Nonlinear Dynamics of Cilia and Flagella

TL;DR

This paper develops a nonlinear wave model for cilia and flagella beating, revealing how nonlinearities influence oscillation amplitude and stability, with numerical solutions aligning with experimental observations.

Contribution

It introduces a nonlinear wave equation for axonemal beating and analyzes the effects of nonlinearities on oscillation amplitude and stability.

Findings

Nonlinear waves approximate linearly unstable modes at observed amplitudes.

Amplitude increases near an oscillatory instability.

Numerical solutions match experimental beat patterns.

Abstract

Cilia and flagella are hair-like extensions of eukaryotic cells which generate oscillatory beat patterns that can propel micro-organisms and create fluid flows near cellular surfaces. The evolutionary highly conserved core of cilia and flagella consists of a cylindrical arrangement of nine microtubule doublets, called the axoneme. The axoneme is an actively bending structure whose motility results from the action of dynein motor proteins cross-linking microtubule doublets and generating stresses that induce bending deformations. The periodic beat patterns are the result of a mechanical feedback that leads to self-organized bending waves along the axoneme. Using a theoretical framework to describe planar beating motion, we derive a nonlinear wave equation that describes the fundamental Fourier mode of the axonemal beat. We study the role of nonlinearities and investigate how the amplitude of oscillations increases in the vicinity of an oscillatory instability. We furthermore present numerical solutions of the nonlinear wave equation for different boundary conditions. We find that the nonlinear waves are well approximated by the linearly unstable modes for amplitudes of beat patterns similar to those observed experimentally.