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 $n\ge 3$ 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)], 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.

Figure 1

(a) Schematic image of the pairwise interactions (5) $V_{{\mathbf{x}}_k,{\mathbf{x}}_m}$ of the $k\mathrm{th}$ monomer (${\mathbf{x}}_k$) with adjacent monomers of the chain with coordinates ${\mathbf{x}}_{k\pm 1},{\mathbf{x}}_{k\pm 2},{\mathbf{x}}_{k\pm 3},...$. Elastic constants $a_{km}$ decay algebraically, depicted by dashed lines with increasing spacing. (b) Up: A triple contact, made by monomers ${\mathbf{x}}_k, {\mathbf{x}}_n, {\mathbf{x}}_m$. Paths $k-n$ and $n-m$ are dependent, and the effective volume of the two loops is less than the product of volumes, occupied by the loops separately. Bottom: Typical conformation of the segment.

Figure 2

Left: Correlation coefficient $r\left(\chi \right)$ between two loops of lengths $s_1$ and $s_2$ as a function of the ratio of their sizes $\chi =s_1/s_2$ as given by (30) for various values of $d_f$: 2.5 (black), 3 (gray), 4 (light gray). Right: The maximal boost of the two-loop probability $f_{\max }\left(\alpha \right)$ in the three-dimensional embedding space ($D=3$) for the case of equal loop sizes $s_1=s_2$ as compared to loops being independent (35).