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Undulation is a form of propulsion in which waves of bending propagate along an elongated, slender body. This locomotor strategy is used by organisms that span orders of magnitude in size and represent diverse habitats and species. Despite this diversity, common neuromechanical phenomena have been observed across biologically disparate undulators, as a result of common mechanics. For example, neuromechanical phase lags (NPL), a phenomenon where waves of muscle contraction travel at different speeds than the corresponding body bends, have been observed in fish, lamprey, and lizards. Existing theoretical descriptions of this phenomenon implicate the role of physical body-environment interactions. However, systematic experimental variation of body-environment interactions and measurement of the corresponding phase lags have not been performed. Using the nematode we measured phase lags across a range of environmental interaction regimes, performing calcium imaging in body wall muscles in fluids of varying viscosity and on agar. A mechanical model demonstrates that the measured phase lags are controlled by the relative strength of elastic torques within the body and resistive forces within the medium. We further show that the phase lags correspond with a difference in the wave number of the muscle activity and curvature patterns. Hence, the environmental forces that create NPL also act as a filter that shapes and modulates the gait articulated by the nervous system. Beyond nematodes, the simplicity of our model suggests that tuning body elasticity may serve as a general means of controlling the degree of mechanical wave modulation in other undulators. 

Self-propelling organisms locomote via generation of patterns of self-deformation. Despite the diversity of body plans, internal actuation schemes and environments in limbless vertebrates and invertebrates, such organisms often use similar traveling waves of axial body bending for movement. Delineating how self-deformation parameters lead to locomotor performance (e.g. speed, energy, turning capabilities) remains challenging. We show that a geometric framework, replacing laborious calculation with a diagrammatic scheme, is well-suited to discovery and comparison of effective patterns of wave dynamics in diverse living systems. We focus on a regime of undulatory locomotion, that of highly damped environments, which is applicable not only to small organisms in viscous fluids, but also larger animals in frictional fluids (sand) and on frictional ground. We find that the traveling wave dynamics used by mm-scale nematode worms and cm-scale desert dwelling snakes and lizards can be described by time series of weights associated with two principal modes. The approximately circular closed path trajectories of mode weights in a self-deformation space enclose near-maximal surface integral (geometric phase) for organisms spanning two decades in body length. We hypothesize that such trajectories are targets of control (which we refer to as “serpenoid templates”). Further, the geometric approach reveals how seemingly complex behaviors such as turning in worms and sidewinding snakes can be described as modulations of templates. Thus, the use of differential geometry in the locomotion of living systems generates a common description of locomotion across taxa and provides hypotheses for neuromechanical control schemes at lower levels of organization. 

Abstract Humanity has long sought inspiration from nature to innovate materials and devices. As science advances, nature-inspired materials are becoming part of our lives. Animate materials, characterized by their activity, adaptability, and autonomy, emulate properties of living systems. While only biological materials fully embody these principles, artificial versions are advancing rapidly, promising transformative impacts in the circular economy, health and climate resilience within a generation. This roadmap presents authoritative perspectives on animate materials across different disciplines and scales, highlighting their interdisciplinary nature and potential applications in diverse fields including nanotechnology, robotics and the built environment. It underscores the need for concerted efforts to address shared challenges such as complexity management, scalability, evolvability, interdisciplinary collaboration, and ethical and environmental considerations. The framework defined by classifying materials based on their level of animacy can guide this emerging field to encourage cooperation and responsible development. By unravelling the mysteries of living matter and leveraging its principles, we can design materials and systems that will transform our world in a more sustainable manner.