Reverse engineering the euglenoid movement

Abstract

Euglenids exhibit an unconventional motility strategy amongst unicellular eukaryotes, consisting of large-amplitude highly concerted deformations of the entire body (euglenoid movement or metaboly). A plastic cell envelope called pellicle mediates these deformations. Unlike ciliary or flagellar motility, the biophysics of this mode is not well understood, including its efficiency and molecular machinery. We quantitatively examine video recordings of four euglenids executing such motions with statistical learning methods. This analysis reveals strokes of high uniformity in shape and pace. We then interpret the observations in the light of a theory for the pellicle kinematics, providing a precise understanding of the link between local actuation by pellicle shear and shape control. We systematically understand common observations, such as the helical conformations of the pellicle, and identify previously unnoticed features of metaboly. While two of our euglenids execute their stroke at constant body volume, the other two exhibit deviations of about 20% from their average volume, challenging current models of low Reynolds number locomotion. We find that the active pellicle shear deformations causing shape changes can reach 340%, and estimate the velocity of the molecular motors. Moreover, we find that metaboly accomplishes locomotion at hydrodynamic efficiencies comparable to those of ciliates and flagellates. Our results suggest new quantitative experiments, provide insight into the evolutionary history of euglenids, and suggest that the pellicle may serve as a model for engineered active surfaces with applications in microfluidics.

Fig. 1.

Quantitative analysis of the movies: method. (A) The frames are segmented and aligned to obtain images of F_a containing information about the shape alone, devoid of translation, rotation, or textures. A B-spline curve, given by its control polygon P_a (black circles), is fitted to the boundary of F_a and is a generating curve of the axisymmetric representation of the pellicle. Original frame image courtesy of Richard E. Triemer. (B) The segmented and aligned frames (i), representative of the shapes adopted by the cell, are embedded in low dimensions by a nonlinear dimensionality reduction technique called Isomap (15). The algorithm maps each frame to a low-dimensional point (circles in ii and iii, color labeling chronological order), so that the intrinsic distance between frames (shape dissimilarity) is preserved as much as possible by the low-dimensional representation. The embedding in the plane (τ₁, τ₂) (ii), shows that the stroke is a closed nonreciprocal path in shape space. Consequently, it can be most compactly described by embedding the frames in a periodic 1D segment (iii), from which we parametrize the stroke as a function of τ by interpolation (iv) with smooth basis functions w_a(τ). At any given τ, the synthetic stroke is a weighted average of the curves fitting video frames whose 1D embedding is in the vicinity of τ. The parameter τ is not proportional to physical time, ignored by the manifold learning algorithm, but rather to arc length in shape space. See SI Text, section Video Processing and Parametrization of the Stroke, for technical details.

Fig. 2.

Quantitative analysis of the movies: results. (A) Two-dimensional embedding of a stroke plotted against video frame number, showing the shape and pace uniformity of the stroke over several realizations. Isomap identifies similar shapes from different realizations (here about four cycles) and yields a single geometric stroke. (B) Pseudo-time parameter τ against normalized physical time t during three full strokes represented in different colors, showing that path in shape space is traveled with a well-defined pace (here, euglenid #3). The black line shows the fit used to reparameterize time. (C) Reduced volume, v, as a function of time during two strokes. We distinguish two distinct motility styles: a volume-changing style for euglenids #1 and #2, and a volume-preserving style for euglenids #3 and #4. See Fig. S3 for the full data.

Fig. 3.

Pellicle kinematics: theory. (A) Surface strain (the 2 × 2 matrix C) is derived assuming simple shear along the strips, γ, acting on a reference pellicle. The reference pellicle conformation (i,iii) is defined by a shape and a pellicle conformation, given by the tangent vector fields s₀ and m₀, along the strips and perpendicular to them, or by the angle α₀ between the pelliscle strips and the surface parallels. (ii) Ultrastructure of the pellicle (transmission electron micrograph from ref. , Copyright 2001 The Society for the Study of Evolution), and depiction of the sliding between adjacent strips causing the pellicle shear γ. (B) Surface strain, C, is now derived by comparing a reference configuration, (iii), given by r₀(λ) and z₀(λ), where λ is the body coordinate, and a deformed configuration (iv), given by r(λ), z(λ) and the azimuthal displacement relative to a fixed direction ψ(λ). The red point denotes a material particle attached to the pellicle, and ()^′ denotes differentiation with respect to λ. Matching the microstructural (A) and the shape-derived (B) expressions for the strain, we find equations relating pellicle shear, pellicle orientation, shape changes, and azimuthal motions.

Fig. 4.

Stroke kinematics: relationship between the local actuation and shape changes. Reference conformation of the pellicle for each euglenid (top), exhibiting very small reduced volumes. Map of the pellicle shear recovered from the observations as a function of time and body coordinate γ(λ,t) (center), shown for two full synthetic strokes. The pellicle shear distributions acting on the reference configurations produce the shapes and pellicle conformations shown below.

Fig. 5.

Stroke hydrodynamics of euglenid #4. (A) Flow pattern around the pellicle at selected instants along the stroke. The fluid velocity field in the symmetry plane is indicated by blue arrows, and the azimuthal component is shown with isocontours, where green is zero, blue negative, and red positive. (B) Relationship between actuation and forward motion, where the vertical axis is the body coordinate λ for the color map of γ(λ,t) and z position (average between the head and the tail) in units of body length for the curve in blue. Both the actuation pattern and the forward motion clearly delineate a power and a recovery phase in the stroke.