Fuzzy temporal random sets with an application to cell biology

Total Internal Reflection Fluorescence Microscopy (TIRFM) greatly facilitates to imaging the first steps of endocytosis, a process whereby cells traffic cargo from the cell surface to endosomes. Using TIRFM, fluorescent-tagged endocytic proteins are observed as overlapped areas forming random clumps of different sizes, shapes and durations. A common procedure to segment these objects consists of thresholding the original gray-level images to produce binary sequences in which a pixel is covered or not by a given fluorescent-tagged protein. This binary logic is not appropriate because it leaves a free tuning parameter to be set by the user which can influence on the conclusions of the statistical analysis. Instead, we have adopted a more realistic approach, in which segmented binary images are modelled as a fuzzy temporal random set. Here, we propose some measures of spatio-temporal interactions based on the fuzzy counterparts of the pair-correlation function and the Ripley K-function. We used a randomization procedure to test for independence. Our results show that this procedure will permit biologists to examine and quantify the interactions between endocytic proteins robustly.