Gait Modulation of Undulatory Microswimmers through the Lens of Optimality

To navigate its habitat, an organism must be able to adapt its locomotory gait to its physical surroundings and behavioral objectives. We analyze the gait performance of lateral undulatory slender limbless microswimmers immersed in a Newtonian fluid environment. Our goal is to investigate the optimality of gait selection and adaptation, considering variables such as speed and energy efficiency. Specifically, we focus on the 1 millimeter long roundworm Caenorhabditis elegans (C. elegans), which adapts its swimming gait to the external load imposed by its surrounding environment. So far, it has not been understood if this gait adaptation can be considered optimal.

Here, we hypothesize that C. elegans selects an undulation frequency and waveform that optimizes energy efficiency. Using a viscoelastic Cosserat rod model to describe the worm's biomechanics, we simulate its undulatory locomotion in fluid environments spanning four orders of magnitude in viscosity. We demonstrate that C. elegans' undulation frequency minimizes its cost of transport and is significantly slower than the predicted fastest frequency. Furthermore, C. elegans frequency adaptation can be understood as an attempt to match its actuation time scale to the response time of its body in a given environment. We show that this adaption is crucial to facilitate efficient undulatory locomotion in more viscous environments and can, therefore, be observed in other microswimmers such as sperm. In low fluid viscosities, we find that the worm's energy cost is dominated by internal friction, favoring long wavelengths and small amplitudes, whereas in high viscosities, external friction takes precedence, favoring shorter wavelengths and larger amplitudes. This trend aligns with experimental observation, which suggests that C. elegans gait adaptation is driven by energy efficiency. Through a quantitative comparison between experimental data and model prediction, we estimate an optimal value for the internal damping coefficient (viscosity) of C. elegans' body material.

Our results show how the interplay between the different friction forces within the model shapes gait optimality. We believe that this interplay could be a more general driver behind gait adaptation in organisms that need to navigate environments with varying levels of external load.