Noise and Synchronization in Pairs of Beating Eukaryotic Flagella

It has long been conjectured that hydrodynamic interactions between beating eukaryotic flagella underlie their ubiquitous forms of synchronization; yet there has been no experimental test of this connection. The biflagellate alga Chlamydomonas is a simple model for such studies, as its two flagella are representative of those most commonly found in eukaryotes. Using micromanipulation and high-speed imaging, we show that the flagella of a C. reinhardtii cell present periods of synchronization interrupted by phase slips. The dynamics of slips and the statistics of phase-locked intervals are consistent with a low-dimensional stochastic model of hydrodynamically coupled oscillators, with a noise amplitude set by the intrinsic fluctuations of single flagellar beats.

Figure 1

Experimental system. (a) A single cell of Chlamydomonas reinhardtii held on a micropipette. (b) Schematic of apparatus for visualization and micromanipulation of cells.

Figure 2

A phase slip in the beating of C. reinhardtii flagella. (a)–(l) Individual image-processed movie frames at times indicated in panel $n$. (m) Phase difference $\Delta (t)$ between flagella as a function of time. (n) Extracted signals from the interrogation areas [seen in (b)] near each of the two flagella.

Figure 3

Dynamics and statistics of phase slips. (a) Long time series of $\Delta (t)$ illustrating forward and backward slips. Inset: sketch of the tilted washboard potential used in the model. (b) Probability distribution of synchronization times and its cumulative distribution function (cdf) showing approximate exponential decay. (c) $\Delta (t)$ for 10 slips events, the average slip (dashed line) and the integrated trajectory from the deterministic part of the model (solid line) for $\delta \nu /\overline{\nu }=0.004$, $\overline{\nu }=47\mathrm{Hz}$, and $ϵ/\overline{\nu }=0.015$; ${\tau }_{\mathrm{slip}}\sim 0.22\mathrm{s}$.

Figure 4

Comparison of single flagellum noise levels before ($y$ axis) and after ($x$ axis) partial deflagellation. In the latter case, the noise can be extracted from the distribution of beating periods (lower inset), and compared to the value estimated from the autocorrelation of the coupled dynamics (upper inset). Only experiments in which the isolated flagellum had a long period of uniform behavior are shown (7 out of 14).