Analysis of sparse molecular distributions in fibrous arrangements based on the distance to the first neighbor in single molecule localization microscopy

Single Molecule Localization Microscopy (SMLM) currently attains a lateral resolution of around 10 nm approaching molecular size. Together with increasingly specific fluorescent labeling, it opens the possibility to quantitatively analyze molecular organization. When the labeling density is high enough, SMLM provides clear images of the molecular organization. However, either due to limited labeling efficiency or due to intrinsically low molecular abundance, SMLM delivers a small set of sparse and highly precise localizations. In this work, we introduce a correlation analysis of molecular locations based on the functional dependence of the complementary cumulative distribution function (CCDF) of the distance to the first neighbor (r₁). We demonstrate that the log(-log(CCDF(r₁))) vs. log(r₁) is characterized by a scaling exponent n that takes extreme values of 2 for a random 2D distribution and 1 for a strictly linear arrangement, and find that n is a robust and sensitive metric to distinguish characteristics of the underlying structure responsible for the molecular distribution, even at a very low labeling density. The method enables the detection of fibrillary organization and the estimation of the diameter of host fibers under conditions where a visual inspection provides no clue.