Transient Superdiffusion and Long-Range Correlations in the Motility Patterns of Trypanosomatid Flagellate Protozoa

<div><p>We report on a diffusive analysis of the motion of flagellate protozoa species. These parasites are the etiological agents of neglected tropical diseases: leishmaniasis caused by <i>Leishmania amazonensis</i> and <i>Leishmania braziliensis</i>, African sleeping sickness caused by <i>Trypanosoma brucei</i>, and Chagas disease caused by <i>Trypanosoma cruzi</i>. By tracking the positions of these parasites and evaluating the variance related to the radial positions, we find that their motions are characterized by a short-time transient superdiffusive behavior. Also, the probability distributions of the radial positions are self-similar and can be approximated by a stretched Gaussian distribution. We further investigate the probability distributions of the radial velocities of individual trajectories. Among several candidates, we find that the generalized gamma distribution shows a good agreement with these distributions. The velocity time series have long-range correlations, displaying a strong persistent behavior (Hurst exponents close to one). The prevalence of “universal” patterns across all analyzed species indicates that similar mechanisms may be ruling the motion of these parasites, despite their differences in morphological traits. In addition, further analysis of these patterns could become a useful tool for investigating the activity of new candidate drugs against these and others neglected tropical diseases.</p></div

Illustration of trajectories and velocities.

<p>A) Typical swimming trajectories of the <i>L. amazonensis</i> and B) the corresponding time series of the radial velocities <i>v</i>(<i>t</i>) are shown. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0152092#pone.0152092.s002" target="_blank">S1 Fig</a> for other protozoa. C) Average velocities and D) standard deviation over all trajectories for each protozoan are represented in the bar plots. The error bars are 95% confidence intervals calculated via bootstrapping.</p

Dataset summary.

<p>Dataset summary.</p

The velocities are not described by a two-dimensional Maxwell-Boltzmann distribution.

<p>A) Typical examples of cumulative distribution functions (CDF) of the velocities <i>v</i>(<i>t</i>) from a single trajectory of the protozoa. The gray continuous lines are the best fits obtained for the two-dimensional Maxwell-Boltzmann distribution (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0152092#pone.0152092.e019" target="_blank">Eq 8</a>), whereas the dashed black lines are the fits for the generalized gamma distribution (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0152092#pone.0152092.e020" target="_blank">Eq 9</a>). The values of parameters <i>α</i>, <i>β</i> and <i>γ</i> were obtained via maximum-likelihood method and are shown in the plots. We also present the <i>p</i>-values of the Kolmogorov-Smirnov test showing that we cannot reject the gamma hypothesis for these particular trajectories. We tested all trajectories and the KS test cannot reject this hypothesis for about 50% of the trajectories (all <i>p</i>-values are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0152092#pone.0152092.s004" target="_blank">S3 Fig</a>). B) Average values of the best fitting parameters over all trajectories. The error bars are 95% confidence intervals calculated via bootstrapping.</p

The transient superdiffusive spreading of the protozoan trajectories.

<p>A) Experimental values of the variance of the radial positions <i>σ</i>²(<i>t</i>) (red dots) for all species studied here (as indicated in the plots) are shown in log-log scale. The dashed lines represent power-law relationships <i>σ</i>²(<i>t</i>)∼<i>t</i>^<i>λ</i>, where the values of <i>λ</i> were obtained by least square fitting a linear model to these log-log relationships (considering <i>t</i> < 10 seconds). The values of <i>λ</i> (and their 95% bootstrap confidence intervals) are shown in plots, and in panel B) they are presented in bar plots, where error bars correspond to the 95% bootstrap confidence intervals. C) Exponents <i>λ</i> calculated within a window of size 30 seconds centralized in <i>t</i>_<i>w</i> as a function of <i>t</i>_<i>w</i>. The different colors represent the four protozoa according to the ones used in Fig 2B. The small shaded regions represent 95% bootstrap confidence intervals and the dashed gray line represents the usual regime (<i>λ</i> = 1).</p