^{1}, Eligiusz Wajnryb

^{2}, Siva A. Vanapalli

^{3}and Jerzy Blawzdziewicz

^{1}

### Abstract

The millimeter-long soil-dwelling nematode Caenorhabditis elegans propels itself by producing undulations that propagate along its body and turns by assuming highly curved shapes. According to our recent study [V. Padmanabhan et al. , PLoS ONE7, e40121 (Year: 2012)10.1371/journal.pone.0040121] all these postures can be accurately described by a piecewise-harmonic-curvature model. We combine this curvature-based description with highly accurate hydrodynamic bead models to evaluate the normalized velocity and turning angles for a worm swimming in an unconfined fluid and in a parallel-wall cell. We find that the worm moves twice as fast and navigates more effectively under a strong confinement, due to the large transverse-to-longitudinal resistance-coefficient ratio resulting from the wall-mediated far-field hydrodynamic coupling between body segments. We also note that the optimal swimming gait is similar to the gait observed for nematodes swimming in high-viscosity fluids. Our bead models allow us to determine the effects of confinement and finite thickness of the body of the nematode on its locomotion. These effects are not accounted for by the classical resistive-force and slender-body theories.

We would like to acknowledge financial support from National Science Foundation (NSF) Grant No. CBET 1059745 (A.B. and J.B.) and National Science Center (Poland) Grant No. 2012/05/B/ST8/03010 (E.W.). S.V. acknowledges NSF CAREER Award Grant No. 1150836.

I. INTRODUCTION

II. NEMATODE KINEMATICS: PIECEWISE-HARMONIC-CURVATURE REPRESENTATION OF NEMATODE GAIT

III. NEMATODE HYDRODYNAMICS

A. Balance of forces and torques acting on the nematode body

B. Bead models

C. Effect of confinement on transverse and longitudinal hydrodynamic forces

IV. EFFECTIVENESS OF LOCOMOTION FOR DIFFERENT CONFINEMENTS AND NEMATODE GAITS

V. HYDRODYNAMICS OF TURNS

VI. CONCLUSIONS

### Key Topics

- Hydrodynamics
- 19.0
- Land transportation
- 18.0
- Hydrological modeling
- 15.0
- Photonic crystals
- 12.0
- Biological movement
- 8.0

## Figures

Typical body postures of C. elegans (a) crawling on agar (W-shape), (b) making Ω-turn on agar, and (c) swimming in water (C-shape). The corresponding right panels show PHC description of the shapes in the left panels. Circles represent the numerical skeletons of the worm images, and the lines show results of a single-mode PHC model (see Sec. II and Ref. 36 ).

Typical body postures of C. elegans (a) crawling on agar (W-shape), (b) making Ω-turn on agar, and (c) swimming in water (C-shape). The corresponding right panels show PHC description of the shapes in the left panels. Circles represent the numerical skeletons of the worm images, and the lines show results of a single-mode PHC model (see Sec. II and Ref. 36 ).

Time progressions for a crawling and a swimming nematode performing the same set of body movements: (a) crawling worm (thick line) slides with velocity v s along a predetermined curve (thin line); (b) swimming worm undergoes translational and rotational slip superposed with the motion along the curve.

Time progressions for a crawling and a swimming nematode performing the same set of body movements: (a) crawling worm (thick line) slides with velocity v s along a predetermined curve (thin line); (b) swimming worm undergoes translational and rotational slip superposed with the motion along the curve.

Curves defined by sinusoidal curvature (5) for several values of the normalized amplitude A/q.

Curves defined by sinusoidal curvature (5) for several values of the normalized amplitude A/q.

Nematode body modeled as a chain of touching spheres. (a) The spheres follow the curve defined by the PHC model with wave-velocity v s . (b) Prescribed individual bead rotations mimic the motion of the interface of the nematode.

Nematode body modeled as a chain of touching spheres. (a) The spheres follow the curve defined by the PHC model with wave-velocity v s . (b) Prescribed individual bead rotations mimic the motion of the interface of the nematode.

Schematic of the flow field generated by an elongated body dragged in a transverse direction through an unconfined fluid. Overall, the scattered flow is in the same direction as the velocity of the body; the resulting resistance force is moderate.

Schematic of the flow field generated by an elongated body dragged in a transverse direction through an unconfined fluid. Overall, the scattered flow is in the same direction as the velocity of the body; the resulting resistance force is moderate.

The elongated-piston effect: The motion of an elongated body dragged in the transverse direction in a parallel-wall channel produces long-range pressure-driven recirculation pattern. The corresponding pressure drop across the body results in a large resistance force. (a) Side view of the system and (b) top view.

The elongated-piston effect: The motion of an elongated body dragged in the transverse direction in a parallel-wall channel produces long-range pressure-driven recirculation pattern. The corresponding pressure drop across the body results in a large resistance force. (a) Side view of the system and (b) top view.

Ratio ζ⊥/ζ|| between the transverse and longitudinal hydrodynamic-resistance coefficients for a linear chain of equal-size spheres vs. the chain length N for unconfined system (dashed line) and parallel-wall channels (solid lines). Channel width normalized by the bead diameter, H/d, is as labeled. The chain moves in the midplane of the channel.

Ratio ζ⊥/ζ|| between the transverse and longitudinal hydrodynamic-resistance coefficients for a linear chain of equal-size spheres vs. the chain length N for unconfined system (dashed line) and parallel-wall channels (solid lines). Channel width normalized by the bead diameter, H/d, is as labeled. The chain moves in the midplane of the channel.

Ratio ζ⊥/ζ|| between transverse and longitudinal hydrodynamic-resistance coefficients vs. normalized gap width H/d. Solid lines represent linear chains of equal-size spheres with chain length N, as labeled. Results of a modified slender-body theory 47 for a confined cylinder of diameter d and infinite length are represented by a dotted line. Inset shows a blowup of the region of the moderate values of H/d.

Ratio ζ⊥/ζ|| between transverse and longitudinal hydrodynamic-resistance coefficients vs. normalized gap width H/d. Solid lines represent linear chains of equal-size spheres with chain length N, as labeled. Results of a modified slender-body theory 47 for a confined cylinder of diameter d and infinite length are represented by a dotted line. Inset shows a blowup of the region of the moderate values of H/d.

Tail trajectories and snapshots of body positions at equally spaced times for a nematode swimming in unconfined fluid (left) and in the midplane of a parallel-wall channel of normalized width H/d = 1.3 (right).

Tail trajectories and snapshots of body positions at equally spaced times for a nematode swimming in unconfined fluid (left) and in the midplane of a parallel-wall channel of normalized width H/d = 1.3 (right).

Normalized swimming velocity γ S vs. wavevector q normalized by the worm length L for a nematode swimming in an unconfined fluid. Normalized amplitude A/q is as labeled. Insets show nematode shapes for parameters corresponding to the points indicated by filled circles.

Normalized swimming velocity γ S vs. wavevector q normalized by the worm length L for a nematode swimming in an unconfined fluid. Normalized amplitude A/q is as labeled. Insets show nematode shapes for parameters corresponding to the points indicated by filled circles.

Normalized swimming velocity γ S vs. wavevector q normalized by the worm length L for a nematode swimming in a midplane of a parallel-wall channel of width (a) H/d = 3 and (b) H/d = 1.3. Normalized amplitude A/q is as labeled. Results obtained by using the CR method are represented by solid lines and those by the HSD approximation are represented by dashed lines.

Normalized swimming velocity γ S vs. wavevector q normalized by the worm length L for a nematode swimming in a midplane of a parallel-wall channel of width (a) H/d = 3 and (b) H/d = 1.3. Normalized amplitude A/q is as labeled. Results obtained by using the CR method are represented by solid lines and those by the HSD approximation are represented by dashed lines.

Normalized swimming velocity γ S vs. normalized amplitude A/q for a nematode swimming in unconfined fluid (as labeled) and in the midplane of a parallel-wall channel for channel width as labeled. Results obtained by using the CR method are represented by solid lines and those by the HSD approximation are represented by dashed lines. The normalized wavevector qL corresponds to the maximal efficiency for the given geometry and amplitude. The dotted line represents normalized velocity for a nematode crawling without slip.

Normalized swimming velocity γ S vs. normalized amplitude A/q for a nematode swimming in unconfined fluid (as labeled) and in the midplane of a parallel-wall channel for channel width as labeled. Results obtained by using the CR method are represented by solid lines and those by the HSD approximation are represented by dashed lines. The normalized wavevector qL corresponds to the maximal efficiency for the given geometry and amplitude. The dotted line represents normalized velocity for a nematode crawling without slip.

Normalized swimming velocity γ S vs. normalized channel width H/d for a nematode swimming in the midplane of a parallel-wall channel for normalized amplitudes A/q as labeled. Results obtained by using the CR method are represented by solid lines and those by the HSD approximation are represented by dashed lines. The normalized wavevector qL corresponds to the maximal efficiency for a given channel geometry and normalized amplitude.

Normalized swimming velocity γ S vs. normalized channel width H/d for a nematode swimming in the midplane of a parallel-wall channel for normalized amplitudes A/q as labeled. Results obtained by using the CR method are represented by solid lines and those by the HSD approximation are represented by dashed lines. The normalized wavevector qL corresponds to the maximal efficiency for a given channel geometry and normalized amplitude.

Nematodes performing turns in different geometries: (a) worm crawling without slip; (b) W-shaped worm swimming in a parallel-wall channel; (c) W-shaped worm swimming in unconfined fluid; (d) C-shaped worm swimming in a parallel-wall channel; (e) C-shaped worm swimming in unconfined fluid. Normalized channel width H/d = 1.3. The normalized wavevector for the W-shaped worm is qL = 9 and for C-shaped worm is qL = 5.5. The turning angle and mode-switching points s 1 and s 2 are marked in (a); dashed lines indicate the direction of motion.

Nematodes performing turns in different geometries: (a) worm crawling without slip; (b) W-shaped worm swimming in a parallel-wall channel; (c) W-shaped worm swimming in unconfined fluid; (d) C-shaped worm swimming in a parallel-wall channel; (e) C-shaped worm swimming in unconfined fluid. Normalized channel width H/d = 1.3. The normalized wavevector for the W-shaped worm is qL = 9 and for C-shaped worm is qL = 5.5. The turning angle and mode-switching points s 1 and s 2 are marked in (a); dashed lines indicate the direction of motion.

Angle of turn θ vs. the normalized point of amplitude change qs 1 for the length of high-amplitude mode (a) qΔs = π/2 and (b) qΔs = π. Crawling without slip is represented by dotted lines. Swimming in unconfined fluid is represented by solid lines, and in parallel-wall channel of width H/d = 1.3 is represented by dashed lines; swimming results are presented for C-shaped worms with qL = 5.5 (thin lines) and W-shaped worms with qL = 9 (heavy lines). The results for a confined system are evaluated using HSD approximation.

Angle of turn θ vs. the normalized point of amplitude change qs 1 for the length of high-amplitude mode (a) qΔs = π/2 and (b) qΔs = π. Crawling without slip is represented by dotted lines. Swimming in unconfined fluid is represented by solid lines, and in parallel-wall channel of width H/d = 1.3 is represented by dashed lines; swimming results are presented for C-shaped worms with qL = 5.5 (thin lines) and W-shaped worms with qL = 9 (heavy lines). The results for a confined system are evaluated using HSD approximation.

Angle of turn θ vs. normalized length of high-amplitude mode qΔs for (a) qs 1 = 0 and (b) qs 1 = π/2; lines are the same as in Fig. 15 .

Angle of turn θ vs. normalized length of high-amplitude mode qΔs for (a) qs 1 = 0 and (b) qs 1 = π/2; lines are the same as in Fig. 15 .

Normalized swimming velocity γ S vs. wavevector q normalized by the worm length L for a nematode swimming in unconfined fluid. The local-curvature model is represented by the dashed–dotted line, the model with smoothed angular velocity is represented by the dashed line, and the model with no interparticle slip is represented by the dotted line. Solid lines represent resistive force theory and no rotation model as labeled.

Normalized swimming velocity γ S vs. wavevector q normalized by the worm length L for a nematode swimming in unconfined fluid. The local-curvature model is represented by the dashed–dotted line, the model with smoothed angular velocity is represented by the dashed line, and the model with no interparticle slip is represented by the dotted line. Solid lines represent resistive force theory and no rotation model as labeled.

A comparison of the HSD approximation (solid lines) with accurate results obtained using the CR method (dashed lines) for the transverse and longitudinal resistance coefficients of linear chains of touching spheres in the midplane of a parallel-wall channel. The (a) transverse and (b) longitudinal resistance coefficients per particle (normalized by the one-particle value) and (c) the resistance-coefficient ratio are shown vs. the chain length N for the normalized channel width H/d, as labeled.

A comparison of the HSD approximation (solid lines) with accurate results obtained using the CR method (dashed lines) for the transverse and longitudinal resistance coefficients of linear chains of touching spheres in the midplane of a parallel-wall channel. The (a) transverse and (b) longitudinal resistance coefficients per particle (normalized by the one-particle value) and (c) the resistance-coefficient ratio are shown vs. the chain length N for the normalized channel width H/d, as labeled.

## Tables

Coefficients , , and of the Hele–Shaw dipole approximation (B8) for active chains of touching spheres, for different values of dimensionless channel width.

Coefficients , , and of the Hele–Shaw dipole approximation (B8) for active chains of touching spheres, for different values of dimensionless channel width.

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