Brownian Motion using a Piezo Actuated Microscope Stage

Abstract

Single particle tracking (SPT) is an important collection of techniques and algorithms for the study of biomolecular transport and understanding the role of the dynamics of small molecules, proteins, and other organic material in health and disease on a cellular level. However successful these techniques are, it is difficult to compare them directly, especially as one is most interested in their performance on a specific experimental setup. Here, we demonstrate through Monte-Carlo simulation and experiment, the ability of a piezo actuated microscope stage to replicate faithfully a discrete-time sampled Brownian motion trajectory. Simulations consisted of an exact discrete model of the piezo stage including closed-loop control, the input Brownian motion trajectory, and feed forward model inverse control. A range of stage response times, discrete time step durations, and diffusion constants were modeled. The output of the system was compared to the input using mean squared displacement (MSD), Kullback-Leibler divergence comparing the single time step displacement distributions, and stage tracking error. We found that MSD was preserved over a large range of stage response times, discrete time step durations, and diffusion constant values. Physical experiments consisted of an epifluorescent microscope taking widefield images of quantum dots fixed to a microscope slide moved using the piezo actuated microscope stage following a Brownian motion trajectory control input. We envision this work will allow for the characterization and evaluation of SPT microscopes and localization algorithms by moving fixed fluorophores through a known Brownian motion trajectory to provide a known ground truth in an experimental setting. Future work includes using other biologically relevant motion models such as confined diffusion and elastic tethering as the motion model, and a comparison of SPT microscope configurations and estimation algorithms through simulated motion with a piezo stage.