Section: New Results

Analysis of the repartition of moving vesicles by spatio-temporal point process models

Participants : Frédéric Lavancier, Thierry Pécot, Charles Kervrann.

Characterizing the spatial repartition of interacting moving proteins is a fundamental step for co-localization and co-expression. Based on the segmentation algorithm [15] , [23] , this challenge amounts to characterizing the repartition or spatial distribution of spots (see Fig. 6 ). This is part of the more general statistical analysis of random geometrical objects, and in particular of random points. Gibbs models form a large class of point process models, that can be used to characterize either complete randomness or attraction or repulsion between points depending on the Gibbs potential at hand.

First in [27] , we focused on infinite range potentials that include the most famous interaction potential arising from statistical physics, namely the Lennard Jones potential. To fit this kind of models to a dataset, the standard inference methods are not applicable. We introduced in [27] a modification of the pseudolikelihood method, with a specific border correction, and we prove that this provides consistent and asymptotically normal estimators. Second, in [26] , we studied an alternative class of models, the determinantal point processes (DPP). They are designed to model repulsion between points and are thus adapted to regular point patterns. These models are becoming very popular in the spatial statistics community due to many appealing properties. We quantified the possible repulsiveness that a DPP can model [26] . In particular, we determined the most repulsive stationary DPP. We finally introduced new parametric families of DPPs that cover a large range of DPPs, from the homogeneous Poisson process (for no interaction) to the most repulsive DPP.

An application of these models to the problem of co-localization between proteins is part of an on-going project. In each protein, the set of vesicles is modeled by a union of random balls, possibly overlapping, and a Gibbs interaction is introduced to take into account the possible interaction in the location of vesicles between two proteins. Our first concern is to test whether the two proteins actually interact, i.e. co-localization occurs, or in other words whether the Gibbs interaction is empty or not. If there is co-localization, the further step is to characterize it through the estimation of the strength of the Gibbs interaction.

References:  [26] [27]

Collaborators: Christophe Ange Napoléon Biscio (LMJL, University of Nantes)

                          Jean-François Coeurjolly (Laboratoire Jean Kuntzmann, Grenoble Alpes University)