The symmetric and asymmetric beats of the Leishmania flagellum. (A–D) One example cell undergoing either (A,B) a symmetric tip-to-base flagellar beat or (C,D) an asymmetric base-to-tip ciliary beat visualised from a 400 Hz high-frame rate phase-contrast video. (A) The symmetric beat, plotted as a kymograph of tangent angle along the flagellum, automatically traced from the phase-contrast video. Yellow indicates clockwise tangent angles relative to the flagellum base, and cyan anticlockwise tangent angles. The waves during the symmetric beat propagate from the flagellum tip to its base (from top right to bottom left), alternating between waves with a negative tangent angle (labelled ‘−’) or a positive tangent angle (labelled ‘+’). Aberrant waves occasionally occur with a different frequency or with additional bends in the flagellum; one example is indicated. (B) Six frames illustrating the propagation of positive and negative tangent angle waves over one beat cycle. Phase-contrast images from the frames indicated with horizontal lines in A. (C) Kymograph of tangent angle for the asymmetric beat. The waves propagate from the flagellum base to its tip (from top left to bottom right), alternating between a wave for a power stroke (small tangent angles, negative in this example) and a wave for a recovery stroke (large tangent angles, positive in this example). (D) Six frames illustrating the power and recovery strokes over one beat cycle as indicated in C. (E) Top, plotted tangent angles at different distances along the flagellum for one symmetric tip-to-base beat cycle from the experiment in A,B. The mean tangent angle at each distance along the flagellum gives the static curvature of the flagellum, represented as the tangent angle, which is very near straight. Bottom, tangent angles with static curvature subtracted. (F) Average static curvature for many flagella undergoing tip-to-base beats (grey), and their mean. (G) Tangent angle plotted as in E, except for one asymmetric base-to-tip beat cycle from the experiment in C,D. (H) Average static curvature for many flagella undergoing base-to-tip beats (grey), and their mean.