RESUMEN
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.
RESUMEN
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.
