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Temporal analysis of relative distances (TARDIS) is a robust, parameter-free alternative to single-particle tracking

Nature Methods (2024)Cite this article

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Abstract

In single-particle tracking, individual particles are localized and tracked over time to probe their diffusion and molecular interactions. Temporal crossing of trajectories, blinking particles, and false-positive localizations present computational challenges that have remained difficult to overcome. Here we introduce a robust, parameter-free alternative to single-particle tracking: temporal analysis of relative distances (TARDIS). In TARDIS, an all-to-all distance analysis between localizations is performed with increasing temporal shifts. These pairwise distances represent either intraparticle distances originating from the same particle, or interparticle distances originating from unrelated particles, and are fitted analytically to obtain quantitative measures on particle dynamics. We showcase that TARDIS outperforms tracking algorithms, benchmarked on simulated and experimental data of varying complexity. We further show that TARDIS performs accurately in complex conditions characterized by high particle density, strong emitter blinking or false-positive localizations, and is in fact limited by the capabilities of localization algorithms. TARDIS’ robustness enables fivefold shorter measurements without loss of information.

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Fig. 1: Overview of the TARDIS algorithm and comparison to tracking algorithms.
Fig. 2: Performance of TARDIS in complex conditions.
Fig. 3: TARDIS enables high-complexity single-particle mobility experiments of intricate diffusive behavior.

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Data availability

All data underlying this study is available at ref. 60. Source data are provided with this paper.

Code availability

The custom TARDIS software used in this manuscript is provided as supplementary data and can be accessed at ref. 39.

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Acknowledgements

This work was financially supported by funding from a VLAG PhD-fellowship (J.H.), start-up funds at Carnegie Mellon University (B.T., K.J.A.M. and U.E.), the NSF AI Institute: Physics of the Future (NSF PHY- 2020295) (U.E.), start-up funds at Bonn University (B.T., K.J.A.M. and U.E.), an Argelander Starter Kit at the University of Bonn (K.J.A.M.) and the Alexander von Humboldt Foundation (K.J.A.M.). We acknowledge the valuable input from group meetings from all members in the U.E. and J.H. laboratories. We thank M. Deserno (Carnegie Mellon University, Pittsburgh, USA) for helpful discussions on TARDIS’ mathematical foundation.

Author information

Authors and Affiliations

  1. Institute for Microbiology and Biotechnology, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany

    Koen J. A. Martens, Bartosz Turkowyd & Ulrike Endesfelder

  2. Department of Physics, Carnegie Mellon University, Pittsburgh, PA, USA

    Koen J. A. Martens, Bartosz Turkowyd & Ulrike Endesfelder

  3. Laboratory of Biophysics, Wageningen University and Research, Wageningen, the Netherlands

    Koen J. A. Martens & Johannes Hohlbein

  4. Microspectroscopy Research Facility, Wageningen University and Research, Wageningen, the Netherlands

    Johannes Hohlbein

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Contributions

Conceptualization: K.J.A.M. Data curation: K.J.A.M. Formal analysis: K.J.A.M. Funding acquisition: K.J.A.M., J.H. and U.E. Investigation: K.J.A.M. and B.T. Methodology: K.J.A.M., B.T., J.H. and U.E. Project administration: K.J.A.M., J.H. and U.E. Software: K.J.A.M. Supervision: K.J.A.M., J.H. and U.E. Visualization: K.J.A.M. Writing—original draft: K.J.A.M. Writing—review and editing: all authors.

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Correspondence to Koen J. A. Martens.

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Nature Methods thanks J. Christof Gebhardt and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Editor: Rita Strack, in collaboration with the Nature Methods team. Peer reviewer reports are available.

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Extended data

Extended Data Fig. 1 Performance of TARDIS and spt tracking methods for a single diffusive population and two diffusive populations at increasing complexity, visualised on a linear x-axis.

Performance of TARDIS is compared to the blind tracking algorithms uTrack-inspired PMMS (piecewise-stationary motion model and iterative smoothing)40,42, TrackMate40,41 and Nearest neighbour analysis, and to the prior-informed methods swift (Endesfelder et al., manuscript in prep.), and Multiple-Hypothesis Tracking (MHT)25 (MHT at the most complex dataset did not run to completion). This data is also presented in Figs. 1c and 2c. (c) Bhattacharyya distance of the distributions in (a) and (b) compared to the ground truth (GT) jump distance distribution, calculated as the negative natural logarithm of the sum of the square root of the product of the distribution value of a method and that of the jump distance ground truth.

Source data

Extended Data Fig. 2 The full dataset as presented Fig. 2a, analysed via TARDIS (blue), TARDIS-JD-extraction (light-blue), TrackMate-LAP40,41 (light green) and nearest-neighbour tracking (dark green).

(a) and (b) represent the same datasets, but visualised on a logarithmic (a) or linear (b) x-axis. Note the changing jump distance x-axis scaling in (b). The TARDIS fit data is also presented in Extended Data Fig. 3a.

Source data

Extended Data Fig. 3 Detailed information on diffusivity and bleach time obtained from TARDIS fitting.

(a) Boxplots showing the obtained diffusion coefficients of datasets presented in Fig. 2a, and (b) the obtained bleaching times of datasets presented in Fig. 2a, showing no bias in either over the complexity range. TARDIS is repeated 10 times on 20.000 simulated localizations for each condition. (c) Individual fit information of data presented in Fig. 2b. For every condition, TARDIS is repeated 10 times on 20.000 simulated localizations with random start positions in TARDIS. The obtained diffusion coefficients are visualised (scatter points represent individual measurements). Note the changing y-axis at 90% removed true positives in (c). Abbreviations used: TP: True Positives, FP: False Positives, fr: frame, locs: localizations. All boxplots show the median as the central mark, with the 25th and 75th percentile as lower and upper edges. Whiskers extend to non-outlier extreme points, and outlier points are plotted as plusses.

Source data

Extended Data Fig. 4 Detailed information on TARDIS fitting results of bleach time of single diffusive population with added noise and blinking chance.

Fit information of bleach characteristics corresponding to the data presented in Fig. 2a,b (main manuscript). For every condition, 10 repetitions were analysed with random start positions in TARDIS. (a,b) RMSE of the diffusion coefficient (a) and bleach half-time (b) for all conditions presented in Fig. 2a (c) The found diffusion coefficient presented in Fig. 2b is visualised (scatter points represent individual measurements). Note the changing y-axis at 90% removed true positives. TARDIS is repeated 10 times on 20.000 simulated localizations for each condition. All boxplots show the median as the central mark, with the 25th and 75th percentile as lower and upper edges. Whiskers extend to non-outlier extreme points, and outlier points are plotted as plusses. Abbreviations used: TP: True Positives, FP: False Positives, fr: frame, locs: localizations.

Source data

Extended Data Fig. 5 Effect of the fraction of inter-particle linkages on diffusion coefficient accuracy.

Analysis of 95% confidence interval (a, b), and fitted diffusion coefficient (c) as a measure of the inter-particle fraction. Dotted lines in (a) are added for clarity. The underlying analysed data is the same as shown in Fig. 2b. Reasons for increased fraction of inter-particle linkages are clarified via marker type (TP removal), marker colour (TP localization density), and marker darkness (FP introduction). Note that a and c have non-linear x-axis (a and b contain the same information, but with different x-axes).

Source data

Extended Data Fig. 6 Diffusion analysis of fluorescent beads at varying frame times.

The same information as presented in Fig. 2d, but with additional frame times in between those shown in the main manuscript. The excitation time on every frame is kept constant. The small decrease in obtained diffusion coefficient as a function of frame time is explained by particles having a higher chance to move outside the field-of-view with larger jump distances.

Source data

Extended Data Fig. 7 TARDIS-JD extraction from data of Chenouard et al.

Tracking data from Chenouard et al.23 has been deteriorated (removing 54% of localizations), analysed via the ‘extract JD’-function of TARDIS, and compared to the ground-truth (GT) data. Four different conditions are analysed: MICROTUBULE, RECEPTOR, VESICLE, and VIRUS, corresponding to [constant velocity], [tethered motion, switching, any direction], [Brownian motion, any direction], and [same direction dynamics, switching between Brownian and linear] dynamics, respectively. Densities are indicated in subplot titles, while the field-of-view is ∼50-by-50 µAU in size. In all scenarios, TARDIS accurately extracts the ground-truth data, and the level of noise is decreasing with decreasing localization removal. The following TARDIS settings were used: Δt bins of 1–3; maximum jump distance of 1e-05 AU; background frames starting at frame-shift of 35, using in total 50 frames; 300 BG bins starting at 3.5e-06 AU.

Source data

Extended Data Fig. 8 E. coli RNA polymerase jump distance analysis after nearest-neighbour tracking.

Jump distance analysis of RNA polymerase in E. coli with (red) and without (black) rifampicin, (the same data presented in Fig. 3b), via nearest-neighbour tracking. Notice the changing peak position and abundance as a function of localization density. The data is shown in a linear X-scale (left) and logarithmic X-scale (right).

Source data

Extended Data Fig. 9 Detailed information on kinetically state-changing particles.

Individual fit information of data presented in Fig. 3c, along with individual fit information for differing binding/unbinding kinetics (titles indicate kon / koff / Diffusion coefficient). For every condition 10.000 ‘true positive’ trajectories were simulated, and 10 repetitions of this were analysed with random start positions in TARDIS, and compared to analysing the same data with anaDDA on the ground-truth trajectory data. TP, Sp and me indicate true positive, spurious, and membrane localizations, respectively. All boxplots show the median as the central mark, with the 25th and 75th percentile as lower and upper edges. Whiskers extend to non-outlier extreme points, and outlier points are plotted as plusses.

Source data

Extended Data Fig. 10 Accuracy of software by Wolf et al44.

The same data analysed by TARDIS in Fig. 2b (a) and Extended Data Fig. 3b (b), analysed via the DANAE software44, which effectively only performs TARDIS-JD extraction. This is then fitted with a diffusive model afterwards. The accuracy, especially at high complexity scenarios, is worse compared to TARDIS. Additionally, DANAE shows bias towards too high values (right), which is caused by imperfect inter-particle distance distribution subtraction. DANAE is repeated 10 times on 20.000 simulated localizations for each condition. All boxplots show the median as the central mark, with the 25th and 75th percentile as lower and upper edges. Whiskers extend to non-outlier extreme points, and outlier points are plotted as plusses.

Source data

Supplementary information

Supplementary Information

Supplementary Notes 1–10 and Figs. 1–15.

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Peer Review File

Supplementary Software 1

TARDIS version 1.20, including Getting Started instructions, a manual, a program installation executable and MATLAB code.

Source data

All source data

Unprocessed video (including localization lists) and simulation data. Unprocessed TARDIS results data.

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Martens, K.J.A., Turkowyd, B., Hohlbein, J. et al. Temporal analysis of relative distances (TARDIS) is a robust, parameter-free alternative to single-particle tracking. Nat Methods (2024). https://doi.org/10.1038/s41592-023-02149-7

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