Motility is one of the main features of living matter, from a single cell to a swarm of birds or a human crowd [1–3]. In the last few decades, the dynamics of motile active agents, both individual and collective behavior, have been intensively studied, giving rise to a rapidly expanding research field in physics bridging nonequilibrium statistical physics, biophysics, and continuum mechanics, now known as active matter and living matter physics. A crucial feature of these systems is that inner activity units convert energy into mechanical forces. In turn, Newton's third law may be violated when we regard it as an open system, with its mechanical energy being injected from microscopic active units. Therefore, the mechanical interactions between the units can be non-reciprocal [4,5].

The concept of reciprocity is also widely used in continuum mechanics. Recently, violation of the Maxwell-Betti reciprocity in elasticity has been discovered in an active system, and termed odd elasticity [6–8]. The elastic matrix in the constitutive stress-strain relation is then allowed to contain nonsymmetric components, and it generates a self-sustained propagating wave. Odd elasticity reflects the nonconservative forces generated by microscopic active units and provides an effective material constitutive relation for active and living matter. This formulation was shown to effectively describe active locomotion as an autonomous system without controlled, tuned actuation [9].

Motile agents at the cellular scale are usually immersed in viscous fluids and are self-propelled by their deformation, as seen in swimming microorganisms [10,11]. The motility of microswimmers, a term used for active agents in a low-Reynolds-number fluid, is, however, only possible when their deformations are non-reciprocal, which is known as the scallop theorem [11–13].

Recent theoretical studies on the swimming dynamics of odd-elastic materials [14,15] revealed the relations between the violation of Maxwell-Betti reciprocity and the non-reciprocal deformation for microswimming around an equilibrium configuration, demonstrating that the swimming velocity is proportional to the magnitude of odd elasticity.

A traveling wave is a typical example of non-reciprocal deformation ubiquitously observed in biological microswimmers. Indeed, many eukaryotic cells use a flexible slender appendage, called a flagellum or cilium, for propulsion by generating a wave. Examples include tail motions of sperm cells and breaststrokes of Chlamydomonas green algae [16]. This evolutionarily conserved filament is actuated by inner molecular motors in coordination, resulting in a periodic traveling wave with a self-organized nature. The flagellar whiplike motion is therefore regarded as a limit cycle oscillator, and the generic form of flagellar swimming is provided by Hopf bifurcation [17]. Recent theoretical and numerical studies using elaborate elastohydrodynamic models also found the emergence of the various flagellar waveform patterns via Hopf bifurcation [18–21]. Moreover, refinements of videomicroscopy of biological flagella have enabled the detailed analyses of waveforms, and found that the flagellar shape dynamics are well described by a noisy limit cycle that reflects internal activity [22–27].

The self-sustained wave for an odd-elastic material, however, is insufficient to describe the flagellar waveform, because the odd-elastic waves are dissipated rather than sustained by the fluid viscosity, similar to the classical (passive) elastic response in a viscous medium [28]. Hence, nonlinearity is required for an odd-elastic system to exhibit a stable limit cycle [15], calling for a more general, nonlinear odd constitutive relation to deal with biological flagellar swimming. In fact, the importance of nonlinear odd elasticity has been reported as a topical challenge within the field of active matter studies [8].

The primary aim of this study is therefore to extend the odd-elastic description of microswimmers to a nonlinear regime to deal with stable periodic deformations, as seen in biological flagellar motion. This theory, which we call odd elastohydrodynamics, therefore provides a unified framework for the study of nonlocal, non-reciprocal interactions of an elastic material in a viscous fluid.

Using this generic formulation, we can access the interactions inside an active elastic material, while these are masked by fluid dynamic coupling when observing flagellar motion under a microscope. To distinguish the non-reciprocal activity from the passive elastic response, we introduce a new concept, the odd-elastic modulus, as a spatial Fourier transform in an extended space. The real and imaginary parts of this complex function possess proper symmetry and characterize the reciprocal and non-reciprocal interactions, respectively.

The secondary aim of this study is then to apply our theory to biological flagellar swimmers. By examining the odd-elastic modulus based on simple mathematical models and biological experimental data, we show the wide applicability of a nonlocal and non-reciprocal description of internal interactions within living materials.

The contents of this paper are summarized as follows. In Sec. II, we provide a setup for the theoretical formulation of odd elastohydrodynamics to describe an active elastic material in a viscous fluid. We also discuss the connection between Hopf bifurcation and nonlinear odd elasticity and express the dynamics of a microswimmer undergoing periodic deformation. In Sec. III, we introduce the concept of the odd-elastic modulus.

In Secs. IV and V, we apply our theory to understand the inner mechanical interactions that biological flagellar motion exhibit. To gain physical intuition regarding nonlocal, non-reciprocal interactions encoded by nonlinear odd elasticity, we start with simple and solvable models in Sec. IV. We also discuss how the odd-elastic modulus captures the inner interactions of these example models. In Sec. V, we numerically investigate the extended bending modulus in model flagellar waveforms for Chlamydomonas and sperm cells, together with experimental data. With these, we propose a new continuum description of living soft matter in a viscous fluid by means of nonlinear odd elasticity. The discussion and conclusions are provided in Sec. VI.

One of the advantages of the odd-elastic description of activity is the application of the autonomous equations of motion. These allow us to analyze some general features of microswimming with periodic deformation, including theoretical formulas for the average swimming velocity. In Appendix A, to complete our general theory of odd elastohydrodynamics, we further extend our framework to encompass fluctuations in shape gaits by internal actuation, following biological observations of a noisy limit cycle in shape space. Exploiting the autonomous structure of the odd-elastic formulation and the gauge-field formulation for microswimming, we investigate the effects of internal active noise on swimming velocity. The role of odd elasticity is further discussed in terms of nonequilibrium thermodynamics.