Spontaneous Emergence of Run-and-Tumble-Like Dynamics in a Robotic Analog of \textit{Chlamydomonas}: Experiment and Theory

TL;DR

A robotic system that comprises dry, self-propelled robots linked by a rigid rod that exhibits RT-like behavior, characterized by sharp, direction-reversing tumbles and exponentially distributed run times, consistent with the real organism.

Abstract

Run-and-tumble (RT) motion is commonly observed in flagellated microswimmers, arising from synchronous and asynchronous flagellar beating. One such example is a biflagellated alga, called \textit{Chlamydomonas reinhardtii}. Its flagellar synchronization is not only affected by hydrodynamic interactions but also through contractile stress fibers that mechanically couple the flagella, enabling adaptable swimming behavior. To explore this, we design a macroscopic mechanical system that comprises dry, self-propelled robots linked by a rigid rod to model this organism. By varying the attachment points of the two ends of the rod on each robot, the model incorporates the effect of fiber contractility observed in the real organism. To mimic a low Reynolds number environment, we program each robot to undergo overdamped active Brownian (AB) motion. We find that such a system exhibits RT-like behavior, characterized by sharp, direction-reversing tumbles and exponentially distributed run times, consistent with the real organism. Moreover, we quantify tumbling frequency and demonstrate its tunability across experimental parameters. Additionally, we provide a theoretical model that reproduces our results, elucidating physical mechanisms governing RT dynamics. Thus, our robotic system not only replicates RT motion but also captures several subtle characteristics, offering valuable insights into the underlying physics of microswimmer motility.

Figures (4)

• Figure 1: (a) Photograph of the experimental system featuring coupled robots. The pivot points, $P_1$ and $P_2$, allow the connecting rod to rotate freely in the horizontal plane. (b) Typical trajectories of free robots executing overdamped AB motion with $v_a$ = 5 cm s$^{-1}$ for three different values of the rotational diffusion constant, $D_r$ (in rad$^2$ s$^{-1}$). (c) A schematic diagram highlighting key variables incorporated into the theoretical model. (d) A typical RT trajectory of the centroid point $C$ in the direction of black arrows, with $v_a$ = 5 cm s$^{-1}$, $D_r$ = 0.06 rad$^2$ s$^{-1}$, $\delta$ = 3 cm, and $\alpha = 90^\circ$. The color bar shows the value of $\beta$. (e) Zoomed-in views of typical tumble events under conditions of high (top) and low (bottom) substrate friction. We increased $v_a$ to 20 cm s$^{-1}$ for the low friction case to better illustrate the smoothening of the tumble event.
• Figure 2: (a, b) Probability distribution of $\beta$ shows a pronounced peak near $\beta$ = 0$^\circ$. Insets: $\beta$ and $V$ are inversely related to each other for both experiment and simulation. Black dashed lines at $\beta$ = 60$^\circ$ set a threshold to differentiate between run and tumble events. (c, d) Typical behaviour of $\beta$ as a function of time. $\tau_\mathrm{t}$ and $\tau_{\text{run}}$ represent tumble duration and run-time respectively. (e, f) In both experiment and simulation, we observe exponentially decaying run times (dashed lines as a guide) and unimodal tumble durations. (g, h) Tumble angle ($\theta_\mathrm{t}$) distributions from experiment and simulation respectively. Inset: A schematic defining $\theta_\mathrm{t}$ as the change in orientation between two successive run events. Hollow and solid symbols correspond to measurements taken at half and one-quarter of the run trajectory length (Lrun), respectively. Distributions are steeper for $L_\text{run}/4$, indicating sharp run reversal events as predominant tumble events. Experimental and simulation parameters are $v_a$ = 5 cm s$^{-1}$, $D_r$ = 0.06 rad$^2$ s$^{-1}$, $\delta$ = 3 cm, and $\alpha = 90^\circ$.
• Figure 3: (a, b) The tumbling frequency, $\lambda$ increases with $D_r$ and $\delta$. (c, d) The phase diagram of $\lambda$ in $\alpha-D_r$ plane for $\delta$ = 3 cm. The region enclosed by the dashed line corresponds to $\lambda \approx 0$. (e, f) For $\alpha > 90^\circ$, $\lambda$ exhibits a critical $D_r$ indicated by a black arrowhead beyond which tumbling emerges in the system. Insets: The run speed, $V_\text{run}\approx v_a$ for $\alpha \leq 90^\circ$ but decreases monotonically for $> 90^\circ$. Here $D_r$ = 0.06 rad$^2$ s$^{-1}$ and error bars represent standard deviation.
• Figure 4: (a) A schematic diagram clarifying the angular coordinates $(\theta_+,\theta_-)$. Here, $\theta_\pm = (\theta_1\pm\theta_2)/2$. (b) Flow diagram of $(\theta_+,\theta_-)$ for $\alpha = 120^\circ$ evaluated from the deterministic part $\mathcal{T}_\pm$ of Eq. \ref{['theq']}. $S_1$$\&$$S_2$ and $U_1$$\&$$U_2$ encircled by dotted lines represent stable and unstable points, respectively. Black dashed lines represent experimental trajectories for $D_r = 0$. Schematic configurations of $S_1$ and $S_2$ are shown at the bottom. Here, $\mathbf{n_1}$ and $\mathbf{n_2}$ are misaligned by an angle of $2\alpha-\pi=60^\circ$ in both cases. (c) Flow diagram for $\alpha = 90^\circ$. Blue dotted lines at $\theta_- = n\pi$ are semi-stable in $\theta_-$ and neutral in $\theta_+$, where $n$ denotes positive integers. Black dashed lines are the experimental trajectories for $D_r = 0$. (d) Flow profile for $\alpha = 60^\circ$. The system exhibits four unstable points ($U_1$, $U_2$, $U_3$, and $U_4$). Solid blue lines at $\theta_- = n\pi$ represent configurations that are stable in $\theta_-$ and neutral in $\theta_+$. The black dashed experimental trajectories for the $D_r = 0$ case. Three among infinitely many run configurations corresponding to $\mathbf{n_1}\parallel \mathbf{n_2}$ are shown at the bottom. The color bars in (b), (c) $\&$ (d) correspond to the magnitude of the vector $(\mathcal{T}_+, \mathcal{T}_-)$ in rad s$^{-1}$.