Dynamic curvature regulation accounts for the symmetric and asymmetric beats of Chlamydomonas flagella

Axonemal dyneins are the molecular motors responsible for the beating of cilia and flagella. These motors generate sliding forces between adjacent microtubule doublets within the axoneme, the motile cytoskeletal structure inside the flagellum. To create regular, oscillatory beating patterns, the activities of the axonemal dyneins must be coordinated both spatially and temporally. It is thought that coordination is mediated by stresses or strains that build up within the moving axoneme, but it is not known which components of stress or strain are involved, nor how they feed back on the dyneins. To answer this question, we used isolated, reactivate axonemes of the unicellular alga Chlamydomonas as a model system. We derived a theory for beat regulation in a two-dimensional model of the axoneme. We then tested the theory by measuring the beat waveforms of wild type axonemes, which have asymmetric beats, and mutant axonemes, in which the beat is nearly symmetric, using high-precision spatial and temporal imaging. We found that regulation by sliding forces fails to account for the measured beat, due to the short lengths of Chlamydomonas cilia. We found that regulation by normal forces (which tend to separate adjacent doublets) cannot satisfactorily account for the symmetric waveforms of the mbo2 mutants. This is due to the model’s failure to produce reciprocal inhibition across the axes of the symmetrically beating axonemes. Finally, we show that regulation by curvature accords with the measurements. Unexpectedly, we found that the phase of the curvature feedback indicates that the dyneins are regulated by the dynamic (i.e. time-varying) component of axonemal curvature, but not by the static one. We conclude that a high-pass filtered curvature signal is a good candidate for the signal that feeds back to coordinate motor activity in the axoneme.

💡 Research Summary

This paper investigates the fundamental feedback mechanisms that coordinate axonemal dynein activity to produce the characteristic beating patterns of cilia and flagella, using the unicellular alga Chlamydomonas as a model system. The authors focus on three candidate feedback modalities: (i) sliding control, where dynein activity is regulated by the relative sliding displacement of adjacent doublet microtubules; (ii) normal‑force control, in which dyneins respond to forces that tend to separate the doublets; and (iii) curvature control, where dynein activity depends on the curvature of the axoneme.

Theoretical framework
The axoneme is modeled as two inextensible filaments separated by a fixed spacing, linked by elastic elements that mimic nexin links and basal compliance. The filament pair’s shape is described by a centerline position r(s,t) and a tangent angle ψ(s,t). A force‑balance equation is derived from a variational energy functional that includes bending rigidity, motor forces, and viscous drag (characterized by normal and tangential friction coefficients). By expanding ψ, the sliding force f, and the tension τ in Fourier modes, the authors separate a static mode (n = 0) from the fundamental dynamic mode (n = 1). The presence of a static curvature C₀ (observed in wild‑type Chlamydomonas) introduces additional coupling terms between the static and dynamic modes, leading to a six‑order system rather than the four‑order system that describes a perfectly symmetric axoneme.

The generic motor‑feedback law is written as
fₙ = χ(nω) Δₙ + β(nω) · ψ̇ₙ + γ(nω) f⊥ₙ,
where χ, β, and γ are complex, frequency‑dependent response coefficients for sliding, curvature, and normal‑force feedback, respectively. For the static mode (n = 0) the coefficients must vanish because the observed constant curvature implies that static forces are confined to the distal tip.

Experimental approach
Isolated axonemes from wild‑type cells (exhibiting a pronounced static curvature and thus an asymmetric beat) and from the mbo2 mutant (which beats nearly symmetrically) were reactivated in vitro. High‑speed phase‑contrast microscopy captured >1000 frames per axoneme with sub‑10 nm spatial precision and sub‑0.1 mrad angular precision. The centerline was tracked, ψ(s,t) was computed, and Fourier analysis yielded the amplitudes and phases of ψ₀(s) and ψ₁(s). Wild‑type axonemes displayed a nearly constant static curvature C₀ ≈ π/L, whereas mbo2 axonemes had C₀ ≈ 0.

Model‑data comparison

1. Sliding control: Simulations showed that for short flagella (L ≈ 10 µm) the sliding‑based feedback cannot generate a propagating wave with sufficient amplitude; the predicted ψ₁ amplitude is far smaller than measured. This confirms that sliding alone cannot account for the observed beats in Chlamydomonas.
2. Normal‑force control: Because normal‑force feedback is proportional to C₀, it becomes active only in asymmetric axonemes. However, the resulting phase relationship between f⊥₁ and ψ₁ is opposite to that required for sustained wave propagation, and the model fails to reproduce the symmetric waveforms of mbo2.
3. Curvature control: When the curvature response β(ω) is taken to be a high‑pass filter (β(0) = 0, β(ω ≠ 0) ≠ 0), the model reproduces both the amplitude and phase of ψ₁ for wild‑type and mbo2 axonemes. The static curvature contributes only to the baseline shape, while the dynamic component ψ̇₁ drives motor regulation. This “dynamic curvature feedback” predicts that dyneins are insensitive to the static bend but respond to the time‑varying curvature, effectively implementing an adaptive filter.

Conclusions and implications
The study provides three key conclusions: (i) sliding‑based feedback is insufficient for short flagella; (ii) normal‑force feedback cannot explain symmetric beats; (iii) a high‑pass‑filtered curvature signal—i.e., dynamic curvature—offers a unified explanation for both asymmetric and symmetric beating patterns. This insight advances our understanding of how mechanical signals are transduced into coordinated motor activity within the axoneme. It also suggests that engineered microswimmers or therapeutic strategies targeting ciliary dyskinesia should consider dynamic curvature as the primary regulatory cue. Future work may extend the model to three dimensions, incorporate central‑pair asymmetries, and explore the molecular basis of the high‑pass filtering observed in dynein regulation.