Precise 3-D localization of fluorescent probes without numerical fitting

We present an analytical algorithm for determining the position of a fluorescent probe in three dimensions (3-D) from a collection of measurements taken at different locations. This algorithm, fluoroBancroft, relies on the fact that the intensity point spread function depends only on the radial distance between the center of the focal point and the position of the fluorescent probe. We present a simulation study comparing the performance of the algorithm to the standard technique of fitting the data to a Gaussian profile. Our results indicate that in the 3-D case, the fluoroBancroft algorithm is able to localize the probe with an accuracy on the order of tens of nanometers using less than ten measurements (pixels). The Gaussian fitting procedure is unable to locate the probe even when using 36 measurements (the maximum number of measurements used in the simulations). Moreover the new technique is typically two orders of magnitude faster than the Gaussian fitting approach in terms of computation time. These results indicate the fluoroBancroft algorithm can be used effectively in a closed-loop controller to track the motion of single fluorescent probes in 3-D in a confocal microscope.