The TrackArt software consists of five modules: SIM for diffusion simulations, IMPORT for importing and filtering trajectories stored in an external file, MSD for classical MSD curve fitting, and CPD and FIT modules for resolving the D s of multiple subpopulations within a data set.
Diffusion simulation
As a method of validating implemented tracking data algorithms, by challenging acquired data with simulation, as well as for teaching and illustrative purposes, a simulation module was developed and included in the TrackArt ensemble. Trajectories are simulated using the Monte Carlo algorithm, for any given number of trajectories and steps, with an optional simulation of localization error (Figure 1A,B). Three simulation modes are available:
Simple diffusion mode - each particle moves with constant D throughout its lifetime. Particles can be divided into two separate populations, each moving with a different D (Figure 1C).
Switching mode - particles switch between two states, each one with a characteristic diffusion coefficients D
1
and D
2
with switch rates k
1-2
and k
2-1
(Figure 1D) according to an equilibrium:
Domains mode - particles diffuse among immobile domains, simulated as circles with a given number per area, diameter, and diameter variation. The probability that a simulated particle will enter or leave a domain is regulated by constant rates k
1-2
and k
2-1
. Depending on these constants, domains are more or less permeable to diffusing particles (Figure 2). In extreme cases, when k
1-2
or k
2-1
is equal to zero, domains act as impermeable obstacles, giving rise to anomalous diffusion (subdiffusion), or as permeable but viscous regions, giving rise to confined diffusion. Moreover, different diffusion coefficients can be defined for particles moving within or outside of domains. A very useful feature is that the domain map can also be imported from a binary image file, as shown in Figure 3A and 3B. This gives a user the option of running diffusion simulation on images derived from, e.g., atomic force microscopy (AFM) or other maps of actual membrane ‘landscapes’ where domains with different shapes and characteristics such as fractal dimensions, percolation, confined or hop diffusion, or grid-patterns occur. For each domain map (either simulated or imported), the pair correlation curve G(r) can also be calculated and plotted (Figure 3C,D). G(r) is the probability of finding a domain at distance r and allows more detailed domain map characterization [19, 20].
Data import
The import module is used to import trajectories into TrackArt from a file, such as an ImageJ plugin: Particle Tracker [21] results, or coordinates exported previously from TrackArt Simulation module. Data from other tracking software also can be imported after formatting (see Additional file 1). Once imported, one can inspect quality of tracks, and filter out unwanted trajectories, including trajectories which are too short, not linked properly, or derived from contaminants or aggregates. Filtering of imported experimental data is essential for meaningful diffusion coefficient estimation thereafter. There are several possible factors contributing to further error: samples (or surface) contamination by other fluorescent contaminants, probes forming aggregates, the existence of an immobile probe fraction (as can happen when probe sticks to a glass surface), spurious particle recognition (noise) or errors in linking coordinates into trajectories due to high particle concentration and/or fluorophore blinking. Although many of these factors can be minimized by careful surface cleaning procedures during sample preparation, the use of sensitive EMCCD cameras, high grade filters, ultra-pure lipids and solvents, and pre-bleaching of contaminants with a strong laser pulse, several exclusion criteria should be used to dismiss trajectories which may be inaccurate. In TrackArt, the aforementioned criteria are applied through various parameters: 1) a minimum number of frames required for each trajectory, 2) a minimum and maximum diffusion coefficient value calculated for each trajectory, 3) a minimum R-squared value of the linear MSD fit for each trajectory, and 4) minimum and maximum average trajectory intensity. The optimal value for each parameter is specific for the measurement conditions, equipment used, type of sample and fluorescent label. Thus, these must be determined empirically for each imaging setup. To aid in that process, TrackArt offers a wide variety of plots and data representations, such as histograms of individual D, trajectory length and intensity, MSD plots, and trajectory preview.
Mean square displacement fits
Analysis of the mean square displacement (MSD) curves is the most common approach for extracting the diffusion coefficient, when only one population of particles with a single characteristic diffusion is observed. There are four models of possible MSD fits implemented in TrackArt, all of which assume particle movement in 2-dimensional space:
Normal (Brownian) diffusion – particles move in an isotropic medium with pure Brownian characteristics. The diffusion coefficient is extracted from the linear MSD fit:
(2)
where α is the intercept and t is time. The fit can be optionally weighted by the inverse of the MSD standard deviation. In analyzing the MSD curve (and thus estimating and minimizing the error in D), it is important to take into account effects such as localization uncertainty, finite camera exposure and the effect of diffusion on the MSD curve. TrackArt simplifies this process by providing on output consisting of MSD standard deviation, parameter errors for weighted and unweighted fits, static and dynamic localization error, reduced localization error, error in D and finally the optimum number of time-lags to use for fitting (minimizing the a and b error). For these calculations, TrackArt implements algorithms and methods described in detail by Michalet et al. [11, 12].
Remaining MSD fitting models include selected models described by Saxton et al. [22]:
Anomalous diffusion (subdiffusion) - particles diffuse among immobile obstacles, which cause deviation from Brownian diffusion;
Free diffusion with flow - particles undergo directional motion due to drift or active transport;
Confined/corralled diffusion –MSD approaches a maximum value for large time-lags, due to limitation of diffusion to within a region of confinement.
Extraction of multiple diffusion coefficients
Analysis of MSD curves is extremely valuable for diffusion coefficient extraction. However the tracked molecules often exist as populations with different diffusion behaviors, e.g. a slow and a fast population, or a single population of particles that switch intermittently between different states. Both situations can be effectively simulated in the TrackArt simulation module. When dealing with relatively large numbers of trajectories, in both cases the overall MSD fit will provide only the mean value of D. Different methods of analyzing tracking data can be used to distinguish between these fractions and extract their individual D s [15, 18, 23]. The one implemented in TrackArt was proposed by Schütz et al. [15] and was successfully used in several diffusion studies [13, 24–27]. Briefly, MSDs for each population and their fraction are extracted as parameters from multi-exponential fits to cumulative probability distribution (CPD) of square displacements. In TrackArt, the CPD module is used for CPD calculation and fits, whereas the FIT module calculates final values of D, the size of fractions diffusing with given Ds, and their errors. More detailed descriptions of each module and their algorithms can be found in Additional file 1 and user manual.