Figure 2. Principal shape modes of sperm flagellar beating.

A. High-precision tracking of planar flagellar centerline shapes (, red) are characterized by their tangent angle as a function of arc-length along the flagellum. B. The time-evolution of this flagellar tangent angle is shown as a kymograph. The periodicity of the flagellar beat is reflected by the regular stripe patterns in this kymograph; the slope of these stripes is related to the propagation of bending waves along the flagellum from base to tip. By averaging over the time-dimension, we define a mean flagellar shape characterized by a tangent angle profile . For illustration, this mean flagellar shape is shown in black superimposed to tracked flagellar shapes (grey). C. We define a feature-feature covariance matrix from the centered tangent angle data matrix as explained in the text. The negative correlation at arc-length distance reflects the half-wavelength of the flagellar bending waves. D. The normalized eigenvalue spectrum of the covariance matrix sharply drops after the second eigenvalue, implying that the eigenvectors corresponding to the first two eigenvalues together account for 97% of the observed variance in the tangent angle data. E. Using principal component analysis, we define two principal shape modes (blue, red), which correspond precisely to the two maximal eigenvalues of the covariance matrix in panel C. The lower plot shows the reconstruction of a tracked flagellar shape (black) by a superposition of the mean flagellar shape and these two principal shape modes (magenta). In addition to tangent angle profiles, respective flagellar shapes are shown on the right. F. Each tracked flagellar shape can now be assigned a pair of shape scores and , indicating the relative weight of the two principal shape modes in reconstruction this shape. This defines a two-dimensional abstract shape space. A sequence of shapes corresponds to a point cloud in this shape space. We find that these point form a closed loop, reflecting the periodicity of the flagellar beat. We can define a shape limit cycle by fitting a curve to the point cloud. By projecting the shape points on this shape limit cycle, we can assign a unique flagellar phase modulo to each shape. This procedure amounts to a binning of flagellar shapes according to shape similarity. G. By requiring that the phase variable should change continuously, we obtain a representation of the beating flagellum as a phase oscillator. The flagellar phase increases at a rate equal to the frequency of the flagellar beat and rectifies the progression through subsequent beat cycles by increasing by . H. Amplitude fluctuations of flagellar beating as a function of flagellar phase. An instantaneous amplitude of the flagellar beat is defined as the radial distance of a point in the -shape space, normalized by the radial distance of the corresponding point on the limit cycle of same phase. A phase-dependent standard deviation was fitted to the data (black solid line). Also shown are fits for additional cells (gray; the position of was defined using a common set of shape modes). J. Swimming path of the head center during one beat cycle computed for the flagellar wave given by the shape limit cycle (panel F) using resistive force theory [18] as described previously [19]. The path is characterized by a wiggling motion of the head superimposed to net propulsion. For a ‘standing wave’ beat pattern characterized by the oscillation of only one shape mode, net propulsion vanishes.