Propulsive forces on the flagellum during locomotion of Chlamydomonas reinhardtii

Abstract

The distributed propulsive forces exerted on the flagellum of the swimming alga Chlamydomonas reinhardtii by surrounding fluid were estimated from experimental image data. Images of uniflagellate mutant Chlamydomonas cells were obtained at 350 frames/s with 125-nm spatial resolution, and the motion of the cell body and the flagellum were analyzed in the context of low-Reynolds-number fluid mechanics. Wild-type uniflagellate cells, as well as uniflagellate cells lacking inner dynein arms (ida3) or outer dynein arms (oda2) were studied. Ida3 cells exhibit stunted flagellar waveforms, whereas oda2 cells beat with lower frequency. Image registration and sorting algorithms provided high-resolution estimates of the motion of the cell body, as well as detailed kinematics of the flagellum. The swimming cell was modeled as an ellipsoid in Stokes flow, propelled by viscous forces on the flagellum. The normal and tangential components of force on the flagellum (f(N) and f(T)) were related by resistive coefficients (C(N) and C(T)) to the corresponding components of velocity (V(N) and V(T)).The values of these coefficients were estimated by satisfying equilibrium requirements for force and torque on the cell. The estimated values of the resistive coefficients are consistent among all three genotypes and similar to theoretical predictions.

Figure 1

(Top row) Video images of wild-type uniflagellate Chlamydomonas cells swimming. (Middle row) Corresponding images after removing rigid-body motion of the cell body, so that the motion of the flagellum is seen with respect to a frame of reference fixed to the cell. (Bottom row) Superimposed curves from the mathematical representation of the flagellum by a smooth surface of its tangent angle, θ (s,τ). See also Movie S1, Movie S2, and Movie S3.

Figure 2

(a) Schematic diagram of the swimming Chlamydomonas showing the laboratory (O_XY) frame of reference and the cell-body-fixed (G_x′y′) frame of reference. The body-fixed frame translates with velocity, ν, and rotates with angular velocity, ω. The body-aligned frame, G_xy, is coincident with G_x′y′, but stationary. Velocity can be expressed with respect to either of the frames, O_XY or G_xy. (b) Image of a swimming Chlamydomonas cell with the measured trajectory of the centroid shown superimposed. The laboratory and body-fixed reference frames are also shown. Scale bar, 5 μm.

Figure 3

Free body diagrams of the cell and flagellum. If inertial effects are neglected, the forces exerted by the flagellum on the cell body are exactly balanced by the viscous drag due to translation and rotation of the body.

Figure 4

(Top row) Flagellar waveforms for typical wild-type (WT), ida3, and oda2 uniflagellate Chlamydomonas cells. (Middle row) Angular position of the cell body plotted versus time for wild-type, ida3, and oda2 cells. (Bottom row) The power spectral density (PSD) function estimated from the corresponding time series above. The mid-range peak in the power spectral density occurs at the characteristic beat frequency.

Figure 5

(a–c) Displacement (X, Y components with respect to the laboratory frame) of the center of the cell body, and the angle of rotation of the cell body relative to the original position and orientation. (d–f) Velocity components (ν_x, ν_y) of the cell body with respect to the body-aligned frame; angular velocity of the cell body, ω. Data are shown as a function of time. (g–i) Linear and angular velocity of the cell, sorted by relative phase in the flagellar beat. Smooth estimates of linear and angular velocity obtained by low-pass filtering are superimposed on the raw (sorted) data.

Figure 6

(a–c) Angular and linear velocity of the cell body (the latter with respect to the body-aligned frame) for a sample cell, shown as a function of relative phase in the flagellar beat cycle. (d) Position of the flagellum at eight equally-spaced phases of the beat aligned with the time series in a–c.

Figure 7

Estimates of velocity (v) and force (f, per unit length) exerted by the fluid on the flagellum of Chlamydomonas in wild-type (WT) and outer-arm-deficient (oda2) cells. Velocity vectors (v) are calculated with respect to the laboratory frame of reference. Velocities are obtained by numerical differentiation of the flagellar coordinates in the body-fixed frame, plus addition of terms due to translation and rotation of the body-fixed frame. Force vectors (f) are estimated by multiplying the normal and tangential velocity components, ν_N and ν_T, by corresponding resistive coefficients C_N and C_T. For the wild-type cell, C_N = 1.53 × 10⁻³ pN·s/μm² and C_T = 0.64 × 10⁻³ pN·s/μm². For the oda2 cell, C_N = 1.62 × 10⁻³ pN·s/μm² and C_T = 0.71 × 10⁻³ pN·s/μm².

Figure 8

Comparison of measured angular and linear velocities of the cell body (solid lines) with values predicted by a mathematical model of an ellipsoid in Stokes flow (dashed lines). In the mathematical model, the net forces and moments on the flagellum are estimated by integrating fluid forces along the length of the flagellum. The net flagellar forces and moments are applied to a prolate ellipsoid with major and minor axes measured in the image plane. The normal and tangential components of the distributed force on the flagellum are estimated from the corresponding velocity components of the flagellum multiplied by resistive-force coefficients. The coefficients that produce the best agreement (smallest mean-squared error) between measured and modeled motion of the cell body are used to produce the dashed curve. Column 1, a wild-type cell (C_N = 1.53 × 10⁻³; C_T = 0.64 × 10⁻³ pN·s/μm²; E = 0.204); Column 2, an ida3 mutant (C_N = 1.01 × 10⁻³ pN·s/μm²; C_T = 0.55 × 10⁻³ pN·s/μm²; E = 0.156); Column 3, an oda2 mutant cell (C_N = 1.62 × 10⁻³ pN·s/μm²; C_T = 0.71 × 10⁻³ pN·s/μm²; E = 0.123).

Figure 9

(a) Mean values (± SD) of the resistive-force coefficients that produce the closest agreement between measured and predicted velocities of the cell body, for wild-type, ida3, and oda2 cells (α = 2.0). Statistical significance (p < 0.05) is denoted by † (different from wild-type) or ^∗ (different from ida3). (b) Normalized mean-squared error between measured and predicted velocities of Chlamydomonas cell bodies. A normalized mean-squared error of E = 0 corresponds to perfect agreement; E = 1 corresponds to poor prediction (mean-squared error equal to the variance of the data). The overall mean (± SD) values of the coefficients are C_N = (1.54 ± 0.50) × 10⁻³pN·s/μm²; C_T = (0.68 ± 0.26) × 10⁻³pN·s/μm²; overall error, E = 0.20 ± 0.08.