Nonlinear dynamics of cilia and flagella

Cilia and flagella are hairlike 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.

Figure 1

Schematic representation of the effective two-dimensional mechanics of planar beats with two elastic rods sliding relative to each other due to the shear forces generated by active elements. Illustrated are the tangential shear forces $f\left(s\right)$ and the local sliding displacement $\Delta \left(s\right)$. Elastic structural elements are indicated as springs.

Figure 2

Geometry of the flagellar deformation in the $x,y$ plane. The shape at a given time is described by the local tangent angle $\psi \left(s\right)$ as a function of the arc length $s$ along the flagellum.

Figure 3

Schematic diagram of the complex plane of motor impedance ${\overline{\alpha }}_{\text{c}}$. For a given dimensionless frequency ${\overline{\omega }}_{\text{c}}$, there exists a critical value ${\overline{\alpha }}_{\text{c}}$ describing an oscillatory instability. The line ${\alpha }_{\text{c}}$ parametrized by ${\overline{\omega }}_{\text{c}}$ is shown in black. In the dashed region to the right of the line the system is quiescent, whereas to the left of ${\overline{\alpha }}_{\text{c}}$ it oscillates. The phase theta describes the orientation of a displacement $\delta \overline{\alpha }$ away from a bifurcation point in the complex plane such that $\delta \overline{\alpha }=\left|\delta \overline{\alpha }\right|e^{i\theta }$.

Figure 4

Examples of beating modes, which are solutions to the wave equation [Eq. (8)] for different boundary conditions. The location of these solutions is indicated in the complex plane of values of the linear response function of motors $\overline{\alpha }$. The chosen examples correspond to a situation where starting from a Hopf bifurcation the amplitude of the unstable mode increases when moving along a path in the $\overline{\alpha }$ plane for which the oscillation frequency is constant. (a) Solutions for clamped head with basal sliding with frequency $\omega /2\pi \simeq 26\text{Hz}$ in the $\overline{\alpha }$ plane (gray line). Depicted is the second brach of unstable modes. The parameters $k_s$ and ${\gamma }_s$ were chosen as determined in [23] such that the beat patterns shown in (b) resemble experimentally observed ones. The amplitude $A$ as a function of $\left|\delta \overline{\alpha }\right|$ is displayed in the inset. (b) Real and imaginary parts of the wave form $\psi$ as a function of dimensionless arc length $\overline{s}$ for different points along the gray line in (a). (C) and (D) same as (A) and (B) but for the clamped head boundary conditions without basal sliding and frequency of $\simeq 28\text{Hz}$. Here we choose the bifurcation on the first branch of unstable modes. (E) and (F) same as (C) and (D) but for a freely pivoting head without basal sliding and frequency of $\simeq 5\text{Hz}$.

Figure 5

Examples of the shape of the nonlinear solutions as they grow from the linearly unstable modes, corresponding to the 26 Hz solution of the second branch of unstable modes in the case of clamped head boundary conditions with basal sliding. Shown are the real and imaginary parts of the rescaled nonlinear solutions with the lighter curves corresponding to larger amplitudes. The amplitudes $A$ (in rad) of the selected modes are indicated by black dots in the diagram that shows how the amplitude grows as a function of the distance $\left|\delta \overline{\alpha }\right|$ to the bifurcation.