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Optimal Diffusion Coefficient Estimation in Single-Particle Tracking (Archived)


Single-particle tracking is a powerful tool for probing the transport behavior and local environmental properties of individual nanoparticles in liquid environments. Unfortunately, common data analysis methods do not properly account for the behavior of typical experiments. We are developing improved analysis algorithms for extracting diffusion coefficients from single-particle tracking data based on rigorous statistical methods. Software for implementing these improved algorithms, including a recently-developed maximum likelihood estimator, is available for download here


In a typical single-particle tracking experiment, optical images of diffusing particles are recorded with a CCD camera, and the series of frames obtained from the camera is analyzed to extract a trajectory. In the simplest case of free Brownian motion, the trajectory is subsequently analyzed to extract a diffusion coefficient. This latter data analysis step, diffusion coefficient estimation, is commonly accomplished using a statistical estimate of the average distance a particle moves, the mean-square displacement (MSD), as a function of time interval. While the MSD is conceptually appealing, because it has a simple and useful physical interpretation, estimating it reliably from experimental data is complicated. The difficulty comes from several technical problems, including "motion blur" during camera integration, localization noise in the single-particle tracking step, and subtle statistical correlations in the data.  

In this project, we have made two significant improvements to current state-of-the-art data analysis techniques for tracking single-particles. First, we derived the exact distribution of measurement results from a camera-based single-particle tracking experiment done on a freely-diffusing particle. This distribution is fundamentally important for interpreting experimental results but had not been previously recorded. Critically, we found that although a freely diffusing particle moves with uncorrelated displacements, motion blur and localization noise artifacts give rise to apparent correlations in a particle's measured displacement.  

Second, we used this distribution to derive an alternate data analysis procedure, a maximum likelihood estimator (MLE) that extracts the diffusion coefficient in a nearly-optimal way. From the exact distribution, we also computed the Fisher Information Matrix and corresponding Cramer-Rao bound, which sets a fundamental limit on the accuracy of any unbiased estimator – including all of those based on MSD. This limit allows any data analysis procedure to be compared to a theoretical optimum. A large simulation study comparing the MLE and another recently-developed algorithm, the optimal least-squares fit (OLSF), is ongoing.  

Selected Publication:

  • Statistics of camera-based single-particle tracking, A. J. Berglund, Physical Review E 82, 011917 (2010).
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Created March 19, 2012, Updated March 10, 2022