Burrowing dynamics of aquatic worms in soft sediments

Observations with L. variegatus.

The projected shapes of a L. variegatus released just above a sedimented bed in a water-filled container, shown schematically in Fig. 1C, is plotted at 1-s time intervals in Fig. 1D. The worm is observed to swim above the bed surface for a few seconds before burrowing rapidly down through the bed, turning, rising, and then further exploring the surface of the sedimented bed. Magnified views with higher time resolution of each phase can be found in SI Appendix, Fig. S1. It can be observed that the worm moves in a narrow sinusoidal path which is not much wider than its body width while burrowing in the sediments. Whereas greater lateral body motion is observed while it is swimming in water near the bed surface. Further, the worm can be also observed to reflect off the side walls and sometimes move backward. Similar behavior is observed as well in a sedimented bed inside thinner quasi-2D containers and wider cuboid containers (SI Appendix, Fig. S2). We also observe that the worms do not crawl up the side walls and slide on glass and acrylic surfaces when fully immersed in water. Thus, the container walls serve as a physical barrier and do not appear to aid the locomotion of the worm. Because the image analysis and tracking are far simpler in 2D, we discuss worm dynamics in a container with internal dimensions Lc=155 mm, Hc=164 mm, and Wc=2 mm, which is sufficiently wide to allow the worm to move freely. The effects of the constraints imposed by lateral walls on the dynamics are further discussed in SI Appendix, Effect of Container Thickness. We focus on the locomotion dynamics when the worm is away from the side walls and when it is essentially traveling forward without turning on itself.

The projected shapes of a worm of length lw=26.6 mm and dw=0.5 mm moving in the sediments are shown in Fig. 2A as it travels approximately its body length. We observe that it traces a narrow path in the medium, similar in form to that observed in the sedimented bed in the Wc=12.7-mm-wide container shown in Fig. 1C. To compare and contrast the observed burrowing dynamics with swimming, we show the worm shapes recorded when constrained between 2 parallel plates separated by 2 mm and immersed in a container filled with water in Fig. 2B. (A schematic can be found in SI Appendix, Fig. S4A.) Because the worm does not swim up very high above the surface unless threatened, we measure the motion when the quasi-2D container is horizontal so that the strokes and the trajectories can be compared while being constrained similarly. Over the same time window, we observe that the worm moves forward only a fraction of its body length while performing somewhat larger undulatory strokes in water. If sediments are also added in this horizontal orientation of the observation plane, corresponding to near-zero overburden pressure as near the surface (SI Appendix, Effect of Sediment Consolidation on Anchoring), we observe the path is narrower compared to that in water, but not as narrow as in the vertical orientation shown in Fig. 2, where the weight of the grains above pushes on the worm to constrain it to move in a still tighter path (SI Appendix, Fig. S4B).

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Fig. 2.

(A and B) Worm trajectory in the laboratory frame of reference in the sediments (A) and water (B). The worm of length lw=26.6 mm is observed to follow a narrow sinusoidal path in the sediments compared to undulating in place in water. (C and D) The corresponding worm shapes in the body frame of reference are also shown shifted up by a fixed distance over each time interval for clarity (Δt=200 ms and T=20 s). (E and F) Measured cumulative displacement parallel D‖ and perpendicular D⊥ to the worm orientation in sediments (E) and water (F) over T=40 s.

To compare the form of the strokes used while burrowing and swimming, we use the body reference frame which is oriented along the average direction in which the worm points and where its origin is located at the worm centroid as shown in Fig. 1B. We plot the same measured shapes in Fig. 2 C and D in the body reference frame, but where the centroid is shifted vertically by time denoted by the vertical axis. We observe that transverse undulations and that the worm elongations and length contraction can be observed in fact in both media. To show that the shortening of the worm indeed arises due its changes, and not simply because of the transverse undulations, we obtain the difference of dynamic worm length from its mean length Δlw(t) and use the color bar to render each snapshot. Periodic changes in the length can be unambiguously observed.

Then, we obtain the component of worm speed over a short time interval Δt parallel to the average body axis v‖=ΔRc⋅x^/Δt and the speed perpendicular to the average body axis v⊥=ΔRc⋅ŷ/Δt. The resulting cumulative displacements of the worm in the direction parallel to the average body axis D‖(t)=∑0tv‖Δt and perpendicular to the average body axis D⊥(t)=∑0tv⊥Δt are plotted in Fig. 2 E and F over a longer observation time T=40 s, with Δt=200 ms. We observe that the worm moves on average along the direction of the body orientation in both media. Hence, the average forward speed of the worm vw≈D‖/T. Fig. 3A shows the worm locomotion speed vw measured over a time interval T≥20 s as a function of lw using 9 different worms in the sediments and 7 different worms in water listed in SI Appendix, Table S1. Besides higher speeds in the sediments, an overall increasing trend with lw is found with significant variations from worm to worm due to behavioral differences. In the following, we focus on understanding the measured speed in terms of the strokes used by the worm and the rheology of the medium.

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Fig. 3.

(A) The measured worm speed versus its length shows significant scatter from worm to worm. However, vw is overall higher in the sediments compared with water. (B) The transverse amplitude AT increases approximately linearly with lw. (C) KT(t) shows a peak at TT (indicated by arrow) corresponding to the transverse undulations timescale. (D) Sw versus Re is observed to scale roughly linearly. (E and F) Calculated Uund versus vw using resistive force theory assuming low amplitude and low-Re conditions in water (E) and the sediments (F). The line slope is 0.94 with goodness of fit R2=0.93 in water and a slope 0.52 and R2=0.49 in the sediments (main text). Peristaltic motion, besides undulation, contributes to vw leading to the lower correlation in the sediments.

Transverse Body Undulations.

We measure the transverse undulations using the root-mean-square (rms) transverse amplitude in the body frame of reference AT=⟨y2⟩, where ⟨..⟩ denotes averaging over the length of the worm. Fig. 3B shows a plot of AT where we observe that it increases roughly linearly with lw and with slightly higher slope in water compared with the sediments (AT/lw=0.069 and goodness of linear regression R2=0.94 versus AT/lw=0.052 and R2=0.67). Further, by characterizing the worm orientation correlation function as discussed in SI Appendix, Worm-Body Orientation Correlations, we find its persistence length to be of the order of its length in both media. Thus, the worm can be considered as rod-like with undulation amplitude AT/lw≪1 in both media.

Then, to determine the relevant regime for its dynamics, we use the Reynolds number Re=ρvwlw/μ, where ρ and μ are the density and viscosity of the medium, respectively, and the swimming number Sw=ρvTlw/μ, where vT is the velocity associated with the transverse oscillations which determines propulsion (18). For 2D sinusoidal oscillations with amplitude B and frequency fT, vT=2πfTB, and we have with B=2AT, Sw=22πfTATlw/μ. To estimate fT, we use the displacement yc(t) corresponding to the midsection of the worm in the body frame of reference and calculate the transverse amplitude correlation function KT(t)=⟨yc(to+t)yc(to)⟩/⟨yc2⟩ averaged over time to. From Fig. 3C, we observe that KT(t) oscillates and shows peaks which correspond to the period of transverse oscillations. We use the time at which the first strong peak occurs with the timescale TT and determine fT=1/TT. Then, we find from the plot of Sw and Re in Fig. 3D that the estimated Re ranges between 1 and 20 in the case of water, corresponding to the crossover regime where viscous and inertia effects can be important. Nonetheless, we find that Sw increases linearly with Re consistent with what may be expected by resistive force theory in the low-Re and low-amplitude regime (11), where the swimming speed of an undulating filamentUund=2π2fTB2λ(ξr−1),[1]where ξr is the ratio of the drag in the perpendicular and parallel directions w.r.t. the rod axis. We plot this estimated swimming speed versus the measured speed in Fig. 3E using λ≈lw and ξr=2 for water (19) and find excellent agreement. Then, multiplying Eq. 1 by ρlw/μ on both sides, we haveSw=lw2πAT(ξr−1)Re,[2]which corresponds to the line drawn in Fig. 3D using AT/lw=0.069 and ξr=2.

Drag-Assisted Propulsion in Sediments.

To find the appropriate drag encountered by the worm while undulating in the sediments, we performed complementary measurements with a thin rod corresponding to the worm body over the typical range of speeds encountered by the worm. As discussed in more detail in SI Appendix, Drag Measurements, we observe a drag which increases sublinearly with speed and depends on the orientation of the rod axis relative to the direction of motion. Over the range of speeds U from 0.1 to 10 mm⋅s−1 relevant to the motion of the worm, we measure the effective viscosity of the medium ηe≈4.0×U−0.63 Pa⋅s and ξr≈6 as shown in SI Appendix, Fig. S6. This measured ξr is similar in magnitude, but somewhat lower than the drag anisotropy of ∼10 reported in dry granular matter (20).

The estimated Re and Sw corresponding to the worm motion in the sediments are also shown in Fig. 3D. They are systematically lower than that for water because the effective viscosity in the medium is essentially 1,000 times higher than that of water, while the worm parameters remain essentially unchanged. We also observe that Sw versus Re can be described by a linear fit. Now, it has been shown that resistive force theory may be applied to sandfish lizards moving through dry sand (20). Thus, notwithstanding the shear thinning nature of the medium, we may expect Eq. 1 to capture the undulatory component of the worm speed. We plot the estimated Uund versus vw with ξr=6 for sediments in Fig. 3D and observe that the data can be described by a linear fit, but with a slope of 0.52 and R2=0.49 indicating the presence of other factors which contribute to the observed vw. Now, if shear thinning effects of the medium were important, it would lead to a lower estimate of Uund (21) which is opposite to the trend needed to capture the lower slope. Thus, to understand the difference, we examine next the contribution of the observed peristaltic motion to the locomotion speed of the worm.

Peristaltic Motion.

To extract the dominant oscillation frequency, we obtain the longitudinal amplitude correlation function KL(t)=⟨Δlw(to+t)Δlw(to)⟩/⟨Δlw2⟩, where ⟨..⟩ denotes averaging over time to, and ⟨Δlw2⟩ is the mean-square fluctuation over T. Fig. 4 A and B show plots of KL(t) in the case of the sediments and water, respectively. Peaks corresponding to the dominant periods can be observed in both media. We associate the longitudinal oscillation period TL with the first peak, which is observed to be stronger and clearer in the sediment example.

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Fig. 4.

(A and B) The length correlation function KL(t) in sediments (A) and in water (B). (C and D) The velocity correlation Kv(t) in sediments (C) and in water (D). The peaks are correlated corresponding to peristaltic motion in the sediments. (E) The measured worm speed vw versus the calculated peristaltic speed vper using the anchor model. The dashed line corresponds to a linear fit with slope 0.5±0.1. (Insets) Illustration of the anchor model (Upper Left) and the fractional change in worm diameter at its center over time (Lower Right).

Then, we calculate the velocity correlation function Kv(t)=⟨vw(to+t)vw(to)⟩/vw2, which measures the correlation of vw at time to with that after a short time t. Because of the normalization by vw2, Kv(t) can be expected to oscillate or approach 1 over time. Kv(t) is plotted in Fig. 4 C and D in the sediments and water, respectively, and shows oscillations about the average value in both media. In the sediments, Kv(t) always remains positive while oscillating and remains strong over the duration plotted. Whereas Kv(t) becomes negative in water, indicating a back and forth motion at short timescales, before decaying rapidly to the mean value. Further, comparing the peaks in KL(t) with those in Kv(t) in the sediments, we observe that the dominant oscillation period TL in KL(t) is twice the period Tv in Kv(t). In comparison, the first peak in KL(t) and Kv(t) appear at the same time in water.

Anchor Model.

To understand these correlations, we consider an idealized anchor model of peristaltic motion (3) as illustrated in Fig. 4E, Upper Left Inset. Here, the worm is represented in the form of a pair of beads connected by a spring which travels forward by elongating its body through a length ϵ while anchoring its tail and then moving forward by contracting its body through the same length ϵ while anchoring its head to regain its initial length, as indicated by the arrows. Then, the distance moved by the worm centroid is ϵ/2 during the expansion as well as the contraction phase, and the net displacement is ϵ in each period of oscillation. On the other hand, if the worm is not anchored, but slips similarly during the expansion and contraction phase, then the net displacement can be expected to be less than ϵ by a factor α, which is between 0 and 1.

Assuming that the primary oscillation of the worm length occurs sinusoidally with period of longitudinal oscillation TL, we have Δlw(t)=AL⁡sin(2πt/Tl)+Δls(t), where Δls(t) captures additional time dependence with ⟨Δls(t)⟩=0. Then, ⟨Δlw(to)Δlw(to+t)⟩=12AL2⁡cos(2πt/Tl)+⟨Δls2⟩. If Δls(t) decays rapidly compared with Tl, then the peaks associated with the sinusoidal mode in Kl(t) can be expected have constant amplitude AL2/2 after the initial rapid decay as is seen in Fig. 4A. Because ϵ=2AL and Tv=TL/2, we can then estimate the speed due to peristaltic motion to be U(t)=Uper(1+cos4πt/TL) to leading order, withUper=2αAL/TL.[3]Obtaining AL from the strength of the first peak in KL(t), we plot the calculated speed Uper versus the measured speed vw in Fig. 4E. We find that Uper versus vw can be fitted by a straight line with a slope Uper/vw=0.5±0.1. In arriving at these values, we have ignored other oscillation frequencies which may contribute to the peristaltic motion. An upper bound Uper/vw=0.85 can be estimated using KL(t) corresponding to t=0 s which includes all length fluctuations. Some of these contributions may offset the fact that the worm does not travel in a straight line, but rather in a sinusoidal path. Hence, we conclude that the peristaltic motion contributes at least equally to the locomotion speed of these worms in the sediments.

Medium Rheology and Locomotion Speed.

L. variegatus is known to dynamically deploy 10- to 20-μm-long hair-like projections called chaetae (22) along with muscle contractions to change the relative friction during the sliding and anchoring phase while crawling on solid surfaces (17). Given the small size of chaetae relative to dw, deploying them has negligible effect on the drag experienced by a worm in a fluid which obeys nonslip boundary conditions. Thus, the anchoring parameter α can be expected to be approximately zero in water and other Newtonian fluids. By contrast, the drag experienced by an intruder in sediments can depend sensitively on the normal force acting between the grains in the medium and its surface (13, 14). Even if the friction between the worm and the grains can be changed, anchoring cannot be achieved if the grains are very loosely packed as when the overburden pressure is nearly zero or when grains simply move out of the way as near the bed surface as discussed in SI Appendix, Effect of Sediment Consolidation on Anchoring.

Thus, the yield-stress nature of the medium is also important to achieving anchoring needed for peristaltic motion. This strength characterized by μo in turn depends on the volume fraction of the sediments ϕ. If ϕ is below a critical value ϕc, the medium can flow and μo can be expected to vanish. When ϕ→ϕc≈0.6, corresponding to the volume fraction of the sedimented bed (13), the granular medium jams and the yield strength increases rapidly within a few percent of this value. Because we observe peristaltic motion, we conclude that the worm can dynamically anchor itself by manipulating ϕ near its body by changing its diameter as shown in Fig. 4E, Lower Right Inset, in addition to deploying chaetae to vary the friction between its body and the medium.

Then, assuming that the locomotion speed of the worm occurs due to a linear superposition of the contributions due to the undulatory and the peristaltic motion, we have Ucal=Uund+Uper. Fig. 5 shows a plot of Ucal versus vw in the sediments and in water corresponding the trails listed in SI Appendix, Table S1. In the case of water and shallow sediment beds, as discussed, α=0 due to absence of anchoring, and only the contribution of the undulatory stroke is included. We observe that all of the data are clustered around the slope 1 line, showing good agreement between the calculated and measured speeds in both media. Thus, we find that the burrowing speed of L. variegatus in water-saturated sediments is determined by a combination of drag-assisted propulsion provided by the transverse undulatory motion and peristaltic motion along its body. Whereas only the transverse undulatory motion is important to its swimming speed in water and near the bed surface where the overburden pressure goes to zero.

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Fig. 5.

The calculated versus the measured speeds corresponding to the trails listed in SI Appendix, Table S1 are in good agreement. A dashed line with slope 1 is drawn for reference.

Dual Strokes in Eisenia fetida.

We further investigate the dynamics of the common composting earthworm Eisenia fetida to examine whether dual peristaltic and undulatory strokes are observed in other organisms as well. The general behavior of the earthworms and their physical characteristics are discussed in SI Appendix, Earthworm Dynamics. Fig. 6A shows the projected shapes of the earthworm as it burrows straight down in a container with Lc=50 cm, Hc=30 cm, and Wc=28 mm filled with the same sediment medium over T=10 s. As in the case of L. variegatus shown in Fig. 2A, we observe that the earthworm burrows in a narrow sinusoidal path which is not much wider than its body except near its head and tail. Then, by plotting the same snapshots as for L. variegatus in the body reference frame in Fig. 2B, we observe that E. fetida also performs undulatory strokes as well as elongation–contraction strokes.

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Fig. 6.

(A and B) Tracked snapshots of the composting earthworm E. fetida of length lw=68.6 mm over T=10 s at Δt=109 ms time intervals in the laboratory reference frame (A) and body reference frame (B) as it burrows through water-saturated granular sediments. The progress of time is denoted by colors according to the color bar in A, and the length relative to the mean length is denoted according to the color bar in B. E. fetida shows peristaltic motion and transverse undulations similar to those observed in L. variegatus in Fig. 2A. The peaks in the length correlation function (C) occur at approximately twice the time interval compared to those in the velocity correlations (D).

To access the period of the longitudinal stroke, we plot KL(t) in Fig. 6C and observe a clear peak at TL=1.6 s. By comparison, a peak in Kv(t) is observed at approximately half of TL corresponding to the signature of peristaltic motion which was also seen in L. variegatus as in Fig. 4 C and E. Accordingly, we calculate Uper=4.5 mm⋅s−1 using Eq. 1 over a time interval where its average speed is measured to be 7.3 mm⋅s−1 or ∼61%, assuming α=1. Then, we obtain B from the transverse undulations to estimate Uund. We estimate TT=15.4±1 s from the peak in KT shown in SI Appendix, Fig. S7C. Then, we obtain Uund=2.4 mm⋅s−1 or ∼33% of the measured speed. Thus, a combination of peristaltic and undulatory strokes contributes to the observed burrowing speed of E. fetida in water-saturated sediments as well. These data, along with 2 other datasets, have been also added to Fig. 5 and are observed to be in overall agreement with the calculated values.