Multistep Track Segmentation and Motion Classification for Transient Mobility Analysis

Abstract

Molecular interactions are often transient and might change within the window of observation, leading to changes in molecule movement. Therefore, accurate motion analysis often requires transient motion classification. Here we present an accurate and computationally efficient transient mobility analysis framework, termed “divide-and-conquer moment scaling spectrum” (DC-MSS). DC-MSS works in a multistep fashion: 1) it utilizes a local movement descriptor throughout a track to divide it into initial segments of putatively different motion classes; 2) it classifies these segments via moment scaling spectrum (MSS) analysis of molecule displacements; and 3) it uses the MSS analysis results to refine the track segmentation. This strategy uncouples the initial identification of motion switches from motion classification, allowing DC-MSS to circumvent the sensitivity-accuracy tradeoff of classic rolling window approaches for transient motion analysis, while at the same time harnessing the classification power of MSS analysis. Testing of DC-MSS demonstrates that it detects switches among free diffusion, confined diffusion, directed diffusion, and immobility with great sensitivity. To illustrate the utility of DC-MSS, we have applied it to single-particle tracks of the transmembrane protein CD44 on the surface of macrophages, revealing actin cortex-dependent transient mobility changes.

Introduction

The analysis of molecule movement, as revealed by live-cell imaging and particle tracking, has helped uncover important information about how molecules interact with their environment (1, 2, 3). As these interactions are often transient and might change within the window of observation, leading to changes in molecule movement, accurate motion analysis often requires transient (i.e., subtrack) motion classification. A prime example are cell surface proteins and lipids, which can exhibit multiple motion types depending on their plasma membrane and juxta-membrane environment (4). They can diffuse freely (4), or become confined, e.g., within actin cortex corrals (5), or get anchored or immobilized if they bind to static intracellular components (6), or exhibit directed motion mediated by cytoskeletal elements (7). Therefore, for a full understanding of the dynamic nature of plasma membrane organization, it is essential to identify not only the different motion types of cell surface molecules, but also the lifetimes of these motion types and transition rates between them.

Most transient motion analysis algorithms employ either rolling window analysis (8, 9, 10, 11) or hidden Markov modeling (HMM) (12, 13, 14). In rolling window approaches, the classification within each window is usually based on mean-square displacement analysis (9, 10) or, in more advanced schemes, machine learning (8, 11). However, whichever the classification scheme, rolling window approaches suffer from a tradeoff between sensitivity to detect motion switches and accuracy of classification, because motion switch sensitivity requires smaller windows, whereas classification accuracy requires larger windows. As a result, rolling window approaches often operate with their smallest classifiable window, i.e., at their worst classification accuracy. They can be also time-consuming computationally, as the analysis is repeated point-by-point.

HMM approaches classify movement by analyzing single-step displacements (12, 13, 14). However, by virtue of being single-step-based, these algorithms have difficulty classifying confined diffusion, which is only apparent at longer timescales, i.e., over multiple steps. Yet confined diffusion of molecules is very common, and it has different biological implications than immobility. Additionally, HMM approaches suffer from an even larger computational demand than rolling window approaches, while at the same time requiring a large number of tracks to learn the motion models accurately (15).

In light of the strengths and weaknesses of the existing analytical methods, and especially given our interest to distinguish among freely diffusing, confined, and immobile cell surface molecules, we developed, to our knowledge, a new transient mobility analysis algorithm, termed “divide-and-conquer moment scaling spectrum” (DC-MSS). DC-MSS uncouples the initial identification of motion switches from motion classification, a novel strategy that, to the best of our knowledge, has not been attempted to date. This allows DC-MSS to circumvent the sensitivity-accuracy tradeoff of traditional rolling window approaches. In the next section, we describe the workflow of DC-MSS. Then, we benchmark its performance and compare it to other transient motion analysis algorithms in terms of its ability to detect switches among free diffusion, confined diffusion, directed diffusion, and immobility. After this we demonstrate its utility via one example application, namely analyzing live-cell single-molecule tracks of the cell surface protein CD44, where DC-MSS revealed actin cortex-dependent transient mobility changes.

Methods

Divide-and-conquer moment scaling spectrum analysis

DC-MSS works in three steps (Fig. 1):

Figure 1.

The three steps of DC-MSS: Initial Track Segmentation, Initial Segment Classification, and Final Segmentation and Classification. Illustration uses a 250-frame simulated track that switches among immobility (brown) and confined (blue), free (cyan) and directed (magenta) diffusion. Red x symbols on track and red bars in timelines indicate detected motion switches in each analysis step. Timeline t is in frames. (Black) Unclassified. (Inset) Zoom of enclosed area.

Initial track segmentation. Using a rolling window of 11 frames, DC-MSS calculates a simple local movement descriptor, namely the maximum pairwise distance (MPD) between particle positions within the window (see Fig. S1, a and b, which illustrate Initial Track Segmentation in detail). MPD reflects the extent of molecule movement, namely MPD (immobile) < MPD (confined) < MPD (free) < MPD (directed), and thus reveals the switches between different motion types. A window of 11 frames was chosen to calculate MPD by examining the frequencies of undersegmentation and oversegmentation and the accuracy of identifying the actual switch point for a range of window sizes (5–17 frames). This analysis indicated that undersegmentation in Initial Track Segmentation was negligible for all window sizes (Fig. S1 c), whereas oversegmentation increased, and switch point identification accuracy improved, with decreasing window size (Fig. S1 d). A window size of 11 frames provided the best balance between the two criteria (Fig. S1 d, bottom row).

The time series MPD(t) is then smoothed using a Gaussian kernel of σ = 2 frames to reduce noise. To identify mobility switch points, the absolute value of the normalized frame-to-frame change in the smoothed time series SMPD(t) is then calculated:

$$\Delta SMPD\left(t\right)=\frac{\left|SMPD\left(t+1\right)-SMPD\left(t\right)\right|}{\left(SMPD\left(t+1\right)+SMPD\left(t\right)\right)/2}.$$

Normalization puts different mobility switches (e.g., immobile to confined versus immobile to free) on the same scale, to facilitate their detection. Finally, the frames at which ΔSMPD has a peak with value in the top 5% are taken as potential mobility switch points.

Because the smoothing Gaussian kernel might blur segment boundaries, short segments lasting fewer than 20 frames (making them unclassifiable via moment scaling spectrum (MSS) analysis; see Initial Segment Classification below), yet are greater than or equal to 16 frames (i.e., 20–2σ), are extended to 20 frames by shifting the boundary into the adjacent segment (as long as this does not shorten the adjacent segment to <20 frames, rendering it unclassifiable; see Fig. S1 b).

As mentioned above, the MPD metric tends to oversegment tracks (Fig. S1 c). We reasoned that initial oversegmentation can be compensated for by merging of adjacent segments, whereas missed switches in the case of undersegmentation are irrecoverable. DC-MSS compensates for oversegmentation at two analysis stages: preclassification in this step, and postclassification in Final Segmentation and Classification below. In this step, the segments identified via ΔSMPD are subjected to a second test, where the distribution of pairwise distances between all positions in each segment is constructed, and then compared between adjacent segments via a Kolmogorov-Smirnov test (Fig. S1 b). If the comparison p > 0.05, then they are merged; otherwise they are kept separate. With this, the initial track segmentation is achieved.

Although Initial Track Segmentation of DC-MSS employs a rolling window to calculate MPD (it is the only step that does so), 11 frames are fewer than the minimum segment duration of 20 frames required for motion classification via MSS analysis. This increases the sensitivity and accuracy of detecting motion switches (Fig. S1, c and d). At the same time, the track segments identified in this step can be of any duration, not limiting segment classification (Initial Segment Classification below) to the minimum classifiable duration of 20 frames, thus increasing segment classification accuracy. Overall, this scheme is expected to circumvent the sensitivity-accuracy tradeoff of traditional rolling window approaches, a premise that is validated by the benchmarking tests of DC-MSS (Results and Discussion).

Initial segment classification. The track segments identified in Initial Track Segmentation with duration ≥20 frames are classified using MSS analysis of molecule displacements (1, 16, 17). A summary of MSS analysis is provided in the Supporting Materials and Methods. Importantly, we derived, to our knowledge, new MSS slope thresholds for motion classification, through which we expanded our MSS analysis algorithm to explicitly identify the immobile state (Fig. S2; and see the Supporting Materials and Methods). These classification thresholds are segment-duration-dependent, and are optimized at each segment duration to yield minimum misclassification probability among adjacent motion types (i.e., confined-immobile, free-confined, and directed-free; see Fig. S2, e and f). Specifically, in the absence of localization error, the new thresholds are associated with a misclassification probability of 0.07–0.18 for the shortest classifiable segments (20 frames; probability depends on motion types), going down to ≤0.01 for longer durations. In the presence of localization error, there is a systematic left-shift in the MSS slope distributions, increasing the probability of misclassifying a motion type as a lower-mobility motion type, and conversely decreasing the probability of its misclassification as a higher-mobility motion type (Fig. S3 and see the Supporting Materials and Methods). All in all, with these new thresholds, MSS analysis can distinguish among four motion types, which may coexist: free diffusion, confined diffusion, directed diffusion, and immobility.

Final segmentation and classification. As mentioned in Initial Track Segmentation, DC-MSS compensates for initial track oversegmentation by testing the merger of adjacent segments. In this third and final step, DC-MSS tests whether to merge adjacent segments for two specific cases. Case 1: If the initial segmentation yields segments that are too short to initially classify, the algorithm tests whether to merge them with adjacent classified segments (Fig. S4 a, left branch). Case 2: If adjacent segments are assigned the same initial classification in Initial Segment Classification, the algorithm tests whether to merge them into one segment (Fig. S4 a, right branch). In both cases, to determine whether to accept the merge, the combined segment is reclassified. If the combined segment is assigned to either the original or a lower mobility class, the merge is accepted. The reasoning behind this is that longer observation times can reveal more confinement. If the combined segment is assigned a higher mobility class, the merge is accepted or rejected based on the classification confidence before and after the merge. Specifically, given the MSS slope values, we use the MSS slope distributions described in the Supporting Materials and Methods to calculate the probability of misclassification (Fig. S4 b) of the combined segment, p_mis(combined), and of the separate segments, defined as follows:

$$p_{\text{mis}}\left(\text{separate}\right)=\{\begin{array}{c}{p}_{\text{mis}}\left(\text{segment}1\right),\text{if}\text{merging}\text{unclassified}+\text{classified}\left(\text{Case}1\right)\\ w_1{p}_{\text{mis}}\left(\text{segment}1\right)+w_2{p}_{\text{mis}}\left(\text{segment}2\right),\text{if}\text{merging}\text{classified}+\text{classified}\left(\text{Case}2\right)\text{,}\end{array}$$

where w₁ and w₂ are the segment weights, proportional to their durations. If p_mis(combined) ≤ p_mis(separate), the merge is accepted together with its new classification. Otherwise, it is rejected, and the separate segments retain their original classifications.

Other methods

Simulation of tracks of varying motion types and compositions. Free diffusion tracks/track segments were simulated as a series of x- and y displacements, each derived from a normal distribution $N\left(0,\sqrt{2D\Delta t}\right)$, where D is the diffusion coefficient and Δt is the time step (= 1 frame). Confined diffusion tracks/track segments were simulated like free diffusion tracks/track segments, except that displacements were reflected off the confinement area boundary, defined as a circle with confinement radius R. Directed diffusion tracks/track segments were simulated as a series of displacements such that each displacement contained two components: the diffusion component, same as for free diffusion above; and the directed component, defined as vcos(θ)Δt (x displacement) and vsin(θ)Δt (y displacement), where v is the drift speed and θ is the drift angle. Immobile tracks/track segments were simulated by generating x- and y coordinates each from a normal distribution N(0,σ_p), where σ_p is the positional standard deviation (localization error).

Rolling window MSS analysis. Using a rolling window of 21 frames, MSS analysis was performed for each window along a track, assigning a classification to the middle frame of each window. Consecutive frames with the same classification were then grouped to yield segments of different classification. This was followed by several steps of postprocessing, similar to those described in (9), to ensure that each classified segment lasts for a minimal duration, thus reducing the probability of erroneous track segmentation and classification.

Results and Discussion

Performance assessment of DC-MSS

We tested the performance of DC-MSS using simulated tracks of varying composition and duration of the four different motion types (immobile, confined, free, and directed). To assess the ability of DC-MSS to detect motion switches without complications from motion classification per se, we first employed tracks (called “reference tracks” below) simulated with the same motion parameters used to derive the classification thresholds (Fig. S2; and see the Supporting Materials and Methods). To assess whether the initial uncoupling of motion switch identification from motion classification improved the algorithm’s sensitivity to detect motion switches, we compared DC-MSS to a classic rolling window approach, termed “RW-MSS”. Additionally, we compared both to a recent HMM approach (13), which ran on all tracks within each dataset together, i.e., with data pooling, for optimal learning of motion models (we will refer to it as “pooled-HMM-Bayes” hereafter). Note that DC-MSS and RW-MSS analyze tracks individually and have no data pooling option. To mimic real experimental data analysis, where the number and types of motion states are not known a priori, we ran pooled-HMM-Bayes with the option to find up to three motion states, which is the maximum it can model. As for DC-MSS and RW-MSS, by construction they can classify any track or track segment as any of the four motion types built into them (there is no option to consider only a subset of motion types). After this detailed analysis with minimized classification complications, we tested the versatility of DC-MSS by applying it to tracks simulated with motion parameters different from those used to derive the classification thresholds.

Performance on reference tracks with a motion switch

Whereas real tracks might switch multiple times and might exhibit more than two motion types, each particular switch in any track involves only two motion types at a time. Therefore, to assess the sensitivity and accuracy of DC-MSS in capturing motion switches, we simulated heterogeneous mobility tracks that consisted of 100 frames each and switched from one motion class to another once within each track. The amount of time spent in each motion class varied from 20 frames of class 1 and 80 frames of class 2, to 80 frames of class 1 and 20 frames of class 2, in steps of 10 frames. Each dataset consisted of 100 tracks of each composition, with half of the tracks starting with class 1 and the other half starting with class 2. To be compatible with pooled-HMM-Bayes, all directed track segments had the same direction (13) (this is irrelevant for DC-MSS or RW-MSS, as they analyze tracks individually without any data pooling). The reported results (Figs. S5, S6, S7, S8, S9, and S10) are from three dataset repeats. The Supporting Materials and Methods describes how the output of pooled-HMM-Bayes was mapped to our classification terminology.

Sensitivity to capture transitions. Apart from free-confined, transitions were captured by DC-MSS with a true positive rate of 79–99% when the lower mobility segment was ≥30 frames long, and with a true positive rate of 47–80% when the lower mobility segment was 20 frames long (Figs. S5, S6, S7, S8, S9, and S10, top row). Performance was worse in the latter case, most likely because 20 frames is our minimum classification duration. Free-confined transitions were the most difficult to capture, with a true positive rate of 45–73%, probably due to the relative similarity of these two motion types.

DC-MSS outperformed RW-MSS in all cases, demonstrating the advantage of first segmenting the tracks based on a local movement descriptor, and then classifying and refining the resulting segments. Additionally, it outperformed pooled-HMM-Bayes for the two most challenging cases, namely free-confined transitions and confined-immobile transitions. For free-immobile transitions, pooled-HMM-Bayes outperformed DC-MSS for tracks with a smaller immobile fraction, but then DC-MSS outperformed pooled-HMM-Bayes for tracks with a larger immobile fraction. The struggle of pooled-HMM-Bayes with tracks containing 80 frames of immobility was reminiscent of its struggle with homogeneous immobile tracks in isolation (Fig. S11). For transitions that involved directed motion, the performance of DC-MSS was in some cases comparable to that of pooled-HMM-Bayes (especially for directed-immobile transitions; Fig. S10), but in others it was worse.

Accuracy of locating the switch point. Apart from free-confined transitions, DC-MSS tended to locate the switch point a few frames away from its real location (median of 3–5 frames; Figs. S5, S6, S7, S8, S9, and S10, middle row), inside the lower mobility class (i.e., a few frames of the lower mobility class were erroneously assigned to the higher mobility class). For free-confined transitions, the identified switch point was on average closer to the real location, although with a larger spread. This again mostly likely reflected the relative similarity of these two motion types on short timescales. DC-MSS was consistently more accurate than RW-MSS in identifying the switch location. Pooled-HMM-Bayes had overall little switch point location bias, with the exception of free-confined transitions where its performance was highly variable. Additionally, in the case of confined-immobile transitions, the identified switch point had a substantially higher spread than for DC-MSS.

Accuracy of segment classification. The motion classification accuracy of individual segments by DC-MSS largely followed the accuracy of classification (Figs. S5, S6, S7, S8, S9, and S10, bottom row) associated with the employed MSS slope thresholds for each segment duration (Fig. S2, e and f). Interestingly, even though RW-MSS used the same thresholds as DC-MSS, its segment classification accuracy was often worse. This demonstrates again the disadvantage of classic rolling window approaches, which operate at their weakest classification accuracy to keep the rolling window as small as possible. It further highlights the advantage of segmenting first and then classifying the initial segments, which could last longer than the minimum classifiable window, thus improving classification accuracy, allowing it to reach its full potential. Pooled-HMM-Bayes performed overall very well when it could detect a switch, except for free-confined transitions, where it often confused the two motion types even when it did detect a switch.

Performance on reference tracks without a motion switch

To assess the false-positive rate of DC-MSS, we simulated homogeneous mobility tracks that had variable duration, specifically 20–80 frames (in steps of 10 frames). Due to the difficulty that pooled-HMM-Bayes faces with classifying immobility in isolation (Fig. S11), the datasets for homogeneous mobility tests contained groups of tracks of a specific duration with different mobility. Specifically, one subset of datasets contained 40 tracks each of immobile, confined, and free, used to assess their respective classifications, and another subset of datasets contained 40 tracks each of immobile, free, and directed, used to assess only the directed classification. The limitation to groups of three was because pooled-HMM-Bayes can only detect up to three mobility classes at a time. The reported results (Figs. S12, S13, S14, and S15) are from three dataset repeats. The Supporting Materials and Methods describes how the output of pooled-HMM-Bayes was mapped to our classification terminology.

In this test, DC-MSS mostly detected that there was no mobility change, with maximum false-positive rates of 8% for directed tracks, 11% for immobile tracks, 16% for confined tracks, and 22% for free tracks (Figs. S12, S13, S14, and S15, top row). Free tracks had the highest false-positive rate, for two reasons. 1) The local movement descriptor used for initial track segmentation, MPD, fluctuates the most when the motion type is free diffusion. Even though we devised procedures to compensate for this (in Initial Track Segmentation and Final Segmentation and Classification), the compensation is not complete. 2) Free diffusion could be misclassified at either end of the MSS slope range, namely as confined diffusion or directed diffusion (Fig. S14, bottom row). The latter is most likely also the reason that confined tracks had the second highest false-positive rate, as confined diffusion also could be mistaken for two other motion types, namely free diffusion or immobility (Fig. S13, bottom row). Immobility and directed diffusion, on the other hand, could only be mistaken in one direction, i.e., immobility as confined diffusion and directed diffusion as free diffusion (Figs. S12 and S15, bottom row).

For all motion types except for immobility, DC-MSS displayed a slightly greater false-positive rate than RW-MSS. However, the false-positive rate of DC-MSS was relatively stable, whereas the false-positive rate of RW-MSS generally increased with track duration. As for pooled-HMM-Bayes, although it had almost no false positives for free and immobile classes (when the immobile type was in the presence of others; compare Fig. S12 to Fig. S11), it was again unable to distinguish the confined type from free diffusion, and had many false-positive switches for directed tracks (Fig. S15), much more than DC-MSS.

Performance on non-reference tracks

To assess the versatility of DC-MSS, next we applied it to tracks simulated with different motion parameters than those used to derive the classification thresholds. As discussed in the Supporting Materials and Methods, the MSS slope distributions of free and immobile tracks are independent of their motion parameters, namely the diffusion coefficient (D) and positional standard deviation (σ_p), respectively ((1), and see Fig. S2 b). In contrast, the MSS slope distributions of confined and directed tracks depend, respectively, on the confinement extent $CE=\sqrt{4D\Delta t}/R$ and the drift extent $DE=v\Delta t/\sqrt{4D\Delta t}$ (R = confinement radius, v = drift speed, and Δt = the time between consecutive frames; see Fig. S2, c and d). Therefore, to assess the versatility of DC-MSS, we performed three tests. 1) We increased CE (i.e., decreased R) of confined tracks, bringing them closer to immobile tracks. 2) We decreased CE (i.e., increased R) for confined tracks, bringing them closer to free tracks. 3) We decreased DE (i.e., decreased v) for directed tracks, bringing them closer to free tracks as well (example tracks can be seen in Fig. S2, a–d).

First, we probed the classification accuracy of DC-MSS by applying it to homogeneous mobility tracks exhibiting each of the motion types individually. As expected, tracks with higher CE were misclassified as immobile more often than the reference tracks, whereas tracks with lower CE or DE were more often misclassified as free (Fig. S16 a). Conversely, decreasing CE reduced the probability of misclassification as immobile, whereas increasing CE or DE reduced the probability of misclassification as free (data not shown). Increasing CE had little effect on the false-positive rate of DC-MSS, whereas decreasing CE or DE slightly increased the rate of false-positive switches (Fig. S16 b). This is most likely because motion gets closer to free diffusion with decreased CE or DE, and free diffusion had the highest false-positive rate in our tests (gray dashed lines in Fig. S16 b).

Second, we probed the sensitivity of DC-MSS to identify motion type switches by applying it to tracks that switch between adjacent motion types, i.e., confined-immobile, free-confined, and directed-free, as those are the most difficult to identify. Varying CE between 0.6 and 0.8 had little effect on the ability of DC-MSS to detect confined-immobile switches, but it had a larger effect on its ability to detect free-confined switches (Fig. S16 c, left and middle panels). Specifically, decreasing CE reduced the ability of DC-MSS to detect free-confined switches, whereas increasing it improved the ability of DC-MSS to detect free-confined switches. Decreasing DE also reduced the ability of DC-MSS to detect directed-free switches, especially when the fraction of the two motion types was not balanced (curve ends in Fig. S16 c, right panel). Interestingly, the effect of parameter variation on the ability of DC-MSS to detect transitions between motion types was overall smaller than its effect on the classification accuracy via MSS analysis (Fig. S16 c versus Fig. S16 a).

Taken together, the above tests demonstrate that DC-MSS has the power to distinguish among four different motion types—free diffusion, confined diffusion, directed diffusion, and immobility—and to capture transitions between them within individual tracks. Its sensitivity and accuracy are overall very good, although they exhibit some dependence on motion parameters (Fig. S16) and localization error (Fig. S3). Given this dependence, a good a posteriori quality control for DC-MSS is to calculate critical track properties, such as confinement extent, drift extent, and localization error, and to interpret the output of DC-MSS in light of their values. Additionally, like any motion analysis algorithm, DC-MSS can only detect and characterize the motion states built into it, and will approximate other motion types with the built-in motion states (as an example, Fig. S17 and the Supporting Materials and Methods discuss the application of DC-MSS to fractional Brownian motion). Therefore, when analyzing tracks of unknown motion type, it is important to test multiple motion models, potentially via multiple analysis algorithms, and to make use of prior knowledge about the studied system to judge the validity of these motion models. Combining DC-MSS with other motion analysis algorithms can be also utilized to enhance the accuracy of motion characterization. For example, the above tests suggest that sometimes the optimal analysis strategy is to first analyze tracks via DC-MSS to identify track parts with motion type combinations that are characterized well by pooled-HMM-Bayes, and then to reanalyze subsets of track parts with pooled-HMM-Bayes to refine the identification of the motion switch points.

Computational demand

All analyses were run with the software MATLAB 2015b (The MathWorks, Natick, MA) using distributed computing on a high-performance computing cluster. Pooled-HMM-Bayes additionally used the supplied C code and parallelization feature to reduce runtime. DC-MSS took ∼20 s per 100-track classification. This was six times faster than RW-MSS and ∼150 times faster than pooled-HMM-Bayes.

Application to single-molecule receptor tracks

With its ability to identify four different types of molecule movement that may occur transiently within a single track, we applied DC-MSS to analyze particle tracking data of the transmembrane protein CD44 on the surface of macrophages (Materials and Methods pertaining to experimental data is given in the Supporting Materials and Methods). CD44 is expressed at high density on the surface of phagocytes (18), and can interact with the actin cytoskeleton by binding to ezrin and/or ankyrin (19). Its movement in the plasma membrane is thus expected to be regulated by the cortical cytoskeleton, leading to multiple motion type possibilities. DC-MSS analysis of single-molecule CD44 tracks on the surface of macrophages revealed that CD44 indeed exhibited three dominant motion types: confined diffusion, free diffusion, and immobility (in order of decreasing probability; Fig. 2 a, upper left, and Fig. 2, b and c). The average localization precision (positional standard deviation) of CD44 molecules was 11 nm, i.e., 0.28 of the step size standard deviation from the free diffusion coefficient (Fig. 2 b), thus in the acceptable accuracy regime of MSS analysis (Fig. S3). Confined tracks had a confinement extent (CE) of 0.1–0.7 (median of 0.4), suggesting that the probability of confined diffusion might be even higher than that estimated by our analysis (Fig. S16). Nevertheless, 70% of the CD44 tracks switched motion type at least once within the 30 s imaging window (imaged at 33 Hz), highlighting the need for transient motion analysis. By analyzing the observed transitions of CD44 between the different motion types (see the Supporting Materials and Methods), we found that transitions occurred among all motion classes, with the immobile-to-confined transition having the highest rate and the free-to-immobile transition having the lowest rate (Fig. 2 a, upper left).

Figure 2.

Differences between CD44 and FcγR mobility are diminished upon actin perturbation. (a) Example tracks (400 frames/12 s duration) and state diagrams describing CD44 and FcγR mobility classes and their interconversion in unperturbed cells and cells treated with latrunculin A (1 μM for 5 min). Each mobility class is shown as a circle with circle area proportional to the class probability (indicated inside/next to circle). Arrows between circles and adjacent numbers indicate the switching rate between mobility classes, with arrow thickness and color strength proportional to the rate. Scale bars, 500 nm. Track and circle colors: cyan, free; blue, confined; brown, immobile; magenta, directed. (b) Boxplots of diffusion coefficient distribution for free and confined track segments for each condition. Boxplot description: red central mark shows median; box edges show 25th and 75th percentiles; dashed whiskers extend to the most extreme data points not designated as “outliers”; and notch emanating from median indicates the 95% confidence interval around the median, shown for visual aid. (c) Boxplots of confinement radius distribution for immobile and confined track segments for each condition. Boxplot description is as in (b). In all panels, results are from N = 1274, 1302, 1485, and 2872 tracks of duration ≥20 frames for unperturbed CD44, CD44+latrunculin A, unperturbed FcγR, and FcγR+latrunculin A, respectively. The tracks for each condition come from 22 to 30 cells, collected over three independent experiments.

We surmised that immobile molecules could be tethered to the actin cortex, whereas confined molecules could be untethered but diffusing within actin-delimited compartments. To shed more light on the role of the actin cortex in establishing these different motion types, we compared the movement of CD44 to that of the Fcγ Receptor (FcγR), which is not believed to bind to actin, in contrast to CD44. We also analyzed the movement of the two receptors after treating the cells with latrunculin A (1 μM for 5 min), which inhibits actin polymerization. FcγR exhibited markedly more free diffusion and less confined diffusion and immobility than CD44, as well as less switching between the different motion types (Fig. 2 a, lower left). Exposure to latrunculin A increased the probability of free diffusion for both receptors, at the expense of confined diffusion and immobility, the latter being almost eliminated (Fig. 2 a, upper and lower right). Interestingly, the effect of latrunculin A was stronger on CD44 than on FcγR, such that it largely eliminated the differences between them in terms of motion type probabilities and transition rates (Fig. 2 a, upper and lower right). Note that the diffusion coefficient increased in all of these cases (Fig. 2 b), thus reducing the localization error relative to the step-size standard deviation to ∼0.17 and increasing the classification confidence (Fig. S3). These results suggest that CD44 undergoes transient immobilization and confinement that are most likely dependent on its interactions with the actin cortex, motivating further experiments and analyses of the molecular underpinnings of these interactions and their functional consequences (20).

Conclusion

We have developed a transient motion analysis algorithm that can identify switches among free diffusion, confined diffusion, directed diffusion, and immobility. The strength of DC-MSS stems from two features. First, it uncouples the initial identification of motion switch points from motion classification, a novel strategy to the best of our knowledge, thus circumventing the sensitivity-accuracy tradeoff of classic rolling window approaches. Second, being based on MSS analysis, it considers displacements over multiple timelags to classify movement, thus allowing it to sensitively distinguish among the abovementioned motion types, in contrast to hidden Markov modeling approaches, which are single-step-based. A limitation of DC-MSS is that it requires a motion type to last for at least 20 frames to identify it as a distinct state within a track, thus it cannot identify potentially short-lived states. Nevertheless, with the continued advancements in fluorescent labeling and imaging techniques, allowing the acquisition of faster and longer molecule tracks, and given the widespread use of mean square displacement analysis to classify molecule movement, we expect DC-MSS to be widely applicable to a large range of biomolecule tracking data.

Software availability

DC-MSS code (including visualization) is available on the Jaqaman lab website (http://www.utsouthwestern.edu/labs/jaqaman/software/). Code to generate synthetic data for DC-MSS validation is supplied as part of the DC-MSS package.

Author Contributions

A.R.V. and K.J. designed the method. A.R.V. implemented the method and performed all tests and data analysis. S.A.F. and S.G. provided experimental single-molecule tracking data. A.R.V. and K.J. wrote the manuscript, with input from S.A.F. and S.G.

Editor: Julie Biteen.

Supporting Citations

References (21, 22, 23, 24, 25, 26, 27, 28) appear in the Supporting Material.