Figure 1.
The model trypanosome and a real trypanosome.
(a) Cell body of the model trypanosome without distortion. The elastic network made from vertices connected by springs defines the surface. The blue line connecting a series of vertices represents the flagellum with the helical half-turn. (b, c) Snapshots of the model trypanosome during simulated swimming motion. (d) 3d volume model of a live trypanosome with fluorescently labeled surface. (e) 3d surface model of the cell in (d), with the flagellum highlighted in blue. (f) The same surface model rotated about the horizontal, in order to get a better view on the left-handed half-turn of the flagellum indicated in red.
Figure 2.
Comparison of the model and a real trypanosome during swimming motion (A–D).
The swimming trajectory and dynamic shape of the simulated model trypanosome (top row) compares well with the forward swimming motion of the real trypanosome (middle and bottom row). Snapshots of the real trypanosome are taken at the indicated times from S4 Video. Fluorescently labelled surface (middle row) or untreated cells (bottom row) were recorded by high speed microscopy (200–500 frames per second). The cells, which exhibit similar speeds and rotational frequencies, show matching cell body conformations at all times over a swimming path of several cell lengths. These periodically repeating shape conformations are initiated by the bending wave passing along the flagellum and determine the trajectory of the swimming parasite.
Figure 3.
Swimming velocity versus sperm number.
Rescaled swimming velocity plotted versus sperm number
(double-logarithmic plot) for to two viscosity values
(squares) and
(triangles) given in MPCD units. Inset: Number of flagellar beats per full cell rotation
versus
. The solid lines indicate power laws in
.
Figure 4.
Swimming velocity versus the ratio of angular frequencies for counterpropagating flagellar waves.
Rescaled swimming velocity plotted versus the ratio
of angular frequencies for simultaneous base-to-tip (
) and tip-to-base (
) propagating flagellar waves. Inset: Orientational correlations of the end-to-end vector,
, for two ratios of frequency.
Figure 5.
Swimming velocity, rotational frequency, torsion, and end-to-end distance for varying flagellar attachment.
(a) Rescaled swimming velocity plotted versus the winding angle
of the helically attached flagellum. The snapshots show the model trypanosomes before applying the bending wave. The flagellum winds counter-clockwise around the cell body up to the angle
. (b) Rescaled rotational frequency of the cell body,
, plotted versus the winding angle
. The inset shows how the amplitude of the imposed flagellar bending wave increases from the broader end of the cell body to the tip by a factor
(red line), 1.25 (blue line), and 1 (black line). (c) Mean torsion
and mean curvature
(inset) of the cell's centerline plotted versus
. (d) Mean end-to-end distance of the cell body versus
. (e) Rescaled swimming velocity
plotted versus reduced distance
from the flagellar pocket, where the helical attachment begins. The snapshots show the model trypanosomes before deformation starts. The flagellum winds around the cell body always by
. (f) Range of end-to-end distances
of the cell body during motion plotted versus
. (g) Helical swimming trajectories of the posterior end indicated by red dots for
(left) and
(right). Snapshots of model trypanosomes during swimming are illustrated.
Figure 6.
Snapshots of several in silico morphotypes during swimming.
(a) The bloodstream form. (b) A possible intermediate morphotype in the tsetse fly, where the total cell length is increased by and the flagellum with the helical half-turn is displaced by
towards the anterior end. (c) A mesocyclic form with a total length of
. The flagellum starts at a distance of
from the posterior end and the winding angle of the helical turn is tuned to
. (d) A model that resembles an epimastigote form with a total length of
, where the attachment of the flagellum is the same as for the mesocyclic form.
Figure 7.
Swimming velocity versus flagellar winding angle for the mesocyclic form.
Rescaled swimming velocity plotted versus winding angle
. The snapshots show the model trypanosome for winding angles
,
, and
. Inset: Rescaled end-to-end distance of the cell body,
, versus
.
Table 1.
The model cell body is constructed with a sequence of circular cross sections indexed by .