Langevin Dynamics Deciphers the Motility Pattern of Swimming Parasites
Title: Langevin Dynamics Deciphers the Motility Pattern of Swimming Parasites
Authors: Vasily Zaburdaev, Sravanti Uppaluri, Thomas Pfohl, Markus Engstler, Rudolf Friedrich, Holger Stark
Source: Physical Review Letters 106, 208103 (2011) 
In this paper, the authors study the random motion of the African trypanosome (Trypanosoma brucei brucei), a parasite known for causing human sleeping sickness. This paper seeks to understand the mechanics behind processes such as the spreading of infections and the healing of wounds by understanding the motility of cells, using this parasite as an example. The African trypanosome is a particularly interesting example because of its peculiar shape (see Figure 1). The cell is long and narrow, with a single flagellum attached at one end. The cell moves by beating the flagellum back and forth, which also causes the rest of its body to deform. The deformation of the body causes the cell to swim in complicated paths, which are investigated and modeled in this paper.
Experimental Setup and Results
In order to study the motion of the trypanosomes, the authors suspended trypanosomes in a medium with viscosity similar to water (1 cP) and trapped the suspension between two glass plates 10 µm apart. The authors then put the slides under a microscope and captured images and video from the microscope. From these images, trajectories for individual cells could be tracked.
The authors noted several interesting things about the trajectories of the trypanosomes. On such a large length scale, the paths of the trypanosomes look like random walks, but on short length scales, they tend to zig-zag back and forth. This back and forth motion is a result of how the cell uses its flagellum to move throughout the medium.
To study this motion further, the authors compare how the speed and direction of motion of a trypanosome at a given time is correlated to the speed and direction of the same trypanosome at some later time. The authors find two characteristic “relaxation times” of the correlation function for a trypanosome changing its direction of travel. The shorter of these two relaxation times is also the only relaxation time found for a trypanosome changing its speed. These relaxation times describe for how long the trypanosome must be moving before its current velocity or direction are no longer correlated with its initial velocity or direction. Based on the inset to Figure 2, the shorter of the two relaxation times was calculated to be 0.12 seconds. This corresponds to the rapid back-and-forth motion mentioned earlier. The authors point out that the period of the cell’s flagellum beating is 50 ms, so after two or three cycles, the cell has changed direction.
The authors also examined the distribution of cell velocities. The authors found that there was a broad, non-Gaussian distribution of velocities. They found that an individual cell trajectory can be characterized by its mean square velocity, and that cells with faster speeds tended to have larger fluctuations in direction. Finally, the authors were able to model the path of the trypanosomes using two Langevin equations, which examine the fluctuations in velocity at each characteristic relaxation time. By changing the amount of noise introduced into each equation, the authors are able to accurately describe the motion of the trypanosome cells.