Statistics of camera-based single-particle tracking

Camera-based single-particle tracking enables quantitative determination of transport properties and provides nanoscale information about material characteristics such as viscosity and elasticity. However, static localization noise and the blurring of a particle’s position over camera integration times introduce artifacts into measurement results even for a particle executing simple diffusion. Common data analysis methods based on the mean-square displacement do not properly account for these effects. In this paper, we analyze the statistics of tracking data for freely diffusing particles in realistic experimental scenarios. We derive a convenient and asymptotically optimal maximum likelihood estimator for the diffusion coefficient and for the magnitude of localization noise together with the corresponding Fisher information, which bounds the performance of all unbiased estimators. We find that the effect of varying the illumination profile during the camera integration time is quantified by a motion blur coefficient, $R$. We also find that a double-pulse illumination sequence maximizes the information content in some common experimental scenarios. Our results provide a rigorous theoretical framework and practical experimental recipe for achieving optimal performance in camera-based single-particle tracking.

Figure 1

(Color online) tenth and 90th percentiles in estimating $D$. For each simulation, $D=1\mu {\text{m}}^2/\text{s}$, $\Delta t=1\text{s}$, $N=500$, and the shutter function $s\left(t\right)$ was constant over the frame time $\left(R=1/6\right)$. For each value of measurement noise $\sigma$, 100 individual simulations were performed and the values of $D$ and $\sigma$ were estimated using the MLE and MSD. For each estimator, the 10th and 90th percentile of $D$ estimates is shown (MLE: filled circles, red online; MSD: filled squares, blue online); in other words, at each value of $\sigma$, the region between the upper and lower point contains 80% of the $D$ estimates. The same percentiles for an optimal unbiased estimator, with covariance matrix equal to ${\mathbf{I}}_{D\sigma }^{-1}$, are also shown (hatched region). A single set of MLE estimator results with $\sigma =1\mu \text{m}$ is shown (open circles, red online). The MLE achieves nearly optimal performance, and significantly outperforms the MSD.

Figure 2

(Color online) Tenth and 90th percentiles in estimating $\sigma$. For each simulation, $\sigma =0.1\mu \text{m}$, $\Delta t=1\text{s}$, and $N=500$. The shutter function $s\left(t\right)$ was constant over the frame time $\left(R=1/6\right)$. For each value of $D$, 100 individual simulations were performed and the values of $D$ and $\sigma$ were estimated using the MLE and MSD. For each estimator, the tenth and 90th percentile of $D$ estimates is shown (MLE: filled circles, red online; MSD: filled squares, blue); in other words, at each value of $D$, the region between the left and right point contains 80% of the $\sigma$ estimates. The same percentiles for an optimal unbiased estimator, with covariance matrix equal to ${\mathbf{I}}_{D\sigma }^{-1}$, are also shown (hatched region). A single set of MLE estimator results with $D=0.1\mu {\text{m}}^2/\text{s}$ is shown (open circles, red online). The MLE achieves nearly optimal performance, and significantly outperforms the MSD.

Figure 3

(Color online) Comparison of maximum-likelihood estimates for different motion blur coefficients $R$. For each case, $N=1000$ data points were simulated with $D=10\mu {\text{m}}^2/\text{s}$, $\Delta t=0.1\text{s}$, and $\sigma =0.1\mu \text{m}$. Three different shutter functions were used (top row) giving, from left to right, $R=0.01$, $R=0.17$, and $R=0.24$. A scatter plot of the resulting maximum-likelihood estimates are in the $D-\sigma$ plane (bottom row). Increasing $R$ has little effect on $D$ estimates, but significantly improves the estimate of $\sigma$.