How Euglena gracilis swims: Flow field reconstruction and analysis

Euglena gracilis is a unicellular organism that swims by beating a single anterior flagellum. We study the nonplanar waveforms spanned by the flagellum during a swimming stroke and the three-dimensional flows that they generate in the surrounding fluid. Starting from a small set of time-indexed images obtained by optical microscopy on a swimming Euglena cell, we construct a numerical interpolation of the stroke. We define an optimal interpolation (which we call synthetic stroke) by minimizing the discrepancy between experimentally measured velocities (of the swimmer) and those computed by solving numerically the equations of motion of the swimmer driven by the trial interpolated stroke. The good match we obtain between experimentally measured and numerically computed trajectories provides a first validation of our synthetic stroke. We further validate the procedure by studying the flow velocities induced in the surrounding fluid. We compare the experimentally measured flow fields with the corresponding quantities computed by solving numerically the Stokes equations for the fluid flow, in which the forcing is provided by the synthetic stroke, and find good matching. Finally, we use the synthetic stroke to derive a coarse-grained model of the flow field resolved in terms of a few dominant singularities. The far field is well approximated by a time-varying Stresslet, and we show that the average behavior of Euglena during one stroke is that of an off-axis puller. The reconstruction of the flow field closer to the swimmer body requires a more complex system of singularities. A system of two Stokeslets and one Rotlet, that can be loosely associated with the force exerted by the flagellum, the drag of the body, and a torque to guarantee rotational equilibrium, provides a good approximation.

Figure 1

A sketch of the swimmer motion $\chi (\mathbf{X},t)$ to highlight its decomposition into prescribed shape changes $s(\mathbf{X},t)$ and a rigid motion $f_r$.

Figure 2

Geometric reconstruction of E. gracilis: 384 cells constitute the cell body and 648 the flagellum. We report on the right the 10 flagellar shapes corresponding to the 10 times $t$ used to discretize the beating period $T_b$ in Ref. [8].

Figure 3

Results of the fitting procedure: (a) and (b) represent linear and angular velocities, respectively. Triangles, squares, and circles refer to the component along $i,j,k$, respectively. Filled markers represent the experimental result reported in Ref. [8] while empty symbols are current BEM computations. Continuous, dashed-dotted and dashed lines are the Fourier continuation of the BEM optimal results we use to compare the trajectories in (c). (c) Comparison between the reference trajectory of Ref. [8] (in light green) and the trajectory from the optimal BEM solution (in dark blue) in a reference frame whose vertical axis is aligned with the average direction of motion.

Figure 4

(a) Micrograph of a swimming Euglena surrounded by tracer particles obtained with $64\times$ magnification and astigmatic optics. (b) Reconstruction of 3D particle positions using the GDPT method. The depth position of tracer particles is determined from the different shape of the defocused-astigmatic particle images. (c) Determination of the depth position of the swimmer by looking at the maximum axial velocity of the fluid in the swimmer wake. (d)–(e) Determination of the time and azimuthally averaged velocity components. All particle positions and velocities are reported in the coordinate frame of the swimmer ($ijk$) and converted in cylindrical coordinates. The time and azimuthally averaged velocity components $v_{\rho }, v_{\zeta }$, and $v_{\varphi }$ are calculated as a function of $\rho$ and $\zeta$.

Figure 5

Fluid flow comparisons in which each row shows results from a different specimen. The first column reports the body linear and angular velocities (lines with triangles, squares, and circles represent $i,j,k$ components in body frame). The remaining three columns depict the different axisymmetrically projected velocities $v_{\rho },v_{\zeta },v_{\varphi }$. The colormap is normalized using the mean velocity of the swimmer during the stroke $v^{*}$.

Figure 6

Sketches of the three different singularity models we analyze for the reconstruction of the velocity field: (a) the Stokeslet doublet, (b) the two Stokeslets one Rotlet model (2S1R), and (c) the three Stokeslets one Rotlet model (3S1R).

Figure 7

Qualitative flow field comparison in the $xz$-plane between the BEM (third column) and two singularity approximations: 2S1R used to approximate the far field (left column) and 3S1R (middle column). The first row represents the mean velocity, while the second reports the velocity computed for $t/T_b=0.9$. The colormap represents the magnitude of the velocity field as projected on the $xz$-plane, while the quivers show the velocity projected on the plane. In all the figures, we superimpose in green the cell body and in blue the flagellar shapes. In particular, all the 10 flagellar shapes are shown in (a), (b), and (c), while only the flagellar shape relative to $t/T_b=0.9$ is reported in (d), (e), and (f).

Figure 8

Qualitative comparison in the $xz$-plane between the BEM tractions and the singularity systems 2S1R (a) and 3S1R (b). The torques represented in black are projections along the $y$ axis. Black arrows on the $xz$-plane complement the representation of the singularity systems. The colormap is for the magnitude $f=\left|\mathbf{f}\right|$ of the BEM tractions $\mathbf{f}$ on the $xz$-plane.