Tunable asymmetric swimming in biflagellate microswimmers

TL;DR

This work identifies and validate a core principle of asymmetric turning in biflagellate microswimmers: propulsive forces interact constructively to drive translation whilst interacting destructively to drive rotation.

Abstract

Many biological microswimmers can modulate their swimming gait to achieve directional control of motility, especially when performing steering towards specific directional cues. This can be achieved without the need for obvious morphological or structural asymmetries in the form of the organism, or in the number or organisation of propulsion-generating appendages such as cilia. In this work, we identify and validate a core principle of asymmetric turning in biflagellate microswimmers: propulsive forces interact constructively to drive translation whilst interacting destructively to drive rotation. We explore the ramifications of this tunable biflagellar swimming mechanism across a range of systems, from a simple, back-of-the-envelope model to a detailed computational representation of an exemplar swimmer. This leads to a markedly general quantitative relation between key drivers of asymmetry, such as ciliary beat frequency, and the curvature of emergent trajectories. We discuss how the model green alga Chlamydomonas reinhardtii, which actuates its two cilia in a symmetric breaststroke for forward swimming, may exploit this feature for phototaxis. Finally, we validate our predictions in a Chlamydomonas-inspired robophysical model, implementing closed-loop control to achieve phototactic turning.

Figures (8)

• Figure 1: Examples of common biflagellate microswimmers found in nature (sizes range from 520). Illustrated are morphologies representative of each genus. A. Chlamydomonas, B. Dunaliella, C. Chlorogonium, D. Spermatozopsis, E. Nephroselmis.
• Figure 2: Two simple models of biflagellar propulsion, drawn to resemble an exemplar swimmer. In (a), we illustrate a minimal model that directly relates rotation rates to differences in propulsive forces, and linear velocities to sums of propulsive forces. In (b), we relax these assumptions of direct proportionality, deriving the equations of motion through a more refined force- and torque-balance argument that includes further geometrical information.
• Figure 3: Exploring frequency asymmetry via a scaling argument. (a) Sample trajectories in the $xy$-plane for two different functions $f$, showcasing the emergence of large-scale trajectories that are approximately circular, with resonant outliers. (b) Plots of curvature, both as computed from full numerical simulations of \ref{['eq: freq: model']} (coloured dots for various functions $f$) and the scaling argument of \ref{['eq: freq: scaling']} (dotted black line). As predicted by the scaling argument, the curvature of the swimmer paths is approximately independent of the details of the forcing function $f(t)$ -- in fact, the dots for various $f$ often coincide almost exactly for a given ratio $k_R/k_L$. Moreover, the scaling law and empirically computed curvatures show excellent agreement, validating the scaling argument. In both (a) and (b), we have taken $A=1$, $\eta_r=\eta_t=1$ and $k_L = 1$, varying $k_R$. In (b), $g$ is defined as $g(t) = 1+0.05 \sin(2t)+0.05 \cos(3t)+0.05 \sin(4t)$ and we have renormalised the curvatures for each $f$ by the curvature for $k_R/k_L = 1.1$.
• Figure 4: Resonance accompanies frequency asymmetries, explaining discrepancies in the predicted scaling law. We plot the integral of $\cos{\theta(t)}$, which contributes to one component of the swimmer translation, for $\beta\ll1$ (black) and $\beta=0$ (grey). The approximation of \ref{['eq: better model: approximate theta']} corresponds to $\beta=0$, which closely matches the exact result with $\beta\ll1$ except near $k_R/k_L\in\{2,3\}$, where resonance occurs and the approximation incurs significant errors. Inset, we show the difference between the two curves, which is small apart from near the resonant frequencies. Here, we have taken $J(z) = \cos(z) + \cos(2z)$ and $\beta/\alpha=0.05$ and integrated from $t=0$ to $t=30\pi$.
• Figure 5: Validating the frequency-asymmetry scaling law with a detailed computational model. Using the detailed model of \ref{['sec: computational model']}, illustrated in (a), we repeat the exploration of \ref{['fig: freq']} for a prescribed beating pattern. (b) Varying $k_R/k_L$, we simulate the motion and compute the resulting path curvatures (black crosses). We also plot the prediction of the scaling law of \ref{['eq: freq: scaling']} with a fitted constant of proportionality (black dotted curve), evidencing excellent agreement with the computational results. As predicted by the analysis of our improved swimmer model, resonance-like deviations from the prediction can be seen for discrete values of $k_R/k_L$.