Motor shot noise explains active fluctuations in a single cilium

TL;DR

The paper demonstrates that active fluctuations in cilia beating can be explained by shot noise from small-number motor binding events. By extending a previously proposed deterministic axoneme model to a stochastic framework with Poisson motor-binding dynamics, the authors show that motor noise is sufficient to reproduce observed phenomena, including transitions between no oscillations, standing waves, and traveling waves, as well as intra-cilium phase defects and finite correlation lengths. They quantify how quality factor $Q$ scales as $oldsymbol{Q}\, ext{ aisebox{0.5ex}{ extipa{–}}}\,oldsymbol{Q} extsubscript{max}$, correlate wave features with motor-number $oldsymbol{N}$, and connect microscopic motor parameters to mesoscopic non-equilibrium behavior. The work provides testable predictions, constrains motor-control theories via data-driven inference, and offers a framework for linking single-motor stochasticity to collective ciliary dynamics in living systems.

Abstract

Mesoscopic fluctuations reveal stochastic dynamics of molecules in both inanimate and living matter. We investigate how small-number fluctuations shape the collective dynamics of molecular motors using motile cilia as model system. We theoretically show that fluctuations in the number of bound motors are sufficient to explain experimentally observed fluctuations, including correlation length and ``phase slips'' of intra-cilium synchronization. Our findings constrain theories of motor control and establish a link between microscopic motor noise and mesoscopic non-equilibrium dynamics.

Figures (13)

• Figure 1: Model of cilia beating from Cass et al.Cass2023, extended to stochastic model.A. Idealized two-filament model of the axoneme, comprising filaments $\mathbf{r}_+$ (green) and $\mathbf{r}_-$ (blue) with constant spacing $a$, and sliding displacement $\Delta$ between the two filaments parameterized by the arc-length $s$ of the centerline (black). Motors attached to each filament can transiently bind to the opposite filament and exert active forces $F_\pm$ (blue/green arrows). B. A linear force-velocity relation relates motor force $F_\pm$ to sliding speed $\dot{\Delta}$. C. Motors unbind with force-dependent rate $\epsilon_\pm(F_\pm)$ with $F_\pm = F_\pm(\dot{\Delta})$. D. The feedback loop represented by the model: an increase in the fraction $n_-$ of minus-motors currently bound to the plus-filament increases the shearing force $F_-n_-$ acting on the filament pair, and hence $\dot{\Delta}$; this increase in $\dot{\Delta}$ increases $n_-$ further by force-dependent unbinding. A similar positive feedback loop applies to $n_+$, with opposite sign. A non-zero sliding speed due to motor activity continuously increases the sliding displacement $\Delta$. This causes elastic restoring forces that drive $\dot{\Delta}$ back towards and even slightly beyond its steady-state value $\dot{\Delta}=0$, which terminates the currently active motor feedback loop and starts the other. E. Typical stochastic realization starting from a straight axoneme with $\Delta\equiv 0$ and $n_\pm \equiv n^\ast$ (red), with $(t,s)$-kymographs of $\Delta/a$ (top), $n_+$ (middle), $n_-$ (bottom). The maximal amplitude is established in less than a beat cycle (blue). Parameters: Table \ref{['tab:params']}, motor number $N = 10^5$.
• Figure 2: Pattern selection is changed by noise.A-D. Computed beat frequency $f_0=\omega_0/(2\pi)$, beat amplitude $A$, wave length $\lambda$ of cilia bending waves, and quality factor $Q$ characterizing frequency jitter, as functions of motor activity $\mu_a$ and motor number $N$ (axis linear in $1/N$). We distinguish distinct regimes of no regular oscillations (NO), standing waves with $\lambda\gg L$ (SW), and traveling waves (TW). Small-number fluctuations change the SW/TW transition boundary (dashed lines). E.$Q$ scales asymptotically as $Q\sim N$. Parameters: Table \ref{['tab:params']}Cass2023; $N_\text{cilium} = 1.7\cdot 10^4$; SEM $\le$ symbol size.
• Figure 3: Partial motor extraction.A-E. To computationally mimic partial motor extraction experiments from Sharma2024 (red), we simulated the stochastic model with a reduced number $N_\text{remain}$ of motors, while keeping the characteristic force per motor $f_0\sim N^{-1}$ constant (black: $N {=} 10^5$, parameters from Cass2023; gray: $N = N_\text{cilium} = 1.7\cdot 10^4$, parameters from Cass2023; blue: $N = N_\text{cilium}$, new parameters; see Table \ref{['tab:params']}). Consistent with the experiment, beat amplitude $A$, quality factor $Q$, and correlation length $\xi$ of local phases $\varphi(s,t)$ decrease for moderate reduction of $N_\text{remain}/N$, while wavelength $\lambda$ stays approximately constant. Solid/dashed curves: TW/SW regime. F. Example of phase defect in kymograph of local phase $\varphi(s,t)$ (experimental data without motor extraction Sharma2024). Inset: rate of phase defects of topological charge $\pm 1$ in experiments (red) and simulations (blue, new parameters) for $N_\text{remain}/N=74\%$ and $100\%$, respectively. Mean$\pm$SEM (omitted if small).
• Figure S1: Phase-diagrams analogous to Fig. 2, yet using white-noise approximation.A-D. Computed beat frequency $f_0=\omega_0/(2\pi)$, beat amplitude $A$, wave length $\lambda$, and quality factor $Q$ as functions of motor activity $\mu_a$ and motor number $N$ (axis linear in $1/N$). Dashed lines serve indicate the SW/TW-transition boundary reported in Fig. \ref{['fig2']} for the stochastic model with motor binding modeled as Poisson jump processes. Parameters: Table \ref{['tab:params']}Cass2023; $N_\text{cilium} = 1.7\cdot 10^4$; SEM $\le$ symbol size.
• Figure S2: Wavelength depends on choice of material frame. We consider synthetic waveform data (columns) represented in different material frames (rows). Example 1 (first column) considers as tangent angle a traveling wave with constant amplitude given by $A\,\cos(2\pi\,s/\lambda - \omega_0 t)$, while Example 2 (second column) considers as tangent angle a traveling wave with linear amplitude profile given by $2A s/L\,\cos(2\pi\,s/\lambda - \omega_0 t)$. For each example, different material frames are discussed as follows. Base gauge.Left: Subsequent cilia shapes for one beat cycle (rainbow color code) represented in a material frame (black dot: basal end). Middle: Arc-length dependent amplitude $a(s)$ according to a fit of Eq. \ref{['eq:gamma_fit']} to the tangent angle $\gamma(s,t)=\gamma_b(s,t)$ in 'base gauge' (black) and reference value $A=0.5$ (blue). Right: Arc-length dependent phase $\Phi(s)$ according to same fit (black), together with linear regression of Eq. \ref{['eq:Phi_fit']} (red), which give the wavelength $\lambda_b \approx 2 L$ at $R^2 \approx 1$ for example 1, and $\lambda_b \approx L$ at $R^2 \approx 1$ for example 2. Co-rotating gauge. Analogous to first row, yet for a slowly co-rotating material frame ('co-rotating gauge'). The motion of the swimming cilium with respect to the laboratory frame was computed using resistive force theory Gray1955, using hydrodynamic friction coefficients from Friedrich2010. The fit of the wavelength gave $\lambda_c \approx 0.925 L$ at $R^2 \approx 0.98$ for example 1, and $\lambda_c \approx 0.947 L$ at $R^2 \approx 0.92$ for example 2. Mean gauge. Analogous to first row, yet for a material frame obtained by subtracting the mean tangent angle ('mean gauge'). The fit of the wavelength gave $\lambda_m \approx L$ at $R^2 \approx 1$ for example 1, and $\lambda_m \approx 1.09 L$ at $R^2 \approx 0.88$ for example 2. Fitted wavelength. Finally, for each example, we determined the apparent wavelengths $\lambda_b$ (blue), $\lambda_c$ (red), $\lambda_m$ (orange) for each of the three choices of material frame, respectively, by a fit of Eqs. \ref{['eq:gamma_fit']} and \ref{['eq:Phi_fit']}, while varying the $\lambda$ parameter in the equation for the tangent angle. $R^2$ from fit of Eq. \ref{['eq:Phi_fit']} indicated by symbol size. Scale bar: $2\,\mu\mathrm{m}$. Parameters: $A=0.5\,\mathrm{rad}$, $\lambda = L = 10\,\mu\mathrm{m}$.