Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella

Abstract

Cilia and flagella are highly conserved organelles that beat rhythmically with propulsive, oscillatory waveforms. The mechanism that produces these autonomous oscillations remains a mystery. It is widely believed that dynein activity must be dynamically regulated (switched on and off, or modulated) on opposite sides of the axoneme to produce oscillations. A variety of regulation mechanisms have been proposed based on feedback from mechanical deformation to dynein force. In this paper, we show that a much simpler interaction between dynein and the passive components of the axoneme can produce coordinated, propulsive oscillations. Steady, distributed axial forces, acting in opposite directions on coupled beams in viscous fluid, lead to dynamic structural instability and oscillatory, wave-like motion. This 'flutter' instability is a dynamic analogue to the well-known static instability, buckling. Flutter also occurs in slender beams subjected to tangential axial loads, in aircraft wings exposed to steady air flow and in flexible pipes conveying fluid. By analysis of the flagellar equations of motion and simulation of structural models of flagella, we demonstrate that dynein does not need to switch direction or inactivate to produce autonomous, propulsive oscillations, but must simply pull steadily above a critical threshold force.

Keywords: cilia; dynein; flagella; instability; oscillations.

Figure 1.

(a) Schematic diagram of a flagellum with shape parametrized by angle . (b) Schematic diagram of axial cross section of the flagellar axoneme showing key components; viewed from the distal end, outer doublet microtubules are numbered clockwise from doublet 1.

Figure 2.

Well-known cases in which steady axial loading or flow leads to instability. (a) A compressive tip load with a fixed direction causes buckling (a static pitchfork bifurcation) when the force exceeds the critical value . (b,c) A compressive tip load that remains tangent to the beam (a non-conservative follower load) leads to oscillatory motion known as flutter, via a dynamic Hopf bifurcation, when the force exceeds a critical value (animation in electronic supplementary material, movie M1). (d) Flutter in a flexible tube conveying water. Instability occurs above a critical flow rate. Panel (d) is from Greenwald & Dugundji [30] reproduced with permission from Paidoussis [31]. (Online version in colour.)

Figure 3.

Simplified mechanical models of the axoneme. Selected longitudinal members (outer doublets) experience distributed axial loads and bending moments due to dynein activity. Colour indicates axial stress. (a,b) 3D model with six outer doublets. (a) Steady, uniform dynein force applied between doublets on one side (e.g. P) of the axoneme produces bending in one direction and a small amount of twist. (b) Steady dynein loading applied between doublet pairs equally on both P and R sides of the axoneme leads to a straight, slightly twisted configuration, with axial stress in doublets. (c) A 2D model showing steady, distributed axial loads and axial stress, in response to steady dynein activity on a single (P or R) side. (d) Axial stress in a 2D model assuming equal activity on both P and R sides. (Online version in colour.)

Figure 4.

Stability analysis of the linearized 1D PDE flagella (two-doublet) model (see figure 3d; electronic supplementary material, figure S2), with steady, uniformly distributed, axial dynein loading of amplitude p (pN µm⁻¹). (a) Real part, α, and (b) imaginary part, ω, of eigenvalues of the linearized equations as the steady dynein load, p, is increased. A dynamic instability occurs near p = 330 pN µm⁻¹ when the real parts of one pair of eigenvalues become positive while the imaginary parts are non-zero. (c,d) Unstable modes (eigenfunctions) of the 1D PDE model with steady, uniform, dynein loading. (c) Least stable mode for p = 350 pN µm⁻¹ (corresponding animation is in the electronic supplementary material, movie M2). (d) Least stable mode for p = 500 pN µm⁻¹. Each curve shows the average flagellar shape at 16 equally spaced phases in the beat, with phase depicted by colour (increasing blue to red). (e,f) Oscillatory behaviour of the 1D PDE model with steady, uniform, dynein loading, obtained by eigenanalysis of the linearized PDEs. Onset and frequency of oscillations depend on both the properties of the doublets and the inter-doublet coupling. (e) Oscillation frequency of the least stable mode versus flagellar length, L, and dynein force density, p, for conditions in which the straight equilibrium configuration is unstable. (f) Frequency of the least stable mode versus coupling stiffness, k_N, and dynein force density, p.

Figure 5.

Time-domain simulations of the nonlinear 1D PDE flagella (two-doublet) model under steady, distally increasing, dynein loading. (a) Angle and angular velocity at the tip of the flagellum as the amplitude of the steady dynein force density is increased from 300 to 500 pN µm⁻¹. (b) Bifurcation diagram showing the maximum/minimum tip displacements versus distally increasing dynein force amplitude, p. (c,d) Flagellar waveforms predicted by time-domain simulation. Each panel shows the average flagella shape (mean of the two doublets) at 16 equally spaced phases of the oscillation, with phase encoded by colour (increasing blue to red). (c) p = 600 pN µm⁻¹and (d) p = 800 pN µm⁻¹. Corresponding animations are in the electronic supplementary material, movies M3–M5. (Online version in colour.)

Figure 6.

(a,b) Snapshots from the time-domain simulation of the 3D FE (six-doublet) model of the flagellum with length L = 12 µm, total flexural rigidity EI_f = 700 pN µm², and steady, distally increasing dynein loading of amplitude p = 184 pN µm⁻¹. Images are shown at six equally spaced phases during one period of oscillation. (a) Side-view (x–z plane). (b) End view (distal-to-proximal, x–y plane). Colour indicates axial stress along the centreline. Red (positive) = tension. Blue (negative) = compression. Corresponding animations are in the electronic supplementary material, movies M6–M7. (c) Flagella tip position versus time predicted by simulation of the 3D FE structural model of the flagellum with steady, distally increasing, distributed dynein loading, p. (i) p = 69 pN µm⁻¹, (ii) p = 93 pN µm⁻¹, (iii) p = 116 pN µm⁻¹ and (iv) p = 139 pN µm⁻¹. Above a critical value of dynein force density the straight equilibrium configuration becomes unstable, and a limit cycle emerges. (Online version in colour.)