Fractional Brownian motion meets topology: statistical and topological properties of globular macromolecules with volume interactions

In the paper we investigate statistical and topological properties of self-avoiding fractional Brownian trajectories. The attention is paid to the sub-diffusive fractal Brownian motion (fBm) producing compact spacial conformations with the fractal dimension $D_f\ge 2$ in the three-dimensional space. The statistical properties of self-avoiding polymers with the fractal dimension $D_f>2$ are analyzed both numerically and via the Flory mean-field approach. Our study is motivated by an attempt to mimic the conformational statistics of collapsed nonconcatenated polymer loops, which are known to form on large scales the compact hierarchical crumpled globules (CG) with $D_f=3$. Due to the nonlocal nature of topological interactions, the analytic and numeric treatment of CG is an extremely difficult problem. Replacing the topologically-stabilized CG by the self-avoiding fBm with properly adjusted fractal dimension, $D_f$, we tremendously simplify the problem of generating the CG-like conformations since we wash out the topological constraints from the consideration. To check whether a replacement of CG by self-avoiding fBm is plausible, we complute numerically, using the Alexander polynomials, the distribution of the knot complexity, $P(\chi)$, of self-avoiding fBm for $D_f =2.0,\, 2.08,\, 2.38$ in the 3D space. We show that with increasing $D_f$, typical fBm polymer conformations become less and less knotted. The combination of Flory arguments with the scaling analysis of self-avoiding fBm permits us to conjecture the dependence of the knot complexity, $\chi$, on the effective Flory critical exponent, $\nu_e$. The found relation $\chi(\nu_e)$ is supported by the Monte-Carlo simulations adjusted for generating the self-avoiding collapsed fBm trajectories.