The effect of loops on the mean square displacement of Rouse-model chromatin

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)].