Odd Elastohydrodynamics: Non-Reciprocal Living Material in a Viscous Fluid

Motility is a fundamental feature of living matter, encompassing single cells and collective behavior. Such living systems are characterized by nonconservativity of energy and a large diversity of spatiotemporal patterns. Thus, fundamental physical principles to formulate their behavior are not yet fully understood. This study explores a violation of Newton's third law in motile active agents, by considering non-reciprocal mechanical interactions known as odd elasticity. By extending the description of odd elasticity to a nonlinear regime, we present a general framework for the swimming dynamics of active elastic materials in low-Reynolds-number fluids, such as wavelike patterns observed in eukaryotic cilia and flagella. We investigate the nonlocal interactions within a swimmer using generalized material elasticity and apply these concepts to biological flagellar motion. Through simple solvable models and the analysis of Chlamydomonas flagella waveforms and experimental data for human sperm, we demonstrate the wide applicability of a nonlocal and non-reciprocal description of internal interactions within living materials in viscous fluids, offering a unified framework for active and living matter physics.

Figure 1

Schematic of general odd-elastic microswimmer. This example swimmer moves in a three-dimensional space ($d=3$). The position and rotation of the swimmer are represented by the relative motions between the laboratory frame $\{{\mathit{e}}_x,{\mathit{e}}_y,{\mathit{e}}_z\}$ and the swimmer-fixed frame $\{{\mathit{e}}_x^{\left(s\right)},{\mathit{e}}_y^{\left(s\right)},{\mathit{e}}_z^{\left(s\right)}\}$. The shape of the swimmer is parametrized by $N$ shape variables $\mathit{\sigma }=({\sigma }_1,...,{\sigma }_N)$. In this schematic, we use displacements of material units from equilibrium positions as the shape variables. In typical linear elastic theory, a recovery force is applied that is proportional to the displacement using a spring constant, as indicated by $k$ in this schematic. In the current study, however, we generalize this elastic force to include internal actuation, which is represented by odd elasticity.

Figure 2

Human sperm swimming as an example of microswimmer with noisy limit cycle. (a) The lowest two principal component analysis (PCA) modes were obtained from experimental data. The horizontal axis indicates the normalized arclength along the flagellum from the head-tail junction. (b) Projections of the shape onto the two-dimensional PCA shape space. Each circle indicates the shape at a different time. (c) Superposed snapshots of a swimming human sperm with a time-periodic beat, obtained by a direct numerical simulation of the Stokes equations. The waveform was extracted from the experimental data for swimming human sperm as a limit cycle in the two-dimensional PCA shape space. Figures reproduced from Ref. [26] with permission under the Creative Commons license [35].

Figure 3

Schematics of elastic matrix $K_{\alpha \beta }$ and its continuum representation $\overline{\kappa }(s,s^{\prime })$ with decomposition into symmetric even elasticity (reciprocal interaction) and antisymmetric odd elasticity (non-reciprocal interaction), (a) $\mathbf{K}={\mathbf{K}}_{\text{e}}+{\mathbf{K}}_{\text{o}}$ and (b) $\overline{\kappa }={\overline{\kappa }}_{\text{e}}+{\overline{\kappa }}_{\text{o}}$. The two-dimensional Fourier transform is associated with a two-dimensional wave vector $({\nu }_s,{\nu }_{s^{\prime }})$. To characterize the non-reciprocal nature of the interactions encoded in the intrinsic elasticity, we consider the Fourier modes along the diagonal components and in the perpendicular direction, respectively indicated by $\nu$ and $\widehat{\nu }$ in the schematics. The odd-elastic modulus, Eq. (18), is defined by the Fourier modes in the $\widehat{\nu }$ direction.

Figure 4

Schematics of example models for intrinsic elasticity. (a) One-dimensional sphere-spring system. Spheres of radius $a$ are connected by linear springs. (b) Elastic filament immersed in a viscous fluid. Assuming a small-amplitude deformation from a straight line, we parametrize the filament shape by the height $h$.

Figure 5

Schematic of coarse-grained representation of elastic filament and typical dynamics in $q_1\text{−}{q}_2$ shape space. We represent the flagellum by $N+1$ rods of length $\mathrm{\Delta }\ell$, which are connected at each end by elastic hinges. The shape configuration is then specified by the relative angles between neighboring rods, denoted by ${\sigma }_{\alpha }$ ($\alpha =1,2,...,N$). The rotation of the filament is described by the angle between the horizontal axis and the first rod denoted by $\theta$. Inset: The shape gait is given by the autonomous system and we show its typical trajectory in the $q_1\text{−}{q}_2$ shape space. The flagellar waveform approaches a periodic pattern described by the stable limit cycle, after starting from the initial point near the origin.

Figure 6

Flagellar waveforms and odd-bending modulus along circle with radius $r=\sqrt{q_1^2+q_2^2}$ in two-dimensional shape space for sinusoidal flagellum with wavenumber $\nu /2\pi =1.5$: (a) superposed waveforms for a swimming flagellum during one beat cycle with its left end initially located at $(x,y)=(0,0)$, (b) $r=0$ at the origin, (c) $r=1$ on the limit cycle orbit, and (d) $r=1.5$ outside the limit cycle. In each panel, the real and imaginary parts are shown by the blue broken curve and the red dotted curve, respectively. The stable limit cycle corresponds to a circle of radius $r=1$.

Figure 7

Flagellar waveforms and odd-bending modulus along stable limit cycle for Chlamydomonas model with proximal end clamped at $(x,y)=(0,0)$. (a) Superposed waveforms during one beat cycle. (b) Real and imaginary parts of the odd-bending modulus.

Figure 8

Flagellar waveforms and odd-bending modulus along stable limit cycle in data-driven human sperm model. (a) Superposed waveforms for a simulated sperm flagellum during one beat cycle with its left end initially located at $(x,y)=(0,0)$. (b) Real and imaginary parts of the odd-bending modulus.

Figure 9

Steady probability distribution and probabilistic current for system with noisy limit cycle with biologically relevant multiplicative noise. Parameters are set as $k^{\text{e}}=-1, k^{\text{ne}}=1, k^{\text{o}}=1, k^{\text{no}}=1$, and $D_r=0.3$. (a) Steady distribution function with a volcano shape. The distribution function has a degenerate peak and reaches zero at the origin and infinity. The function is normalized so that the integrated probability becomes unity. (b) Distribution function projected onto the $q_1\text{−}{q}_2$ plane and superposed with probabilistic current vectors. The circular current has a maximum strength around the peak of the distribution.