Analysis of transient behavior in complex trajectories: application to secretory vesicle dynamics

Abstract

Analysis of trajectories of dynamical biological objects, such as breeding ants or cell organelles, is essential to reveal the interactions they develop with their environments. Many previous works used a global characterization based on parameters calculated for entire trajectories. In cases where transient behavior was detected, this usually concerned only a particular type, such as confinement or directed motion. However, these approaches are not appropriate in situations in which the tracked objects may display many different types of transient motion. We have developed a method to exhaustively analyze different kinds of transient behavior that the tracked objects may exhibit. The method discriminates stalled periods, constrained and directed motions from random dynamics by evaluating the diffusion coefficient, the mean-square displacement curvature, and the trajectory asymmetry along individual trajectories. To detect transient motions of various durations, these parameters are calculated along trajectories using a rolling analysis window whose width is variable. The method was applied to the study of secretory vesicle dynamics in the subplasmalemmal region of human carcinoid BON cells. Analysis of transitions between transient motion periods, combined with plausible assumptions about the origin of each motion type, leads to a model of dynamical subplasmalemmal organization.

FIGURE 1

Schematic setup of the TIRF microscope. A translation of the mirror modifies the evanescence depth without changing the illuminated area at the glass/cell interface.

FIGURE 2

Resolution anisotropy, , versus S/N for fluorescent beads and vesicles (see text).

FIGURE 3

Measurement of the asymmetry parameter Asym along a vesicle trajectory. To estimate Asym for a given point of the trajectory (indicated by the arrowhead), this parameter was calculated for various window widths (marked in bold) centered on the point of interest. Only the window width that maximized Asym value was retained (boxed). The asymmetry profile was obtained by repeating this procedure point by point along the trajectory (see Fig. 4).

FIGURE 4

Profiles of the three motion classification parameters calculated for a sample vesicle trajectory. (A) Analyzed vesicle trajectory. The solid dot indicates the beginning of the trajectory. (B) Profile of the Asym parameter that quantifies the trajectory asymmetry. (C) Profile of the Dev parameter used to evaluate MSD curvature. (D) Profile of the diffusion coefficient D. (The tracked vesicle was from a BON cell expressing NPY-GFP; evanescence depth = 200 nm; acquisition rate = 10 Hz.)

FIGURE 5

Profiles of the three motion classification parameters calculated for a simulated noisy Brownian trajectory with a diffusion coefficient of 20 × 10⁻⁴ μm²/s. (A) Analyzed trajectory. The solid dot indicates the beginning of the trajectory. (B) Profile of the Asym parameter that quantifies the trajectory asymmetry. (C) Profile of the Dev parameter used to evaluate the MSD curvature. (D) Profile of the diffusion coefficient D.

FIGURE 6

Statistical behavior of the parameters Dev and Asym that quantify MSD curvature and trajectory asymmetry, respectively, for simulated 3D Brownian trajectories. (A) Histograms of the number of simulated 3D Brownian trajectories for Dev for different lengths of trajectories N. Thresholds Dev_0.99 chosen such that Dev > Dev_0.99 for 99% of the simulated tracks are indicated as vertical dotted lines. Values of Dev_0.99 are shown above each histogram. (B) Dev_0.99 plotted as a function of N on semilogarithmic scale. (C) Thresholds Asym_0.99, chosen such that 99% of the simulated trajectories had Asym < Asym_0.99 (at fixed N) plotted as a function of N on semilogarithmic scale.

FIGURE 7

Percentages of 3D simulated Brownian trajectories exhibiting constrained (left) or directed (right) periods for different thresholds and minimal durations. Left panel shows the influence of the minimal duration below threshold for different Dev_r thresholds (0.05, −0.01, and −0.05); 7% of the simulated random walks exhibit at least one period of constrained motion with the chosen threshold (−0.01, heavy line) and minimal duration (20 points), indicated by a solid circle. Right panel shows the influence of the minimal duration above threshold for different Asym thresholds (0.95, 1.00, and 1.05); 1% of the simulated random walks exhibit at least one period of directed motion with the chosen threshold (1.00, heavy line) and minimal duration (10 points), indicated by a solid circle. The simulated random trajectories included 300 points, a value that matched the mean duration of experimental trajectories obtained from vesicle tracking.

FIGURE 8

Motion analysis of a vesicle trajectory. (A) Analyzed vesicle trajectory. The segment marked in black corresponds to directed motion and the one in gray to constrained motion, as detected by the algorithm. The black dot indicates the beginning of the trajectory. (B) MSD plot for Δt < 5s calculated for the entire trajectory shown in A. The dashed line corresponds to the weighted linear fit of the 5 initial MSD points (see text). (C) Profile of the Asym parameter that quantifies the trajectory asymmetry as a function of time, calculated for the trajectory shown in A. The dashed line corresponds to the threshold used to distinguish directed and diffusive motions (Asym > 1.00). The black bar indicates the position of the detected directed period. (D) Profile of the Dev_r parameter that evaluates the MSD curvature as a function of time, calculated for the same trajectory. The dashed line corresponds to the threshold used to discriminate between constrained and diffusive motion (Dev_r < −0.01). The gray bar indicates the position of the detected constrained period. (Tracked vesicle from a BON cell expressing NPY-GFP; evanescence depth = 200 nm; acquisition rate = 10 Hz.)

FIGURE 9

Vesicles withdraw from the plasma membrane during transitions from constrained to diffusive motions. Shown is a representative vesicle trajectory in the (x, z) plane. The segment marked in gray corresponds to constrained motion and the one in black to diffusive motion, as detected by the algorithm. The solid dot indicates the beginning of the trajectory.