Since Einstein’s seminal work in 1905, the main means of characterizing stochastic processes has been the mean square displacement (MSD). However, this order parameter fails to capture many features of dynamics at the forefront of science today, ranging from glassy relaxation to active transport in biological cells. Although there have been several studies seeking to go beyond the MSD, such studies have not made full use of the information available in individual trajectories in two (or more) dimensions, as are now commonly obtained in particle tracking experiments. Here, we introduce an approach that quantifies directional properties of complex motions and discover striking correlations in a number of condensed phase systems.

This research is a collaboration between 4 MRSEC research groups: Margaret Gardel, Stuart Rice, Norbert Scherer, and Aaron Dinner.

Analyses of random walks traditionally use the mean square displacement (MSD) as an order parameter characterizing dynamics. We show that the distribution of relative angles of motion between successive time intervals of random walks in two or more dimensions provides information about stochastic processes beyond the MSD. We illustrate the behavior of this measure for common models and apply it to experimental particle tracking data. For a colloidal system, the distribution of relative angles reports sensitively on caging as the density varies. For transport mediated by molecular motors on filament networks in vitro and in vivo, we discover self-similar properties that cannot be described by existing models and discuss possible scenarios that can lead to the elucidated statistical features.

Schematic of the construction of relative angles. (*Upper*) We sample a Brownian motion (gray line) at regular time intervals (red circles); these points are connected by vectors and angles between successive vectors are calculated. One time interval () is indicated by the solid blue arrows, and another, which is twice as long (), is indicated by the dashed blue arrows. (*Lower*) Calculation of the angle given the vectors in the first two (shorter) time intervals in the upper panel.

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Caging and escape in quasi-2D colloidal suspensions. (*A*) MSD for packing fractions (blue circles) and (red squares). The full black line () fits the data at all times, whereas for the case, a transition to linear behavior, black dashed line (), occurs after about 2 s. (*B*) Relative angle distributions for (*Upper*) and (*Lower*). The curves correspond to temporal coarse-grainings of frames (blue squares), frames (red circles), and frames (black triangles) in both cases. Data are from ref. 21; the total measurement time in both cases is 33.3 s, with a frame rate of 30 Hz.

Statistical measures of insulin granule transport in a pancreatic cell line (MIN6). (*Upper*) MSD as a function of lag time (*Left*) and measurement time (*Right*) for three constant Δ values: s (blue), s (red), and s (black). (*Lower*) profiles for frame (red), frames (black), frames (cyan), frames (blue), frames (brown), and frames (green). MIN6 cells were cultured and imaged in DMEM at 37 °C under 5% (vol/vol) CO2 gas. MIN6 cells were transfected with proinsulin-GFP (24) using Lipofectamine 2000 (Life Technologies 11668) with the manufacturer’s recommended conditions. Imaging was done on a spinning-disk confocal microscope (3I Marianas system) with a 100×, 1.45 NA objective (alpha Plan-Fluar; Zeiss 421190–9900-000), at 1 Hz. Particle tracking was done with Diatrack 3.03 (Semasopht).

Statistical measures of the motion of myosin thick filaments on a disordered actin network. (*Upper*) MSD as a function of lag time (*Left*) and measurement time (*Right*) for three constant Δ values: s (blue), s (red), and s (black). (*Lower*) profiles for frame (red), frames (black), frames (green), frames (blue), and frames (brown). The frame rate is 5 Hz.

**Distribution of directional change as a signature of complex dynamics**

Stanislav Burov, S. M. Ali Tabei, Toan Huynh, Michael P. Murrell, Louis H. Philipson, Stuart A. Rice,Margaret L. Gardel, Norbert F. Scherer, and Aaron R. Dinner, PNAS 2013 110 (49) 19689-19694. doi:10.1073/pnas.1319473110