Measuring Spatiotemporal Dependencies in Bivariate Temporal Random Sets with Applications to Cell Biology

Analyzing spatio-temporal dependencies between different types of events is highly relevant to numerous biological phenomena (e.g. signalling and trafficking) especially as advances in probes and microscopy have facilitated imaging of dynamic processes in living cells. For many types of events, the segmented areas can overlap spatially and temporally forming random clumps. In this paper, we model binary image sequences of two different event types as a realization of a bivariate temporal random set and propose a non-parametric approach to quantify spatial and spatio-temporal interrelations using the pair-correlation, cross-covariance and the Ripley ${\mathbb K}$ functions. Based on these summary statistics we propose a randomization procedure to test independence between event types by applying random toroidal shifts and Monte Carlo tests. A simulation study assessed the performance of the proposed estimators and showed that these statistics capture the spatio-temporal dependencies accurately. The estimation of the spatio-temporal interval of interactions was also obtained. The method was successfully applied to analyze the interdependencies of several endocytic proteins using image sequences of living cells and validated the procedure as a new way to automatically quantify dependencies between proteins in a formal and robust manner.