Although the highly dynamic and mosaic organization of the plasma membrane is well-recognized, depicting a resolved, global view of this organization remains challenging. We present an analytical single-particle tracking (SPT) method and tool, multiple-target tracing (MTT), that takes advantage of the high spatial resolution provided by single-fluorophore sensitivity. MTT can be used to generate dynamic maps at high densities of tracked particles, thereby providing global representation of molecular dynamics in cell membranes. Deflation by subtracting detected peaks allows detection of lower-intensity peaks. We exhaustively detected particles using MTT, with performance reaching theoretical limits, and then reconnected trajectories integrating the statistical information from past trajectories. We demonstrate the potential of this method by applying it to the epidermal growth factor receptor (EGFR) labeled with quantum dots (Qdots), in the plasma membrane of live cells. We anticipate the use of MTT to explore molecular dynamics and interactions at the cell membrane.
Knowledge of the dynamic organization of plasma membrane components is crucial for a better understanding of cellular functions. One fundamental question remains as to how the spatiotemporal organization of the membrane regulates signaling upon physiological stimulation. The development of experimental approaches used to study these structure-function relationships has remained challenging1,2,3,4, with presently available techniques requiring a compromise between temporal and spatial resolution.
Among classical approaches, fluorescence recovery after photobleaching is based on ensemble-averaging measurements within a small area of the cell membrane2. Fluorescence correlation spectroscopy (FCS) provides a good alternative as it can be used to quantify translational diffusion with high temporal resolution and statistical accuracy4. Recently, we increased the strength of FCS by obtaining spatial information from measurements performed at variable observation volumes5. This approach can be used to detect the dynamic partitioning into nanodomains or the presence of a meshwork in live cells6. FCS observation, however, is still limited to a small area of the plasma membrane.
Single-molecule techniques such as SPT provide high spatial resolution, with molecular motion not masked by ensemble averaging3,7. When connecting moving tags, however, fast particles become difficult to connect at high densities: nearest neighbors are usually favored when ambiguities arise resulting, for instance, from particles appearing, disappearing or crossing paths. This drawback can be reduced both experimentally, using a high-quality, dedicated apparatus (optics, electron-multiplying charge-coupled device (CCD) camera, biomaterials and others), and analytically, by optimizing the algorithms. The introduction of fluorescent Qdots with high photostability, brightness and biolabeling versatility8 has considerably increased the sensitivity of detection and greatly facilitated biological investigations of cellular processes9,10. The ability to trace a large number of targets would expand the potential of single-molecule approaches; the resulting larger number of observations would concurrently improve statistical relevance. Although it would be challenging, an approach that allows tracking particles at high densities and with fast acquisition speed is required to achieve an appropriate spatiotemporal resolution. Unfortunately, overcrowded datasets increase the occurrence of false detection or connection, and reducing illumination time directly degrades the signal-to-noise ratio (SNR).
Here we present a method for high-density labeling, MTT. Classically, SPT analyses are performed in two independent stages: peak detection and estimation, and trace reconnection11,12. In MTT, appropriate statistical tools provide optimal peak characterization. Introducing a deflation loop by subtracting detected peaks allows detection of additional peaks that were previously masked because of their lower intensity. Another improvement comes from the intrinsic link between detection and connection: analyses are alternately performed using information obtained at both stages. Therefore, accurate localization and exhaustive detection, combined with integrated historical information of trajectories, provide efficient multiple-target reconnection. Robustness is achieved by including minimum hypotheses on the underlying diffusion phenomena and by having to adjust a minimal set of parameters. Reconnection, in turn, allows validation of detected particles. To illustrate the tracking results, we detected for each trajectory possible confinement within diffusive events, a key parameter of membrane dynamics. Combining analytical tools allowed us to track particles at higher densities and collect snapshots and movies of cell membrane organization over a short period of time (∼10 s). This new conceptual approach introduces a general method of studying molecular dynamics for various constraints of speed or scale. We tested MTT by analyzing the membrane dynamics of EGFR13 naturally expressed in COS-7 cells. Our work demonstrates the potential usefulness of MTT in analyzing membrane dynamics under physiological conditions.
Schematic principle of the MTT algorithm
SPT is based on the ability to accurately identify the center of a peak. By imaging isolated fluorescent molecules, spatial resolution close to the molecular scale can be achieved, and it is possible to construct trajectories of these fluorescent molecules (which we call reconnection) as long as they remain visible. But it is challenging to do this at high particle density and low SNR. In this work we overcame these challenges and simultaneously tracked a large number of molecules. In MTT, the detection, estimation of positions and reconnection are done sequentially (Fig. 1). First, MTT provides efficient detection of particles, and second, it takes into account past information for the reconnection. A mathematical description is available in the Supplementary Note online.
Individual fluorophores behave as point-source emitters and generate blurred images described by the point-spread function (PSF) of the microscope. Classically, the image of each fluorophore can be described by the fluorescence level Xi,j at pixel (i,j), which results from the sum of a fluorescence peak, offset and noise11,12:
where I is the intensity of the particle, G is a bidimensional Gaussian modeling the PSF centered in (i0,j0) with a radius r0 (Supplementary Note), m corresponds to the mean background intensity and N models the background noise of s.d., σ (Supplementary Table 1 online).
For any given image, I is unknown and represents a random variable with a probability distribution including the phenomena of photon noise, bleaching and blinking. Bleaching is negligible for fluorescent markers such as Qdots because their lifespan is longer than the acquisition time8. Blinking is more critical as it is present at all time scales from nanoseconds to seconds14. There are three possible scenarios: (i) the particle may be constantly 'on' during one (or several) image(s); (ii) it may have blinked (one or several times) on time scales smaller than the acquisition time, leading to an intermediate fluorescence intensity; or (iii) the particle may have disappeared, having been turned off (fully blinked) during one (or several) image(s) (Supplementary Fig. 1 online). For a completely 'on' particle, the experimental distribution and variability correspond to a Gaussian distribution centered on an average intensity 〈Ion〉. When particles are partially turned off, we add a second term (uniform law) describing the equal probability for all blinking values. The third case (particle turned off) has no relevant contribution to the likelihood of reconnection as no signal is detected.
According to equation 1, the background of an image is a mix of thermal noise, photon noise and autofluorescence from the sample. It is modeled as independent background m in addition to a Gaussian noise N of unknown power σ2. The parameters m and σ2 are assumed to be locally constant (Supplementary Fig. 1).
MTT iteratively reconstructs trajectories up to time t (Fig. 1). Past statistics are used to accurately reconnect trajectories at the next time point, t + 1. For each trajectory, in addition to the last position, MTT characterizes the parameters of the following probability laws, (i) the local diffusion coefficient Dlocal (Supplementary Table 1), (ii) the mean and s.d. of the intensity and (iii) the photophysical state relative to blinking (Supplementary Fig. 1).
Exhaustive particle detection and estimation by deflation
At high particle densities, exhaustive detection is critical as missing or false particles will directly compromise all the later steps in the analysis. Therefore, MTT has to minimize errors. Detection and estimation are completed by a deflation loop.
Particle detection is classically performed using a bi-dimensional Gaussian approximating the microscope PSF15. For each image, first, an optimal detection test is applied to screen for the presence of Gaussian peaks. The usual statistical test is a generalized likelihood ratio test (GLRT; Supplementary Note). The image intensity is analyzed in a sliding window in which the respective probabilities of particle presence (H1) or absence (H0) are compared, taking into account the image noise: targets are no longer identified by the fluorescence intensity but by the H1/H0 ratio (Fig. 2a). This ratio is high when peaks are significantly separated from noise as evaluated by the algorithm. A window size ws of 7 × 7 pixels ensures a correct estimation of the background while limiting possible long-range background variations. The accuracy of the decision is set by fixing a constant false-alarm rate. To ensure a probability of false alarm (PFApeak) of less than 1 for an image of 512 × 512 pixels, we set the GLRT with a constant PFApeak of 10−6. This strategy provides a high confidence level for all detected particles, insuring that they do not arise from the noise. Notably, the PFApeak is independent of the intensity and background, and consequently of the experimental image acquisitions. Finally, ws and PFApeak are the only two parameters required for detection.
This step only provides a binary detection. Identified peaks then need to be properly estimated for each relevant parameter such as intensity, subpixel position, width and offset. They are evaluated by a multi-parametric Gaussian fit on the signal intensity. Filtering tests remove putative aberrant points (that is, negative intensity or positions out of the sliding window).
Particles with low intensity can potentially be misdetected owing to neighboring particles of stronger intensity within the window, mainly leading to background overestimation (Fig. 2b). We therefore carried out a deflation process. Starting from Pframe particles already detected in the current frame, we subtracted their respective estimated Gaussian peaks from the raw image Xin as follows:
We reiterated the detection and estimation processes on the resulting image (Fig. 2c) until we detected only noise, illustrating the efficiency of detection.
Optimized reconnection using past information
We selected an approach that avoids both the limitation of a local strategy and the lengthy computation time of a fully global strategy. For each trajectory, MTT searches either for an association with a particle detected in the subsequent frame or for possible blinking. To limit the factorial growth of the reconnection combinatorics, we restricted the set to trajectories and particles with sufficient reconnection probabilities. Conflicts between trajectories are then optimally resolved using statistical information from each trajectory.
For each trajectory Tk, a reconnection domain was defined by the area potentially explored by a particle after one time lag. To delineate the domain, we assume a Brownian motion with a maximum diffusion coefficient Dmax. This provides a limit for reconnection without making assumptions about the real diffusion mode of the particles. Thus, the domain is a disk, defined by a probability law, which is a bidimensional Gaussian with a s.d. rmax. A cutoff set at 3rmax ensures 99% confidence of finding particles (Supplementary Fig. 2 and Supplementary Video 1 online). In the most favorable case, the reconnection is unambiguous with only one peak within an isolated reconnection domain. However, when trajectories are crossing, conflicting overlap may occur between reconnection domains sharing at least one particle. Thus, MTT evaluates the probability of each possible combination attributing Tdomain trajectories to Pdomain particles, including possible blinking. In the simplest case where Tdomain = Pdomain (Fig. 3a), Tk will not blink (or will turn on again) at time t + 1. If Tdomain > Pdomain (Fig. 3b), one or more candidate trajectories blink at time t + 1. On the contrary, Tdomain < Pdomain signifies that new trajectories appear. In all cases, we consider that blinking, acting on all time scales, will occur with the same probability for all particles.
MTT then validates the most probable reconnection among all possible combinations (Fig. 3c). Starting from a set of particles and trajectories, MTT searches for the best combination based on a comparison of probabilities between each (Ti,Pj) association. The probability of each association is the product of the statistics computed for their respective positions, intensities and blinking. Here the descriptors of trajectories are not built only from the last frame. Instead, the use of several frames allows an evaluation of their probability laws. To handle possible variations over time, all descriptors are estimated within a sliding temporal window (t – wt, t) and readjusted at each iteration. A length wt fixed at 5 frames ensures a balance between the precision of parameter estimation (that is, Dlocal) and sensitivity to its temporal variation (that is, transient confinement). MTT assigns default typical statistical values to initiate reconnection (from frame 1 to wt) until past information is available. This does not strongly affect the accuracy of the results as trajectories are long enough (230 frames on average) compared to wt (5 frames). As reconnection requires information about peak positions, it directly benefits from the detection efficiency. Reciprocally, reconnection strengthens detection efficiency by validating potential new trajectories. Indeed, peaks are first detected with a PFApeak of 10−6, which avoids missing particles. Then, for un-reconnected, orphan peaks, another run of detection is performed with a more stringent PFAorphan of 10−7 to provide confidence in putative new trajectories as done for the initial frame.
The informative descriptors used for the reconnection test are defined as follows: (i) a local diffusion characterized with minimal assumption on the mode of diffusion and without excluding possible abrupt variations. The subsequent law adds two probabilities accounting for local and maximum (unconstrained) diffusions, respectively; (ii) an intensity law taking into account blinking (Supplementary Fig. 1); and (iii) a disappearance probability for trajectories ended owing to long-lasting blinking or particle leaving the image field, defined by an exponential decay τoff, with a threshold set at 3τoff to stop reconnection. Considering each physical phenomenon as independent, the global likelihood is the product of the probabilities associated with each phenomenon (Supplementary Note).
Evaluation of MTT sensitivity and reliability
We validated the MTT input parameters using simulated data to benchmark the performance of MTT at each step (detection, estimation, reconnection). As fluorescence signals are spread over a large range of intensities, a decibel scale was more appropriate. The fraction of detected peaks was close to 100% (probability of peak detection, PDpeak ≈1) for SNR higher than ∼20 dB (Fig. 4a). This threshold was consistent with the typical experimental values of 20–35 dB (Supplementary Fig. 1). Detection was also weakly sensitive to the peak density (Fig. 4a). The deflation increased the detection efficiency, substantially approaching the theoretical diffraction limit (Supplementary Note) even for densities as high as 10% (10 peaks per 10 × 10 pixels). The error on location (∼0.04 pixel = 6 nm) was optimal for all detected particles (down to ∼20 dB), reaching the theoretical detection limit (Fig. 4b and Supplementary Note). For comparison, a three-point estimator16 appeared inadequate, because its location accuracy was already degraded at 30 dB. Signal integration by the CCD pixels, not included in the model, had almost no effect on location precision (data not shown)17. We estimated the tracing accuracy for increasing peak density. Deflation substantially improved the accuracy, which was robust with regard to the underlying diffusion model (Fig. 4c). Notably, by reducing the number of critical issues and better handling the remaining ones, MTT is less likely to create artifactual reconnections to nearest neighbors. Use of past statistics is also helpful to adequately resolve crossing particle trajectories (Supplementary Fig. 2).
Detection of confinement
Membrane dynamics are expected to involve an interchange of Brownian diffusion and confinement in subdomains1,4. We therefore detected confined events on each trajectory according to the equations presented in references 18–22. We computed confinement level Lconf as:
where Dmax is the unconstrained diffusion coefficient (0.2 μm2/s for EGFR) and var(r) is the variance of the displacements within a sliding temporal window of length wconf (Supplementary Table 1). A length wconf of 4 frames is both short enough to detect transient confinement and long enough to ensure a reliable detection. Confinement can be statistically assessed when Lconf remains over the threshold Lmin for a sufficient time tmin (Fig. 5a).
Using simulated data, we aimed to identify threshold values that minimize false detections (low PFAconf) without decreasing the identification of true confinement (high PDconf). A robust method to objectively set the confinement thresholds is to perform a receiver operating characteristics analysis (Supplementary Fig. 3 online): the values maximizing the global score PDconf × (1 − PFAconf) were 5 for Lmin and 3 frames for tmin. Hence, PDconf can be kept high (0.78) and PFAconf reasonably low (0.025). Notably, as these values came from stable parts of the curves, they have little impact on the overall detection efficiency.
We evaluated confinement detection against various models: transient slowdown, permeable domains, anomalous diffusion or free diffusion (Supplementary Fig. 4 online). Our confinement detector searched for local disparities instead of global, large-scale variations as for anomalous diffusion. Consistent with this hypothesis, MTT detected slowdowns and domains better than anomalous behavior, which rendered results comparable to free diffusion.
Cartography of EGFR dynamics in live cells
We tested the principle of cartography by generating a snapshot of the membrane dynamics for the EGFR at the surface of live COS-7 cells. We projected up to 1,200 trajectories with ∼230 frames on average recorded over a short period of time (10 s, 300 frames/stack) onto the differential interference contrast image of the cell (Fig. 5b; pixel size 160 nm). In a static three-dimensional (3D) representation, the peak height equals the strength of confinement, whereas valleys relate to few or no confinement (Fig. 5b). For a more dynamic view, we investigated the confinement frame by frame over longer periods of time (Fig. 5c, and Supplementary Videos 2 and 3 online). This temporal analysis simultaneously revealed stable and evanescent confined areas. We also noticed rare linear motion (<2%), as expected for endocytic vesicles23.
We observed transitions between various regimes of diffusion, characterized by a wide range in duration (0.1–10 s), size (0.01– 1 μm) and strength (1–100) (Fig. 6a and Supplementary Fig. 3). These values corroborate the transient confinement zones observed using a similar confinement detection method18 for a glycosyl-phosphatidylinositol–anchored protein Thy1 and a glycosphingolipid, GM1. Compared to free-diffusion events, confined periods are characterized by smaller submicrometer sizes (70 ± 1 nm, n = 5 cells), although with similar subsecond duration (250 ± 15 ms). The fraction of confined time accesses the fraction αconf of confined receptors (0.40 ± 0.05). Similarly, the fraction of confined area was 0.08 ± 0.02. We next quantified the confinement diversity using statistic descriptors of confinement levels calculated from the values displayed in the movies. A dot plot of variance-to-mean ratio (VMR) versus mean revealed a broad distribution of both descriptors (Fig. 6b). Regimes continuously ranged from free diffusion (low mean) to high confinements (high mean), both of which were stable over time (low VMR). However, most events displayed intermediate confinement with high VMR, suggesting dynamic structures. We located these different regimes by mapping the descriptors in color-encoded maps (Fig. 6c). The dynamics of confinement structures can also be assessed by determining their stability over time. A straightforward evaluation is to quantify the revisit frequency of confined areas. Counting the number of confined events associated to each pixel revealed highly revisited areas (Fig. 6c). Considering variance, frequently revisited areas mostly correlated with intermediate confinement. We probed a representative population of EGFR (∼1%). Thus, a rate of 50 visits within a 36-s recording (1.4 visits/s) would correspond for the whole population to more than 100 visits/s. This confirms subsecond dwell time ranges (<10 ms) for the most dynamic events.
This representation of confinement relies on spatiotemporal characteristics, resulting from the density of targets and the overall exposure time (camera readout). Both parameters must balance experimental limitations and biological relevance. We therefore made measurements at increased acquisition speed (Supplementary Fig. 5 online). MTT remained efficient down to 5 ms/frame, with an acceptable SNR (∼25 dB), allowing valuable investigations.
Single EGFR tracking has previously been performed to study signal transduction10,24 and membrane dynamics25. The EGFR cartography built using MTT provides a global view of molecular dynamics. Notably, the statistics built from individual observations are in excellent agreement with SPT25. This demonstrates that, on top of providing a snapshot of the spatiotemporal dynamics of large cell membrane areas, each trajectory is accurately reconnected by MTT. Confinement owing to submembrane cytoskeletal fences6,26, lipid domains6,27 and direct19 or long-range28 protein interactions is a major factor of membrane organization1,4. Our measurements reveal a partial confinement into sub-micrometer domains, consistent with transient confinement zones18, reminiscent of a preferential partition in lipid-dependent domains29,30. Notably, the confinement profile presents a broad distribution of strength (Supplementary Fig. 3). This may reflect an underlying diversity and complexity of confining processes. Nevertheless, this apparent diversity may also arise from a large spread of affinities, leading to a large distribution of escape times21. Owing to our higher spatiotemporal resolution, the observations for EGFR on confinement sizes, dwell times and mean square displacement refine previous SPT observations25. We found transient confinement (∼250 ms) into small domains (<100 nm) (Fig. 6), which are consistent with local measurements obtained by FCS (F. Conchonaud & D.M., unpublished data). For confined events, the mean square displacement consistently displays a negative deviation1. The decrease in diffusion directly relates to the local viscosity. The biochemical nature of the underlying heterogeneities could derive from coexisting lipid-dependent and actin-mediated domains. EGFR transient confinement was as efficiently modeled by alternating fast and slow diffusion as by permeable domains, in agreement with either dynamic partition or corralling. Possible anomalous diffusion could be investigated by a generalized approach to detect putative obstacles at various length scales. This would further minimize our initial assumptions for the confinement detection as described for the detection, evaluation and reconnection steps.
This project is ultimately aimed at elucidating how the dynamic organization of the plasma membrane influences cellular responses. This requires improving performance related to the two major SPT limitations7: the acquisition speed (relates to SNR), and single-fluorophore discrimination (particle density) (Fig. 4). Fast SPT acquisition is critical for inherently fast movements like lipid diffusion27, rapid transient confinement26 and 3D motion such as virus trafficking31. MTT has proven to be sensitive at fairly low SNR, allowing working with reduced light intensity (limited photodamage), less bright dyes, or increased acquisition speed (Supplementary Fig. 5). Evaluating tracking performances is linked to the biological question. For instance, a minimal length of errorless trajectories is required to discriminate anomalous diffusion from confinement. The ultimate limitation will occur at the transition between individual labeling and uniform staining corresponding to a density of ∼10%, when fluorescent peaks start to strongly overlap (Supplementary Note). This theoretical limit can be tackled considering new photoactivation-based techniques (photoactivation localization microscopy (PALM), fluorescence PALM (FPALM) and stochastic optical reconstruction microscopy (STORM))32 offering a promising alternative for exhaustive and resolved imaging. With sptPALM33, substantial improvement in the rate of imaging (20 frames/s) allowed tracking of photoactivated molecules appearing during the course of hundreds of seconds. A diffusion map of membrane proteins has similarly been established based on the longest trajectories (>15 frames), which represent 3% of all molecules33. Using a more conventional setup, MTT provides a robust and complementary analytical tool for refined diffusion analysis. Additional developments, including multicolor imaging, will expand MTT to a new spectral dimension. Simultaneous in situ measurements of different signal transduction partners will enable investigation into spatiotemporal distributions and kinetic interactions. This new and sensitive approach will provide a detailed view of membrane dynamics, linking local characteristics and heterogeneities with physiological implications.
We analyzed the data by custom algorithms written with Octave (GNU) and Matlab (MathWorks). A detailed description of the MTT algorithm is available in the Supplementary Note. The source code of the MTT software is available online (http://www.ciml.univ-mrs.fr/lab/he-marguet.htm); academic and nonprofit use is free of charge.
Methods for Monte Carlo simulations, cell culture, Qdot staining, and equipment and settings are available in Supplementary Methods online.
Note: Supplementary information is available on the Nature Methods website.
We thank H.T. He, L. Leserman, P. Réfrégier, F. Conchonaud, S. Mailfert and M. Fallet for helpful discussions and suggestions, and E. Witty (AngloScribe) for language editing. This project was supported by grants from Institut National de la Santé et de la Recherche Médicale, Centre National de la Recherche Scientifique, Agence Nationale de la Recherche and European regional development fund.
MTT computation on experimental data. The left part is the raw image corresponding to the cell depicted in Fig. 5b and the right part illustrates the MTT process, corresponding to the 120 × 80 pixels area zoomed in on in Supplementary Fig. 2. Two circles (with pseudo-colors for clarity purpose) delimit the concentric reconnection domains associated to local and maximal diffusion of each particle, eventually increasing after blink (Fig. 3). Images were acquired as described, at 36 ms acquisition time (28 fps).
Spatial variations of EGFR confinement. Each frame of the video was independently computed as in Fig. 5c, from an acquisition stack of 1,000 frames (corresponding to 36 s) as a contour plot of the local confinement level. Contours were calculated by cubic interpolation through (x, y, Log(Lconf)) points using Matlab. Each pixel can account for up to tens of trajectories (Fig. 6c). In these cases, the mean value of Lconf was used. To generate a movie instead of a map, contours were generated frame by frame and recorded at 2× reduced speed (14 fps). Using high-density labeling, non-labeled regions corresponded to a reasonably low fraction of the membrane. They were either interpolated or left as gaps in the surface, according to their surrounding values. Temporal smoothing over 3 frames was performed to improve visual quality. Non-specific binding to the coverslip around the cell was discarded by computing the cell contour using the DIC image. Image size is 61.5 × 51.25 μm, at 100× magnification, using the same color-scale as in Fig. 5c.
3× zoom of Supplementary Video 2. This video was recorded at 2× reduced speed (14 fps). Image width is 20.5 μm.