The TILD method for detecting the moment (instance) when a diffusing molecule in the PM undergoes the hop movement from a compartment to an adjacent one in the PM. We developed an improved method for detecting the hop moment (instance). This method detects the Transient Increase of the effective Local Diffusion (TILD) in a single-molecule trajectory. TILDs are likely to occur when a molecule hops between two membrane compartments, but our analysis itself remains model-independent of any particular model. During an intercompartmental hop, a molecule experiences two compartments instead of one within the time window that includes the hop instance, thus leading to an increase in the effective local diffusion coefficient. To identify TILDs in a given trajectory, consider a window with a size of n frames (n steps), starting from a frame m (m ∼ m + n). Within this window, the center of the n recorded coordinates (n points) is defined, the radial displacements for all of these n points from this center are calculated, and then their maximal value R_MAX(m,n) is determined. Next, the relative diffusion coefficient for this window is defined as $Drel(m,n)≡RMAX2(m,n)4nδtD2−4,$ where δt is the time step in the trajectory (inverse of the frame rate), and D₂₋₄ is the average short-term diffusion coefficient determined for the time interval between 2δt and 4δt (averaged over the whole trajectory; Kusumi et al., 1993), which is included for normalization. For free Brownian diffusion, D_rel(m,n) is ≈ 1 (allowing for statistical variations), independent of m or n. If a molecule is temporarily trapped in a finite domain, then as the window size n increases, D_rel(m,n) decreases due to entrapment. When the window size has increased sufficiently to include the release of the diffusing particle from the finite domain, there will be a sharp increase in D_rel(m,n), due to the extended range of diffusion. To flag releases, the function $H(m,n)=|1/Drel(m+1,n)−1/Drel(m,n)|$ was employed, and by scanning all possible m and n pairs in the full trajectory, a map of H(m,n) was produced. Releases from the trapped domain are flagged by sharp peaks in H(m,n) for both the special starting position (e.g., if position m is before a hop and position m+1 is after a hop, then for all window sizes n, D_rel(m,n) will be greater than D_rel(m+1,n)) and for the combination of a starting position and a window size (e.g., if the trajectory, starting from position m with a window size n, and ending at point p = m + n, is wholly within an entrapped domain, and if extending the window size by 1 includes release from the entrapped domain, then D_rel(m,n+1) will be greater than D_rel(m,n) for all m and n values, such that m + n = p). The minimum number of points within an entrapped domain required to allow the detection of the moments of TILDs is determined by the stochastic nature of diffusion (+ the diffusion coefficient within the domain and the domain size) and the noise present in the position determination. As such, due caution was exercised to avoid choosing a plethora of small compartments erroneously. Typically, the total length of each trajectory analyzed here was 1,000 steps. The n value was varied from 21 to N−m, where N is the total number of steps in a trajectory (= 1,000). In addition to this anterograde direction of analysis over a single trajectory, the same trajectory was analyzed in the retrograde direction in the same way as for the anterograde direction, in order to determine whether the sudden increase in H(m,n) can also be observed for the same m in the anterograde analysis. Therefore, for a single m value, H(m,n) was calculated for [N−40] windows ([N−m−20] + [m−20]). Here, we first describe the results of testing the TILD method using Monte-Carlo simulated hop and simple-Brownian trajectories with a single-molecule localization precision of 50 nm (error for Cy3 in the apical PM, which is the worst precision in the present report; A). Second, we describe the application of the TILD method to detect the hop moments in single-molecule trajectories of Cy3-DOPE, observed in the intact apical PM as well as in the actin-depleted blebbed PM (B). (A) Testing the TILD detection method using computer-generated hop and simple-Brownian diffusion trajectories, and examination of the exponential distributions of the residency times within a compartment. (A a) Typical hop-diffusion (left) and simple-Brownian (right) trajectories generated by Monte Carlo simulation (every 0.1 ms; the initial 400-steps of the 1,000-step-long trajectories are shown here, whereas the TILD analysis was performed for the full 1,000 steps for all of the trajectories). The moments of TILDs determined by the developed protocol are shown by the thick, short red subtrajectories (three-frame trajectories defined by the TILD moment ±1 frame; note their tilde-like shapes). This particular hop-diffusion trajectory (400-frames long) on the left contained three TILDs. The simple-Brownian trajectory (400-frames long) on the right exhibits no TILD. Hop-diffusion trajectories were generated as described previously (Ritchie et al., 2005), except that the unit time step was 0.1 µs. Using a two-dimensional square array of partially permeable barriers separated by 100 nm (L), with a probability of transmission per attempt of 0.0005, the experimental D_MACRO for Cy3-DOPE in the intact apical PM (0.30 µm²/s; Table 2) was reproduced (D_micro was set at 9 µm²/s, as shown in Fujiwara et al., 2002). A single-molecule localization error of 0, 25, or 50 nm was added as the Gaussian noise to each x- and y-coordinate in the hop trajectories (here, the trajectories including a 50-nm localization error are shown). Among 100 trajectories generated by the simulation (1,000 frames per trajectory), 97, 94, and 80 trajectories were statistically classified into the suppressed diffusion mode in the presence of single-molecule localization errors of 0, 25, and 50 nm, respectively. Test trajectories of simple-Brownian particles were generated by Monte Carlo simulations, using a diffusion coefficient of 6 µm²/s, as experimentally obtained in blebbed PMs (Table 2; employing 9 µm²/s made virtually no difference), and Gaussian localization errors were added. (A b) Typical plots of H(m,n) vs. m (only the results with window sizes of n = 100, 200, and 300 for the trajectories shown in A a. The sharp changes (peaks) in H(m,n) are likely to represent the hop movements (transient increase of local diffusion coefficient), and spurious peaks from statistical variations and noise can mostly be distinguished because they only appear in the displays for limited numbers of windows. Briefly, for the same frame number m, H(m,n) was calculated for all possible n’s (N−40 windows), and when the percentage of windows in which H(m,n) ≥ 1 was >20% among a total of [N−40] windows, the molecule was regarded as undergoing the process of intercompartmental hopping. In this figure, the peaks that satisfied these thresholds are highlighted by vertical pink bars, indicating the occurrences of TILDs, i.e., hop events. Other peaks in this display did not satisfy the threshold conditions explained here, and do not represent TILDs. In the application of the TILD detection method to individual trajectories, a three-frame running average (replacing the position of the kth frame with the position averaged for the k−1, k, and k+1 frames) was first applied to each trajectory, to minimize the effect of apparently large displacements that stochastically occurred due to the 50-nm single-molecule localization error. The detectability percentages of hop events (given by the simulation program) for simulated trajectories classified into suppressed diffusion were 82, 76, and 66%; and the accuracies of the predicted hop events were 75, 66, and 60% at localization errors of 0, 25, and 50 nm, respectively. In our standard conditions for long-term single-molecule tracking experiments (1,000 steps; 0.1-ms resolution; a 532-nm laser excitation laser power of 23 µW/µm² at the sample), the single-molecule localization error was 49 nm in the apical PM. (A c) Distributions of the residency lifetimes within a compartment for Monte-Carlo simulated test particles undergoing hop diffusion. The residency time of a particle within a compartment was obtained as the duration between two consecutive TILDs. Top (Ground Truth Distribution Given by the Simulation): The correct distribution of the residency times determined from the hop events (given by the simulation program), which occurred in 100 simulated 1,000-frame long hop trajectories. The residency times shorter than 40 frames (4 ms) were neglected, due to various uncertainties in the short time ranges. The distribution could be fitted with a single exponential function with a decay time constant of 9.0 ± 0.088 ms (mean ± SEM; SEM is provided as the fitting error of the 68.3% confidence interval). The exponential distribution of the residency times found for simulated hop-diffusion trajectories can actually be predicted theoretically, as summarized in Supplemental theory 1 in the Supplemental text. The theory also predicts that its decay time constant can be described by L²/4D_MACRO. In the present simulation, the average dwell time calculated using this equation was 8.3 ms (L = 100 nm, the average D_MACRO obtained from the simulation was 0.3 µm²/s). This value agrees quite well with the dwell lifetimes obtained by simulated hop-diffusion trajectories (9.0 ± 0.088 ms). Bottom (TILD Detection Test): The residency time distribution, determined by the TILD-detection method from 100 simulated 1,000-frame long hop trajectories that included a single-molecule localization precision of 50 nm (residencies in 763 compartments with durations longer than 4 ms). The decay time constant was 9.0 ± 0.17 ms. This agrees well with the correct distribution, suggesting that the developed protocol is useful for evaluating the residency lifetime, although at the level of individual hops, our software misses hops (66% detectability) and incorrectly detects hops (60% accuracy). The number of detected TILDs per 1,000-frame simulated simple-Brownian trajectories, using a diffusion coefficient of 6 µm²/s and a localization precision of 50 nm, was only 0.35/trajectory (n = 100 trajectories; 0.33 when a diffusion coefficient of 9 µm²/s was assumed). (B) Detecting TILDs in Cy3-DOPE and TfR trajectories obtained in intact and actin-depleted blebbed PMs. (B a) Typical Cy3-DOPE trajectories (0.1-ms resolution) in the intact apical PM (left, the 40-ms-long initial part of the trajectory shown in Fig. 3 C) and in the actin-depleted blebbed apical PM (right; typical among 50 and 20 trajectories, respectively). The moments of TILDs, detected as shown in B b, are shown by the thick, red three-step subtrajectories. The trajectory obtained in the intact apical PM contained three TILDs, whereas that in the actin-depleted blebbed apical PM exhibited no TILDs. In the trajectory obtained in the intact apical PM (left), the colors of the trajectories are changed across the short red TILD trajectory, and this convention was used throughout this report. (B b) The plots of H(m,n) vs. m for the trajectories shown in B a. TILDs were detected in all of the experimental trajectories of Cy3-DOPE and TfR obtained in the intact apical PM (see Fig. 3 D; 50 trajectories with a length of 1,000 frames for Cy3-DOPE at a frame rate of 10 kHz and for TfR at a frame rate of 6 kHz). The average numbers of detected TILDs (with intervals longer than 2 ms) per 1,000-frame long trajectory classified into the suppressed-diffusion mode were 7.8 (= 297 events/38 trajectories) for Cy3-DOPE and 4.4 (= 175 events/40 trajectories) for TfR. Meanwhile, in the actin-depleted blebbed PM, where more than 90% of the trajectories were statistically classified into the simple-Brownian diffusion mode (20 trajectories were examined for both Cy3-DOPE and TfR; Table 2), the numbers of detected TILDs per 1,000-frame long trajectory classified into the simple-Brownian diffusion mode were only 0.22 (= 4 events/18 trajectories) for Cy3-DOPE and 0.68 (= 13 events/19 trajectories) for TfR. (B c) Distributions of the dwell times within a compartment for Cy3-DOPE (50 trajectories; 337 residencies) in the intact apical PM, with the best-fit exponential curves and dwell lifetimes of 9.2 ± 0.34 ms. The exponential shape of this distribution is consistent with the hop-diffusion model (under strong-type confinements; see Supplemental theory 1). Furthermore, the exponential residency lifetime found for Cy3-DOPE (9.2 ms) agrees well with that found for the simulated hop-diffusion trajectories, using parameters similar to the experimentally determined values for Cy3-DOPE (9.0 ms; A c, top).