Limit of the accuracy of parameter estimation for moving single molecules

Abstract

This thesis presents the research work done in determining a benchmark from which the accuracy of the parameter estimates of the attributes of moving molecules imaged by single-molecule fluorescence microscopy can be measured against. The focus is on moving single molecules since a sound theoretical basis for determining the limit of the accuracy of the parameter estimates is currently lacking. Since accuracy in parameter estimation is closely related to the Fisher information matrix, we have derived expressions of the Fisher information matrix showing its dependence on the trajectories of the molecules. From these Fisher information matrices, we have obtained the Cramer-Rao Lower Bound which provides us with the limit of the accuracy of the parameter estimates. We have considered cases where images of a moving molecule are acquired with and without time discretization of the imaging process. In addition, the case where two molecules move in proximity to one another is also considered. Issues of prior knowledge of the parameters are also addressed. Other than the theoretical work done, simulations to study how the limit of the accuracy is affected by experimental conditions and parameters that describe the trajectory of the molecule are carried out. Even though the results obtained here are essentially for the single-molecule fluorescence microscopy, they can be extended to the general problem of tracking an object using quantum limited detectors.