Principle of our method.

a: The track-generation steps with our motion model shown here for directed motion. Each time step is decomposed into sub-steps, namely diffusion and anomalous motion (either directed or confining) with ri the real positions, zi an intermediate position, hi the anomalous variable, and the generation of observed (measured) positions, ci. The variables ci − ri and ri+1 − ri follow Gaussian distributions with mean 0 and standard deviations σ and d, respectively. The sub-step from zi to ri+1 is deterministic and the anomalous variable hi can also evolve with a standard deviation q. b: Graph representation of aTrack showing the motion model (left), analytical integration (middle), and outputs (right). To compute the track probability, we integrate over the hidden variables. This results in an analytical recurrence formula that is used to determine the type of motion and to estimate the parameters of the motion. c: Examples of tracks that can be produced with our motion model.

Determining the motion type using a likelihood-ratio test.

a-b: Probability distributions of the difference between the log of the maximum likelihood of the alternative hypothesis (either confinement Lc or diffusion Ld) and the null hypothesis (Brownian diffusion Lb) for single tracks (10,000 tracks). Confinement factor l = 0.25 and velocity v = 0.02 µm·Δt−1. a: Effect of the number of time points in a track on its log difference (Lc − Lb for confined tracks) and (Ld − Lb for directed tracks). b: The ability to distinguish confinement and directed motion from diffusion as a function of the confinement factor and particle velocity, respectively. c: heatmaps of the likelihood ratios lb/lc (confined) or lb/ld (diffusive) varying both the anomalous diffusion parameter and the track length. Mean of 10,000 tracks. a-c: When not stated otherwise, the track parameters were as following. Localization error σ = 0.02 µm. Confined tracks: diffusion length d = 0.1 µm. Directed tracks: d = 0.0 µm (no diffusion), constant speed and orientation.

Characterizing confinement with aTrack.

a-d: Confinement of tracks with a fixed potential well. a: Examples of simulated tracks with different confinement factors. b: Histograms of the estimated parameters for individual tracks of 200 time points varying the number of time points in tracks. c-d: Heatmaps of the mean estimated confinement factor and confinement radius depending on the track length and the confinement factor or radius respectively. D : Confinement of tracks with a moving potential well (Brownian motion). Left: simulated tracks with different diffusion length of the potential well. Right: histograms of the estimated diffusion length of the the potential well and confinement varying the actual diffusion length of the potential well. Confinement factor = 0.1. a-d: 10,000 tracks per condition. d = 0.1 µm, Localization error σ = 0.02 µm. See Fig. S3 for complementary results.

Characterizing directed motion with aTrack.

a-c: Tracks with linear motion (constant speed and orientation). a: Examples of simulated tracks in directed motion. b: Histograms of the estimated velocity of individual tracks of 30 time points. 10,000 tracks per histogram. True parameters d = 0. µm, Localization error σ = 0.02 µm (fixed). The next panels use the same parameters unless specified otherwise. c: Heatmap of the relative biases on the estimated velocity . d-f: Tracks with constant speed but changing orientation. d: Simulated directed tracks with rotational diffusion. Here, the rotational diffusion angle coefficient is defined as the standard deviation of the change of orientation at each time step (analogous to the diffusion length). v = 0.02 µm·Δt−1. e: Heatmap of the error on the rotational diffusion angle for a range of velocities and rotational diffusion angles. Tracks of 200 time points. f: Distributions of the estimated rotational diffusion angle for a range of rotational diffusion angles. Tracks of 200 time points with v = 0.1 µm·Δt−1. g: Tracks simultaneously undergoing both linear motion and diffusion with varying levels of diffusion. Heatmaps of the likelihood ratio, bias on the diffusion length in µm, and estimated velocity depending on the number of time points per track and on the diffusion length d. a-g: Where not stated otherwise, the track parameters were as following. Localization error σ = 0.02 µm, d = 0.0 µm, velocity v = 0.1 µm·Δt−1, constant speed and orientation.

Characterizing populations of multiple states.

Analysis of tracks with 5 sub-populations of set diffusion length d, confinement factor l, velocity v for directed tracks, and anomalous change parameter q (diffusion length of the potential well for confined tracks and changes of speed for directed tracks). Tracks are 300 time point long. a: Track examples from each of the 5 states with the corresponding state parameters. b: Estimated parameters for the 5 states (using a 5-state model). c: Log likelihood of the model depending on the number of states assumed by the model. The log likelihood was normalized by the number of tracks, offset by the log likelihood assuming 10 states.

Model robustness with other motion types.

a-b: Example tracks and corresponding distributions used to determine the type of motion for aTrack and Randi (41). aTrack uses the difference between the likelihood assuming super-diffusion and the likelihood assuming sub-diffusion (bottom-left). To classify tracks using Randi, we used the estimated anomalous diffusion coefficient. The accuracy is the fraction of correctly labeled tracks in a data set composed of 5,000 sub-diffusive or superdiffusive tracks and 5,000 Brownian tracks. Classifications were done using the thresholds that best divide the distributions. a: Analysis of tracks with 200 time steps following fractional Brownian motion with anomalous coefficients of 0.5 (subdiffusive), 1 (diffusive), and 1.5 (superdiffusive). b: Analysis of tracks with 200 time steps following our motion model. Confined tracks: diffusion length d = 0.1 µm, localization error σ = 0.02 µm, confinement force l = 0.2, fixed potential well. Brownian tracks: d = 0.1 µm, σ = 0.02 µm. tracks in both directed and diffusive motion: d = 0.1 µm, σ = 0.02 µm, directional velocity v = 0.1 µm·Δt−1. Directed tracks: d = 0. µm, σ = 0.02 µm, v = 0.1 µm·Δt−1, angular diffusion coefficient 0.1 Rad2·s−1. c: Analyzing tacks confined by hard boundaries using aTrack. A simulated track with 200 time points diffusing on disks of different sizes. Top panel: Log likelihood difference Lc − LB and fraction of significantly confined tracks (likelihood ratio lB/lc < 0.05) depending on the confinement radius. Middle panel: Estimated confinement depending on the true confinement radius. Bottom: estimated confinement radius depending on the track length. Blue areas: standard deviations of the estimates.

Experimental demonstrations.

a-c: Analysis of spindle pole body (SPB) tracks. Each track has 200 time points. a: Illustration showing the role of the SPB in the cell cycle (MT: microtubule). During mitosis, the elongation of the interpolar microtubules induces a directed movement of the SPB that divides the nucleus. b: Fraction of tracks significantly directed or confined with a confidence interval of 95% according to our likelihood ratio test on single tracks. Mean and standard deviation of 3 biological replicates of at least 400 tracks. The 95% confidence interval implies that we expect 5% of false positives in a fully Brownian population. c: Example SPB tracks and state labeling from our individual-track analysis. Red tracks are significantly directed (95% confidence interval), and blue tracks are non-significantly directed. d-e: Analysis of nanoparticle (NP) tracks in the presence of motile bacteria (50 time points per track), where some NPs adhere to cells. d: NP tracks colored according to their state of motion classification using aTrack’s single-track statistical test. e: Maximum likelihood (per track) of the population of tracks depending on the number of states (standardized by the likelihood assuming 10 states). f-g: Analysis of tracks for 1 µm beads trapped using optical tweezers. The data set is composed of 100 tracks of 100 time points. f: Track examples. g: Log likelihood difference (Ldirected − Lconfined).

Distributions of the log likelihood differences (Lc − Lb and Ld − Lb) for tracks in Brownian motion (column 1), confined motion (column 1), or linear motion (column 3) using the confinement motion test (row 1) or the directed motion test (row 2) for tracks with different number of time points. 10,000 tracks per distribution. The simulated track parameters were as following: localization error σ = 0.02 µm; confined tracks: diffusion length per step d = 0.1 µm, confinement factor l = 0.25; linear tracks: d = 0.0 µm, velocity v = 0.02 µm·Δt−1, constant speed and orientation.

Distributions of the likelihood ratios lb/lc and lb/ld corresponding to Fig. S1. As expected, the distributions are skewed toward 0 only when the proper test is applied.

a: Histograms of the estimated diffusion length per step of the potential well depending the confinement factor corresponding to Fig 3b. b-c: Heatmaps of the estimated diffusion lengths per step of the potential well (b) and of the estimated diffusion lengths or the particle depending on the track length and on the confinement factor corresponding to Fig 3c. d: Distributions of the estimated confinement factor depending on the track length (same conditions than Fig 3c with a confinement factor of 0.25). e: Heatmap of the relative biases on the estimated confinement factors depending on the confinement radius and track length corresponding to Fig 3d. f: Distributions of the estimated confinement radius depending on the true confinement radius (same conditions as in Fig 3d with tracks of 150 time points). g: Distributions of the log likelihood difference (Lc − Lb), estimated confinement factor and estimated diffusion length of the particle depending on the diffusion length of the potential well corresponding to Fig 3d.

a: (Complement of Fig 4b) Rainbow plots of the log likelihood difference, estimated diffusion length, and estimated change of velocity for tracks in perfect linear motion (with localization error) for different linear motion velocities. 10,000 tracks per condition. localization error: σ = 0.02 µm, tracks of 30 time points. b: (Complement of Fig 4c) Heatmaps of the likelihood ratio, estimated diffusion length and change of velocity (average) for tracks in perfect linear motion varying the track length and the directed motion velocity. σ = 0.02 µm.

a: (Complement of Fig 4e) Study of the impact of the rotational diffusion of tracks in directed motion with changing orientation. Heatmaps of the likelihood ratio, of the absolute error on the rotational diffusion angle, and of the estimated diffusion length when varying the directed motion velocity and the rotational diffusion angle. d = 0.0 µm, σ = 0.02 µm. b: (Complement of Fig 4g) Characterization of the motion parameters of particles with both diffusive and directed motion. Mean absolute error on the diffusion length and on the velocity of the linear motion varying the number of time points in each track. Directed motion velocity: v = 0.1 µm·Δt−1, σ = 0.02 µm. a-b: mean values from 10,000 tracks.

Effect of the number of tracks on the different parameters of the tracks: the likelihood, the root mean squared error, the standard deviation and the bias on the estimates of the diffusion length and anomalous parameter (velocity or confinement factor) for both directed motion and confined motion. All tracks were composed of 50 time points and 50 replicates were performed to estimate the error for each number of tracks. Directed tracks: persistent motion velocity v = 0.02 µm·Δt−1, angular diffusion coefficient : 0.1 Rad2·Δt−1, d = 0.0 µm, σ = 0.02 µm. Confined tracks: confinement factor 0.2, d = 0.1 µm, σ = 0.02 µm.

AIC, BIC, and corrected BIC corresponding to the log likelihood shown in Fig 5c depending on the number of states assumed by the model and on the number of tracks per data set. The corrected BIC corresponds to the BIC with an additional penalization term of 0.02kn with n the number of tracks and k the number of parameters. Under the AIC, BIC, and corrected BIC curves, we plotted the optimal number of states (the one that minimizes the criterion) for each data set.