  Recent advances in molecular biology and fluorescence microscopy imaging have
made possible the inference of the dynamics of single molecules in living
cells. Such inference allows to determine the organization and function of the
cell. The trajectories of particles in the cells, computed with tracking
algorithms, can be modelled with diffusion processes. Three types of diffusion
are considered : (i) free diffusion; (ii) subdiffusion or (iii) superdiffusion.
The Mean Square Displacement (MSD) is generally used to determine the different
types of dynamics of the particles in living cells (Qian, Sheetz and Elson
1991). We propose here a non-parametric three-decision test as an alternative
to the MSD method. The rejection of the null hypothesis -- free diffusion -- is
accompanied by claims of the direction of the alternative (subdiffusion or a
superdiffusion). We study the asymptotic behaviour of the test statistic under
the null hypothesis, and under parametric alternatives which are currently
considered in the biophysics literature, (Monnier et al,2012) for example. In
addition, we adapt the procedure of Benjamini and Hochberg (2000) to fit with
the three-decision test setting, in order to apply the test procedure to a
collection of independent trajectories. The performance of our procedure is
much better than the MSD method as confirmed by Monte Carlo experiments. The
method is demonstrated on real data sets corresponding to protein dynamics
observed in fluorescence microscopy.
