Move, Dither, Move, Dither. On the Structure of Random Walks and Single-Particle Trajectories

Abstract

The fundamental principle in interpreting single-particle trajectories is that a pure random walk is the control and null hypothesis. In order to make any claim about a putative physical or biological event in an observed single-particle trajectory, one must evaluate the probability that the event could have occurred by chance in the corresponding pure random walk. By eye, a random walk often shows alternating periods of “moving” and “dithering.” We quantify this apparent structure by Monte Carlo calculations of pure random walks in which various measures of apparent directed motion ("moving") and apparent confinement ("dithering") are compared. A key point is that long dithering or moving events are rare in themselves. But in examining a trajectory for these events, we do not specify the starting point of the event, and all that we specify about the duration is that it be above some minimum threshold of detectability. The large number of potential starting points and durations of these events compensates for their rarity. An essential feature of the approach is to separate three distinct aspects of the problem. First, we separate characterization from trajectory segmentation, and treat only characterization here. Second, we separate characterization from the effect of noise, and assume that the particle positions are exactly known. In practical applications to experimental single-particle trajectories, the random noise in the position measurement must be taken into account. The move-dither analysis will ultimately form the basis for new tests to identify directed and confined motion in single-particle trajectories, and to distinguish anomalous from normal diffusion. Supported by NIH grant GM038133.