Nonlinear instability in flagellar dynamics: a novel modulation mechanism in sperm migration?

Abstract

Throughout biology, cells and organisms use flagella and cilia to propel fluid and achieve motility. The beating of these organelles, and the corresponding ability to sense, respond to and modulate this beat is central to many processes in health and disease. While the mechanics of flagellum-fluid interaction has been the subject of extensive mathematical studies, these models have been restricted to being geometrically linear or weakly nonlinear, despite the high curvatures observed physiologically. We study the effect of geometrical nonlinearity, focusing on the spermatozoon flagellum. For a wide range of physiologically relevant parameters, the nonlinear model predicts that flagellar compression by the internal forces initiates an effective buckling behaviour, leading to a symmetry-breaking bifurcation that causes profound and complicated changes in the waveform and swimming trajectory, as well as the breakdown of the linear theory. The emergent waveform also induces curved swimming in an otherwise symmetric system, with the swimming trajectory being sensitive to head shape-no signalling or asymmetric forces are required. We conclude that nonlinear models are essential in understanding the flagellar waveform in migratory human sperm; these models will also be invaluable in understanding motile flagella and cilia in other systems.

Figure 1.

Imaging frames for non-rolling human sperm cells migrating within a viscous medium, containing 2% methyl-cellulose, in a capillary chamber of 400 µm depth, captured at the same focal plane: (a) swimming trajectories (arrows) for three cells, indicated in white, yellow and cyan and plotted at the same time interval, 0.4 s (electronic supplementary material, movie S1). (b) A circularly swimming cell further along the capillary chamber; the imaging sequence is superimposed at equal time intervals of 2 s, with the swimming trajectory given by the yellow curve. For further details about the microscopy materials and methods, see electronic supplementary material.

Figure 2.

A schematic of the sliding filament mechanism. Relative to a laboratory fixed frame {x̂, ŷ}, the vector X(s,t) describes the position of the point which is an arclength s along the flagellum neutral line (dashed curve) at time t. The internal shear force f(s,t) is acting tangentially and in opposite directions on each sliding filament r±(s) (solid grey curves) causing the flagellum to bend. The distance between the centre of mass of the sperm head and the flagellum junction is denoted by a and the flagellar axoneme diameter is b.

Figure 3.

Snapshots of the flagellar evolution for the clamped and pivoting head boundary conditions, plotted at equal time intervals (darker curves for later times). The internal sliding force is given by equation (2.3) for π ≤ k ≤ 10π and 4 ≤ Sp ≤ 19, as indicated, and the dimensionless force magnitude A is chosen to produce a maximum flagellum amplitude of 0.1L in equation (2.4) (electronic supplementary material, figure S1). (a–l) Comparison of the time evolution for the linear (Lin) and nonlinear (Non) theory. Note that the linear theory fails to predict the flagellar shape for cases (c,i,f,l), and that the pivoting boundary condition is more sensitive to the influence of the nonlinear dynamics. (m–x) Typical symmetry-breaking shapes, characterized by an ‘S’ for clamped boundary conditions (Sp = 19) and a ‘C’ for pivoting boundary conditions (Sp = 15). Furthermore, all beating patterns are periodic in time, despite their appearance.

Figure 4.

The breakdown of linear theory, the symmetry-breaking instability and its consequences for the free head swimmer. (a,b) The discrepancy measure, D_max = max_t,s|X_N − X_L|, is illustrated for varying internal shear wavenumber k, and sperm compliance parameter, Sp, in both the pivoting head and clamped head cases. (c,d), the maximum of the linear theory prediction of curvature, κ_max, is presented. In all these four plots, the solid black contour marks where D_max = 0.1, noting that significantly larger values are observed to characterize poor agreement between the linear and nonlinear theories. (e,f) The transient features of the symmetry-breaking bifurcation to asymmetric waveforms for the pivoting head boundary condition, when k = 6π and Sp = 15 (figure 3q). (e) Time sequence once the flagellar buckling instability can be readily observed, with waveforms overlaid at equal time intervals with the point of attachment in blue. The initial waveform (t = 18.18) is illustrated in light grey and the final waveform (t = 150.76) in black, with a progression in darkness with time. (f) The associated absolute tension |T| as a function of time t and arclength s. (g) The influence of the symmetry-breaking instability on the overall trajectory and flagellar beating pattern of the free swimming cell, plotted at equal time intervals. Smoothed trajectories are plotted for two different sperm head geometries: a spherical head (dashed blue) and a ‘human-like’ head morphology (dashed red), both with the same human-like head volume (Smith et al. 2009a). Here, the internal shear density is given by equation (2.3) with k = 10π and Sp = 20, and the dimensionless force magnitude A is 75% of the force amplitude used for the pivoting case when the maximum displacement is 0.1L (equation (2.4) and electronic supplementary material, figure S1). The buckling transition induces an asymmetric waveform that drives swimmers towards permanent circular paths; the swimming direction is inverted if f → −f in equation (2.1).