The emergent dynamics of double-folded randomly branching ring polymers

TL;DR

This work investigates the emergent dynamics of double-folded, randomly branching ring polymers by implementing a dynamic Monte Carlo framework on an elastic lattice model that explicitly represents both ring monomers and an underlying tree structure. By switching on and off three local dynamic modes—reptons along fixed trees, hairpin-driven side-branching, and Brownian motion of tree nodes—the authors classify dynamics into three canonical regimes and analyze their combinations across ideal, self-avoiding, and melt systems. A key contribution is the demonstration that, while a simple additive null model describes mixed dynamics for noninteracting rings, interacting rings in melts exhibit a nontrivial acceleration due to coupling between local modes, captured by a coupling term δD that scales with √(D_A D_B) and p_A p_B. The results connect to biologically relevant chromatin and DNA dynamics by showing a crumpled-ring monomer subdiffusion exponent around 0.4 in melts, paralleling bacterial chromosome behavior, and provide a quantitative framework for interpreting mixed dynamical processes in complex polymers. The study offers a mesoscale modeling paradigm to dissect how local mobility and topological constraints jointly shape macroscopic diffusion in polymeric systems.

Abstract

The statistics of randomly branching double-folded ring polymers are relevant to the secondary structure of RNA, the large-scale branching of plectonemic DNA (and thus bacterial chromosomes), the conformations of single-ring polymers migrating through an array of obstacles, as well as to the conformational statistics of eukaryotic chromosomes and melts of crumpled, non-concatenated ring polymers. Double-folded rings fall into different dynamical universality classes depending on whether the random tree-like graphs underlying the double-folding are quenched or annealed, and whether the trees can undergo unhindered Brownian motion in their spatial embedding. Locally, one can distinguish (i) repton-like mass transport around a fixed tree, (ii) the spontaneous creation and deletion of side branches, and (iii) displacements of tree node, where complementary ring segments diffuse together in space. Here we employ dynamic Monte Carlo simulations of a suitable elastic lattice polymer model of double-folded, randomly branching ring polymers to explore the mesoscopic dynamics that emerge from different combinations of the above local modes in three different systems: ideal non-interacting rings, self-avoiding rings, and rings in the melt state. We observe the expected scaling regimes for ring reptation, the dynamics of double-folded rings in an array of obstacles, and Rouse-like tree dynamics as limiting cases. Of particular interest, the monomer mean-square displacements of $g_1\sim t^{0.4}$ observed for crumpled rings with $ν=1/3$ are similar to the subdiffusive regime observed in bacterial chromosomes. In our analysis, we focus on the question to which extent contributions of different local dynamical modes to the emergent dynamics are simply additive. Notably, we reveal a non-trivial acceleration of the dynamics of interacting rings, when all three types of local motion are present.

Figures (8)

• Figure 1: (a) An example of a (tightly wrapped) double-folded ring polymer and (b) the corresponding tree on a trigonal lattice. Small loops represent reptons (bonds of zero length, see Section \ref{['sec:Model and Method']}), for which adjacent monomers along the ring occupy identical lattice sites.
• Figure 2: (a)-(b):A monomer move involves hopping a randomly selected monomer into the nearest neighbor lattice site. (a) The Repton move transports stored length along the ring and locally redistributes it without altering the tree structure. (b) The Hairpin move is responsible for the creation or annihilation of side branches. The top row shows the double-folded ring, and the bottom row shows the underlying tree structure. (c)-(d) Two examples of tree node trial moves, where a randomly selected tree node attempts to move to one of the twelve nearest neighbors of the face-centered cubic (FCC) lattice used in our simulations, subject to a double-folding preserving constraint. For clarity, a two-dimensional schematic is shown here, while the actual simulations are performed in three dimensions.
• Figure 3: Conformational statistics for isolated ideal rings, isolated self-avoiding rings, and rings in the melt state when subjected to Quenched Tree dynamics (QTree), ring in an array of obstacle-like (RiAO) dynamics, and the combination of the two (mixed, with $p_m=0.5$). Insets show the local slopes of the data; horizontal dashed lines correspond to the expectation scaling exponents for each system (summarized in Table \ref{['tab:exponents']}). The conformational statistics should be independent of the dynamics if the system condition remains the same and the results in the figure strongly confirm this assertion. Error bars are the same size or smaller than the symbols.
• Figure 4: Overview of emergent dynamics for different ring polymer systems: ideal (top row), isolated self-avoiding (middle row), and double-folded ring polymers in melts (bottom row). Results are shown for: monomer MSD, $g_1(t)$ (left column), CM MSD , $g_3(t)$, (middle column) and the size dependence of the asymptotic CM diffusion constants, $D(N_n)$, (right column). Dynamics considered include: hairpin move, slithering dynamics, ring-in-an-array-of-obstacles (RiAO)-like dynamics, quenched tree dynamics (QTree), and a mix of RiAO and QTree dynamics with $p_m=0.5$. Insets show the local slopes of the data, with dashed lines indicating the expected exponents for QTree and RiAO dynamics (summarized in Table \ref{['tab:exponents']}). Error bars are smaller than or comparable to the symbol sizes.
• Figure 5: Scaling analysis of the slithering dynamics of ideal, randomly branching, double-folded rings emerging from the exclusive application of repton moves in our model. The inset graphs in each panel share the same x-tick positions as the main panel they accompany.