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Motor shot noise explains active fluctuations in a single cilium

This paper demonstrates that stochastic fluctuations in the number of bound molecular motors are sufficient to explain the experimentally observed beat fluctuations, oscillation precision, and synchronization defects in a single cilium, thereby linking microscopic motor noise to mesoscopic non-equilibrium dynamics.

Original authors: Maximilian Kotz, Veikko F. Geyer, Benjamin M. Friedrich

Published 2026-05-04

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CC BY 4.0

Original authors: Maximilian Kotz, Veikko F. Geyer, Benjamin M. Friedrich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tiny, hair-like whip called a cilium (or flagellum) attached to a microscopic swimmer, like a single-celled algae. This whip doesn't just wiggle randomly; it beats in a rhythmic, wave-like motion to push the cell through water. For decades, scientists thought this motion was like a perfectly tuned machine: a clockwork mechanism where every part moved with mathematical precision.

However, this new paper suggests that the rhythm isn't perfect. It's actually a bit "noisy," and that noise comes from the tiny engines driving the whip.

Here is the story of the paper, broken down into simple concepts:

1. The Engine Room: A Crowd of Tiny Motors

Inside the cilium, there are thousands of tiny molecular motors (called dyneins). Think of them as a team of rowers in a long boat.

The Old View: Scientists used to think of these rowers as a single, giant, super-strong entity. If you have 10,000 rowers, you just calculate the average force they produce.
The New View: The authors realized that because there are only a finite number of rowers (about 17,000 in a typical cilium), there is always some randomness. Sometimes a rower takes a break; sometimes they pull a little harder. This is called "shot noise."

The Analogy: Imagine a crowd of 10,000 people clapping. If they are perfectly synchronized, it sounds like a single, steady boom. But if you only have 10 people clapping, you hear a chaotic, uneven rhythm. The paper argues that the cilium's motors are more like the small group of 10 than the massive crowd of 10,000. The "noise" of their individual, random actions creates the wobble we see in the beat.

2. The "Dry" Experiment

To prove this, the scientists built a computer model of the cilium. They made a simplifying assumption: they ignored the water resistance (hydrodynamics) for a moment.

Why? They calculated that the water pushing back on the cilium is actually very weak compared to the internal forces of the motors and the elastic structure of the whip. It's like trying to hear a whisper in a quiet room; the water is just background static. The real story is happening inside the whip.
By ignoring the water, they could focus purely on how the motors interact with the structure.

3. The "Motor Extraction" Test

The researchers compared their noisy computer model to real-world experiments where scientists chemically removed some of the motors from the cilium (like taking rowers out of the boat).

What happened? As they removed motors, the beat became "messier." The rhythm wasn't as precise, and the quality of the wave dropped.
The Match: The computer model, which included the "shot noise" of the remaining motors, predicted this messiness perfectly. The model showed that fewer motors meant more noise, just like a smaller rowing team would have a more chaotic rhythm.

4. The "Glitch" in the Wave

One of the most fascinating findings is about phase defects.

The Analogy: Imagine a wave of people doing "the wave" in a stadium. Usually, the wave travels smoothly from one end to the other. But sometimes, the wave stops in the middle, or a new wave starts in the middle, creating a glitch where the rhythm breaks.
The Discovery: The paper found that in both their noisy computer simulations and real experiments, these "glitches" (where the wave starts or stops unexpectedly) happen more often when there are fewer motors. The noise from the small number of motors is strong enough to occasionally break the synchronization of the whole whip.

5. What This Tells Us About the Motors

By fitting their noisy model to the experimental data, the authors could estimate how strong and fast these individual motors are.

They found that the motors are surprisingly strong and fast—stronger than some previous estimates suggested.
Crucially, they found that the motors are "busy" much more of the time than previously thought. Because so many motors are actively working at once, the "noise" is actually reduced, allowing the cilium to beat with high precision (a high "quality factor").

The Bottom Line

This paper changes how we view the cilium. It's not a perfectly smooth, deterministic machine. Instead, it is a biological oscillator driven by a noisy crowd of tiny engines.

The "imperfections" (the noise) aren't just errors; they are a fundamental feature of how these tiny machines work. The fact that the cilium can still beat so precisely despite this noise tells us that the system is incredibly robust. The "shot noise" of the motors explains why the beat wobbles, why it sometimes glitches, and how the precision of the beat changes when you remove the engines.

1. Problem Statement

Motile cilia and flagella are essential for cellular locomotion and fluid transport, driven by the coordinated activity of thousands of molecular motors (dyneins) within the axoneme. While deterministic models have successfully described the average beating patterns of cilia, they fail to account for active fluctuations (noise) observed in experiments.

The Gap: Experimental data shows that cilia oscillations possess a finite "quality factor" ( $Q \approx 10-100$ ), indicating frequency jitter and phase defects. Previous theories largely ignored the stochastic nature of motor binding/unbinding, assuming a continuous mean-field force.
The Hypothesis: The authors propose that the intrinsic shot noise arising from the finite number of motors ( $N \approx 1.7 \times 10^4$ ) binding and unbinding stochastically is the primary source of these active fluctuations. They aim to test if this "small-number" noise is sufficient to explain experimental observations, including the precision of oscillations and synchronization defects.

2. Methodology

The authors developed a theoretical framework combining stochastic dynamics with hydrodynamic and elastic mechanics.

Model Extension: They generalized the deterministic "dry axoneme" model by Cass et al. (which ignores hydrodynamics) into a stochastic model.
- Two-Filament Model: The axoneme is idealized as two parallel filaments connected by motors.
- Stochastic Dynamics: Instead of mean-field equations for motor fractions ( $n_\pm$ ), the model treats motors as independent Poisson jump processes with binding rate $\pi_0$ and load-dependent unbinding rate $\epsilon_\pm$ .
- Force Balance: The equation of motion balances bending elasticity, sliding elasticity, active motor forces, and internal sliding friction. Hydrodynamic forces were shown to be negligible ( $<1.5\%$ ) for reactivated axonemes, justifying the "dry" limit.
Parameter Tuning Strategy: To isolate the effect of noise, the authors varied the motor number $N$ while rescaling the single-motor force $F_0 \propto N^{-1}$ . This kept the mean-field motor activity ( $\mu_a$ ) constant while selectively tuning the magnitude of small-number fluctuations ( $\propto N^{-1/2}$ ).
Simulation-Based Inference (SBI): To connect theory with experiment, they used Gaussian Process-based SBI to fit the stochastic model to experimental data from Sharma et al. (2024). This dataset involved partial motor extraction, where dynein motors were chemically removed from Chlamydomonas axonemes, systematically reducing $N$ while leaving other structures intact.
3D Generalization: A minimal 3D model was also constructed to account for the nine doublet microtubules (DMTs) of the axoneme, allowing for more accurate inference of microscopic parameters (lever arm length, force, speed).

3. Key Contributions

Theoretical Link: Established a direct quantitative link between microscopic motor shot noise and mesoscopic non-equilibrium dynamics in cilia.
Validation of Noise Source: Demonstrated that fluctuations in the number of bound motors are sufficient to explain the experimentally observed quality factors ( $Q$ ) and phase defects without invoking external noise sources.
Pattern Selection Mechanism: Revealed that motor noise acts as a control parameter that shifts the transition boundary between standing waves (reciprocal, non-propulsive) and traveling waves (propulsive).
Parameter Inference: Used controlled perturbation (motor extraction) to infer microscopic motor parameters ( $F_0$ , $v_0$ ) that are consistent with biological constraints but differ from previous deterministic fits.

4. Key Results

A. Impact of Noise on Oscillation Quality

Quality Factor ( $Q$ ): The model predicts that $Q$ scales inversely with the motor number ( $Q \propto N$ ). As $N$ decreases (due to extraction), noise increases, and $Q$ drops. This matches experimental trends perfectly.
Amplitude and Wavelength: The model correctly predicts that beat amplitude ( $A$ ) decreases with motor extraction, while the wavelength ( $\lambda$ ) remains approximately constant.
Frequency Discrepancy: Interestingly, the model predicts an increase in beat frequency ( $f_0$ ) as motors are removed, whereas experiments show a decrease. The authors suggest this implies cilia beating may combine features of both oscillatory and excitable media, a nuance not captured by the current minimal model.

B. Phase Defects and Synchronization

The stochastic simulations naturally produced phase defects (topological charges of $\pm 1$ ) where local synchronization breaks down.
These defects were observed in both the simulations and the re-analyzed experimental data.
The frequency of these defects increased with motor extraction (higher noise), confirming that noise-induced amplitude fluctuations can lead to local synchronization failure.

C. Inferred Microscopic Parameters

Using SBI on the motor extraction data, the authors inferred new microscopic parameters:

Characteristic Motor Force ( $F_0$ ): $\approx 24 \pm 4$ pN (3D model). This is significantly higher than previous estimates ( $\sim 5$ pN) but consistent with the need for a higher duty ratio to reduce noise.
Motor Speed ( $v_0$ ): $\approx 15.9 \pm 1.2$ $\mu$ m/s.
Duty Ratio ( $\eta$ ): The inferred duty ratio is $\approx 20\%$ , much higher than the $\sim 1\%$ in previous deterministic fits. A higher duty ratio means more motors are active simultaneously, reducing the relative impact of shot noise and explaining the high experimental $Q$ .

D. Energy Dissipation

The model estimates an average dissipation rate of $\approx 200$ fW per beat cycle.
Internal dissipation dominates over hydrodynamic dissipation, validating the "dry axoneme" approximation.
The system operates far from the thermodynamic limit ( $Q \ll Q_{max}$ ), consistent with biological efficiency constraints.

5. Significance

Paradigm Shift: The paper moves beyond deterministic descriptions of cilia beating, proving that stochasticity is not just a perturbation but a fundamental feature governing the precision and synchronization of biological oscillators.
Methodological Advance: It demonstrates the power of Simulation-Based Inference (SBI) combined with controlled perturbation experiments (motor extraction) to rigorously constrain complex biological models. This approach resolves the "under-constrained" nature of fitting models to single-condition data.
Biological Insight: The findings suggest that cells may regulate the duty ratio of dynein motors to optimize the trade-off between force generation and oscillation precision. The presence of phase defects highlights the fragility of synchronization in noisy, active matter.
Future Directions: The discrepancy in frequency trends upon motor extraction points to a need for more complex models that integrate excitable media dynamics, offering a roadmap for future theoretical developments in active matter physics.

In summary, this work successfully explains the "noisy" nature of cilia beating as a direct consequence of the discrete, stochastic nature of molecular motors, providing a robust theoretical foundation for understanding active fluctuations in living systems.