Intrachain distances in a crumpled polymer with random loops

Crumpled polymer further folded into random loops has been proposed as a minimal model of chromosome organization. How do loops affect spatial distances in such a polymer? Here we investigate the statistics of intrachain distances, $R(s)$, at different length scales s in the ensemble of polymer configurations with frozen (quenched) disorder of loops. We delineate the effect of the loops by solving the model analytically for the crumpled polymer chain, which was long suggested as a null model of chromatin organization. As we show, the chain compacts across scales upon folding into loops and features a characteristic “toe” in $R(s)$ at the length scale of several loop sizes $\lambda$. Quantitatively comparing $R(s)$ with the behavior of the contact probability function, ${{P}_{c}}(s)$, computed in our previous works K. Polovnikov and B. Slavov, Phys. Rev. E 107, 054135 (2023) [1] and K. E. Polovnikov, B. Slavov, S. Belan, M. Imakaev, H. B. Brandáo, and L. A. Mirny, bioRxiv: 2022.02.01.478588 [2], we further demonstrate breaking of the famous mean-field relation between the two observables. The latter result is a striking manifestation of the non-Gaussianity of the polymer ensemble, induced by the loops disorder. Altogether, our theoretical findings pave the way towards quantitative inference of parameters of loopy chromosomes from the microscopy data in vivo and warn researchers against using Gaussian methods of analysis of population-averaged conformation capture datasets (e.g., Hi-C).