New Inverse Problems of Super-Resolved Microscopy

Super-Resolved Fluorescence Microscopy (SRFM) and in particular Single Molecule Localization Microscopy (SMLM) have revolutionized the field of Biology by allowing for recording microscopic images of biological probes with a resolution of about 1 to 20 nanometers. This high resolution allows for visualization of single proteins, which play a key role in the transmission of diseases: For instance the SARS-CoV-2 virus makes use of its spike glycoprotein to gain entry into the host cells. The basic experimental setup consists in chemically loading particular proteins with fluorescent dyes. SRFM records several microscopic images of immobilized samples and utilizes the complex interaction of laser light with fluorescent dyes for visualization via statistical processing. This proposal is concerned with mathematical and computational aspects related to SRFM. Our developed mathematical analysis and computational algorithms are based on sophisticated mathematical models which take into account light propagation of fluorescent light in in-homogenuous media and describe the complex interaction of laser light with dye-molecules. The ultimate goal of this proposal is to extract in-homogenuous material parameters (like the density and permeability in the cell) from microscopic image sequences recorded with ultrahigh resolution imaging techniques. The key observation for starting our research is that current SRFM experiments do not use all potentially available measurement data, and we will make use of them to show pathways to compute material parameters, which are not imaged up to date. This proposal is a mathematical one, which is anticipated to serve as starting point for a translational or interdisciplinary project based on the outcome of the developed mathematical analysis and the computer simulations, prior to physical experiments: We formulate new mathematical inverse problems of SRFM, where studies on its uniqueness and stability indicate feasibility of computational imaging in microscopic applications. In particular we want to find out the role and the effect of material in- homogenities on the accuracy of reconstructions of dyes locations. In contrast to existing computational techniques in SRFM, which reduce to image processing and statistical tasks for accurate localization of centers of dyes, we consider nonlinear inverse problems for reconstructing the material parameters of the probe.