Detecting directed motion and confinement in single-particle trajectories using hidden variables
- Department of Engineering Physics, Polytechnique Montréal, Canada
- École Polytechnique, France
- Institute of Biochemistry, ETH Zürich, Switzerland
Figures
Figure 1
Principle of aTrack.
(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 the real positions, an intermediate position, the anomalous variable, and the generation of observed (measured) positions, . The variables and follow Gaussian distributions with mean 0 and standard deviations and , respectively. The sub-step from to is deterministic and the anomalous variable can also evolve with a standard deviation . (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.
Figure 2 with 2 supplements
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 or directed ) and the null hypothesis (Brownian diffusion ) for single tracks (10,000 tracks). Confinement factor and velocity ν = 0.02 μm·Δt−1. (a) Effect of the number of time points in a track on its log difference ( for confined tracks) and ( 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 (confined) or (directed) 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 follows. 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.
Figure 2—figure supplement 1
Log-likelihood differences as a function of the track length.
Distributions of the log likelihood differences ( and ) 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 follows: localization error; σ = 0.02 μm; confined tracks: diffusion length per step, d = 0.1 μm confinement factor; ; linear tracks: d = 0.0 μm, velocity ν = 0.02 μm Δt−1 , constant speed and orientation.
Figure 2—figure supplement 2
Likelihood ratios as a function of the track length.
Distributions of the likelihood ratios and corresponding to Figure 2—figure supplement 1. As expected, the distributions are skewed toward 0 only when the proper test is applied.
Figure 3 with 1 supplement
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 (per time step) or radius, respectively. (e) Confinement of tracks with a moving potential well (Brownian motion). Left: simulated tracks with different diffusion lengths of the potential well. Right: histograms of the estimated diffusion length of the potential well and confinement radius varying the actual diffusion length of the potential well. Confinement factor = 0.1 per time step. (a-d) 10,000 tracks per condition. d = 0.1 μm, Localization error σ = 0.02 μm . See Figure 3—figure supplement 1 for complementary results.
Figure 3—figure supplement 1
Parameter estimates for confined motion model.
(a) Histograms of the estimated diffusion length per step of the potential well depending on the confinement factor corresponding to Figure 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 Figure 3c. (d) Distributions of the estimated confinement factor depending on the track length (same conditions as Figure 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 Figure 3d. (f) Distributions of the estimated confinement radius depending on the true confinement radius (same conditions as in Figure 3d with tracks of 150 time points). (g) Distributions of the log likelihood difference (), estimated confinement factor, and estimated diffusion length of the particle depending on the diffusion length of the potential well corresponding to Figure 3e.
Figure 4 with 2 supplements
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), ν = 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 ν = 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 . (a-g) Where not stated otherwise, the track parameters were as follows. Localization error σ = 0.02 μm, d = 0.0 μm, velocity ν = 0.1 μm Δt−1, constant speed and orientation.
Figure 4—figure supplement 1
Linear motion model estimations.
(a) (Complement of Figure 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 Figure 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.
Figure 4—figure supplement 2
Impact of directional changes in directed motion models.
(a) (Complement of Figure 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 Figure 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: ν =0.1 μm Δt−1, σ = 0.02 μm. (a-b) Mean values from 10,000 tracks.
Figure 5 with 2 supplements
Characterizing populations of multiple states.
Analysis of tracks with five sub-populations of set diffusion length , confinement factor , velocity for directed tracks, and anomalous change parameter (diffusion length of the potential well for confined tracks and changes of speed for directed tracks). Tracks are 300 time points long. (a) Track examples from each of the five states with the corresponding state parameters. (b) 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. (c) Estimated parameters for the five states (using a five-state model).
Figure 5—figure supplement 1
Dataset size and parameter estimate error.
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, ν = 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.
Figure 5—figure supplement 2
Estimating the number of states.
AIC, BIC, and corrected BIC corresponding to the log likelihood shown in Figure 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 with the number of parameters and the log likelihood. 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.
Figure 6 with 1 supplement
Model robustness with other motion types.
(a-b) Example tracks and corresponding distributions used to determine the type of motion for aTrack and Randi (Argun et al., 2021). 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 exponent. The accuracy is the fraction of correctly labeled tracks in a data set composed of 5000 sub-diffusive or super-diffusive tracks and 5000 Brownian tracks. Classifications were done using the thresholds that best divide the distributions. (a) Analysis of tracks with 100 time steps following fractional Brownian motion with anomalous exponent of 0.5 (sub-diffusive), 1 (diffusive), and 1.5 (super-diffusive). (b) Analysis of tracks with 100 time steps following our motion model. Confined tracks: diffusion length d = 0.1 μm, localization error σ = 0.02 μm, confinement force , 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 = 0.02μm , directional velocity ν = 0.1 μm·Δt−1. Directed tracks: d = 0. μm, σ = 0.02 μm , ν = 0.1 μm·Δt−1, angular diffusion coefficient 0.1 Rad2s-1. (c) Analyzing tracks 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 ) depending on the confinement radius. Middle panel: Estimated confinement radius () depending on the true confinement radius. Bottom: estimated confinement radius depending on the track length. Blue areas: standard deviations of the estimates.
Figure 6—figure supplement 1
Effect of dynamic and static localization error on estimated motion parameters.
Simulated populations of Brownian tracks with continuous exposure and estimated the (population-wise) diffusion length per step and localization error. At each step, the position is estimated as the average position of 200 sub-steps with a static localization error of 0.02 μm (per time-step). (a-c) Simulations of 5000 tracks of 99 time points with varying diffusion lengths. (a) Estimated diffusion length as a function of the true diffusion length. (b) Estimated diffusion length as a function of the true diffusion length (log-log plot). (c) Estimated localization error as a function of the true diffusion length. (d) Estimated diffusion lengths for 5000 tracks with varying number of time steps and fixed diffusion length per time step of 1 μm.
Figure 7 with 1 supplement
Experimental demonstrations.
(a) Illustration showing the interaction of the budding yeast spindle pole body (SPB) with actin via microtubules (MT). Actin-dependent motors are responsible for moving the nucleus toward the bud neck during S-phase. (b-c) Analysis of spindle pole body (SPB) tracks. The analysis was carried out on tracks of 100 time points. (b) Examples of tracks classified by aTrack to be either significantly directed or non-significantly directed. A random selection of tracks colored by their associated likelihood ratios with and without LatA treatment can be found in Figure 7—figure supplement 1b. (c) Mean fraction of directed tracks from three biological replicates for the WT and two biological replicates for the latrunculin-resistant mutant. Each replicate contains at least 640 tracks for an average of 4180 tracks per replicate. Error bars: standard deviation. *: significant difference according to a t-test with independent variables (p-value = 0.00165). (d-e) Analysis of gold 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 and the log likelihood difference () of all tracks. Tracks are considered significantly directed if the likelihood ratio (which is an overestimate of the p-value) is lower than 0.05 (type I error) divided by the number of tracks (85) according to the Bonferroni correction (=log likelihood difference > 7.44). (e) Maximum likelihood (per track) of the population of tracks depending on the number of states (minus the likelihood assuming 10 states). (f-g) Analysis of tracks for 1 μm beads trapped using optical tweezers with different laser powers. (f) Illustration of the optical trap and example tracks of 100 time points for different laser powers. (g) Fitting a single-state confined diffusion model on a population of 300 tracks with 20 time points.
Figure 7—figure supplement 1
Confinement characterization in Saccharomyces cerevisiae.
(a) Mean squared displacements (MSD) as a function of the number of time steps in the WT strain without treatment (DMSO). (b) Random selections of tracks of 100 time points from the WT strain without and with LatA treatment (resp. − LatA and + LatA) colored according to their likelihood ratio (red if and blue otherwise).
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Detecting directed motion and confinement in single-particle trajectories using hidden variables
eLife 13:RP99347.
https://doi.org/10.7554/eLife.99347.3
