Chromosomal locus tracking with proper accounting of static and dynamic errors

The mean-squared displacement (MSD) and velocity autocorrelation (VAC) of tracked single particles or molecules are ubiquitous metrics for extracting parameters that describe the object's motion, but they are both corrupted by experimental errors that hinder the quantitative extraction of underlying parameters. For the simple case of pure Brownian motion, the effects of localization error due to photon statistics (“static error”) and motion blur due to finite exposure time (“dynamic error”) on the MSD and VAC are already routinely treated. However, particles moving through complex environments such as cells, nuclei, or polymers often exhibit anomalous diffusion, for which the effects of these errors are less often sufficiently treated. We present data from tracked chromosomal loci in yeast that demonstrate the necessity of properly accounting for both static and dynamic error in the context of an anomalous diffusion that is consistent with a fractional Brownian motion (FBM). We compare these data to analytical forms of the expected values of the MSD and VAC for a general FBM in the presence of these errors.

Figure 1

MSD for chromosomal loci in yeast. All error bars are S.E.M. from 100 bootstrapped samples of the ensemble. Each panel depicts data from the green channel MSD (green or light gray), red channel MSD (red or darker gray), and “cross-MSD” (black). (a) Dots are data points computed from experimental data. Solid lines are fits to Eq. (8). (Inset) Example image of 2D-projected locus track in white light image of a yeast cell. This example is from a diploid, but all cells analyzed here were haploids. Scale bar: 1 μm. (b) Naïve estimates of α from fitting to a straight line in log-log space. Each point corresponds to fitting all points from $n=1$ through the marked time point. (c) Estimates of α taking only static error into account. Only estimates corresponding to five points or more are shown since the nonlinearity requires several points to be reliably fit (see Fig. S11 [46] and discussion in main text). (d) Estimates of α taking both static and dynamic error into account; plotted points start at $n=5$ for the same reason as in (c).

Figure 2

(a) Example simulated images of particle diffusing within a single frame with $\alpha =0.6,b=0$, and $t_E=10$ ms. From left to right $D^{*}$ is ${10}^{-3},{10}^{-2},{10}^{-1},{10}^0,\mathrm{and}{10}^1$. Pixel size = 160 nm. (b) Resulting MSDs from 2D simulation of FBM for various α (top to bottom rows: $\alpha =0.2,0.4,0.6$) and $b$ (left to right columns $b=0,50,100$), with $\overline{p}=500$ and (from bottom to top in each panel:) $D^{-}={10}^{-3}$ (blue), ${10}^{-2}$ (cyan), ${10}^{-1}$ (green), ${10}^0$ (yellow), and ${10}^1$ (red) $\mu m^2/{\mathrm{s}}^{\alpha }$. Dots are simulated data, solid lines are theoretical predictions, and dashed lines are the predictions in the absence of errors.

Figure 3

Computed scaled VACs across various time scales for chromosomal locus data from the green channel [(a) and (d)], the red channel [(b) and (e)], and the cross-correlated data [(c) and (f)]). Bottom row depicts the same curves from the first 2 s as in the top row, zoomed in and with a different color scale. The color scales for the top and bottom rows are shown on the right. For reference in gray scale version, larger δ corresponds to a curve with negative peak further to the right.

Figure 4

Analysis of VACs from chromosomal loci data. All error bars are S.E.M. from 100 bootstrapped samples of the ensemble. Each panel depicts data from the green channel VAC (green or light gray), red channel VAC (red or darker gray), and cross-correlation (black). (a) Estimates of α resulting from naïve fits of the VACs without accounting for static or dynamic error. (b) Estimates of α resulting from taking static error into account, but not dynamic error. (c) Estimates of α taking both static and dynamic error into account. (d) Example computed and fit VACs for $n=1$. Dots with error bars are computed from data; solid lines are fits taking static and dynamic error into account.

Figure 5

Scaled VAC from simulated 2D data for various $n$, $b$, and α. Here $t_E=10$ ms, $\overline{p}=500$ photons, $D^{*}=0.1\mu {\mathrm{m}}^2/{\mathrm{s}}^{\alpha }$. Each black rectangle corresponds to a different $b$ (from left to right: 0, 50, 100). Each row corresponds to a particular α (from top to bottom 0.2, 0.6, 1.0). Within a rectangle and row the left panel shows comparison of the data to the naïve VAC of Eq. (17) which does not take errors into account. The right panel shows comparison of the same data to the VAC in the presence of errors described in Eqs. (20))–(22). For reference in gray scale version, larger $n$ corresponds to a curve with negative peak further to the right.

Figure 6

Illustrative example of $n=1$ VAC analysis of 2D simulated data for $D^{*}=0.01\mu {\mathrm{m}}^2/{\mathrm{s}}^{\alpha }$, $t_E=10$ ms, $\overline{p}=1000$, and $b=0$. The ensemble consisted of 100 tracks of length 100 frames each. Offsets due to static error have been removed by first estimating ${\sigma }_0$ for the specified photon statistics. Error bars reflect the S.E.M. from 100 bootstrapped samples of the ensemble. In both panels the gold dashed line corresponds to the VAC without inclusion of dynamic error, the magenta dashed-dotted line corresponds to the intermediate treatment of dynamic error as described in the text, and the cyan solid line corresponds to the full treatment of dynamic error. (a) Example computed VAC for $\alpha =0.2$ (black dots and error bars) and the corresponding predicted curves for each treatment of dynamic error. The point at $m=1$ is not plotted for the sake of scaling the figure and since each scaled VAC is equal to 1 here by definition. (b) Bias in the estimate of α for various α.