The cell as a system of many components, governed by the laws of physics and chemistry drives molecular functions having an impact on the spatial organization of these systems and vice versa. Since the relationship between structure and function is an almost universal rule not only in biology, appropriate methods are required to parameterize the relationship between the structure and function of biomolecules and their networks, the mechanisms of the processes in which they are involved, and the mechanisms of regulation of these processes. Single molecule localization microscopy (SMLM), which we focus on here, offers a significant advantage for the quantitative parametrization of molecular organization: it provides matrices of coordinates of fluorescently labeled biomolecules that can be directly subjected to advanced mathematical analytical procedures without the need for laborious and sometimes misleading image processing. Here, we propose mathematical tools for comprehensive quantitative computer data analysis of SMLM point patterns that include Ripley distance frequency analysis, persistent homology analysis, persistent 'imaging', principal component analysis and co-localization analysis. The application of these methods is explained using artificial datasets simulating different, potentially possible and interpretatively important situations. Illustrative analyses of real complex biological SMLM data are presented to emphasize the applicability of the proposed algorithms. This manuscript demonstrated the extraction of features and parameters quantifying the influence of chromatin (re)organization on genome function, offering a novel approach to study chromatin architecture at the nanoscale. However, the ability to adapt the proposed algorithms to analyze essentially any molecular organizations, e.g., membrane receptors or protein trafficking in the cytosol, offers broad flexibility of use.

Institute of Biophysics of the Czech Academy of Sciences Královopolská 135 612 00 Brno Czech Republic

Kirchhoff Institute for Physics Heidelberg University Im Neuenheimer Feld 227 69120 Heidelberg Germany

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