  Several recent experiments, including our own in the fission yeast, S. pombe,
have characterized the motions of gene loci within living nuclei by measuring
the locus position over time, then proceeding to obtain the statistical
properties of this motion. To address the question of whether a population of
single particle tracks, obtained from many different cells, corresponds to a
single mode of diffusion, we derive theoretical equations describing the
probability distribution of the displacement covariance, assuming the
displacement is a zero-mean multivariate Gaussian random variable. We also
determine the corresponding theoretical means, variances, and third central
moments. Bolstering the theory is good agreement between its predictions and
the results obtained for various simulated and measured data sets, including
simulated particle trajectories of simple and anomalous diffusion, and the
measured trajectories of an optically-trapped bead in water, and in a
viscoelastic solution. We also show that, for sufficiently long tracks, each
covariance distribution in these examples is well-described by a skew-normal
distribution with mean, variance, and skewness given by theory. For
experimental S. pombe gene locus data, however, we find that the first two
covariance distributions are wider than predicted, although the third and
subsequent covariances are well-described by theory. This suggests that the
origin of the theory-experiment discrepancy is associated with localization
noise, which influences only the first two covariances. Thus, we hypothesize
that the discrepancy is caused by locus-to-locus heterogeneity in the
localization noise. Further simulations reveal excess covariance widths can be
largely recreated on the basis of heterogeneous noise. We conclude that the
motion of gene loci in fission yeast is consistent with a single mode of
diffusion.
