  Many researchers have been encouraged to describe the dynamics of chromosomal
loci in chromatin using the classical Rouse model of polymer dynamics by the
agreement between the measured mean square displacement (MSD) versus time of
fluorescently-labelled loci and the Rouse-model predictions. However, the
discovery of intermediate-scale chromatin organization, known as topologically
associating domains (TADs), together with the proposed explanation of TADs in
terms of chromatin loops and loop extrusion, is at odds with the classical
Rouse model, which does not contain loops. Accordingly, we introduce an
extended Rouse model that incorporates chromatin loop configurations from
loop-extrusion-factor-model simulations. Specifically, we extend the classical
Rouse model by modifying the polymer's dynamical matrix to incorporate extra
springs that represent loop bases. We also theoretically generalize the
friction coefficient matrix so that the Rouse beads with non-uniform friction
coefficients are compatible with our Rouse model simulation method. This
extended Rouse model allowes us to investigate the impact of loops and loop
extrusion on the dynamics of chromatin. We show that loops significantly
suppress the averaged MSD of a chromosomal locus, consistent with recent
experiments that track fluorescently-labelled chromatin loci in fission yeast
[M. L. P. Bailey, I. Surovtsev, J. F. Williams, H. Yan, T. Yuan, S. G. Mochrie,
and M. C. King, Mol. Biol. Cell (in press)]. We also find that loops slightly
reduce the MSD's stretching exponent from the classical Rouse-model value of
0.5 to a loop-density-dependent value in the 0.45-0.40 range. Remarkably,
stretching exponent values in this range have also been reported in recent
experiments [S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Phys. Rev. Lett.
104, 238102 (2010) and Bailey et al., Mol. Biol. Cell (in press)].
