Confined Brownian Motion Tracked With Motion Blur: Estimating Diffusion Coefficient and Size of Confining Space

Abstract

Mesoscopic environments and particles diffusing in them are often studied by tracking such particles individually while their Brownian motion explores their environment. Environments may be, e.g., a domain in a cell membrane, an interior compartment of a cell, or an engineered nanopit. Particle trajectories are typically determined from time-lapse recorded movies. These are recorded with sufficient exposure time per frame to be able to detect and localize particles in each frame. Since particles move during this exposure time, particles image with motion blur. This motion blur can compromise estimates of diffusion coefficients and the size of the confining domain if not accounted for correctly. We do that here. We give explicit and exact expressions for the variance of measured positions and the mean-squared displacement of a Brownian particle confined in, respectively, a 1D box, a 2D box, a 2D circular disc, and a 3D sphere. Our expressions are valid for all exposure times, irrespective of the size of the confining space and the value of the diffusion coefficient. They apply also in the common case where the exposure time is smaller than the time-lapse due, e.g., to “dead time” caused by the readout process in the camera. These expressions permit determination of diffusion coefficients and domain sizes for given movies for the simple geometries we consider. More important, the trends observed in our exact results when parameter values are varied are valid also for more complex geometries for which no exact analytical solutions exist. Wherever the underlying physics is the same, the exact quantitative description of its consequences provided here is portable as a qualitative and semi-quantitative understanding of its consequences in general. The results may also be useful for other types of reflected Brownian motion than those occurring in single-particle tracking, e.g., in nuclear magnetic resonance imaging techniques. For use in that particular context, we briefly discuss the effects of confinement on anisotropic Brownian motion imaged with motion blur.