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.
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 behaviours 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 muscular 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 .
Nematodes and some polychaetes, however, lack expansible anteriors used for burrow extension by fracture, and instead move through muddy sediments by undulation . For these organisms, kinematics are similar to those of sandfish lizards, which use body undulations rather than limbs to generate thrust and fluidize desert sands . 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 discrete grains against gravitational forces, in contrast to elastic muds, which are held together by adhesion and cohesion of the intra-granular organic matrix .
These different responses of sands to undulatory burrowers are quantified using wave efficiency, η = vx/vw, the ratio of the animal's forward velocity, vx, to the velocity of the posteriorly travelling undulatory wave, vw. Wave efficiencies of η ≈ 0.5 for burrowing sandfish lizards are consistent with fluidization of granular sand , and of η ≈ 1 for sand lances and eels are consistent with a solid response of sand [7,8]. For swimming animals, wave efficiency varies considerably and depends on Reynolds number (Re) , with high values of greater than 0.7 for swimming eels [11,12] down to 0.23 for sperm . For swimming leeches, η is proportional to size and drops from 0.55 to 0.43 with a 10-fold increase in viscosity . Crawling animals have high η, greater than 0.8 compared with 0.19–0.29 for swimming nematodes , 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 . 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 longitudinal muscles, enabling lateral bending and undulatory movements (figure 1). Armandia brevis also swims by undulation, enabling direct comparison of wave efficiency while burrowing and swimming. Wave efficiency values for burrowing A. brevis close to 1 and higher than that of swimming worms would indicate solid response of the medium, whereas 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 . Though dispersal is one explanation, most swimming A. brevis are reproductive, and spawning occurs only once before death . That swimming 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/λ), when transitioning to burrowing from running and swimming, respectively [6,7]. Larger A, λ and frequency have been observed for nematode worms swimming in water compared with media with higher resistance, but positive linear relationships indicate constant A/λ 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/λ, 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 , 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  and a best-fit sine function has been used for measuring amplitude, wavelength and wave speed , curvature 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, θ, and distance along the path, s, fits a sine curve . A sine fit to θ–s is a better approximation of snake shape than an x–y sine fit, consistent with waves of alternating muscle contraction travelling along the length of the animal changing the body curvature . Direct measurement of kinematic parameters such as amplitude, wavelength, radius of curvature and vw from profiles enables comparisons, but lacks a mechanistic basis; fit to a meander curve would be consistent 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 , (ii) A. brevis fluidizes muddy sediments, indicated by , and (iii) the muddy sediment deforms plastically through grain rearrangement, indicated by η ≈ 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 cohesive elastic muds through which worms burrow by fracture [1–3], with theoretical predictions based on linear elastic fracture mechanics (LEFM). Because A. brevis lives in surface sediments , we developed an additional analogue for weak surface muds comprising 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.
Armandia brevis were collected from shallow subtidal sediments in Mission Bay, San Diego, CA, and from sediments in the flowing seawater tanks at Scripps Institution of Oceanography, La Jolla, CA. For experiments to determine whether worms burrowed by fracture, glass aquaria were filled with seawater gelatin (28.35 g l−1), and worms were placed in pre-made cracks and filmed following methods of Che & Dorgan . For elastic aggregate burrowing experiments, concentrated seawater gelatin (85 g l−1) was chopped in a food processor to fragments roughly 200–500 μm 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 perpendicular 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 substantially 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 sequentially 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 calculated. For consistency, midlines were shortened to total body length, Ls, 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 θ–s coordinates using a cartesian-to-polar coordinate conversion for each segment ds of the midline. Best-fit sine function in x–y coordinates was the sine fit, in θ–s coordinates 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 R2, but subsampling did not effectively remove the autocorrelation or change the R2. To better assess the fit of the two models, we also compared the body angle, sin(θ), and curvature, dθ/ds, profiles to those predicted by a sine and meander fit (see the electronic supplementary 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 η = vx/vw, where the wave velocity in θ–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 vw. Sinuosity was inflated for swimming worms by yaw, seen as side-to-side movement of the COM, so rather than using vx calculated from the smoothed COM path, velocity was calculated along the unsmoothed COM path (greater than vx for swimming, but approx. vx for burrowing) and this corrected velocity was used in wave efficiency calculations.
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 fracture resistance, elastic resistance to sediment deformation and friction. The work to extend a burrow by fracture is the sum of the work of fracture, WCr = Gc(Δx)wcr, and the elastic work, WEl = 2σw ww(Δx)h . Gc is the fracture toughness (J m−2), Δx is the distance the crack extends (m), wcr is the crack width (m), σw is the internal pressure of the worm (Pa), ww is the width of the worm (m) and h is the half-thickness measured dorsoventrally (m). A factor of 2 is included because elastic displacement occurs along both dorsal and ventral crack surfaces. Thrust force, FCr + FEl, required to drive a wedge-shaped worm forward a distance Δx must balance the resistance: 2.1For 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, FFr = 2 μσwwwL, with normal force based on elasticity, 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 FEl, but in gelatin, this term is small and can be ignored.
Rather than travelling in a straight line, undulatory movement occurs in a two-dimensional plane. Resistive forces occur along the body axis, s, and we calculate total external resistive forces as 2.2
During undulation, thrust, FTh, is applied normal (n) to the body axis, and here, we assume that these normal forces, FTh(n), are limited by material resistance rather than the amount of muscular force the worm can generate. Assuming that burrow extension follows LEFM and the burrow is a planar crack compressing the worm dorsoventrally , 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 resistance, the crack length is the axial length of the worm, L, rather than wcr: 2.3
In reality, thrust force is not applied along the entire length of the worm or at the maximum possible magnitude, so a scaling factor is incorporated into equation (2.3). This thrust force, FTh(n)T(s), results in an added (to Fres) frictional resistance term proportional to the thrust and distributed along L on the outer curved side of the body, FFr_added(s) = μFThT(s).
Axial resistive forces and lateral thrust forces are balanced by the curvature of the body (figure 2a). Converting to x–y coordinates with the x-axis aligned with the s-axis at L = 0 (when the head is oriented parallel to velocity, with θσ the body angle), and assuming steady state, 2.4
The internal pressure of the worm, σw, depends on the material stiffness, E, and the elastic displacement, here h, the half-thickness 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 . Elastic modulus, E, relates stress to strain in a material, but for burrowers, relating displacement (h) to strain (ɛ = h/length scale) is confounded by what length scale to use. We can calculate from C. moorei results a scale factor as a function of the geometry, f(g) (m−1), 2.5Finite-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 σw and E held 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. 7b in ).
Applying the assumption in equation (2.5), assuming that the crack width is related to worm width as wcr = aww, and rearranging to obtain non-dimensional terms, the force balance (equation (2.4)) becomes 2.6where 2.7The right-hand side of equation (2.6) is the ratio of non-dimensional resistive forces to non-dimensional maximum thrust forces, ψ, 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, elasticity and friction for A. brevis, we use values for worm geometry and material properties to calculate approximate values for the non-dimensional terms in equation (2.6). Measured for A. brevis, L = 14 mm, ww = 0.7 mm, h = 0.35 mm; for gelatin, E = 7100 Pa , Gc = 0.4 J m−2 [3,23], calculated from C. moorei, f1(g) = 50 m−1  and a = 2 , and we assume μ = 0.3. For sinuosity of approximately 1.3, ∫cos(θ)ds ≈ 0.75L. Next, the length along the body over which thrust must be applied for forces to balance and the corresponding added friction were calculated numerically for simulated worms with ratios of amplitude to wavelength, A/λ, varying from 0.1 to 0.5. Worms were modelled as cosine-derived ideal meander curves using custom Matlab scripts. For a meander curve derived from a cosine curve, for which the head of the worm was oriented at θ, the balance of resistance, Fres(s), and thrust, FTh(n), was calculated 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 2.8and in the y-direction, perpendicular to the COM path, 2.9Maximum possible thrust was applied (T(s) = 1) along an increasing 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, FFr_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 photoelastic 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 ds to increase T(s). T(s) was incrementally increased at the position at which θ 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 determines the normal force upon which ventral friction depends): if lateral resistance is low relative to friction, the body is more curved . 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 numerator, equation (2.6)) are: fracture component, aGc/Eh2f1(g) = 18; elastic work associated with burrow extension, 2; friction component, 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 . 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/λ, maximum thrust forces along the entire length of the worm were insufficient to balance resistive forces; as A/λ increased, thrust forces applied along decreasing percentages of the body could balance resistance (figure 2b). The relative thrust force (left side of equation (2.8), dashed lines in figure 2b inset) increased as T(s) increased, reaching the resistive forces relative to maximum thrust (right side of equation (2.8); solid lines) for A/λ > 0.1. Added friction from thrust forces increased the resistance forces (solid lines in figure 2b inset) substantially, to approximately 2.5× the resistance force with no added friction for A/λ = 0.1. As A/λ 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 θ 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 2b inset) was close to zero and thrust (left side of equation (2.9), dashed line in figure 2b inset) fluctuated around 0. T(s) was more variable for the largest A/λ, 0.5, owing to the many positions at which θ ≈ π/2, and indicates that forces smaller than maximal forces could be distributed along this region of the body. Even at intermediate A/λ, the substantial portion of the body required to exert the maximum possible force indicates that the LEFM model is only barely mechanically feasible. Our model applies maximum forces; exceeding these would result in lateral crack extension and reduced lateral 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/λ (figure 2b). Only at very high A/λ does this mechanism seems mechanically feasible, but at high A/λ 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, fracture toughness and stiffness are low in the top approximately 2–3 cm of sediments [24,25], corresponding to the depth distribution of A. brevis . At the surface, fracture toughness approaches zero, indicating that surface muds are non-cohesive, 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 supplementary material, movie S2; figure 3b). Wave efficiencies were significantly higher for burrowing than for swimming worms (figure 4a). For burrowing worms, η = 1.00 ± 0.10 (mean ± s.d.), and for swimming worms, η = 0.58 ± 0.11, similar to values of 0.50–0.58 for the related Ophelina sp.  and of approximately 0.5 for burrowing sandfish lizards that fluidize sands . Calculating wave efficiency from smoothed vx values rather than from velocity along the unsmoothed COM path results in η = 0.48 ± 0.10 for swimming worms and η = 0.97 ± 0.11 for burrowing worms. Faster swimming worms slipped less, consistent with dependence on Re, whereas no relationship was observed between η and velocity for burrowing worms (figure 4a). Velocity showed a strong dependence 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). R2 was high, however, and more variable across individuals for both meander (0.94 ± 0.03 for burrowing; 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,d and electronic supplementary material, table S1). More substantial differences between sine and meander fits were found for body angle profiles, with R2 for meander fits 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 λ/L than when swimming, and we re-analysed the meander fit of each half of the body length of burrowing worms (because λ/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 amplitudes and wavelengths and at higher cycle frequencies (f) than burrowing worms (all ANOVA p < 0.001; electronic supplementary material, table S2 and figure S2). Higher f for swimming worms corresponds with higher velocities, and there are no obvious discontinuities for burrowing and swimming, with similar distances travelled per cycle in both media (figure 4b). Moreover, body shapes, quantified by A/λ, 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 . Similar body shapes (A/λ) are consistent with the hypothesis that burrowing and swimming worms use the same gait [17,18], but body shape depends primarily on external forces . For undulating fishes, increasing amplitude from head to tail corresponds with lags between waves of body curvature and muscle activity , but constant amplitude along the length of A. brevis enables use of change in curvature as a first-order approximation of muscle activity. The relationship between muscle activation and body curvature is complex , and changes in curvature are also influenced by internal elasticity and external forces ; however, similar changes in curvature in the two media are consistent with similar muscle activity patterns. As a meander curve, the relationship between θ at a fixed location on the worm and time is sinusoidal, as is the relationship between dθ/dt and time. Amplitude, maximum dθ/dt per cycle (see the electronic supplementary material, figure S3), shows a strong linear relationship with f with 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 contraction is similar and probably sinusoidal along the length of the body for both swimming and burrowing worms. Larger A and λ 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 .
Whereas transitions from swimming to burrowing [7,8] and from running to burrowing  involve substantial changes in locomotory behaviour, undulatory burrowing worms change only the frequency of movement when transitioning to swimming, a derived mode of locomotion in this group . Use of a burrowing undulatory gait for swimming could explain why neither A. brevis, most nematodes , nor larval lampreys  swim with undulatory waves increasing in amplitude, in contrast to most elongate undulatory swimmers, e.g. snakes , eels  and amphioxus . Muscle contraction patterns and body stiffness contribute to this increasing amplitude [30,31], which reduces yaw, probably resulting in greater efficiency . 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 movement. This sinusoidal change in angle along a path or over time describes shapes of rivers , snakes , flagella , 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, similarity 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 the ratio of work of fracture (∝Gc, fracture toughness) to elastic work (∝Eh, stiffness and body thickness) . For peristaltic burrowing, work of fracture is approximately 10× elastic work , but for undulatory burrowers, this large friction component that in elastic gels depends primarily on stiffness, E, combines with elastic work, altering this ratio. Compared with its behaviour with peristaltic burrowers, the same material would seem much stiffer. For smaller undulatory burrowers like the nematode C. elegans, Gc/Eh is much higher than for A. brevis, possibly explaining their ability to burrow through gelatin  and agar. Similarly, in soft sediments with higher Gc/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 mechanics 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, are small, live close to the sediment–water interface, and exhibit undulatory or non-peristaltic movements. Reduction of friction by alternating expansion and contraction during peristalsis suggests higher efficiency than undulatory burrowing in compacted 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 undulatory 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 burrowers to crack  exert smaller normal forces and less frictional resistance on small undulatory burrowers. These different mechanisms of burrowing in muds—plastic deformation 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 discussions; T. Tucker and A. Francoeur for helping with data analysis; and S. Woodin and an anonymous reviewer for constructive feed-back on the manuscript.
- Received December 10, 2012.
- Accepted February 5, 2013.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.