Meandering worms: Mechanics of undulatory burrowing in muds

Abstract

Recent work has shown that muddy sediments are elastic solids through which animals extend burrows by fracture, whereas non-cohesive granular sands fluidize around some burrowers. These different mechanical responses are reflected in the morphologies and behaviours of their respective inhabitants. However, Armandia brevis, a mud-burrowing opheliid polychaete, lacks an expansible anterior consistent with fracturing mud, and instead uses undulatory movements similar to those of sandfish lizards that fluidize desert sands. Here, we show that A. brevis neither fractures nor fluidizes sediments, but instead uses a third mechanism, plastically rearranging sediment grains to create a burrow. The curvature of the undulating body fits meander geometry used to describe rivers, and changes in curvature driven by muscle contraction are similar for swimming and burrowing worms, indicating that the same gait is used in both sediments and water. Large calculated friction forces for undulatory burrowers suggest that sediment mechanics affect undulatory and peristaltic burrowers differently; undulatory burrowing may be more effective for small worms that live in sediments not compacted or cohesive enough to extend burrows by fracture.

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, 20122948, published 27 February 2013280 2013 Proc. R. Soc. B
Kelly M. Dorgan, Chris J. Law and Greg W. Rouse
muds
Meandering worms: mechanics of undulatory burrowing in
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Research
Cite this article: Dorgan KM, Law CJ, Rouse
GW. 2013 Meandering worms: mechanics of
undulatory burrowing in muds. Proc R Soc B
280: 20122948.
http://dx.doi.org/10.1098/rspb.2012.2948
Received: 10 December 2012
Accepted: 5 February 2013
Subject Areas:
biomechanics, ecology, environmental science
Keywords:
burrowing, gait, biomechanics, kinematics,
polychaete, annelida
Author for correspondence:
Kelly M. Dorgan
e-mail: kdorgan@ucsd.edu
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rspb.2012.2948 or
via http://rspb.royalsocietypublishing.org.
Meandering worms: mechanics of
undulatory burrowing in muds
Kelly M. Dorgan, Chris J. Law and Greg W. Rouse
Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093–0202, USA
Recent work has shown that muddy sediments are elastic solids through
which animals extend burrows by fracture, whereas non-cohesive granular
sands fluidize around some burrowers. These different mechanical responses
are reflected in the morphologies and behaviours of their respective inhabi-
tants. However, Armandia brevis, a mud-burrowing opheliid polychaete,
lacks an expansible anterior consistent with fracturing mud, and instead
uses undulatory movements similar to those of sandfish lizards that fluidize
desert sands. Here, we show that A. brevis neither fractures nor fluidizes sedi-
ments, but instead uses a third mechanism, plastically rearranging sediment
grains to create a burrow. The curvature of the undulating body fits meander
geometry used to describe rivers, and changes in curvature driven by muscle
contraction are similar for swimming and burrowing worms, indicating that
the same gait is used in both sediments and water. Large calculated friction
forces for undulatory burrowers suggest that sediment mechanics affect undu-
latory and peristaltic burrowers differently; undulatory burrowing may be
more effective for small worms that live in sediments not compacted or
cohesive enough to extend burrows by fracture.
1. Introduction
Mechanical interactions between organisms and their environments are integral
to locomotion, but mechanical responses of soils and sediments to forces applied
by burrowing organisms are poorly understood. How morphologies and beha-
viours of infauna affect burrowing performance is important in understanding
the evolution of burrowing animals and the ecology of sediment communities.
Many worms with diverse morphologies and behaviours extend burrows
through muddy sediments by fracture, using eversible mouth parts and muscu-
lar expansions to apply dorsoventral forces to burrow walls that are amplified at
the burrow tip [1– 3]. Fracture of muddy sediments results not only from direct
forces applied by burrowers, but also from hydraulic pumping by infauna during
burrow irrigation [4].
Nematodes and some polychaetes, however, lack expansible anteriors used
for burrow extension by fracture, and instead move through muddy sediments
by undulation [5]. For these organisms, kinematics are similar to those of sand-
fish lizards, which use body undulations rather than limbs to generate thrust and
fluidize desert sands [6]. Fluidization of the medium is indicated by backward
slipping of the animal and bulk transport of suspended grains in the opposite
direction of locomotion. Mechanical responses differ, however, among burrowers
in non-cohesive granular sands: fishes such as sand lances and eels burrow in
saturated marine sands with no slipping [7,8]. Kinematics are similar to terrestrial
crawling, and burrowing results in small discrete movements of sand grains
rather than bulk transport. In both cases, movement involves displacement of dis-
crete grains against gravitational forces, in contrast to elastic muds, which are
held together by adhesion and cohesion of the intra-granular organic matrix [9].
These different responses of sands to undulatory burrowers are quantified
using wave efficiency,
h
¼v
x
/v
w
, the ratio of the animal’s forward velocity, v
x
,
to the velocity of the posteriorly travelling undulatory wave, v
w
. Wave efficiencies
of
h
0.5 for burrowing sandfish lizards are consistent with fluidization of gran-
ular sand [6], and of
h
1 for sand lances and eels are consistent with a solid
response of sand [7,8]. For swimming animals, wave efficiency varies
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considerably and depends on Reynolds number (Re)[10],with
high values of greater than 0.7 for swimming eels [11,12] down
to 0.23 for sperm [13]. For swimming leeches,
h
is proportional
to size and drops from 0.55 to 0.43 with a 10-fold increase in
viscosity [14]. Crawling animals have high
h
, greater than
0.8 compared with 0.19–0.29 for swimming nematodes [5],
approaching 1 on surfaces with sufficient friction in which
no slipping occurs.
Undulatory burrowing has not yet been quantitatively
explored in muddy sediments, which are elastic solids that
differ mechanically from non-cohesive sands [9]. Armandia
brevis (Moore, 1906), an opheliid polychaete that inhabits
muddy to sandy sediments, lacks both eversible mouthparts
and circular muscles needed for peristalsis. Rather, it has
bands of oblique muscles that act antagonistically to longitudi-
nal muscles, enabling lateral bending and undulatory
movements (figure 1). Armandia brevis also swims by undula-
tion, enabling direct comparison of wave efficiency while
burrowing and swimming. Wave efficiency values for burrow-
ing A. brevis close to 1 and higher than that of swimming
worms would indicate solid response of the medium, whereas
h
1 would indicate fluidization.
For A. brevis, swimming is a derived behaviour—apart
from this taxon and other members of its own subfamily
Ophelininae, no other opheliid swims [15]. Though dispersal
is one explanation, most swimming A. brevis are reproductive,
and spawning occurs only once before death [16]. That swim-
ming seems to be a secondary mode of locomotion raises the
question of whether swimming behaviours are distinct from
burrowing, i.e. whether the same gait is used in both media
with kinematic differences attributable to mechanics of solids
compared with fluids.
Discrete gaits are characterized by discontinuous changes
in movement patterns. Both sandfish lizards and sandlance
fishes substantially change their body shapes, increasing the
amplitude relative to wavelength (A/
l
), when transitioning
to burrowing from running and swimming, respectively
[6,7]. Larger A,
l
and frequency have been observed for nema-
tode worms swimming in water compared with media with
higher resistance, but positive linear relationships indicate con-
stant A/
l
and reveal no discrete transition indicative of gait
change [17,18]. Although the transition from fluid to elastic
mud is inherently discontinuous, similar body shape of
A. brevis, characterized by A/
l
, would be consistent with the
same gait used in both media. As body shape is determined
from discrete time points and is affected by external forces
[19], we also compared changes in body angle over a cycle
of undulatory movement as well as patterns of body curvature
to determine whether burrowing and swimming gaits are the
same. Whereas undulatory movements have been described as
sinusoidal [10] and a best-fit sine function has been used for
measuring amplitude, wavelength and wave speed [6], curva-
ture of the path of A. brevis appeared flatter and broader than a
sine function. Similar geometry was first described for paths of
meandering rivers, which can be approximated as a sine-
generated curve; the relationship between direction angle of
the path,
u
, and distance along the path, s, fits a sine curve
[20]. A sine fit to
u
–sis a better approximation of snake
shape than an x–ysine fit, consistent with waves of alternating
muscle contraction travelling along the length of the animal
changing the body curvature [21]. Direct measurement of kin-
ematic parameters such as amplitude, wavelength, radius of
curvature and v
w
from profiles enables comparisons, but
lacks a mechanistic basis; fit to a meander curve would be con-
sistent with sinusoidal muscle contraction and relaxation,
whereas deviations indicate non-sinusoidal contraction or
non-uniform external forces.
We explored the burrowing and swimming behaviour of
A. brevis and combined experiments with theory to assess
three hypotheses for the mechanism of undulatory burrow
extension in muds: (i) A. brevis extends burrows by fracture
[1], (ii) A. brevis fluidizes muddy sediments, indicated by
h
1[6], and (iii) the muddy sediment deforms plastically
through grain rearrangement, indicated by
h
1 [7,8].
2. Material and methods
To address whether A. brevis extends burrows by fracture, we
combined experiments on worms in gelatin, an analogue for cohe-
sive elastic muds through which worms burrow by fracture [1– 3],
with theoretical predictions based on linear elastic fracture mech-
anics (LEFM). Because A. brevis lives in surface sediments [16], we
developed an additional analogue for weak surface muds com-
prising organic-mineral aggregates, fragments of concentrated
gelatin, in which kinematics were analysed to determine whether
fluidization or solid grain rearrangement occurs. Kinematics of
burrowing and swimming worms were compared to assess
whether A. brevis uses the same gait in the different media.
(a) Kinematics
Armandia brevis were collected from shallow subtidal sediments
in Mission Bay, San Diego, CA, and from sediments in the flow-
ing seawater tanks at Scripps Institution of Oceanography, La
Jolla, CA. For experiments to determine whether worms bur-
rowed by fracture, glass aquaria were filled with seawater
gelatin (28.35 g l
21
), and worms were placed in pre-made
cracks and filmed following methods of Che & Dorgan [3]. For
elastic aggregate burrowing experiments, concentrated seawater
gelatin (85 g l
21
) was chopped in a food processor to fragments
OM
LM
LM
c
g
OM
es
p
(a)
(b)(c)
Figure 1. (a) Morphology of Armandia brevis.(b) Ventral view of oblique
muscles (OM) shown using polarized light microscopy with segments distin-
guished by parapodia (p) and segmental eye spots (es). (c) Cross-section of
internal anatomy showing large bands of oblique (OM) and longitudinal
muscle (LM), gut ( g), and cuticle (c). (Online version in colour.)
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roughly 200– 500 mm diameter. Gelatin pieces were slowly added
to a plexiglass ant farm aquarium (7 71.5 cm) with seawater
and were gently mixed to remove air bubbles. Worms were
placed below the surface of the aggregates equidistant from the
two walls of the aquarium, and movements were recorded.
Only video segments in which the worms moved in a plane per-
pendicular to the camera angle (in focus within an approx. 1 cm
focal plane) and did not reach either wall were used for analysis.
Worms were filmed swimming in a Petri dish (10 2 cm) with a
Canon T2i DSLR camera at 60 fps. For both burrowing and
swimming, wall effects may occur, but eliminating or substan-
tially reducing wall effects was logistically difficult because of
visualization of burrowing worms and the speed of swimming
worms. Videos were first subsampled using QUICKTIME PRO v. 7
(swimming) or LABVIEW v. 7.1 (burrowing), and processed in
IMAGEJ v. 1.44 to obtain body outlines and head and tail positions
for each frame.
To characterize gait, first body shape was analysed by
measuring amplitude and wavelength. From outline coordinates
and head and tail positions, midlines were calculated and further
analysis done using custom MATLAB (R2010B) scripts. To measure
amplitude and wavelength, curvature of the path of travel
(smoothed centre of mass (COM) path) was removed by sequen-
tially rotating the midline for each frame and later frames about
the COM for that frame. Next, maximum deviations of midlines
from that straightened path and distances between peaks were cal-
culated. For consistency, midlines were shortened to total body
length, L
s
, of 95 per cent of the shortest midline in a sequence.
We compared fits with a meander curve and sine curve for
both burrowing and swimming worms. Shortened midlines
were converted from x–y to
u
–scoordinates using a cartesian-
to-polar coordinate conversion for each segment dsof the
midline. Best-fit sine function in x–y coordinates was the sine
fit, in
u
–scoordinates was the meander fit. Because both sine
and meander curves were fit to a line of points, residuals were
not randomly distributed and the autocorrelation of midline
points likely resulted in inflated R
2
, but subsampling did not
effectively remove the autocorrelation or change the R
2
.To
better assess the fit of the two models, we also compared the
body angle, sin(
u
), and curvature, d
u
/ds, profiles to those
predicted by a sine and meander fit (see the electronic supplemen-
tary materials for details). The sine and meander curves were fit to
individual frames rather than the path, for which both amplitude
and wavelength varied considerably, making modelling the path
as a single travelling sine wave infeasible.
Wave efficiency was calculated as
h
¼v
x
/v
w
, where the wave
velocity in
u
–s coordinates calculated from cross correlation of
subsequent frames was transformed to x–y coordinates by the
sinuosity, the ratio of the body length (before shortening) to
the shortest distance from head to tail, to obtain v
w
. Sinuosity
was inflated for swimming worms by yaw, seen as side-to-side
movement of the COM, so rather than using v
x
calculated from
the smoothed COM path, velocity was calculated along the
unsmoothed COM path (greater than v
x
for swimming, but
approx. v
x
for burrowing) and this corrected velocity was used
in wave efficiency calculations.
(b) Model
LEFM theory was used to calculate a force balance for burrow
extension by fracture for an undulating worm. Forces resisting
forward movement in linear elastic muds include cohesive frac-
ture resistance, elastic resistance to sediment deformation and
friction. The work to extend a burrow by fracture is the sum of
the work of fracture, W
Cr
¼G
c
(Dx)w
cr
, and the elastic work,
W
El
¼2s
w
w
w
(Dx)h[22]. G
c
is the fracture toughness (J m
22
),
Dxis the distance the crack extends (m), w
cr
is the crack width
(m), s
w
is the internal pressure of the worm (Pa), w
w
is the
width of the worm (m) and his the half-thickness measured
dorsoventrally (m). A factor of 2 is included because elastic dis-
placement occurs along both dorsal and ventral crack surfaces.
Thrust force, F
Cr
þF
El
, required to drive a wedge-shaped
worm forward a distance Dxmust balance the resistance:
ðFCr þFElÞDx½GcðDxÞwcr þ2
s
wwwðDxÞh¼0:ð2:1Þ
For peristaltic burrowing, friction is ignored because the normal
force on narrow moving segments is greatly reduced by nearby
dilated stationary segments, an assumption that does not apply
to more rigid-bodied undulatory burrowers. Here, we include a
friction force, F
Fr
¼2ms
w
w
w
L, with normal force based on elas-
ticity, which acts along L, the entire length of the worm (m). In a
crack-shaped burrow, friction acts on both dorsal and ventral
surfaces (requiring a factor of 2). In sediments with density
greater than that of the worm, overlying weight also contributes
to friction and F
El
, but in gelatin, this term is small and can
be ignored.
Rather than travelling in a straight line, undulatory move-
ment occurs in a two-dimensional plane. Resistive forces occur
along the body axis, s, and we calculate total external resistive
forces as
FresðsÞ¼FCr ðsÞþFElðsÞþFFr ðsÞ
¼Gcwcr þ2
s
wwwhþðL
0
2
ms
wwwds:ð2:2Þ
During undulation, thrust, F
Th
, is applied normal (n) to the body
axis, and here, we assume that these normal forces, F
Th
(n), are
limited by material resistance rather than the amount of muscu-
lar force the worm can generate. Assuming that burrow
extension follows LEFM and the burrow is a planar crack com-
pressing the worm dorsoventrally [1], material resistance to
these lateral forces is limited by the lateral fracture resistance.
Maximum lateral thrust force, therefore, depends on the lateral
work of fracture, elastic work and friction, the same components
as axial resistance but differing in geometry. For lateral resist-
ance, the crack length is the axial length of the worm, L, rather
than w
cr
:
FThðnÞ¼FCr ðnÞþFElðnÞþFFr ðnÞ
¼ðL
0
GcdsþðL
0
2
s
whdsþðL
0
2
ms
wwwds:ð2:3Þ
In reality, thrust force is not applied along the entire length of the
worm or at the maximum possible magnitude, so a scaling factor
TðsÞ[[0;1] is incorporated into equation (2.3). This thrust force,
F
Th
(n)T(s), results in an added (to F
res
) frictional resistance term
proportional to the thrust and distributed along Lon the outer
curved side of the body, F
Fr_added
(s)¼
m
F
Th
T(s).
Axial resistive forces and lateral thrust forces are balanced by
the curvature of the body (figure 2a). Converting to x–y coordi-
nates with the x-axis aligned with the s-axis at L¼0 (when the
head is oriented parallel to velocity, with
us
the body angle),
and assuming steady state,
FresðsÞcos
u
þFThðnÞTðsÞsin
u
¼0:ð2:4Þ
The internal pressure of the worm, s
w
, depends on the material
stiffness, E, and the elastic displacement, here h, the half-thick-
ness of the worm, as well as the crack geometry. For the
peristaltic burrowing polychaete Cirriformia moorei (Blake,
1996), this value was determined from measured displacements
along the length of the body and material stiffness using a
two-dimensional finite-element model [22]. Elastic modulus, E,
relates stress to strain in a material, but for burrowers, relating
displacement (h) to strain (1¼h/length scale) is confounded
by what length scale to use. We can calculate from C. moorei
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results a scale factor as a function of the geometry, f(g)(m
21
),
s
w¼E1Ehf ðgÞ:ð2:5Þ
Finite-element modelling shows that f(g) depends primarily on
the distance from the worm to the lateral crack edge and that
only a twofold increase in body thickness occurred as the crack
tip was extended from the lateral side of the worm out to a
large distance at which thickness reached an asymptote, with
s
w
and Eheld constant. These results suggest that differences
in f(g) between C. moorei and A. brevis do not exceed a factor of
2 and are probably much smaller (cf. fig. 7bin [2]).
Applying the assumption in equation (2.5), assuming that the
crack width is related to worm width as w
cr
¼aw
w
, and rearrang-
ing to obtain non-dimensional terms, the force balance (equation
(2.4)) becomes
ðL
0
TðsÞsinð
u
Þds
ww
¼
aGc=Eh2fðgÞþ2þ2
m
ðL
0
cosð
u
Þds=hþ
mc
ðL
0
TðsÞcos
u
ds=ww
Gc=Eh2fðgÞþ2þ2
m
ww=h;
ð2:6Þ
where
c
¼Gc
Eh2fðgÞþ2þ2
m
ww
h:ð2:7Þ
The right-handside of equation (2.6) is the ratio of non-dimensional
resistive forces to non-dimensional maximum thrust forces,
c
,and
the left-hand side indicates over what length of the body those
thrust forces must be applied for forces to balance.
To compare the relative importance of fracture resistance, elas-
ticity and friction for A. brevis,weusevaluesforwormgeometry
and material properties to calculate approximate values for the
non-dimensional terms in equation (2.6). Measured for A. brevis,
L¼14 mm, w
w
¼0.7 mm, h¼0.35 mm; for gelatin, E¼7100 Pa
[3], G
c
¼0.4 J m
22
[3,23], calculated from C. moorei,f
1
(g)¼50 m
21
[22] and a¼2 [3], and we assume
m
¼0.3. For sinuosity of approxi-
mately 1.3, Ðcos(
u
)ds0.75L. Next, the length along the body over
which thrust must be applied for forces to balance and the corre-
sponding added friction were calculated numerically for simulated
worms with ratios of amplitude to wavelength, A/
l
, varying from
0.1 to 0.5. Worms were modelled as cosine-derived ideal meander
curves using custom MATLA B scripts.Forameandercurvederived
from a cosine curve, for which the head of the worm was oriented
at
u
, the balance of resistance, F
res
(s), and thrust, F
Th
(n), was calcu-
lated by converting from s–n to x–y coordinates, with the x-axis
oriented along the COM path. This force balance in the x-direction is
ðL
0
TðsÞsinð
u
Þds
ww
¼
aðGc=Eh2fðgÞÞcos
u
0þ2cos
u
0þ2
m
ðL
0
cosð
u
Þds=hþ
mc
ðL
0
TðsÞcos
u
ds=ww
Gc=Eh2fðgÞþ2þ2
m
ww=h
ð2:8Þ
and in the y-direction, perpendicular to the COM path,
ðL
0
TðsÞcosð
u
Þds
ww
¼
aðGc=Eh2fðgÞÞsin
u
0þ2sin
u
0þ2
m
ðL
0
sinð
u
Þds=hþ
mc
ðL
0
TðsÞsin
u
ds=ww
Gc=Eh2fðgÞþ2þ2
m
ww=h:
ð2:9Þ
0 0.002 0.004 0.006 0.008 0.010 0.012
−2
−1
0
1
2
×10−3 m×10
−3 m
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
2
0100
% body length
dimensionless force
A/l = 0.1
A/l = 0.15
A/l = 0.3
A/l = 0.2
A/l = 0.5
A/l = 0.1
A/l = 0.5
m
Fr
El
linear elastic
fracture
elastic–plastic
fracture
OM OM
plastic
elastic–plastic
Th
Fr
Cr+El
Fr
Cr+El
Cr
(a)(b)
(c)
Figure 2. (a) Schematic dorsal view of forces resisting movement (solid line) and forces generating thrust (arrows) against lateral crack edges, calculated from linear elastic fracture
mechanics (LEFM), with lengths proportional to calculated magnitudes. Thrust forces applied normal to the curved body must balance axial resistance from friction (Fr) and anterior
fracture (Cr) and elastic (El) resistance to burrow (scale bar, 2 mm). (b)ModelledArmandia brevis with ideal meander shapes (A/
l
¼0.1 to 0.5; black lines) with distribution of
normal forces (arrows) necessary based on LEFM to overcome resistive forces along the body. (inset) Relative thrust (upper dashed lines) and resistance (upper solid lines) in the
x-direction and in the y-direction (lower dashed and solid lines, respectively) are plotted as a function of increasing % body length over which T(s)¼1 is applied. Modelled results
for y-direction forces are shown only for A/
l
¼0.1 but are representative of all values of A/
l
.(c) Schematic cross-section view of forces generating thrust against lateral crack
edges. Maximum thrust (Th) is limited primarily by fracture resistance under LEFM, but plastic deformation, here illustrated on the ventral side, could increase lateral resistance
through work to plastically deform sediment with additional elastic–plastic resistance resulting from the deformed geometry (hypothetical forces shown as dotted lines; oblique
muscles (OM) shown connecting lateral and ventral grooves). (Online version in colour.)
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Maximum possible thrust was applied (T(s)¼1) along an increas-
ing percentage of the body until thrust forces and resistive forces
balanced (left side of equation (2.8) exceeded the right side). We
chose this approach rather than using a constant T(s) along the
length of the worm both to minimize the added friction term,
F
Fr_added
(s), and to be consistent with qualitative observations of
focused forces applied by the related Ophelina acuminata (Orsted,
1843) moving in a pre-made crack in gelatin visualized using photo-
elastic stress analysis (K. M. Dorgan 2009, unpublished data). Force
was applied in the direction normal to the modelled midline and
opposite the direction of forward movement. The force balance in
the y-direction was used to determine at what position dsto
increase T(s). T(s) was incrementally increased at the position at
which
u
was closest to the ratio of the discrepancy in the y-direction
force balance (difference between right and left sides of equation
(2.9)) relative to the discrepancy in the x-direction force balance
(difference between right and left sides of equation (2.8)) until the
x-direction thrust force exceeded resistance.
3. Results and discussion
(a) Armandia brevis does not extend burrows by linear
elastic fracture
Worms placed in different orientations in pre-formed cracks of
varying widths in gelatin, an analogue for elastic muds, exhibited
undulatory movements with small-amplitude head wiggling,
but no worms extended the burrow even when curved with
the posterior braced against the crack edge (n.10). Shapes of
crawling snakes depend primarily on external forces, specifically
the ratio of lateral resistance to gravitational force (which deter-
mines the normal force upon which ventral friction depends): if
lateral resistance is low relative to friction, the body is more
curved [19]. Armandia brevis, oriented dorsoventrally compressed
in a crack, experienced low lateral resistance compared with
dorsoventral friction augmented by fracture resistance.
Modelling results showed that relative magnitudes of
dimensionless components of axial resistance (right side numer-
ator, equation (2.6)) are: fracture component, aG
c
/Eh
2
f
1
(g)¼18;
elastic work associated with burrow extension, 2; friction com-
ponent, 18; and added friction depends on T(s), and is
calculated numerically. Fracture resistance higher than elastic
resistance is consistent with calculations for C. moorei of a
work of fracture approximately 10elastic work [22]. Relative
magnitudes of dimensionless components of lateral thrust
(right side denominator, equation (2.6)) are: fracture component,
9; elastic work, 2; and friction, 1 (figure 2a). That friction plays a
substantial role in resisting axial movement but does little to
preventing lateral slipping is consistent with the elongate
shape of the worm.
For small A/
l
, maximum thrust forces along the entire
length of the worm were insufficient to balance resistive
forces; as A/
l
increased, thrust forces applied along decreas-
ing percentages of the body could balance resistance
(figure 2b). The relative thrust force (left side of equation
(2.8), dashed lines in figure 2binset) increased as T(s)
increased, reaching the resistive forces relative to maximum
thrust (right side of equation (2.8); solid lines) for A/
l
.0.1.
Added friction from thrust forces increased the resistance
forces (solid lines in figure 2binset) substantially, to approxi-
mately 2.5the resistance force with no added friction for
A/
l
¼0.1. As A/
l
increased, initial resistance force decreased
owing to a decrease in the x-component of friction, and the
added friction from thrust increased less steeply because as
u
increases, more of the added friction acts in the y-direction.
T(s) was increased at positions along the length of the worm to
maintain equilibrium in the y-direction—the resistance (right
side of equation (2.9), solid line in figure 2binset) was close
to zero and thrust (left side of equation (2.9), dashed line in
figure 2binset) fluctuated around 0. T(s) was more variable
for the largest A/
l
, 0.5, owing to the many positions at
which
u

p
/2, and indicates that forces smaller than maxi-
mal forces could be distributed along this region of the
body. Even at intermediate A/
l
, the substantial portion of
the body required to exert the maximum possible force indi-
cates that the LEFM model is only barely mechanically
feasible. Our model applies maximum forces; exceeding
these would result in lateral crack extension and reduced lat-
eral resistance. Applying smaller forces by reducing T(s),
however, would require force to be applied along a greater
percentage of the body, increasing the added friction, which
is already substantial at low-to-intermediate A/
l
(figure 2b).
Only at very high A/
l
does this mechanism seems mechani-
cally feasible, but at high A/
l
efficiency is low and
manoeuverability may be limited.
(b) Non-cohesive granular media exhibits solid response
to undulatory burrowing
Based on LEFM, lateral fracture resistance is only barely
sufficient for A. brevis to overcome anterior resistance, but
dorsoventral plastic deformation of sediments could increase
resistance to lateral slipping (figure 2c). In natural muds, frac-
ture toughness and stiffness are low in the top approximately
2–3 cm of sediments [24,25], corresponding to the depth dis-
tribution of A. brevis [16]. At the surface, fracture toughness
approaches zero, indicating that surface muds are non-cohe-
sive, high-porosity aggregates and that linear elastic fracture
occurs only below the surface layer. Armandia brevis was
able to burrow through an analogue of non-cohesive elastic
fragments of gelatin simulating surficial, unconsolidated
sediments comprising organic-mineral aggregates (see the
electronic supplementary materials, movie S1). Elastic– plastic
fracture with dorsoventral plastic deformation is, however,
theoretically feasible and, assuming a gradient in sediments
from surface aggregates to cohesive elastic mud, would increase
worms’ depth limit. Ventral and lateral grooves increase the
angle of contact between the worm and sediment, probably
increasing lateral resistance and facilitating elastic– plastic
fracture (figure 2c).
Worms burrowed through this analogue material with a
non-slipping undulatory wave (figure 3a), consistent with the
hypothesis of sediment exhibiting solid behaviour with plastic
reorganization of grains. By contrast, worms swimming through
a fluid medium clearly slip backwards (see the electronic sup-
plementary material, movie S2; figure 3b). Wave efficiencies
were significantly higher for burrowing than for swimming
worms (figure 4a). For burrowing worms,
h
¼1.00 +0.10
(mean +s.d.), and for swimming worms,
h
¼0.58 +0.11,
similar to values of 0.50– 0.58 for the related Ophelina sp. [26]
and of approximately 0.5 for burrowing sandfish lizards that
fluidize sands [6]. Calculating wave efficiency from smoothed
v
x
values rather than from velocity along the unsmoothed
COM path results in
h
¼0.48 +0.10 for swimming worms
and
h
¼0.97 +0.11 for burrowing worms. Faster swimming
worms slipped less, consistent with dependence on Re,whereas
no relationship was observed between
h
and velocity for
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burrowing worms (figure 4a). Velocity showed a strong depen-
dence on cycle frequency for both burrowing and swimming
(figure 4b). Swimming worms travelled the same distance per
cycle as burrowing worms (figure 4b), suggesting that inertia
may balance backward slipping.
(c) Undulatory kinematics fit a meander curve
Meander curves showed good fit, significantly better than x–y
sine curve fits for both burrowing (12/15 worms) and
swimming (17/17 worms) when comparing frames for
individual worms (see figure 3c,d and electronic supplementary
material, table S1). R
2
was high, however, and more variable
across individuals for both meander (0.94 +0.03 for bur-
rowing; 0.99 +0.01 for swimming; mean +s.d.) and sine
(0.91 +0.03 for burrowing; 0.95 +0.03 for swimming) fits,
probably inflated by autocorrelation of midline points (see
figure 3c,dand electronic supplementary material, table S1).
More substantial differences between sine and meander fits
were found for body angle profiles, with R
2
for meander fits
510152025
4
6
8
10
12
14
16
18
20
22
24
46810 12 14 16 18 20 22 24
2
4
6
8
10
12
14
16
mm mm
mm
mm
(a)(b)
(c)
(d)
0 2 4 6 8101214161820
−2
0
2
mm
mm
*
12 14 16
−1
0
1
mm
mm
*
51015202530
−4
−2
0
2
4
mm
mm
10 12 14 16
−4
0
4
mm
mm
*
*
14 12 10 8 6 4 2 0
−1
0
1
position along body (mm)
*
76543210
−1
0
1
position along body (mm)
q (rad)
q (rad)
*
Figure 3. Midlines of A. brevis (a) burrowing (at 0.067 s intervals) and (b) swimming (at 0.017 s intervals), coloured sequentially from green to black with cor-
responding centre of mass as solid circles (dotted black line is smoothed COM path). Raw images superimposed are shown in insets (scale bar, 2 mm). For both
(c) burrowing and (d) swimming, a sine fit (solid black) through midlines (solid line with asterisk (*) at head) does not match curvature (residuals in dotted line) as
well as a meander fit to body angle,
u
, as a function of body position, s(black dotted line shows the converted sine fit). (Online version in colour.)
0
velocity (mm s−1)
f (cycles s−1)
10
20
30
40
50
60
70
2 4 6 8 10 12
0
velocity (mm s−1)
0.2
0.4
0.6
0.8
1.0
1.2
(a)(b)
10 20 30 40 50 60 70
wave efficiency
Figure 4. Parameters quantifying gait for burrowing (open circles) and swimming (fillled squares). (a) Wave efficiency is correlated with velocity for swimming
(R
2
¼0.52; p,0.01), but not for burrowing worms. (b) Distances travelled per cycle for burrowing (5.7 +2.0 mm cycle
21
, mean +95% CI; R
2
¼0.74) and
swimming (6.9 +2.6 mm cycle
21
;R
2
¼0.67) do not differ (combined data, solid line; 5.4 +0.8 mm cycle
21
,R
2
¼0.87).
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of 0.84 +0.06 and 0.96 +0.02, and for sine fits of 0.45 +0.08
and 0.47 +0.07 for burrowing and swimming, respectively
(see the electronic supplementary material, figure S1 and
table S1). For swimming worms, curvature profiles also fit a
meander curve significantly better than a sine curve (see the
electronic supplementary material, table S1). Midline profiles,
as well as body angle and curvature, of swimming worms
showed a better fit than burrowing worms (see the electronic
supplementary material, table S1). However, worms burrowed
with smaller
l
/Lthan when swimming, and we re-analysed the
meander fit of each half of the body length of burrowing worms
(because
l
/L0.5); this increased the fit for burrowing
worms, removing differences in fits of body shape and
curvature between burrowing and swimming worms and
substantially reducing the difference for body angle (see the
electronic supplementary material, table S1).
(d) The same gait is used for burrowing and swimming
Swimming worms undulated with significantly larger ampli-
tudes and wavelengths and at higher cycle frequencies ( f)
than burrowing worms (all ANOVA p,0.001; electronic sup-
plementary material, table S2 and figure S2). Higher ffor
swimming worms corresponds with higher velocities, and
there are no obvious discontinuities for burrowing and swim-
ming, with similar distances travelled per cycle in both media
(figure 4b). Moreover, body shapes, quantified by A/
l
,were
the same, 0.18 +0.03 (mean +s.d.) for burrowing and
0.19 +0.05 for swimming worms, similar to nematodes
[5,17] and burrowing sandfish [6]. Similar body shapes (A/
l
)
are consistent with the hypothesis that burrowing and swim-
ming worms use the same gait [17,18], but body shape
depends primarily on external forces [19]. For undulating
fishes, increasing amplitude from head to tail corresponds
with lags between waves of body curvature and muscle activity
[27], but constant amplitude along the length of A. brevis
enables use of change in curvature as a first-order approxi-
mation of muscle activity. The relationship between muscle
activation and body curvature is complex [28], and changes
in curvature are also influenced by internal elasticity and exter-
nal forces [19]; however, similar changes in curvature in the two
media are consistent with similar muscle activity patterns. As a
meander curve, the relationship between
u
at a fixed location on
the worm and time is sinusoidal, as is the relationship between
d
u
/dtand time. Amplitude, maximum d
u
/dtper cycle (see the
electronic supplementary material, figure S3), shows a strong
linear relationship with fwith no discontinuities that would
indicate a gait transition (figure 5). Linearity indicates that
the rate of body bending, presumably resulting from the rate
of muscle contraction, used in both modes of locomotion
is directly proportional to the cycle frequency, which is in
turn directly proportional to velocity (figure 4b): the pattern
of movement does not change with speed. More importantly,
this pattern of movement does not differ between burrowing
and swimming. Similar fit to a meander curve of body shape,
body angle and curvature (see the electronic supplementary
material, figure S1 and table S1) indicates that muscle contrac-
tion is similar and probably sinusoidal along the length of the
body for both swimming and burrowing worms. Larger Aand
l
for swimming worms can be attributed to low resistance of
water to internal elastic forces. For the nematode C. elegans,
crawling worms experience comparable external loads and
internal elastic forces, whereas for swimming worms external
forces are much smaller than internal elasticity [18].
4. Conclusions
Whereas transitions from swimming to burrowing [7,8] and
from running to burrowing [6] involve substantial changes in
locomotory behaviour, undulatory burrowing worms change
only the frequency of movement when transitioning to swim-
ming, a derived mode of locomotion in this group [15]. Use of
a burrowing undulatory gait for swimming could explain why
neither A. brevis, most nematodes [5], nor larval lampreys [29]
swim with undulatory waves increasing in amplitude, in con-
trast to most elongate undulatory swimmers, e.g. snakes [30],
eels [11] and amphioxus [31]. Muscle contraction patterns
and body stiffness contribute to this increasing amplitude
[30,31], which reduces yaw, probably resulting in greater effi-
ciency [5]. This side-to-side movement of swimming A. brevis
(figure 3b) may reduce swimming efficiency but does not
affect burrowing, their primary mode of locomotion.
Fit to a meander curve explicitly links the shapes of
elongate animals to the muscular activity that drives move-
ment. This sinusoidal change in angle along a path or over
time describes shapes of rivers [20], snakes [21], flagella
[32], A. brevis and probably other biological and abiotic
elongate patterns as well. In animal locomotion, deviations
from this meander model shape may indicate non-uniform
external forces or alterations of behaviour. For A. brevis, simi-
larity in meander fit supports our hypothesis that swimming
worms use the same muscle contraction patterns but at
higher frequency than burrowing worms.
Our finding that LEFM is an unlikely mechanism for
undulatory burrowing in muds is based on the size of
A. brevis and limited data for mechanical properties of
muds. Behaviours of burrowers using fracture depend on
0
f (c
y
cles s−1)
20
40
60
80
100
120
2 4 6 8 10 12
.
.
.
0.15 0.20 0.25 0.30
8
10
12
14
16
A/l
(dq/dt)/f
q
q
q
max(dq/dt) (rad s−1)
Figure 5. The amplitude of the sinusoidal relationship of d
u
/dtas a function of
time is strongly correlated with cycle frequency (R
2
¼0.95) with slopes not sig-
nificantly different for burrowing (open circles), swimming (closed squares), and
combined data (at 95% CI). For swimming worms (inset), some variability can
be explained by a positive correlation (R
2
¼0.61, p,0.05) between slopes cal-
culated for individual worms [max(d
u
/dt)/f(rad s
21
)/(cycles s
21
)] and A/
l
—
faster muscle contraction per cycle results in larger A/
l
. Sequential images of a
burrowing worm (0.067 s apart) show changing
u
(upper inset; scale bar, 2 mm).
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the ratio of work of fracture (/G
c
, fracture toughness) to elas-
tic work (/Eh, stiffness and body thickness) [23]. For
peristaltic burrowing, work of fracture is approximately
10elastic work [22], but for undulatory burrowers, this
large friction component that in elastic gels depends primar-
ily on stiffness, E, combines with elastic work, altering this
ratio. Compared with its behaviour with peristaltic bur-
rowers, the same material would seem much stiffer. For
smaller undulatory burrowers like the nematode C. elegans,
G
c
/Eh is much higher than for A. brevis, possibly explaining
their ability to burrow through gelatin [17] and agar. Simi-
larly, in soft sediments with higher G
c
/E than gelatin,
friction would be less important and fracture a more feasible
mechanism of undulatory burrowing. Friction increases not
only the apparent stiffness of muds, but also total work
to burrow, potentially exceeding the muscle capacity of
A. brevis in deeper sediments (a possible explanation for the
large longitudinal muscle bands, figure 1c). Friction is
probably high regardless of sediment mechanics: in weak
sediments with less elastic cohesion, the normal force
depends primarily on overlying weight, and friction would
increase with depth, similar to in elastic muds with increasing
E. Sediments with heterogeneous mixtures of sand and mud
are common habitats for burrowers and probably have mech-
anics that fall between those of sands and muds. The similar
granular responses of surface muds on small scales and of
sands to larger burrowers (figure 6) and the potential use
of elastic–plastic fracture suggest interesting hypotheses
about burrowing in heterogeneous sandy muds, which may
involve a combination of elastic and granular mechanisms
that depend on burrower size, morphology and behaviour,
as well as small-scale differences in sediment mechanics.
Muddy sediments are ubiquitous and are inhabited by
diverse animals, many of which, such as A. brevis,aresmall,
live close to the sediment– water interface, and exhibit undula-
tory or non-peristaltic movements. Reduction of friction by
alternating expansion and contraction during peristalsis
suggests higher efficiency than undulatory burrowing in com-
pacted sediments. The limited distance over which forces can
be applied during peristalsis, however, may be insufficient to
overcome fracture resistance or even to anchor small worms
in less consolidated sediments. Mechanics indicate that undu-
latory burrowing is more effective in these weak surface
sediments and that these differences are greater for small
worms—sediments that are too tough for small peristaltic bur-
rowers to crack [3] exert smaller normal forces and less
frictional resistance on small undulatory burrowers. These
different mechanisms of burrowing in muds—plastic defor-
mation or elastic–plastic fracture for undulatory burrowers
versus elastic fracture for peristaltic burrowers (figure 6)—
suggest habitat partitioning and different functional roles of
infauna based on sediment mechanics and body size.
This project was funded by NSF OCE grant no. 1029160. We thank
C. Hermans, P. Jumars, M. Koehl and M. Pruett for helpful discus-
sions; T. Tucker and A. Francoeur for helping with data analysis;
and S. Woodin and an anonymous reviewer for constructive
feed-back on the manuscript.
grain size
z
z
0
Armandia brevis swimming
sand fluidization
plastic grain rearrangement
plastic grain rearrangement
Armandia
brevis
e.g. sandfish lizard
e.g., sand lances, eels
burrow extension by fracture
depth, sediment compaction
granular
solid
elastic solid
fluid
mud sand
Figure 6. Scheme of different mechanisms of burrowing in idealized muds (left) and sands (right). Dotted line indicates a later time, and the differing mechanics of
the media are indicated. Plastic grain rearrangement by Armandia brevis (upper left) is consistent with descriptions of kinematics of non-slipping burrowing in sands
[7,8] (lower right), although in muds, aggregate grains have lower density and are deformable, suggesting that friction may be more important and gravity less
important than in sands. This solid granular response differs from sand fluidization (upper right), which has thus far only been observed in dry terrestrial sands by
lizards considerably larger than A. brevis [6]. Depth or sediment compaction is hypothesized to distinguish between the two mechanisms in each sediment type, but
more research is needed on the mechanical responses of natural sediments to burrowing behaviours on spatial scales corresponding to burrower morphologies. Many
natural sediments are heterogeneous mixtures of sand and mud and probably exhibit properties of both media. Lower left panel of burrow extension by fracture
adapted from Dorgan et al. [1] with permission.
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