Trajectory-Profile-Guided Single Molecule Tracking for Assignment of

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Trajectory-Profile-Guided Single Molecule Tracking for Assignment of One-Dimensional Diffusion Trajectories Kevin C. Robben, Khanh-Hoa Tran-Ba, Takashi Ito,* and Daniel A. Higgins* Department of Chemistry, Kansas State University, 213 CBC Building, Manhattan, Kansas 66506-0401, United States S Supporting Information *

ABSTRACT: A variety of algorithms exist for optical single molecule tracking in two and three dimensions. One general class of algorithms employs costfunctionals to link the individual fluorescent spots, produced by a molecule in sequential video frames, into trajectories. This method has also been used to track one-dimensional (1D) molecular motions for relatively low diffusion rates (i.e., D < 1 μm2/s). At high diffusion rates, the cost-functional approach often fails to accurately reproduce 1D trajectories, particularly when the molecules are closely spaced. In this paper, we present a new algorithm called trajectory-profile-guided (TPG) tracking that is designed specifically for 1D trajectories. TPG tracking involves an initial search for one-dimensionally aligned fluorescent spots (i.e., candidate molecules). Qualifying candidates are subsequently identified and linked into trajectories based on several criteria. We test the TPG algorithm’s accuracy and precision against cost-functional based tracking using both simulated and experimental video data. The results show that TPG tracking more accurately reproduces the actual 1D trajectories, particularly at higher diffusion rates. TPG tracking is also shown to produce longer trajectories and more accurate estimates of trajectory aspect ratios (i.e., their dimensionality), molecular diffusion coefficients, and order parameters for aligned 1D trajectories over a wide range of diffusion coefficients. he tracking of fluorescent single particles and single molecules in high-resolution optical microscopes is now being used to obtain unique insights into both the microscopic structure and mass transport dynamics of a variety of biological and technological materials. For example, single molecule tracking (SMT) has been applied in cell biology to study the diffusion of lipids and proteins within cells and on cellular membranes.1−4 SMT has also been used to study partitioning, diffusion, and adsorption in lipid films,5,6 in and on silica materials,7,8 within polymer gels9 and polymer films,10,11 and at solid−liquid interfaces.12 The majority of SMT studies reported to date emphasize understanding the two- and three-dimensional (2D and 3D) motions of probe molecules. An exciting new direction that has just recently emerged13,14 is the application of SMT to investigations of confined molecular motions in both biological15,16 and technological17,18 1D nanomaterials. Broad interest in 1D nanomaterials stems partly from their ubiquitous applications in drug delivery,16 chemical separations,19−23 chemical sensing,21,23−25 catalysts,26 fuel cells,27 and batteries.28 To date, anisotropic diffusion has been observed and characterized by SMT in lipid tubules,15,16 liquid crystals,29 metal organic frameworks,30 surfactanttemplated mesoporous silica,17,18 and in phase-separated block copolymers.31,32 The aforementioned applications of 1D nanomaterials rely upon their ability to confine and guide the motions of reagents or analytes along a predefined direction. SMT investigations of 1D diffusion in these materials promise to provide new insights into their nanoscale morphology and mass transport properties, allowing them to be better optimized for their intended applications. Accurate

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© 2014 American Chemical Society

representations of the single molecule motions in time and space must be obtained to achieve this goal. The first step in SMT involves the recording of videos depicting the locations and motions of the individual molecules.33 These molecules appear in each video frame as round fluorescent spots with Gaussian intensity profiles. Image analysis software is used to identify the spots and to determine their locations, usually by fitting to Gaussian functions.34,35 Low signal-to-noise ratios and frame-to-frame variations in single molecule emission often limit this process. Many approaches have been used to overcome such difficulties; these include dynamic multiple-target tracing,36 image filtering,37 and location estimation and refinement.38 The precision of single molecule localization depends on the signal level, background counts, and the image pixel size.35,39 However, subdiffraction and subpixel precision is routinely obtained.40 After the individual molecules have been located, they are subsequently linked into trajectories. Early forms of automated spot linking involved connecting frame-consecutive particles that were closest to each other.15,41 Such methods have since been enhanced for more efficient linking,42 in many cases by implementation of cost-functionals in the linking process.38,43 Multiple-hypothesis tracking (MHT)44,45 has also been employed. MHT is different from cost-functional algorithms in that it globally considers all possible links and choses the Received: August 1, 2014 Accepted: October 10, 2014 Published: October 10, 2014 10820

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videos was kept relatively high, ∼2.7 times that of the experimental data included with this report (see the Supporting Information for these videos). However, the signal-to-noise ratio was even higher in recent experimental results from our group.52,53 The molecules in the simulated videos also moved by jumping between frames. These two conditions enhanced ImageJ particle detection, providing better circumstances for comparison of the cost-functional and TPG linking methods while yielding videos that still closely approximate experimental ones obtained under common conditions. Trajectory-Profile-Guided (TPG) Tracking. The TPG method works equally well for tracking 1D motions of individual molecules and particles. Therefore, the more generic “single particle” terminology is employed in much of the discussion below. Definitions. Here, the particle pool is defined as the collection of particles detected in a finite sequence of video frames that have not yet been linked into a trajectory. Candidate particles represent a subset of the particle pool whose positions fall within a small-threshold orthogonal distance, dε, of an arbitrarily chosen line (i.e., a possible 1D diffusion pathway). Each of these lines is called a candidate pathway. Note that a candidate pathway may or may not describe the coordinates over which a given particle diffuses in 1D. Linked particles represent the collection of particle positions that have been grouped together to make a framecontinuous trajectory. A trajectory domain is a finite segment along a candidate pathway where a particle trajectory profile apparently exists. One or more trajectory domains may exist along a candidate pathway. TPG Tracking Method. Prior to linking the particles by the TPG algorithm, a list of all particle positions and their frame times was obtained using the aforementioned ImageJ plugin.50 For simplicity, the TPG program was written to rescale the coordinates in each video to 128 × 128 pixels, while the time unit was taken as 1 frame. This scaling has no effect on the accuracy of particle linking and the results are easily rescaled to the original coordinates and frame time after linking is complete. The TPG algorithm repeats the procedure outlined below until no new trajectories are found in a user-specified number of repetitions. The algorithm is depicted schematically as a flowchart in Figure S1 in the Supporting Information. The number of nonproductive repetitions required for termination was set to be equivalent to 6 times the number of frames in the video being analyzed. The first step in the TPG procedure is to gather all particle locations from a continuous subseries of video frames (typically 50 or 100 frames) selected at random from the full video; the reasoning for use of a subseries is presented below. Figure 1A depicts the particle positions (i.e., the particle pool within the subseries) from a simulated video. Next, a candidate pathway (a line) is generated as defined in eq 1.

option with the fewest conflicts. An improved form of MHT that accounts for the reliability of molecule positioning has been reported by Wöll et al.37 Another linking method by Yoon et al. applies Bayesian inference for improved tracking of blinking and low intensity molecules.46 A recent review by Meijering et al.47 provides more details on these and other tracking methods, along with relevant comparisons of their advantages and limitations. While some of the aforementioned methods have been applied in studies of diffusion in 1D nanostructured materials,29,32,48,49 they are not well suited to tracking strongly anisotropic motions. Most 1D SMT studies to date have employed cost-functional methods with varying success. Problems with these methods commonly arise in the production of 1D trajectories in dense-molecule scenarios. In this paper, we present a linking algorithm that is specifically designed for 1D trajectories. This algorithm employs an initial 1D search to find groupings of well-aligned fluorescent spots in consecutive video frames and subsequently creates trajectory profiles to use as a guide for spot linking. We have termed this method trajectory-profile-guided (TPG) tracking. The TPG algorithm is described herein and applied to both simulated and experimental 1D trajectories. The results are compared to those from a well-known cost-functional algorithm provided as a plugin to the ImageJ software package38,50 that has been widely utilized in a number of previous 1D single molecule tracking studies.29,32,48,49,51 The discussion includes a presentation of the advantages and limitations of the TPG algorithm in comparison to the cost-functional method.

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EXPERIMENTAL SECTION Experimental Procedures. A dye-doped cylinder-forming polystyrene-poly(ethylene oxide) diblock copolymer (PS-bPEO) film32 was used to obtain experimental SMT data.48 Detailed procedures for sample preparation and SMT measurements using a wide-field fluorescence microscope are given in the Supporting Information. Simulation Procedures. Video simulations of Brownian diffusing molecules were obtained using a program written in house in the LabView environment. Each simulation began by randomly seeding a 128 pixel × 128 pixel frame, representing a 15 μm × 15 μm region, with 40 molecules. Each molecule was also assigned a trajectory angle from a random distribution having a width of ∼21°, centered at 5° above horizontal, in the image plane. Each molecule was given a Gaussian intensity profile of 200 nm width. Background noise and particle bleaching affects were added to mimic experimental data. Each video was 500 frames in length. Exact molecular locations and trajectories were extracted in parallel to the simulated videos. The former were employed as known “standards” to assess the efficacy of the SMT routines. A series of simulations were performed using different diffusion coefficients, as defined below. Spot Identification and Linking Procedures. The locations of the molecular spots in both the experimental and simulated videos were determined using an ImageJ plugin.50 Linking of the spots was performed by two different methods. The first employed the same ImageJ plugin, in which costfunctional tracking was used.38 The original routine was modified to remove the dependence on spot intensity. The second was based on the TPG algorithm and was executed as a C program. For the purpose of examining this algorithm strictly as a linking method, the signal-to-noise ratio of the simulated

y(x) = tan θ(x − x0) + y0

(1)

A random number generator was used to produce initial values for θ, x0, and y0, where these parameters represent the angle of the line in the image plane and its central position along the x and y axes, respectively. The candidate pathway then becomes a tool to search for particle positions falling along a line within the subseries of frames. The candidate pathway is subsequently fit to the available particle data by adjusting θ, x0 and y0 to 10821

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trajectory domains (colored red, green and blue). The left most particles (red) appear intermittently on the plot, indicating that this trajectory runs across (rather than along) the candidate pathway (see Figure 1A,B). It is the apparent continuous appearance of the middle (green) and right (blue) particles that serves as good evidence that each follows this candidate pathway, which is necessary for trajectory linking. Figure 1D depicts the integral distribution (black line) of particle positions for the data shown in Figure 1C. This distribution is broadened and smoothed by convolution of the particle positions (originally delta functions) with a Gaussian that accounts for diffusive broadening of the particle positions (see the Supporting Information). The three peaks shown in Figure 1D are individual trajectory profiles, which comprise the distribution. Also shown in Figure 1D are the first and second derivatives of the distribution determined by Savitzky−Golay filters.55,56 The zeros of the first derivative identify the beginning, middle and end of a trajectory domain along the candidate pathway. The second derivative is used to identify which zeros represent the peaks and ends of each trajectory domain. Next, the candidate particles are assigned to individual trajectory domains based on their positions between the end points of each, as approximately represented by the shaded regions in Figure 1D. The process of linking runs in ascending order of frames throughout each trajectory domain. In order for two particles to be linked together, two criteria must be met: (1) The two particles must be assigned to the same trajectory domain, and (2) a particle in frame τ may only be linked to a single particle in frame τ ± 1. The first criterion is necessary to avoid linking across two different trajectory domains. The second ensures that linking only occurs between consecutive frames. Expanded implementation of this method could allow for missing frames so that blinking particles may also be linked.42 In the rare case that two or more particles are present within the same frame and trajectory domain, a selection between the different choices must be made (see the Supporting Information). As it is difficult to retrieve meaningful mass transport data from trajectories 0.02, the accuracy of TPG tracking appears to become steadily worse, while the accuracy of cost-functional tracking remains largely constant. The apparent constant accuracy of costfunctional linking is explained by the biased nature of the method caused by its reliance on the maximum particle displacement parameter during linking. In the present costfunctional analyses, the particle displacement parameter used was twice the expected mean particle displacement between frames for each simulation. The results for DARs ≤ 0.01 show that TPG tracking yields more accurate D values even for slow diffusion, while cost-functional methods continue to underestimate D due to the aforementioned bias.

Figure 4. Plots of (A) trajectory aspect ratios, (B) mean diffusion coefficients, and (C) relative error in the mean diffusion coefficients as a function of DAR. Black circles depict the values measured from the known trajectories, blue triangles those obtained from TPG tracking, and red squares those from cost-functional tracking. The error bars in panel B depict the 90% confidence intervals.

In a series of publications, we have employed SMT methods to quantify the in-plane orientation and organization of 10824

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cylindrical nanostructures that support 1D diffusion.29,32,48,49,51 Such assessments require the accurate measurement of the inplane orientations of 1D trajectories. Therefore, trajectory angle accuracy also represents an important parameter upon which to judge the efficacy of TPG tracking. Figure 5 shows trajectory

Figure 6. Trajectory plots for SRB dye in a PS-b-PEO film retrieved using (top row) TPG tracking and (bottom row) cost-functional tracking. The three TPG trajectory plots were obtained by running the analysis on the same data set three times, using the same parameters. The cost-functional analysis were performed using displacement values of 0.53, 0.66, and 0.79 μm, respectively, from left to right. The frame dimensions are 9.38 μm × 9.38 μm.

three sets of trajectories were obtained by repeatedly running the TPG routine on the same set of particle positions using the same parameters. As is readily apparent from these plots, a few trajectories are missing from certain runs. This difficulty is easily overcome by running the TPG routine three or more times on each set of data and then compiling the trajectories that are identified two or more times to form a complete set. Cost-functional methods also miss trajectories but for different reasons. For example, because the diffusion coefficient for a dye molecule in a real sample is initially unknown, a range of displacement parameters must be employed to obtain an initial set of trajectories. As shown in Figure 6 (bottom row), different displacement parameters lead to detection of different numbers of trajectories (≥13 frames in length). The input parameters may then be refined in an iterative process to obtain trajectories that best fit the observed particle motions, as judged by an “expert human”. Note that unlike the simulated data above, the “correct” trajectories can never be known with complete certainty for real experimental data. Table 1 provides quantitative results compiled from analyses of the trajectories in Figure 6. Most of the parameters vary little, if at all, between the separate TPG runs. The main drawback of TPG tracking comes in determining the diffusion coefficient when a few trajectories are missed in each run (see Figure 6, top row). These missing trajectories lead to variations in the D value, giving a standard relative error of 9.2%. Again, such errors can be avoided by compiling the trajectories found in a series of replicate runs. The characteristics of cost-functional linking for the same experimental data are consistent with those of the simulated videos. Using a mean diffusion coefficient of 0.82 μm2/s determined by TPG tracking and a video frame width of 9.38 μm, we estimate the DAR to be 0.028. On the basis of Figure 4A, the cost-functional routine should pick up fewer 1D trajectories and generate more 2D-like trajectories in this range. This prediction is confirmed by the larger σδ values found for cost-functional tracking (see Table 1). In further support of this assessment, the σδ values were observed to increase with increasing displacement values. Cost-functional linking also

Figure 5. Histograms of trajectory angles for (A) known, (B) TPG, and (C) cost-functional trajectories compiled from simulated data at the higher two DARs.

angle histograms obtained from simulated videos generated at two higher DARs (0.043, 0.064). These histograms show that the TPG method most closely reproduces the angle distribution of the known trajectories, while the histogram from the costfunctional analysis is broader. A common measure of 1D trajectory alignment is the order parameter, ⟨P⟩ = 2⟨cos2(Δθ)⟩ − 1.48 Here, Δθ = θ̅ − θ, θ is the angle of a given trajectory and θ̅ is the average orientation of the entire population. The value of ⟨P⟩ usually ranges from zero, for completely disordered, to one, for perfectly ordered populations. The order parameter for the known trajectories in Figure 5A was 0.73. The value obtained from the TPGgenerated trajectories was 0.72, demonstrating a high degree of accuracy. In contrast, the cost-functional trajectories yielded ⟨P⟩ = 0.50. The depressed order parameter in the latter comes from several erroneously large trajectory angles. These are typically found at higher DARs where cost-functional linking fails to properly reproduce the 1D trajectories. At smaller DARs (25%, compared to the TPG results, in the first two cases and by ∼13% for the largest displacement value. However, the latter value may have come at a cost of overestimating the diffusion coefficient and the localization precision, when compared to TPG linking. TPG tracking does have a few limitations beyond the missed trajectories mentioned above. Most importantly, the algorithm is currently limited to linear 1D trajectories and does not describe the curvilinear trajectories sometimes encountered in aligned mesoporous silica48 and block copolymer films.51 Adapting this method to curvilinear trajectories can be done by assuming a general parametrized equation to describe these trajectories (i.e., replacing eqs 1 and 2). Additionally, the present implementation of the TPG method requires framecontinuous trajectories during linking, but allowance for “missing” particles can be easily added to the routine. The TPG method may also fail to link particles that make relatively large lateral jumps between adjacent 1D pathways. These trajectories will be broken up into shorter segments. However, particles that jump between pathways separated by less that the candidate pathway width, dε, will still be linked into continuous trajectories.

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences of the U.S. Department of Energy (Grant DE-FG0212ER16095) for financial support of this work.

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REFERENCES

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CONCLUSION In conclusion, we have presented a new method for 1D single particle tracking. This algorithm is distinct from alternative methods in that it uses trajectory profiles to guide the linking process. The results show that TPG tracking succeeded in finding legitimate 1D trajectories across a wide range of DARs, whereas the cost-functional method to which it was compared failed to do so at high DARs. On the basis of the analysis of both the simulated and experimental videos, we showed that TPG tracking is most effective for DARs ≤ 0.043 and that the TPG routine found longer trajectories on average and was more accurate in determining diffusion coefficients and order parameters for aligned 1D trajectories. We also discussed the limitations of TPG tracking when using random search methods to find candidate pathways, which occasionally led to missed trajectories and some variation in the measured diffusion coefficients. However, the TPG method avoids the bias in diffusion coefficients that can arise in cost-functional linking. TPG based tracking methods are certain to find utility in the analysis of strongly anisotropic diffusive motions occurring in 1D nanostructured materials.

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AUTHOR INFORMATION

ASSOCIATED CONTENT

* Supporting Information S

Additional information on sample preparation, wide-field imaging, the TPG program, data analysis, and representative videos. This material is available free of charge via the Internet at http://pubs.acs.org. 10826

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