Impact of Microscopic Motility on the Swimming Behavior of Parasites: Straighter Trypanosomes are More Directional

Microorganisms, particularly parasites, have developed sophisticated swimming mechanisms to cope with a varied range of environments. African Trypanosomes, causative agents of fatal illness in humans and animals, use an insect vector (the Tsetse fly) to infect mammals, involving many developmental changes in which cell motility is of prime importance. Our studies reveal that differences in cell body shape are correlated with a diverse range of cell behaviors contributing to the directional motion of the cell. Straighter cells swim more directionally while cells that exhibit little net displacement appear to be more bent. Initiation of cell division, beginning with the emergence of a second flagellum at the base, correlates to directional persistence. Cell trajectory and rapid body fluctuation correlation analysis uncovers two characteristic relaxation times: a short relaxation time due to strong body distortions in the range of 20 to 80 ms and a longer time associated with the persistence in average swimming direction in the order of 15 seconds. Different motility modes, possibly resulting from varying body stiffness, could be of consequence for host invasion during distinct infective stages.


Model
We consider a Pearson random walk [8,17] with randomly drawn exponential distributed displacements δ and turn angles θ. Let the probability distribution function (pdf) for the displacements and the pdf for the turning angles where −π ≤ θ ≤ π; c N being the normalization constant c N = γ(1 − e −πγ ).
Each time step t = 1, 2, . . ., the displacement r t and the turn angle θ t are chosen. Thereafter the 2D displacement vector is added to the actual position vector, It is interesting to note that first, any symmetric peaked shape of the turn angle pdf with well defined variance, can serve as a generating pdf for a Pearson random walk.
Second, for any γ < ∞ the Pearson walk becomes a normal random walk in the limiting case t → ∞. Thus the mean-square displacement (MSD) is asymptotical linear in time, R 2 t ∼ t. However, for intermediate time scales t ≈ 1/γ, the Pearson walker exhibits directional, or so-called persistent, motion, being an The 3D direction correlation function, also called cosine correlation function, for symmetric displacement pdfs with finite variance, is given by where the turn angles θ are taken between successive displacements to a time scale t. For a 3D motion the correlation function can be derived as the mean cosine of the turning angles c to the power of t [17], We calculate the mean cosine of turning angles for their pdf, Eq.
(2), as where coth(x) ≡ (e x + e −x )/(e x − e −x ). Finally, we readily obtain from Eq. (5) and Eq. (6) the 2d direction correlation function, given Eq. (2), as where is the persistence time. Note that Eq. (7) is independent of the spatial scale 1/λ. Notably, for γ = ∞ the turn angle pdf becomes a delta function, and C(t) = c = 1 whereas the normal random walk case γ = 0 is represented in an uniform turn angle pdf implying a delta shaped correlation function C(t) = δ(t).
The 44 trypanosome trajectories display an exponential displacement distribution with mean value δ = 1.26µm. For the model we therefore assume the overall displacement distribution P (δ) = λ exp (−λδ) with λ = 1/δ. We plugged the fitted values for the persistence times t IW p , for intermediate walkers, and t PW p , for persistent walkers ( Table 1 in the main manuscript), into Eq. (8). As explained in the main manuscript, the persistence time for the tumbling walker class t RW p is heavily determined by the fast rotation motion. Here we use, however, the fitted value t RW p ≈ 0.60s for illustration. Finally, for the three motility modes, solving Eq. (8) for γ yields γ RW = 1.21, γ IW = 6.55, and γ PW = 8.19, respectively.
Complementary to Fig.1 and Table I in the main text, we exhibit in Fig.1 below the turn angle distribution for each motility mode. The turn angle distribution for the tumbling walker class is already very close to true random walkers that would display a perfectly flat curve in Fig.1 (corresponding to the trivial value t RW p = 0). Experimental trajectories were categorized using empirically found thresholds for the spread of the turn angle distribution.