A quantitative comparison of single-dye tracking analysis tools using Monte Carlo simulations

Abstract

Single-particle tracking (SPT) is widely used to study processes from membrane receptor organization to the dynamics of RNAs in living cells. While single-dye labeling strategies have the benefit of being minimally invasive, this comes at the expense of data quality; typically a data set of short trajectories is obtained and analyzed by means of the mean square displacements (MSD) or the distribution of the particles' displacements in a set time interval (jump distance, JD). To evaluate the applicability of both approaches, a quantitative comparison of both methods under typically encountered experimental conditions is necessary. Here we use Monte Carlo simulations to systematically compare the accuracy of diffusion coefficients (D-values) obtained for three cases: one population of diffusing species, two populations with different D-values, and a population switching between two D-values. For the first case we find that the MSD gives more or equally accurate results than the JD analysis (relative errors of D-values <6%). If two diffusing species are present or a particle undergoes a motion change, the JD analysis successfully distinguishes both species (relative error <5%). Finally we apply the JD analysis to investigate the motion of endogenous LPS receptors in live macrophages before and after treatment with methyl-β-cyclodextrin and latrunculin B.

Figure 1. Principle of the mean-square displacement (MSD) and jump distance (JD) analyses.

(a) A typical single trajectory of receptor-bound LPS recorded in live macrophages (Δt = 33 ms). Trajectory data can be analyzed either by MSD or JD analysis. For MSD analysis, an average of the mean-square displacement is calculated over all points of the individual trajectory for multiples of the smallest resolved time intervals (Δt, 2Δt, 3Δt etc). The MSD plot over nΔt is linear and the gradient is directly proportional to the diffusion coefficient. (b) Diffusion coefficients are obtained for all single trajectories and typically presented in histograms. For a random walk with a single diffusion coefficient and long trajectories this distribution is centered around the diffusion coefficient. Multiple mobility populations can be resolved in principle, however a reliable dissection requires a fairly large data set (Note S3 in File S1). The JD analysis plots a histogram of all particle displacements within a fixed time interval Δt for all trajectories. (c) Fitting Eq. 7 to the distribution of the displacements yields the minimum number of diffusion coefficients needed to describe the motion of the particles in the system.

Figure 2. Validation of tracking function on simulated data.

The fraction of detected particles and fully reconstructed tracks (number of true positives/ground truth (GT)) as a function of the ratio of the average nearest neighbor distance over the average displacement a/d. Image stacks with 150 particles were simulated over 30 frames. In the first image frame particles were located at a distance 5×radius of Airy disk from each other and then moved assuming Brownian motion in subsequent frames. d was varied. Each data point shows the mean value of 50 repetitions.

Figure 3. Illustration of the parameter β.

Depending on the SNR, a particle can be localized within a certain localization precision σ (red circle). After a certain time interval the particle has traveled a distance d. If σ is a substantial fraction of the distance d (i.e. β is small), the measurement of d is imprecise, leading to errors in the determination of the diffusion coefficient. Increasing the time interval Δt and thus β allows a more precise measurement of d.

Figure 4. Comparison of JD analysis and MSD analysis on simulated data.

Simulations with varying mobile and immobile populations were used to compare the performance of the JD analysis (black curves) to the standard MSD approach (red curves). (a,b,d) The results shown are the relative error |D_m, (output)−D_m,(ground truth, GT)|/D_m (GT) in the diffusion coefficient of the mobile population as a function of the parameter β and (c) the relative error |f_m (output)−f_m(GT)|/f_m (GT) in the mobile fraction as a function of the parameter β. All results are the mean values of 10 repetitions, with each repetition based on the analysis of 750 simulated trajectories. Error bars represent ±1 standard deviation (a) All particles belong to the mobile population. (b,c) 50% of the particles are mobile, and 50% immobile (d) 50% of particles undergo a motion change (mobile → immobile).

Figure 5. Cumulative histograms of jump distances of LPS-bound receptors in the plasma membrane (blue line).

Best fits according to Eq. 7 are shown (cyan). (a) The cyan line represents the fit result assuming one species is present. The residual shows a clear deviation from the data. (b) The same data fitted assuming two species are present (components of the fit are shown in green for the mobile and red for the immobile population). The residual shows no systematic deviation of the fit from the data. Note that the deviation of the fit from the histogram towards larger displacements is also found in our simulated data (Fig. S7 and Note S3 in File S1) and thus does not indicate a deviation of the data from Brownian motion. (c) MβCD treated and (d) latrunculin B treated macrophages fitted assuming two mobility populations. The values for the diffusion coefficients and fractions can be found in Table 1.