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. random walks | angular correlation | cytoskeleton C omplex dynamics often emerge from ensembles of interacting constituents. Trajectories that are obtained by tracking individual constituents contain information beyond the evolution of ensemble properties, and these data can thus reveal new mechanistic features of the system studied. Examples cut across disciplines and include quantum dots (1), colloidal beads (2), features in cells (3, 4), fish in schools (5), birds in flocks (6), and primates in social groups (7, 8) . These data (individual trajectories) demand theoretical frameworks for characterizing and interpreting them. The standard reporter for different forms of motion is the mean square displacement (MSD) where brackets and overlines denote ensemble and time averages, respectively. In simple Brownian motion (9), the MSD grows linearly with the separation in time between two observation points (the lag time, Δ) and does not depend on the amount of data included in averages (the measurement time, T)-i.e., there is ergodicity. Anomalous (i.e., non-Brownian) dynamics can arise from correlations in the walk steps. Correlations in the step sizes [e.g., as in fractional Brownian motion (FBM)] (10) give rise to nonlinear scaling of the MSD with lag time, retaining ergodicity (11). By contrast, a power-law distribution of dwell times [e.g., as in a continuous time random walk (CTRW)] (12) is associated with linear scaling with lag time but nonergodicity (13, 14). The two types of correlations can exist together (4, 15). Despite the success of the MSD as an order parameter for dynamics, it is essentially a 1D measure. We expect random walks in two and more dimensions to contain information beyond the MSD, and various alternative analyses have been suggested (16) (Conclusions). In this paper, we introduce a statistical measure of such information. Specifically, we consider the relative angle, which quantifies the direction of motion over successive time intervals. We show that different models of stochastic processes give rise to different distributions of relative angles and how the intervals can be varied to probe contributing time scales. We apply our order parameter to 2D experimental data obtained for mesoscopic systems. We examine colloidal suspensions at two densities and show that the distribution of relative angles is a sensitive means of detecting and quantifying caging. Two different cytoskeletal systems are considered: insulin-containing vesicles (granules) in a pancreatic cell line (4) and in vitro mixtures of purified myosin motors and actin filaments. These active systems exhibit a common signature of directional motion that cannot be understood in terms of existing models. We propose possible scenarios that could lead to the observed statistics. Together, these examples show that the distribution of relative angles is a straightforward probe of complex dynamics that provides information beyond the MSD. In this section, we introduce our order parameter for quantifying directional change. As shown in Fig. 1 , each trajectory comprises a set of recorded positions of the particles XðtÞ. Given these data, we form the vectors that connect positions separated by Δ steps, Vðt; ΔÞ = Xðt + ΔÞ − XðtÞ. We define the relative angle θðt; ΔÞ as the angle between Vðt; ΔÞ and Vðt + Δ; ΔÞ cos θðt; ΔÞ = Vðt; ΔÞ · Vðt + Δ; ΔÞ jVðt; ΔÞjjVðt + Δ; ΔÞj : In analogy to Eq. 1, the parameter Δ plays the role of a lag time; it controls the degree of temporal coarse-graining. For a given Δ, we compute θðt; ΔÞ across all times in each trajectory and then across all particles. We build a histogram of these values and normalize such that it integrates to one. The resulting probability density function, Pðθ; ΔÞ, serves as our order parameter. We repeat this procedure for different Δ and examine how the order parameter changes. The quantity that we calculate, Eq. 2, is reminiscent Significance 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.
Burov et al. REF analyze random walks considering the distribution of relative angles of motion between successive time intervals, which provides information about the underlying stochastic processes.
Distribution of directional change as a signature of complex dynamics
Proceedings of the National Academy of Sciences of the United States of America
Proceedings of the National Academy of Sciences of the United States of America
Physics Medicine