Non-backtracking walks reveal compartments in sparse chromatin interaction networks.

Chromatin communities stabilized by protein machinery play essential role in gene regulation and refine global polymeric folding of the chromatin fiber. However, treatment of these communities in the framework of the classical network theory (stochastic block model, SBM) does not take into account intrinsic linear connectivity of the chromatin loci. Here we propose the polymer block model, paving the way for community detection in polymer networks. On the basis of this new model we modify the non-backtracking flow operator and suggest the first protocol for annotation of compartmental domains in sparse single cell Hi-C matrices. In particular, we prove that our approach corresponds to the maximum entropy principle. The benchmark analyses demonstrates that the spectrum of the polymer non-backtracking operator resolves the true compartmental structure up to the theoretical detectability threshold, while all commonly used operators fail above it. We test various operators on real data and conclude that the sizes of the non-backtracking single cell domains are most close to the sizes of compartments from the population data. Moreover, the found domains clearly segregate in the gene density and correlate with the population compartmental mask, corroborating biological significance of our annotation of the chromatin compartmental domains in single cells Hi-C matrices.

Many-body contacts in fractal polymer chains and fractional Brownian trajectories.

We calculate the probabilities that a trajectory of a fractional Brownian motion with arbitrary fractal dimension d_{f} visits the same spot n≥3 times, at given moments t_{1},...,t_{n}, and obtain a determinant expression for these probabilities in terms of a displacement-displacement covariance matrix. Except for the standard Brownian trajectories with d_{f}=2, the resulting many-body contact probabilities cannot be factorized into a product of single-loop contributions. Within a Gaussian network model of a self-interacting polymer chain, which we suggested recently [K. Polovnikov et al., Soft Matter 14, 6561 (2018)1744-683X10.1039/C8SM00785C], the probabilities we calculate here can be interpreted as probabilities of multibody contacts in a fractal polymer conformation with the same fractal dimension d_{f}. This Gaussian approach, which implies a mapping from fractional Brownian motion trajectories to polymer conformations, can be used as a semiquantitative model of polymer chains in topologically stabilized conformations, e.g., in melts of unconcatenated rings or in the chromatin fiber, which is the material medium containing genetic information. The model presented here can be used, therefore, as a benchmark for interpretation of the data of many-body contacts in genomes, which we expect to be available soon in, e.g., Hi-C experiments.

Effective Hamiltonian of topologically stabilized polymer states.

Topologically stabilized polymer conformations in melts of nonconcatenated polymer rings and crumpled globules are considered to be a good candidate for the description of the spatial structure of mitotic chromosomes. Despite significant efforts, the microscopic Hamiltonian capable of describing such systems still remains unknown. We describe a polymer conformation by a Gaussian network - a system with a Hamiltonian quadratic in all coordinates - and show that by tuning interaction constants, one can obtain equilibrium conformations with any fractal dimension between 2 (an ideal polymer chain) and 3 (a crumpled globule). Monomer-to-monomer distances in topologically stabilized states, according to available numerical data, fit very well the Gaussian distribution, giving an additional argument in support of the quadratic Hamiltonian model. Mathematically, the polymer conformations are mapped onto the trajectories of a subdiffusive fractal Brownian particle. Moreover, we explicitly show that the quadratic Hamiltonian with a hierarchical set of coupling constants provides the microscopic background for the description of the path integral of the fractional Brownian motion with an algebraically decaying kernel.

Fractal Folding and Medium Viscoelasticity Contribute Jointly to Chromosome Dynamics.

Chromosomes are key players of cell physiology, their dynamics provides valuable information about its physical organization. In both prokaryotes and eukaryotes, the short-time motion of chromosomal loci has been described with a Rouse model in a simple or viscoelastic medium. However, little emphasis has been put on the influence of the folded organization of chromosomes on the local dynamics. Clearly, stress propagation, and thus dynamics, must be affected by such organization, but a theory allowing us to extract such information from data, e.g., on two-point correlations, is lacking. Here, we describe a theoretical framework able to answer this general polymer dynamics question. We provide a scaling analysis of the stress-propagation time between two loci at a given arclength distance along the chromosomal coordinate. The results suggest a precise way to assess folding information from the dynamical coupling of chromosome segments. Additionally, we realize this framework in a specific model of a polymer whose long-range interactions are designed to make it fold in a fractal way and immersed in a medium characterized by subdiffusive fractional Langevin motion with a tunable scaling exponent. This allows us to derive explicit analytical expressions for the correlation functions.