Toward quantitative super-resolution microscopy: molecular maps with statistical guarantees

Microscopy, 2024, 73(3), 287–300 DOI: https://doi.org/10.1093/jmicro/dfad053 Advance Access Publication Date: 21 November 2023 Article Toward quantitative super-resolution microscopy: molecular maps with statistical guarantees Katharina Proksch1,† , Frank Werner 2,†,* , Jan Keller–Findeisen3,† , Haisen Ta4 and Axel Munk5,6 1 Faculty of Electrical Engineering, Mathematics and Computer Science, Universiteit Twente, Zilverling 2098, Enschede 7500, The Netherlands Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, Würzburg 97074, Germany 3 ̈ Naturwissenschaften, Am Fassberg 11, Göttingen 37077, Department of NanoBiophotonics, Max-Planck-Institut für multidisziplinare Germany 4 Center for Hybrid Nanostructures, Universitaẗ Hamburg, Luruper Chaussee 149, Hamburg 22607, Germany 5 Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstraße 7, Göttingen 37077, Germany 6 Felix Bernstein Institute for Mathematical Statistics in the Bioscience, University of Göttingen, Goldschmidtstraße 7, Göttingen 37077, Germany 2 † To whom correspondence should be addressed. E-mail: These authors contributed equally Abstract Quantifying the number of molecules from fluorescence microscopy measurements is an important topic in cell biology and medical research. In this work, we present a consecutive algorithm for super-resolution (stimulated emission depletion (STED)) scanning microscopy that provides molecule counts in automatically generated image segments and offers statistical guarantees in form of asymptotic confidence intervals. To this end, we first apply a multiscale scanning procedure on STED microscopy measurements of the sample to obtain a system of significant regions, each of which contains at least one molecule with prescribed uniform probability. This system of regions will typically be highly redundant and consists of rectangular building blocks. To choose an informative but non-redundant subset of more naturally shaped regions, we hybridize our system with the result of a generic segmentation algorithm. The diameter of the segments can be of the order of the resolution of the microscope. Using multiple photon coincidence measurements of the same sample in confocal mode, we are then able to estimate the brightness and number of molecules and give uniform confidence intervals on the molecule counts for each previously constructed segment. In other words, we establish a so-called molecular map with uniform error control. The performance of the algorithm is investigated on simulated and real data. Key words: asymptotic normality, counting, family-wise error rate, multiplicity adjustment Introduction Super-resolution microscopy In fluorescence microscopy, structures of interest inside a specimen are labeled with fluorescent markers and then imaged using visible light illumination. Only the fluorescence itself and thus the labeled structures are detected, making it possible, for example, to investigate details inside living cells with unrivaled contrast. The tremendous development of super-resolution fluorescence microscopy in recent decades has extended spatial resolution beyond the diffraction limit of conventional microscopy to the nanometer scale. All super-resolution light microscopy concepts rely on distinguishing fluorophores locally by consecutively transferring them between a dark (non-fluorescent) and a bright (fluorescent) state using light to induce these transitions [1,2]. The transitions between these states can be performed in either a spatially controlled or stochastic manner, with the latter denoted here as single-molecule switching microscopy [3]. In both approaches, only a small subset of molecules is left in the bright state at each measurement step and the final image is assembled by repeating the experiment many times. A well-established spatially controlled method uses stimulated emission depletion (STED) [4,5]. Thereby, a red-shifted light spot featuring a central intensity minimum is co-aligned with the excitation light spot. It induces strongly saturated stimulated emission, effectively inhibiting the fluorophores from emitting fluorescence in the periphery of a focused excitation light spot. The very small spot of effectively allowed fluorescence emission can be scanned over the sample, where each scanning position in a rectangular grid corresponds to a pixel in the final image. For example, the STED principle has been used in the past to reveal the distribution of synaptic proteins in living mice [6] or the dynamics of membrane lipids in living cells [7]. Recently, a combination of stochastic switching and excitation light patterns with at least one isolated intensity zero, called minimal photon fluxes, was used to achieve isotropic resolution on the order of a few nanometers [8]. From a statistical perspective, the recovery of spatial intensity and specimen distribution from super-resolution fluorescence microscopy images leads to sophisticated convolution models with Poisson or Binomial data distributions, which in themselves present a number of challenges (see, e.g. [9–12] and references therein). In many biological contexts, however, it is not only the precise localization of structures that is of interest but also Received 21 April 2023; Revised 2 October 2023; Editorial Decision 27 October 2023; Accepted 3 November 2023 © The Author(s) 2023. Published by Oxford University Press on behalf of The Japanese Society of Microscopy. All rights reserved. For permissions, please e-mail: * 288 the determination of the exact number of fluorescent markers at a given location, especially if this number can be related to the local number of proteins or other biological targets of interest. Such quantitative knowledge of target structures at the nanoscale has the potential to greatly improve the understanding of many biological processes. Knowledge of the absolute number of molecules can provide the basis for structural models of protein complexes or determine thresholds for the number of molecules required to produce a particular effect. For example, estimating the number of constituent proteins in kinetochores reported unexpectedly high numbers of proteins present [13], whereas quantifying the number of proteins used for flagellar regeneration helped refine models for flagellar assembly [14]. In general, the mean recorded fluorescence signal in a microscope is proportional to the number of active fluorescent markers. If we denote by f N a spatial function assigning each location (or pixel) the corresponding number of markers, and by f p their corresponding brightness ( i.e. probability to emit a photon after an excitation pulse was applied), then the observed quantity is mathematically given by a spatial convolution of the product fN ⋅ fp with the so-called point spread function (PSF) h, which is determined by the microscope (see the ‘Modeling, notation and prerequisites’ section for details). This (...truncated)