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Kinematics of flagellar swimming in Euglena gracilis: Helical trajectories and flagellar shapes

  1. Antonio DeSimonea,2
  1. aMathLab, International School for Advanced Studies, 34136 Trieste, Italy;
  2. bInstitute of Fluid Mechanics and Aerodynamics, Bundeswehr University Munich, 85577 Neubiberg, Germany;
  3. cSezione Oceanografia, Istituto Nazionale di Oceanografia e di Geofisica Sperimentale, 34151 Trieste, Italy
  1. Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 16, 2017 (received for review May 16, 2017)

  1. Fig. 2.

    (A) Three-dimensional representation of Euglena’s body, with trajectories of the body center (blue) and of the eyespot (red) reconstructed from the fitting procedure. The reference axes <mml:math><mml:mi>??</mml:mi></mml:math>??, <mml:math><mml:mi>??</mml:mi></mml:math>??, and <mml:math><mml:mi>??</mml:mi></mml:math>?? are chosen such that <mml:math><mml:mi>??</mml:mi></mml:math>?? is the axis of the helical trajectory. (B) Projection of the centroid trajectory <mml:math><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mi>??</mml:mi></mml:mrow></mml:math>Π?? and (C) of the eyespot trajectory <mml:math><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mi>??</mml:mi></mml:mrow></mml:math>Π?? on the focal plane. (D) Projection of the major axis of the cell body <mml:math><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mi>??</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>cos</mml:mi><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>sin</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>Π??=(cosθ,sinθ) on the focal plane. (EG) Comparison between experimental measurements and model best-fitting curves for the quantities <mml:math><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mi>??</mml:mi></mml:mrow></mml:math>Π??, <mml:math><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mi>??</mml:mi></mml:mrow></mml:math>Π??, and <mml:math><mml:mi>θ</mml:mi></mml:math>θ. Note that the experimental data for <mml:math><mml:mrow><mml:mi mathvariant="normal">Π</mml:mi><mml:mi>??</mml:mi></mml:mrow></mml:math>Π?? are available only when the eyespot is visible.

  2. Fig. 3.

    Stereographic reconstruction of the flagellum. (A) Fixed a phase <mml:math><mml:mi>τ</mml:mi></mml:math>τ between 0 and the beating period <mml:math><mml:msub><mml:mi>T</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:math>Tb, images taken at times <mml:math><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi>τ</mml:mi><mml:mo>+</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math>tk=τ+kTb, show the flagellum in the same configuration with respect to the cell body, but from different viewpoints (Top). At each instant the Euglena’s position and orientation with respect to the focal plane are known from 3D cell body motion reconstruction results (Bottom). (B) The flagellum is recovered by finding the 3D curve whose projections (on the respective planes) are the closest possible to the experimental projections.

  3. Fig. 4.

    Grid of experimental images from four different swimming cells (columns) taken at different times (rows). The reconstructed time history of flagellar shapes from cell 1 (green, first column), conveniently scaled in both space and time, is attached to the other cells’ bodies (blue). A good overlap between the projections of the attached flagella and the experimental images emerges (see Movie S4 and Supporting Information, section 4 for more details). This shows that the flagellar beat of cell 1, presented here in the main text, is a valid representative of a common beating style.

  4. Fig. 5.

    (AD) Representation of Euglena’s kinematics. The dimensions of the cell body are not to scale with displacements for visualization purposes. (A) Side view. The Euglena follows a right-handed helical path. Full or transparent bodies are shown at the beginning of each beat. (B) Top view. The body center trajectory <mml:math><mml:mrow><mml:mi>??</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>??(t) is rendered using a color map to highlight the absolute instantaneous velocity. (C) Body motion as seen from a reference frame of axes <mml:math><mml:mi>??</mml:mi><mml:mo>′</mml:mo></mml:math>??′, <mml:math><mml:mi>??</mml:mi><mml:mo>′</mml:mo></mml:math>??′, and <mml:math><mml:mi>??</mml:mi><mml:mo>′</mml:mo></mml:math>??′ moving on the circular helix <mml:math><mml:mrow><mml:msub><mml:mi>??</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>??h(t) while rotating by <mml:math><mml:mrow><mml:msub><mml:mi>??</mml:mi><mml:mi>??</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>????(ωt) (top view). Full or transparent bodies are shown at instants equally spaced within one beat. (D) Quasi-conical surface spanned by the major symmetry axis of the cell during one beat. (E) Sequence of flagellar shapes for 10 consecutive instants (phases) within one beat. Each phase is labeled with a different color. Body reference axes <mml:math><mml:mi>??</mml:mi></mml:math>??, <mml:math><mml:mi>??</mml:mi></mml:math>??, and <mml:math><mml:mi>??</mml:mi></mml:math>?? are represented in red, green, and blue, respectively. (F) Translational velocity <mml:math><mml:mi>??</mml:mi></mml:math>?? and (G) angular velocity <mml:math><mml:mi>??</mml:mi></mml:math>?? of the cell, represented in body reference frame coordinates. Velocities are shown for each flagellar phase and color-coded accordingly. (H) Polar angle and (I) azimuth angle of the unit tangent vector to the flagellum, as functions of the arc length and time.

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