Role of activity and dissipation in achieving precise beating in cilia: Insights from the rower model

TL;DR

This work quantifies precision of oscillation using a quality factor, identifying its scaling with activity and oscillation amplitude, and finding precision maximization at an optimal amplitude using an exact analytic expression for the precision quality factor.

Abstract

Cilia and flagella are micron-sized filaments that actively beat with remarkable precision in a viscous medium, driving microorganism movement and efficient flow. We study the rower model to uncover how cilia activity and dissipation enable this precise motion. In this model, cilia motion is represented by a micro-bead's Brownian movement between two distant harmonic potentials. At specific locations, energy pumps trigger potential switches, capturing cilia activity and generating oscillations. We quantify precision of oscillation using a quality factor, identifying its scaling with activity and oscillation amplitude, finding precision maximization at an optimal amplitude. The data collapse is not accurate for noisy oscillations. An exact analytic expression for the precision quality factor, based on first passage time fluctuations, and derived in the small noise approximation, explains its optimality and scaling. Energy budget analysis shows the quality factor's consistency with the thermodynamic uncertainty relation. Finally, we demonstrate that asymmetric beating reduces oscillation precision compared to the symmetric model: although the optimal amplitude remains unchanged, the overall scaling of the quality factor depends explicitly on the asymmetry parameter.

Figures (5)

• Figure 1: Oscillations in the rower model for cilia. (A) A schematic of the realistic beating of a cilium is depicted. (B) It illustrates the oscillation mechanism in the rower model. A micron-sized bead in a fluid medium moves in the downhills of two harmonic potentials (represented by $\sigma=\pm1$) with stiffness $k$ and separated by a distance $\mu$. The motion is restricted between $-\mathcal{A}$ to $\mathcal{A}$ with $2\mathcal{A}<\mu$. Once the bead reaches a terminal position ($\pm\mathcal{A}$), switching between the potential happens, implying the pumping of energy $\mu k\mathcal{A}$. (C) and (E) Typical oscillating trajectories for high noise ($D=0.02\mu m^2/s$) and low noise ($D=0.005\mu m^2/s$) strengths are shown. (D) and (F) Auto-correlation functions for trajectories for subplots (C) and (E) are shown. Other parameters used: $\mu=1 \mu m$, $\mathcal{A}=0.25 \mu m$, and $k=1.5pN/\mu m$.
• Figure 2: The quality factor results for various activity and dissipation parameters are shown. (A) For small noise strengths: Scaled $Q$ vs $\mathcal{A}_s$ for five different parameter sets collapse onto a single curve with the line representing the analytical formula in Eq. \ref{['eqn:g']}. Parameters sets ($D$ in $\mu$m$^2$ s$^{-1}$, $\eta$ in mPas, $\mu$ in $\mu$m, and $k$ in pN/$\mu$m): (i) $0.02$, $7.4$, $3.0$, and $4.0$ ($\bigcirc$), (ii) $0.017$, $10.0$, $2.0$, and $6.0$ ($\triangle$), (iii) $0.017$, $10.0$, $2.0$, and $3.5$ ($\bigtriangledown$), (iv) $0.015$, $10.0$, $3.0$, and $2.0$ ($\square$), and (v) $0.01$, $7.4$, $2.0$, and $4$ ($\pentagon$). (B) For large noise strengths: Scaled $Q$ vs. $\mathcal{A}_s$ for large noise strengths, showing a deviation from data collapse. Parameters: $\eta{=}7.4$ mPas, $\mu{=}2\mu$m, and $k{=}4$pN/$\mu$m.
• Figure 3: The precision quality factors from two definitions are plotted against the scaled amplitude $\mathcal{A}_s$. Multiplying $\mathcal{Q}$ by 0.1 aligns the data, indicating the definitions differ only by a proportionality constant. Parameters: $\mu = 2 \, \mu m$$k = 4 \, pN/\mu m$. The proportionality is more accurate at low noise (A) than at higher noise levels (B).
• Figure 4: Relative variance of FPT, $CV^2(\mathcal{T})$: (A) Scaled $CV^2(\mathcal{T})$ vs. $\mathcal{A}_s = 2\mathcal{A}/\mu$ for all parameters, with the black line showing the analytical formula (Eq. \ref{['eqn:g']}). (B) Deviation from SNA results for higher noise strengths in the scaled plot. The dashed lines are obtained solving the exact formula given by Eq. \ref{['eqn:exact_cv2']}. Parameters as in Fig. \ref{['Fig:2']}.
• Figure 5: Energy dissipation rate $\dot q_{avg}$ versus scaled amplitude $\mathcal{A}_{s}$. Symbols denote simulation results, and the line represents the analytical result (Eq.\ref{['eqn:qavg']}). Parameters: $\mu = 2 \ \mu$m, $k = 4 \ pN/\mu$m, and $D=0.02\mu m^2 s^{-1}$.