Direct Measurement of Inertial Impact and Propulsive Force in a Eukaryotic Swimmer

TL;DR

The study tackles how force transduction from a biological motor translates to motion in swimmers near $Re \sim 1$, where inertia matters. By directly deconvolving inertial impact from propulsive force in two Volvox species, the authors reveal a ~30 Hz pulsatile motor output associated with metachronal ciliary waves. The smaller Volvox carteri exhibits strong velocity fluctuations at this frequency, while the larger Volvox ferrisii shows inertial damping that yields a smooth trajectory, consistent with a dynamic model $m\,\dfrac{dv}{dt} = F(t) - 6\pi\eta rv$. This demonstrates that, beyond the Stokes regime, organismal inertia decouples motor dynamics from swimming kinematics, with implications for active-matter theories and the evolution of multicellularity.

Abstract

The transduction of force into motion for microswimmers at intermediate Reynolds numbers ($Re \sim 1$), where inertia becomes relevant, is a fundamental problem in active matter. Using the multicellular alga \textit{Volvox} as a model physical system, we perform the first direct measurements that deconvolve a swimmer's inertial impact force from its motor's propulsive force. We discover a $\sim$30 Hz propulsive pulse, the mechanical signature of collective ciliary action. This high-frequency motor output drives a fluctuating velocity in the low-$Re$ \textit{V. carteri}, but is mechanically filtered by the inertia of the larger \textit{V. ferrisii}, resulting in a smooth swimming trajectory. Our work demonstrates that for swimmers beyond the Stokes regime, kinematics are not a direct proxy for the underlying motor dynamics, a foundational assumption in the study of microscopic motility.

Figures (6)

• Figure 1: Force traces measured with an optical lever system during collision reveal a $\sim$30 Hz propulsive oscillation. (a) Representative force trace for V. carteri (Vc). (b) Power spectrum of the trace in (a). (c) Representative force trace for V. ferrisii (Vf). (d) Power spectrum of the trace in (c).
• Figure 2: Free-swimming kinematics, measured by high-speed microscopy, show inertial damping of the force pulse. (a) Time-lapse images of Vc swimming. (b) Time-lapse images of Vf swimming. Scale bars, 100 $\mu$m. (c) The velocity of Vc oscillates strongly, corresponding to (d) a distinct peak at $\sim$30 Hz in its power spectrum. (e) In contrast, the velocity of Vf is nearly constant, and (f) the $\sim$30 Hz peak in its power spectrum is strongly attenuated.
• Figure 3: Inertial impact force is a signature of Vf motility. (a,b) Representative force traces showing the onset of collision for (a) Vc and (b) Vf. (c) Magnified view of the collision onset in (b), defining the peak impact force ($F_{\text{imp}}$) and the peak of the first subsequent propulsive cycle ($F_{\text{prop}}$). (d) The ratio $F_{\text{imp}}/F_{\text{prop}}$ is $\sim$1 for Vc but significantly $>$1 for Vf.
• Figure 4: Hydrodynamic simulation confirms inertial damping. (a) Schematic of the 1D model. (b,c) Simulated velocity traces for (b) Vc and (c) Vf, subjected to a 30 Hz force pulse. The model correctly predicts significant velocity oscillations for Vc and a damped, smooth trajectory for Vf.
• Figure 5: Experimental setup and signal validation. (a) Photomicrographs of V. carteri (Vc) and V. ferrisii (Vf). Scale bars, 100 $\mu$m. (b) Schematic of the optical lever system. (c) Representative raw signals demonstrating the protocol for validating genuine collision events. A genuine collision (Type I, blue shaded region) is characterized by a clean deflection signal (D) with stable total laser intensity (I). An optical artifact (Type II) is identified by a sharp drop in intensity and a large torsional signal (T).