PSF decomposition of nanoscopy images via Bayesian analysis unravels distinct molecular organization of the cell membrane

The spatial organization of membrane receptors at the nanoscale has major implications in cellular function and signaling. The advent of super-resolution techniques has greatly contributed to our understanding of the cellular membrane. Yet, despite the increased resolution, unbiased quantification of highly dense features, such as molecular aggregates, remains challenging. Here we describe an algorithm based on Bayesian inference of the marker intensity distribution that improves the determination of molecular positions inside dense nanometer-scale molecular aggregates. We tested the performance of the method on synthetic images representing a broad range of experimental conditions, demonstrating its wide applicability. We further applied this approach to STED images of GPI-anchored and model transmembrane proteins expressed in mammalian cells. The analysis revealed subtle differences in the organization of these receptors, emphasizing the role of cortical actin in the compartmentalization of the cell membrane.

probability of containing features that can be associated to marker fluorescence emission (upper row on Supplementary Fig. S2). The routine calculates the likelihood that the pixel intensity values I i,j in a square box of size b w have a Gaussian (noise-like) distribution with background offset m and variance s 2 : where the hat symbols indicate maximum likelihood estimators (MLEs). MLEs are obtained by partial derivation of the likelihood function, giving: The calculation is performed over the whole image by sliding the box pixel by pixel. A box size of length b w = 4×FWHM+1 was found to correctly estimate the presence of emitters and limit false positive identifications. Boxes partly lying outside the image area are excluded. In order to minimize computational time, the calculation of L noise and MLEs is performed via Fourier transform. In this way, we obtain a likelihood map of the image ( Supplementary Fig. S2). The pixel position showing the maximum negative likelihood -L noise (corresponding to the maximum probability of having features that do not correspond to noise) is singled out and the box centered at this pixel position is passed to the next algorithm step, consisting in the box-BIC evaluation ( Supplementary Fig. S2, middle row). In this step, the intensity map of the box region is attempted to be reconstructed as the sum of n PSFs. Each PSF corresponds to a two-dimensional Gaussian function: where x 0 and y 0 are the peak location coordinates, I 0 is the peak intensity and σ is the radius at which the PSF decays at ~60% of its maximum. For a given image, σ is kept fixed at the value where the FWHM is independently measured from images of sparse markers adsorbed on a coverslip and sets the resolution of the STED image (Supplementary Fig. S1a-b). The peak coordinates and intensity are used as free parameters and the fitting is performed via minimization of the Bayesian Information Criterion: In the last equation, the first product in the logarithm represents the likelihood of the model in reconstructing the image and is given by: (eqn. S7) with M and S 2 being the local estimation for noise background offset and variance.
The second term in the logarithm describes the likelihood for the PSF peak intensities to belong to the marker intensity distribution: where g I is the probability distribution of marker peak intensities experimentally determined from the image of sparse antibodies ( Supplementary Fig. S1a). To allow calculation of g I for every value of I k , the peak intensities histogram obtained from isolated markers was interpolated by means of a Gaussian kernels density distribution ( Supplementary Fig. S1c).
The third term in Eqn. S6 introduces a penalty for the addition of further PSFs, thus preventing overestimation of the particle number. The number of PSFs used to model the intensity distribution within the box is sequentially increased and the corresponding BIC

Quantification of receptors spatial organization
Receptor spatial organization was quantified by calculating the nnd 2 and the paircorrelation function 3,4 , from the localization coordinates obtained via application of the algorithm to the STED images and taking into account the related localization accuracy.
Calculation of the nnd was performed by generating a distance matrix r i,j from the localization coordinate list (x i ,y i ), with elements given by The minimum of each column for i<j represents the distance between the j-th localization and its closest neighbor i * and was used to build the nnd. The distribution was calculated by taking into account that, due to the finite localization accuracy, the distance between localized receptors has an error Δr i,j given by ! "r i, j = "r i 2 + "r j   The recall fraction R f as a function of molecular density for simulated images with FWHM=90 nm at varying imaging parameters. For the compress sensing analysis we used a patch size of 7x7 pixels, ε=1.5 and a 12x12 subdivision (corresponding to 2.1 nm per pixels). In order to allow comparison with our method, compressed sensing results were converted into lists of molecular positions 8 .