Hydrodynamic origins of symmetric swimming strategies

TL;DR

The results suggest that the prevalence of symmetric and alternating gaits in nature reflects not merely a developmental constraint, but a physical optimality principle for locomotion in viscous environments, complementing developmental and neural constraints.

Abstract

Efficient locomotion is important for the evolution of complex life, yet the physical principles selecting specific swimming strokes often remain entangled with biological constraints. In viscous fluids, the scallop theorem constrains the temporal organization of strokes, but no analogous principle is known for their spatial structure, leaving the prevalence of symmetric gaits across diverse organisms without a physical explanation. Here we show that spatial symmetry acts as an emergent organizing principle for efficiency in viscous fluids. By analysing deformable swimmers whose strokes are not constrained to any particular symmetry class, we identify a hydrodynamic duality: symmetric and anti-symmetric strokes are dynamically equivalent, yielding identical speeds and efficiencies, which we prove are optimal among all strokes. We validate this using numerical simulations of Stokes flow, demonstrating that these symmetry rules persist even in three-dimensional body plans. Our results suggest that the prevalence of symmetric and alternating gaits in nature reflects not merely a developmental constraint, but a physical optimality principle for locomotion in viscous environments, complementing developmental and neural constraints.

Figures (7)

• Figure 1: Swimming strokes and their symmetry decomposition. (a) A bilaterally symmetric swimmer, with the sagittal (reflection) plane defined by the swimming direction and the dorsal--ventral axis. (b) Three classes of swimming strokes illustrated by montages of the body shape over one cycle, with representative animal silhouettes: symmetric strokes, anti-symmetric strokes, and non-symmetric strokes. (c) Fourier decomposition of the swimmer boundary. The radial displacement $R(\theta,t)$ and tangential coordinate $\Theta(\theta,t)$ are expanded in cosine and sine harmonics, with symmetric (even) modes ($\alpha_n$, $\beta_n$; blue) and anti-symmetric (odd) modes ($\gamma_n$, $\delta_n$; orange).
• Figure 1: Optimal swimming efficiency as a function of mode number. Maximum swimming efficiency $\eta^*_n$ for a two-mode stroke using Fourier modes $n$ and $n{+}1$, computed from the eigenvalue problem (Eq. \ref{['eq:eigprob']}). The efficiency increases monotonically with mode number and converges to the limiting value $\eta^*_\infty = \sqrt{2}/(8\pi) \approx 0.0563$ (dashed line), first obtained by Shapere and Wilczek shapere1989efficiencies. Closed-form expressions for $\eta^*_2$ and $\eta^*_3$ are given in Eqs. \ref{['eq:eta_star']} and \ref{['eq:eta_star_n3']}. By hydrodynamic duality, the same efficiency values apply to the corresponding anti-symmetric strokes.
• Figure 2: Symmetry, hydrodynamic duality, and swimming efficiency. (a) Schematic of the duality. (b) Dimensionless efficiency $\eta=\langle U\rangle/\langle P\rangle$ comparing random swimmers using $n=2,3$ modes, with and without the constraint of $\langle \Omega\rangle=0$, with the symmetrized swimmers. The dotted line corresponds to the theoretical maximum efficiency $\eta^*_2$ under the constraint that the swimmer uses $n=2,3$ modes. (c) Optimal straight swimmer within the symmetric (equivalently, anti-symmetric) subspace when restricted to the $n=2,3$ modes, or the $n=3,4$ modes, illustrating the corresponding shape cycle.
• Figure 2: Regularization parameter selection for the boundary integral solver. Both panels show results for three stroke classes (symmetric $n{=}2$, anti-symmetric, and combined), evaluated at deformation amplitude $\varepsilon=0.01$ with coarse resolution $M_c=36$ and fine resolution $M_f=72$ boundary vertices and $T=256$ time steps per period. (a) Mesh-convergence indicator $|U_f/U_c - 1|$, measuring the fractional change in computed speed when doubling the spatial resolution from $M_c$ to $M_f$, as a function of the Stokeslet regularization factor $\varepsilon_{\mathrm{reg}}$. (b) Theory error $|(U_{\mathrm{num}}-U_{\mathrm{th}})/U_{\mathrm{th}}|$, measuring the deviation of the fine-grid numerical result from the $O(\varepsilon^2)$ analytical prediction. Dashed vertical lines indicate the value of $\varepsilon_{\mathrm{reg}}$ that minimises the mesh-convergence indicator for each stroke class; this criterion was adopted to determine the per-stroke regularization parameters throughout the paper.
• Figure 3: Large deformations and extension to three-dimensional swimmers. (a) Ratio of numerically computed to analytically predicted mean swimming speed ($\mathrm{Numerics}/\mathrm{theory}$) as a function of deformation amplitude $\varepsilon$, for five stroke classes. At large deformations the boundary parameterisation becomes non-monotonic and the shape approaches a singularity, making the swimming velocity ill-defined; this is why calculations for some strokes are truncated at $\varepsilon \sim 0.2$--$0.3$. (b) Swimming speed of the symmetric stroke versus that of its anti-symmetric dual for $256$ random samples of $M=576$ two-mode stroke pairs, coloured by the static ellipticity $e = (1+\varepsilon_d)/(1-\varepsilon_d)$ of the reference body, where $\varepsilon_d$ parameterises the undeformed shape as $R(\theta) = a(1 + \varepsilon_d \cos 2\theta)$. Deformation amplitude was fixed at $\varepsilon=0.15$. The unit of speed is $a/\tau$. Points on the dashed diagonal confirm equal speeds (hydrodynamic duality). Outliers deviating by more than a factor of $3/2$ are classified by their behaviour in an $\varepsilon$-sweep (Extended Data Fig. \ref{['figS4']}): triangles denote near-cancellation of the leading-order velocity (zero crossing), squares denote irregular divergence of the numerics from the perturbative prediction. Inset: probability of such a deviation as a function of $|\varepsilon_d|$. (c) Schematic of the three-dimensional "ellipsoid slice" swimmer. Left: ellipsoidal body with the equatorial ($x$--$y$) swimming plane. Top right: cross-sections of the deformed body in the swimming ($x$--$y$) plane at several $z$-levels. Bottom right: the $x$--$z$ cross-section showing the body silhouette and slice positions. (d) Normalised swimming speed $|\langle U_x\rangle|/\varepsilon^2$ (in the unit of $L_x/\tau$) versus body aspect ratio $L_z/L_x$ for Pair 1 ($n=2,3$ modes, solid) and Pair 2 ($n=3,4$ modes, dashed), each in symmetric (blue, open markers) and anti-symmetric (orange, filled markers) variants, at deformation amplitude $\varepsilon=0.15$.