Geometric phase predicts locomotion performance in undulating living systems across scales
Self-propelling organisms locomote via generation of patterns of self-deformation. Despite the diversity of body plans and the complex challenges of integrating electrical, chemical, and mechanical signals to achieve such shape changes, in steady forward movement, limbless systems across scales and environments often use travelling waves of axial body bending. However, delineating how parameters (wave amplitudes, frequencies) lead to high locomotor performance (e.g. speed, stability, energy) remains challenging. Here we show that a geometric framework developed in the last decades by theoretical physicists and control engineers that replaces complex calculation with a diagrammatic scheme is well suited to discovery and comparison of optimal patterns of wave dynamics in living systems across scales. We focus on a regime of undulatory locomotion, that of highly damped environments which is applicable not only to small organism movement in viscous fluids, but also animals moving in frictional fluids (sand) and on frictional ground. We find that the travelling wave dynamics used by mm-scale nematode worms and cm-scale desert dwelling snakes and lizards can be described by sinusoidal time-variation of the amplitudes of two sinuouslike modes. These near circular paths in a self-deformation space enclose near-maximal surface integral (geometric phase) across two decades in body length. Thus, the introduction of techniques from differential geometry into living systems can assist in the growth of comparative neuromechanics, allowing comparison of self-deformation dynamics across taxa and providing hypotheses for function at lower levels of organization.