Dispersion Relations for Undulatory Locomotion in Nematodes

Nematodes modify their gait parameters (wavelength λ<!--[if gte msEquation 12]> style='mso-bidi-font-style:normal'>λ, frequency f, and curvature amplitude κ<!--[if gte msEquation 12]> style='mso-bidi-font-style:normal'>κ) in response to changing environmental rheologies. We found that for a broad range of viscous and viscoelastic fluids as well as granular and gel environments, the different gaits approximately collapse onto a single curve with <!--[if gte msEquation 12]> style='mso-bidi-font-style:normal'>f∝ style='font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style:
normal'>λ ^{style='mso-bidi-font-style:normal'>2}f∝λ², implying that a single ‘dispersion relation’ governs undulatory wave propagation despite differences in the form of external resistive forces. To explain this, we derived a theoretical dispersion relation for swimming in a fluid by regarding the nematode as a viscoelastic beam in a low-Re, Newtonian fluid, following Fang-yen et al (PNAS, 2010). Our model shows that to maintain force balance, waves must obey the observed scaling relation. Furthermore, experimentally derived model parameters lead to good quantitative agreement with the model. Surprisingly we find that the addition of elastic terms to the external drag has no effect on the real part of the dispersion relation, instead modifying the imaginary part. This implies that the presence of elasticity does not affect the f-λ scaling relation but can influence the 'evanescent' decay of undulations over time. Overall, we find that mechanical constraints arising from the structure of the equations of motion lead to common dispersion relations across different environments. This may help nematodes maintain consistent speeds by managing the tradeoff between wave efficiency (set by λ) and undulation speed (set by f) across terrains.