A geometrically informed permutation test for dependency in spatiotemporal patterns of protein species in microscopy images

Understanding the spatiotemporal dependencies between different protein species, as observed through fluorescent confocal laser microscopy, is crucial for gaining insight into their biological functions. We introduce an estimator of their bulk movement patterns between time points on a space, Ψ, using the earth mover's distance. We propose a test statistic that combines these bulk movement patterns over a partition, Ψ^w, of Ψ into w subregions and compares them between two samples. At the core of our approach lies a novel null hypothesis framework, consisting of statements regarding between- and within-sample independence of bulk movement patterns, alongside a statement of distributional invariance under the operation of a geometrically defined subgroup of permutations acting on Ψ^w. This framework yields a geometrically informed permutation (GIP) test designed to quantify the significance of dependencies between bulk movement patterns. We validate the approach using synthetic data spanning a range of independent and dependent scenarios with varying geometrical properties. Finally, we apply the GIP test to experiments involving the microtubule-associated proteins EB3 and TACC3, obtaining evidence that reinforces previous biological findings on the colocalisation of these proteins. More broadly, our proposed methodology is applicable to a wide range of spatiotemporal molecular data problems in which geometric structure is fundamental, particularly when the surrounding cellular or tissue environment informs the dynamics.