ground at the time the hindlegs
first began their propulsive movements. In the examples shown, the
gap from the ventral surface of the trochanter and the ground ranged
from 200–520m (Fig.4A–D). The sequence of joint movements
during these jumps was the same as when the trochantera were in
contact with the ground initially. The first propulsive movements
of the hindlegs were a depression of the trochanter about the coxa
Fig.2. Drawings (A) and photographs (B) of the
proximal joints of a right hindleg of a flea to show
the rotation of the coxa about the thorax, and of the
depression of the trochanter and femur about the
coxa in movements that are used during a jump.
The top row shows the hindleg in the fully cocked
position that it adopts before a jump. The next two
rows show the progressive depression of the
trochanter about the coxa. The femur and trochanter
do not move much relative to each other so that
they can be modelled as one link.
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840
that was progressively accompanied by an extension of the tibia
about the femur (Fig.6, supplementary material Movie 2). The effect
of these movements was to raise the body, accelerate it forwards
and to cause both the front and middle legs to lose contact with the
ground before take-off. The time when these legs left the ground
depended on the initial attitude of the body relative to the ground,
but could occur as much as 1ms before take-off. By the point of
take-off, the hind coxa had rotated backwards relative to the thorax,
the trochanter had rotated through 100deg so that it was at its most
depressed position, the tibia was extended by an angle of 130deg
about the femur and the tarsus was depressed about the tibia with
G. P. Sutton and M. Burrows
all its segments aligned. After take-off, the hindlegs remained
outstretched and trailed underneath and behind the body.
If the initial thrust for a jump was delivered through the trochanter,
then beginning with the trochanter off the ground would reduce the
initial thrust so that the flea should take longer to reach take-off
and there would be a differently shaped relationship between
velocity and time than if the trochanter began in contact with the
ground (Fig.7). No differences, however, were seen in the
acceleration times whether jumping occurred with the trochanter
initially on or off the ground. The acceleration time was 1.4±0.20ms
for jumps starting with the hind trochantera off the ground (one
Fig.3. Photographs showing the regions
of blue fluorescence under UV
illumination that indicate the presence of
resilin. (A)View of the right hindleg and
metathorax from inside. A single patch of
blue fluorescence is present in the
thorax. (B)The same region at higher
magnification. (C)View of the ventral
metathorax viewed from inside with the
ventral midline at the centre (dashed
line) and patches of blue fluorescence
associated with the left and right
hindlegs.
Fig.4. Images of four
A. erinacei
(A–D) captured at
5000framess–1during natural jumping and at the
moment when their hindlegs first began to move to propel
a jump. The thick vertical magenta bar shows that the
trochanter was not in contact with the ground in any of
these jumps. The hind tarsus (magenta arrows) was in
contact with the ground in each jump. The position of the
segments of the hindlegs are indicated by thin yellow
lines.
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841Jump of a flea
jump randomly taken from each of five fleas) and 1.4±0.24ms for
jumps starting with them on the ground (one jump randomly taken
from each of ten fleas). The values are not significantly different
(P0.86, t0.18, d.f.10.3, Welch two-sample t-test).
The take-off velocity for jumps that began with the trochantera
off the ground was 1.3±0.25ms–1 (N5), whereas the velocity for
jumps that began with the trochantera on the ground was
1.3±0.22ms–1 (N10). The values were again not significantly
different (P0.67, t0.44, d.f.6.5, Welch two-sample t-test).
The shape of the relationship between velocity and time for jumps
that started with the trochantera off the ground (Fig.7A) was
qualitatively similar to that for jumps starting with the trochantera on
the ground (Fig.7B). Both velocity curves are S-shaped, rising to a
peak just before take-off and falling when the hind tarsi had left the
ground, at which point force could no longer be delivered through
them to the ground. Acceleration times were also comparable.
Moreover, in those jumps where the trochantera were initially on the
ground, there was no decrease in acceleration when they left the
ground before take-off (Fig.7B). If force were applied through the
trochantera, the velocity should fall at the time when they lost contact
with the ground. Instead, in all 51 jumps the flea continued to
accelerate from the first depression of the trochantera until take-off,
regardless of whether the starting position of the trochantera was on
or off the ground, indicating that force was transmitted through the
tibiae and tarsi, and not through the trochantera.
Take-off elevation
The elevation trajectories at take-off for 51 jumps by 10 fleas were
all within a narrow 24deg window (Fig.8A). The minimum angle
was 28deg and the maximum was 52deg, with a mean of 39±6.1deg.
A histogram of the jump elevations (Fig.8B) shows that A. erinacei
most commonly took-off with an elevation of 36deg. No jumps had
the trajectories close to vertical as predicted by the Rothschild model.
In this model, the trochanter would be pushed downward onto the
ground and therefore the resulting ground reaction force would be
vertical. By contrast, the observed range of low elevation angles in
natural jumping were consistent with the Bennet-Clark model and
could easily be directed if forces were directed to the ground through
the tibia and tarsus.
The narrow 24deg range of elevation angles used by fleas
contrasts with the much wider 80deg range used by locusts when
jumping [fig. 9A in Sutton and Burrows (Sutton and Burrows,
2008)]. The ranges are significantly different (P0.0011, ratio of
variances=0.093, d.f.=9; R var.test function). Data on take-off
velocity and elevation are not available for other species of fleas,
but the horizontal distance that seven species jump has been
reported (Krasnov et al., 2003). For these fleas, the minimum
distance of a jump was between 39 and 73% (Fig.8C) of the
maximum jumping distance (Krasnov et al., 2003). Although we
did not directly measure the distance jumped by the fleas in our
experiments, we have used the physics of a ballistic projectile to
answer the following question: does the restricted elevation of A.
Fig.5. Photographs of a natural jump of
A. erinacei
in which the
trochanter began close to the ground. The yellow arrow indicates
the position of the hind coxa at the point when the hindlegs first
move to propel a jump. Selected images at the times indicated
are arranged in two columns with the bottom left hand corner as
a constant point of reference. The positions of the segments of
the right hindleg are shown above each image and are
superimposed in yellow on the first image. The angle of the body
relative to the ground is shown for four images. The yellow
arrows in the images at –0.6 and –1.2ms indicate that the
trochanter was clear of the ground before and at the first
movement of a hindleg. Take-off occurred at 0ms. Images in this
and Fig.6 were captured at 5000framess–1.
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842
erinacei result in restricted variation in possible jump distance? The
jumping distance of A. erinacei can be estimated from Eqn1:
D2V2sin() cos() / g, (1)
where Dis distance, Vis velocity, is the angle of elevation at
take-off and gis the acceleration due to gravity.
From our measurements, an A. erinacei jumping with a constant
velocity would have a minimum jump distance (at 28deg elevation)
that is 83% of the maximum distance (at 45deg) (Fig.8C). The
window for jump elevation in A. erinacei is thus insufficient to
generate jump distances that are as variable as those in other species
of flea. If velocity is taken into consideration, however, A. erinacei
will have variations in jump distance that are similar to those of
other fleas. The window of jump velocities for A. erinacei ranges
between 0.9 and 1.85ms–1 (Fig.9A), which means that the minimum
jump distance will be 25% that of the maximum (Fig.9B). This
suggests that jump distances in A. erinacei are a little more variable
G. P. Sutton and M. Burrows
than those in other species of flea [Fig.10B, compare lines i and iii
(Krasnov et al., 2003)] and that it generates these differences more
by altering its take-off velocity than its angle of elevation. It is not
known which of these two parameters determine jump distances in
other fleas.
Kinetic modelling
To model the Rothschild hypothesis, which proposed that ground
forces are directed through the trochanter, the flea was considered
as a mass with a spring that applies its force through the coxa and
trochanter directly to the ground (Fig.10A). This model has similar
dynamics to a mass–spring system. To model the Bennet-Clark
hypothesis, which proposed that ground forces are directed through
the hind tibia and tarsus, the flea was considered as a three-link
mechanical system (Fig.10B): one link represented the coxa, the
second the femur and trochanter (Fig.2) and the third the tibia. The
first question was to determine if either model could duplicate the
Fig.6. Photographs of a natural jump of
A.
erinacei
in which the trochanter began off the
ground. Selected images are annotated as in
Fig.5.
THE JOURNAL OF EXPERIMENTAL BIOLOGY

843Jump of a flea
kinematics observed by Rothschild et al. (Rothschild et al., 1975).
We chose this kinematic data in an effort to give the Rothschild
hypothesis every advantage over the Bennet-Clark hypothesis.
The kinetic model of the Rothschild hypothesis (Fig.10A)
predicted a final velocity that was identical to that observed in natural
jumping (Fig.11B). First, the displacement of the centre of mass of
the body predicted by the model described well the trajectory of a
natural jump (Fig.11A). Second, the velocity at take-off predicted
by this model was also consistent with the observed velocity (both
are 1.35ms–1). The time course by which this end result was
achieved was very different between the Rothschild model and the
kinematics observed in a natural jump (Fig.11B). The model
predicted that there was a very fast change from stationary to the
final take-off velocity, whereas in natural jumping the velocity
increased over an acceleration period of approximately 1ms. The
model predicted that the acceleration should peak rapidly at the high
value of 22,000ms–2, when the hindlegs first moved, and should
then decrement to zero 0.7ms before take-off (Fig.11C). In natural
jumps, however, the initial acceleration as the hindlegs started to
move was much smaller, at 500ms–2, rose steadily to 1500ms–2,
but then dropped to zero at take-off (Fig.11C). This model therefore
failed to predict the time course of velocity and acceleration of a
natural jump.
The kinetic model of the Bennet-Clark hypothesis (Fig.10B), in
contrast, predicted displacements, velocities and accelerations that
were similar to those observed during natural jumps (Fig.11A–C).
First, the displacement of the centre of mass of the body predicted
by the model described the trajectory of a natural jump well
(Fig.11A). Second, the model accurately predicted both the time
course and the final take-off velocity (Fig.11B). The model
accurately predicted the time course of acceleration; it predicted an
acceleration of 450ms–2 when the hindlegs first moved, a peak at
1800ms–2, and then a return to zero just before take-off (Fig.11C).
This form of acceleration resulted in an S-shaped curve for velocity
(Fig.11B), in good agreement with the data from natural jumps
(Rothschild et al., 1975) and our data (Fig.7A).
The gradually rising and rapidly falling shape of the acceleration
curve predicted by the Bennet-Clark model is a direct consequence
Displacement (mm)
Velocity (m s–1)
Time (ms)
0.5
1
1.5
0
0.5
1.0
1.5
0
Hind trochanter
off ground
Take-off
Hind trochanter
off ground
Take-off
–1. 5–1–0.50.50
–1. 5 –1 – 0. 5 0. 50
0.5
1.0
1.5
2.0
2.5
0
0.5
1
1.5
2
0
A
B
Fig.7. Graphs of the displacement (unfilled circles) and velocity (filled
circles) of the centre of mass of the body for the jumps shown in Fig.5
when the hind trochanter was close to the ground at the start and left the
ground at –0.6ms (A, orange vertical line) and Fig.6 when the trochanter
started off of the ground (B, orange vertical line). The solid line in B shows
the velocity kinematics for
A. erinacei
derived from the Bennet-Clark model
and the dashed line shows the position kinematics. The profiles of each
parameter are the same for each jump.
12345678910
100
80
60
40
20
Individual
Locusts
N=15
N=15
All fleas
20 40 60 800
1.0
0.8
0.6
0.4
0.2
0
i
ii
iii
Take-off elevation (deg)
Take-off elevation (deg)
Take-off elevation (deg)
C
Jump distance/max.
distance
A
B
303540455025 55
2
4
6
8
0
Occurrrence
N=6
N=5
N=5
N=5
N=6
N=3
N=6
N=3
N=9
N=3
Fig.8. Control of jump elevation. (A)Histogram to show the distribution of
take-off angles in the 51 jumps by 10
A. erinacei
. (B)Box plot of the jump
elevations for all the fleas in this study, shown individually (1–10) and
together (all fleas). The horizontal grey lines define the upper and lower
limits of the 24deg window of take-off angles. For comparison, the box plot
on the right shows the larger range of take-off angles for locusts (Sutton
and Burrows, 2008). (C)Projected distances of jumps by fleas. The vertical
bars show the minimum and maximum distances that
A. erinacei
would
jump using: (i) the range of take-off angles in A and assuming a constant
take-off velocity, (ii) the minimum and (iii) maximum distances jumped by
some other fleas based on data from Krasnov et al. (Krasnov et al., 2003).
The shaded area of the graph marks the area of jump elevations observed
for
A. erinacei
.
THE JOURNAL OF EXPERIMENTAL BIOLOGY

844
of the mechanics of a spring–moment arm system. At the start of
a jump, the force in the spring will be high, but the moment arm is
small, thus the product of the force and the moment arm starts small.
This results in a small ground reaction force. Because the coxo-
trochanteral joint depresses to propel the jump, the force in the spring
will decrease linearly, but the moment arm will increase as a sine
wave function. The moment arm initially increases faster than the
force decreases, so that the product of the two increases (Fig.11C).
Finally, as the sine wave function of the moment arm reaches a
maximum, the spring force decreases faster than the moment arm
increases, so that the product of the two decreases (Fig.11C). The
initial rise and subsequent fall of the acceleration curve is consistent
with a combination of an elastic mechanism and a rotating lever
arm system.
Parameters of the Bennet-Clark model were varied to reflect the
morphology of A. erinacei (Fig.7A) and other possible morphologies
(data not shown). Take-off velocity and take-off times could be
changed by altering the model parameters. The qualitative shape of
the acceleration curve, however, was insensitive to changes in the
model parameters, because the basic geometry of a moment arm
combined with an elastic mechanism constrains the acceleration time
curve to have this form. In particular, the shape of the acceleration
curve was insensitive to the stiffness of the spring. The model was
re-run with a series of linear springs, which all stored sufficient
kinetic energy to achieve a take-off velocity of 1.3ms–1, and all of
these springs returned an acceleration vs time curve that had an initial
rise followed by a fall. The Rothschild model, however, was unable
to generate time courses of acceleration that were similar to our
kinematic analysis of A. erinacei. This leads us to the conclusion
that the Bennet-Clark model is more able to duplicate the in vivo
kinematics than the Rothschild model.
DISCUSSION
This paper has provided five lines of evidence to support the
hypothesis proposed by Bennet-Clark and Lucey (Bennet-Clark and
Lucey, 1967) that fleas jump by transmitting force developed by
muscles in the thorax through their hind tibiae and tarsi to the ground:
(1) extensive structures are present on the tarsus and the end of the
tibia, but not on the trochanter, which could be used to grip a
substrate; (2) fleas can jump even when the trochantera are not in
contact with the ground; (3) the acceleration of a jump does not
G. P. Sutton and M. Burrows
decrease when the trochantera lose any contact they may have with
the ground; (4) jumps have large horizontal components; and (5)
the acceleration of a jump is consistent with a kinetic model that
applies force to the ground through the tarsus and tibia but is not
consistent with a model that applies force through the trochanter.
0.80.6 1.00.2 0.40
1.0
0.8
0.6
0.4
0.2
0
i
ii
iii
B
A
1.0 1.2 1.4 1.6 1.8
0.82.0
2
4
6
8
10
12
0
Take-off velocity/max. velocity
Take-off velocity (m s–1)
Jump distance/max.
distance
Occurrence
Fig.9. Jump velocity. (A)Histogram of the take-off velocities that ranged between 0.9 and 1.85ms–1in 51 jumps by 10
A. erinacei
. (B)Projected distances
that fleas would jump with a constant elevation of 39deg. (i)Ratio of minimum and maximum distance predicted by fleas jumping with a constant elevation.
(ii)Minimum and (iii) maximum ratio for the data of Krasnov et al. (Krasnov et al., 2003). The shaded area of the graph marks the area of jump velocities
observed for
A. erinacei
.
TibiaCoxa
Femur–trochanter
BBennet-Clark model
Mass
Resilin pad
ARothschild model
Coxa–trochanter
F
F
Fig.10. (A)Mechanical model for the hypothesis that a flea directs forces to
the ground with the trochanter (Rothschild model). The resilin pad is
modelled as a spring between the ground and the centre of mass.
(B)Mechanical model for the hypothesis that a flea directs forces to the
ground with the tibia and tarsus (Bennet-Clark model). The leg is modelled
as a three link system; tibia, femur–trochanter and coxa, with the resilin
pad acting as a spring that generates a torque about the coxo-trochanteral
joint. In both models, the centre of mass was placed at the end of the
coxa.
THE JOURNAL OF EXPERIMENTAL BIOLOGY

845Jump of a flea
The only evidence to support the hypothesis of Rothschild and
Schlein (Rothschild and Schlein, 1975) that the force is transmitted
to the ground by the hind trochantera is that: (1) during a majority
of jumps the trochantera start in close proximity to the ground; and
(2) there are short hairs on the trochantera that could possibly be
used to grip the ground. Although both the Bennet-Clark and the
Rothschild models propose viable mechanisms to propel a jump,
only the Bennet-Clark model proposes a mechanism that is
consistent with the accelerations, velocities, initial position
kinematics, take-off trajectories and the attachment mechanisms
observed in vivo. Moreover, the kinematic data presented here and
the data presented by Rothschild et al. are both consistent with the
Bennet-Clark hypothesis (Rothschild et al., 1975).
From the data presented here it is not possible to differentiate
between the hypotheses stated by Bennet-Clark and by Rothschild
about the locking mechanisms for the leg joints. Bennet-Clark
hypothesized that the joints were locked by the moment arm of the
trochanteral depressor muscle going ‘over centre’, and thus the
resilin ‘spring’ would act initially as a trochanteral levator [a similar
mechanism has been proposed for the claw snap of an alpheid shrimp
(Ritzmann, 1974) and for the femoro-tibial joint of the locust hindleg
(Cofer et al., 2010)]. Subsequently, activation of another muscle
pulls the apodeme over so that the spring acts as a depressor, thus
triggering the jump (Bennet-Clark and Lucey, 1967). Rothschild
hypothesized that the joints were locked via a series of cuticular
‘catches’ along the hindlegs and the body, and that the release of
these locks triggered the jump (Rothschild and Schlein, 1975). To
distinguish between this aspect of the two hypotheses requires a
different sets of experiments that we have not attempted, and thus
the release mechanism remains unresolved.
Control of jump direction and distance
The fact that the flea exerts force to the ground through the tibia
means that its leg mechanics are very similar to those of the
froghopper (Hemiptera, Auchenorrhyncha, Cercopidae) (Burrows,
2006a; Sutton and Burrows, 2010). Froghoppers and fleas both have
an elastic storage mechanism that applies a large torque about the
coxo-trochanteral joint, a trochanter that is mechanically linked
closely to the femur, and very little torque is applied about their
femoro-tibial joints. These insects do differ in that the legs of a flea
are orientated in the sagittal plane of the animal, in contrast to the
legs of a froghopper that are orientated in the coronal plane. The
mechanics predict that the legs of both froghoppers and fleas
generate forces that are parallel to their hind tibiae. From this, it
follows that the mechanisms that froghoppers use to control jump
azimuth could be used by fleas to control jump elevation. A. erinacei,
however, does not vary greatly the elevation of its jumps. The 51
jumps analysed here were all within a 24deg window of elevation.
By contrast, the jumps of locusts (Schistocerca) and flea beetles
(Alticinae) both have almost a 90deg window of elevation
(Brackenbury and Wang, 1995; Sutton and Burrows, 2010), and the
jumps of froghoppers have a 72deg window of elevation (Burrows,
2006b).
The small window of jump elevations used by fleas is puzzling
because it would seem to be in its best interest to jump in a wide
range unpredictable directions. It has been observed that some
invertebrates have a limited number of preferred escape trajectories
(Domenici et al., 2008), but A. erinacei appears to be different in
that it has only one small window of escape trajectories. A. erinacei
can vary its jumping distance, and it does so not by adjusting its
take-off elevation, but by adjusting its take-off velocity. This
variation in jump distance is consistent with the variation in jump
0.2
0.4
0.6
0.8
1.0
1.2
0
1.4
1.0 –– 0.5 0 0.5
1.0 –– 0.5 0 0.5
1000
2000
3000
4000
0
m m m
0.5
1.0
1.5
0
1.0 –– 0.5 0 0.5
First movement of hindlegs
A
B
C
Displacement (mm)
Time (ms)
Bennet-Clark
model
Rothschild
model
Bennet-Clark model
Experiemental data
Rothschild model
Velocity (m s–1)
Acceleration (m s–2)
Take-off
Fig.11. Comparison of the outcomes of the Rothschild model for jumping
(grey line), and the Bennet-Clark model (black line) with data from natural
jumps of
Xenopsylla cheopis
(Rothschild et al., 1975) (black dots). The
displacement (A), velocity (B) and acceleration (C) of the centre of mass
are shown. Take-off occurred at time zero (vertical yellow line) with the first
movement of the hindlegs occurring at different times (vertical teal and
magenta lines) for the two models. In the Rothschild model, the predicted
acceleration extended above the
y
-axis to 22,000ms–2. Insets in C show
the positions of the femur and coxa in the Bennet-Clark model illustrating
that the rise and fall in acceleration (black line) is caused by a combination
of a decreasing spring force (downward pointing arrows) and an increasing
moment arm (marked by ‘m’). See text for details.
THE JOURNAL OF EXPERIMENTAL BIOLOGY

846 G. P. Sutton and M. Burrows
distance seen in other species of flea (Krasnov et al., 2003) and
strongly suggests that fleas in general vary the distance they jump
by changing jump velocity and not jump elevation. This makes the
amount of energy stored before a jump a crucial control parameter
for flea jumps. Although it is possible that the high-density foam
platform used here restricts the directions in which the flea is able
to jump, we do not think this is probable, because other species of
insect have had little difficulty generating different jump trajectories
from this surface (Burrows, 2010; Burrows, 2009; Burrows, 2003).
It is ironic that a famous jumper like a flea has such a restricted
jump direction relative to insects less renowned for their jumping
prowess, such as locusts (Bennet-Clark, 1975; Santer et al., 2005;
Sutton and Burrows, 2008), froghoppers (Burrows, 2003; Sutton
and Burrows, 2010), flies (Drosophila) (Card and Dickinson, 2008)
or flea beetles (Alticinae) (Brackenbury and Wang, 1995). Because
fleas also use jumps to reach prey, it is also reasonable to compare
restricted jump trajectory to fast predatory strikes. The flea jump is
more constrained in direction than frog tongue strikes (Mallet et al.,
2001; Monroy and Nishikawa, 2009), squid tentacle strikes (Van
Leeuwen and Kier, 1997) or spider predatory jumps (Weihmann et
al., 2010). Applying force to the ground via the tibia gives the flea
a theoretical ability to control its take-off elevation, but it does not
use this mechanism.
Generality of proposed mechanisms in other fleas
Are the mechanisms that we report for A. erinacei representative
of other species of flea such as Xenopsylla cheops (Rothschild and
Schlein, 1975), Spilopsyllus cuniculus (Bennet-Clark and Lucey,
1967) and several species examined by Krasnov et al., including
Xenopsylla conformis (Krasnov et al., 2003)? The average velocity
at take-off (1.3±0.22ms–1) in A. erinacei is slightly faster the average
for Spilopsyllus cuniculus [1.0ms–1 (Bennet-Clark and Lucey,
1967)]. The average jumping range from this take-off velocity,
169mm (from Eqn1, at an elevation of 39deg), places A. erinacei
in the upper range of performance for the flea species measured by
Krasnov (Krasnov et al., 2003). A. erinacei also has a similar
variability in the range of its jumps compared with other flea species
(Fig.9). A. erinacei also begins a jump with an initial position
indistinguishable from the initial position taken by Xenopsylla
cheops (Fig.4) (Rothschild et al., 1975). Finally, the body plan of
A. erinacei is very similar to the body plan of other fleas (Grimaldi
and Engel, 2005). Based on these data, we suggest that the
mechanical conclusions about flea jumping based on jumps by A.
erinacei are applicable to other species of flea.
APPENDIX 1
The equations of motion for the Rothschild model of flea jumping
were developed based on the hypothesis that the cuticular ‘spring’
applied forces directly to the ground through the hind trochanter.
In this model, the recoil of the spring pushes directly down on the
trochanter and onto the ground.
The equation of motion is thus:
MCOM Ÿ–KSpring Y, (A1)
where MCOM is the mass of the flea, Ÿis the vertical acceleration
of the flea, KSpring is the spring stiffness and Yis the vertical position
of the flea. The spring is constrained only to be able to provide
positive forces. Take-off is when the spring displacement0.
Parameters for the model are presented in TableA1.
APPENDIX 2
The equations of motion for the Bennet-Clark model of flea jumping
were developed based on the hypothesis that the cuticular ‘spring’
applied forces through the hindlegs as a system of lever arms, with
the final ground reaction forces being directed through the tibia.
This model has a larger set of parameters, based on the properties
of the spring and on the properties of the leg segments. These
parameters are presented in TableA1. The model output was almost
completely insensitive to the mass and moment of inertia of the leg
segments, but these values were included because running the model
with massless leg segments created numerical instability in the
simulation.
Equations of motion for each segment were generated using a
Lagrangian formulation (Greenwood, 1988), which derives the
equations of motion for the torques on each link (TTibia, TFemur, TCoxa)
by differentiating the energy (E) of the system by the angle of each
link (Tibia, Femur, Coxa).
The kinetic energy of the system was then calculated in terms of
the masses of the body, coxa, femur and tibia (MCOM, MCoxa, MFemur
and MTibia, respectively), the velocities of the body, coxa, femur
and tibia (VCOM, VCoxa, VFemur and VTibia, respectively), the moments
of inertia of the body, coxa, femur and tibia (ICOM, ICoxa, IFemur and
Table A1. Model parameters for the Rothschild and Bennet-Clark
models, based on
Xenopsylla cheopis
Parameter Value Units Source*
M
COM 0.2 mg a
I
COM 0.144 mgmm2b
M
Coxa 0.004 mg c
I
Coxa 5.3⫻10–5 mgmm2d
L
Coxa 0.4 mm a
M
Femur 0.04 mg c
I
Femur 5.3⫻10–5 mgmm2d
L
Femur 0.4 mm a
M
Tibia 0.004 mg c
I
Tibia 1.2⫻10–4 mgmm2d
L
Tibia 0.6 mm a
K
Spring 55.0 Nm–1 e
L
Initial 0.35 mm e
L
M0.055 mm a
Initial 

Coxa 80.0 deg f
Initial



Femur –90.0 deg f
Initial



Tibia 30.0 deg f
Initial ␪Coxa 0 degs–1 f
Initial ␪Femur 0 degs–1 f
Initial ␪Tibia 0 degs–1 f
I
, inertia;
K
, stiffness;
L
, length;
M
, mass; , angle of elevation at take-off;
␪,y.
*Sources: a, Bennet-Clark and Lucey (Bennet-Clark and Lucey, 1967) and
Rothschild and Schlein (Rothschild and Schlein, 1975). b, The moment of
inertia of the center of mass was estimated by approximating the flea’s
body as a uniform disc. c, The mass of the coxa, tibia and femur were
estimated, but had little affect on the output of the simulation. They were
given mass because modelling the legs as massless resulted in equations
of motion that were unstable (see Appendix 2). d, Moments of inertia for
the coxa, femur and tibia were estimated by approximating the legs as
uniform rods. e, The stiffness and resting length of the spring were
estimated by assuming that the initial energy stored in the spring (1/2
stiffness ⫻spring compression2) was equal to the final kinetic energy of
the jump (1/2 mass ⫻velocity2). Many values of spring stiffness and initial
spring length were modelled, but the qualitative shape of the output was
not dependent on this parameter. f, The starting leg angles were
determined from the leg kinematics of
A. erinacei
because starting leg
kinematics were hard to determine from the photos in Rothschild et al.
(Rothschild et al., 1975).
THE JOURNAL OF EXPERIMENTAL BIOLOGY

847Jump of a flea
ITibia, respectively) and the angular velocities of the coxa, femur and
tibia (␪Coxa, ␪Femur and ␪Tibia, respectively). The model’s centre of
mass was rigidly attached to the coxa and thus had the same angular
velocity as the coxa:
The equations of motion in terms of the torques (TFemur, TTibia,
TCoxa) on the links were generated using Lagrange’s equations
(Greenwood, 1988):
For small insects, the gravity terms are negligible and thus were
omitted (Scholz et al., 2006).
The torques were calculated by modelling the resilin lump as a
spring connecting the centre of mass with the end of a moment arm
(LM) on the trochanter–femur link. This produced equal and opposite
torques on the femur and the coxa, which generated an expression
for TCoxa and TFemur: in terms of the stiffness of the spring (KSpring),
the length of the spring (LSpring) and the moment arm of the trochanter
(ArmSpring):
TCoxa –TFemur KSpring LSpring ArmSpring(A6)
(KSpring is identical to the stiffness for the Rothschild model). LSpring
is the distance between the attachment point of the spring and the
centre of mass minus the resting length of the spring in terms of
the geometry of the leg. As the legs rotate, this distance changes
with the function:
where LCoxa and LMare the lengths of the coxa and the trochanteral
moment arm for the spring, respectively. Coxa and Femur are the
angles of the coxa and femur. Linitial is the resting length of the
spring.
ArmSpring is the component of the moment arm perpendicular to
the force applied by the spring:
Lastly, we assumed that the small musculature connecting the
femur to the tibia allowed no torques to be applied at that joint, i.e.
TTibia0.
At each timestep the ten equations were solved for the angular
accelerations of the links (␪Coxa, ␪Femur and ␪Tibia). The equations
were solved with the Mathematica 5.2 NDSolve function. The model
parameters are shown in TableA1.
E=1
2(MCOMV
COM
2+MCoxaV
Coxa
2+MFemurV
Femur
2
+MTibiaV
Tibia
2+ICOM

θCoxa
2+ICoxa

θCoxa
2
+IFemur

θFemur
2+ITibia

θTibia
2) . (A2)
TFemur =D
Dt
∂E
∂
θFemur
⎛
⎝
⎜⎞
⎠
⎟ , (A3)
T
Tibi a =D
Dt
∂E
∂
θTibi a
⎛
⎝
⎜⎞
⎠
⎟ , (A4)
T
Coxa =D
Dt
∂E
∂
θCoxa
⎛
⎝
⎜⎞
⎠
⎟ . (A5)
L
Spring =L
Coxa
2+LM
2−2L
Coxa LMcos(θCoxa −θFemur )−L
initial , ( A 7 )
ArmSpring =L
Coxa L
m
L
Coxa
2+L
m
2−2L
Coxa LMcos(θCoxa −θFemur )
sin (θFemur −θ
Coxa ) . (A8)
ACKNOWLEDGEMENTS
G.P.S. was funded by the Marshall Sherfield Commission and the Human
Frontiers Research Program. We thank Cambridge colleagues for their many
helpful suggestions during the course of this work and for their comments on the
draft manuscript.
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