Monte Carlo simulation of a lattice model for the dynamics of randomly branching double-folded ring polymers

Supercoiled DNA, crumpled interphase chromosomes, and topologically constrained ring polymers often adopt treelike, double-folded, randomly branching configurations. Here we study an elastic lattice model for tightly double-folded ring polymers, which allows for the spontaneous creation and deletion of side branches coupled to a diffusive mass transport, which is local both in space and on the connectivity graph of the tree. We use Monte Carlo simulations to study systems falling into three different universality classes: ideal double-folded rings without excluded volume interactions, self-avoiding double-folded rings, and double-folded rings in the melt state. The observed static properties are in good agreement with exact results, simulations, and predictions of Flory theory for randomly branching polymers. For example, in the melt state rings adopt compact configurations and exhibit territorial behavior. In particular, we show that the emergent dynamics is in excellent agreement with a recent scaling theory and illustrate the qualitative differences with the familiar reptation dynamics of linear chains.

Figure 1

(a) Branched tree on a trigonal lattice and (b) a corresponding (tightly wrapped) double-folded ring polymer. Small loops represent bonds of zero length, where adjacent monomers along the ring occupy identical lattice sites. (c) Example of a nonlocal “amoeba” Monte Carlo move [37, 38] altering the tree structure. The dashed brown line shows the location of a branch prior to the MC move, and the solid brown line shows an arbitrary location where the branch could be reattached to the tree. (d) Examples of local MC moves for the present model of double-folded ring polymers. Dots represent monomers, and black lines represent an allowed conformation of the double-folded chain. The allowed (forbidden) moves are indicated by the green (red) color. R: the Repton move. H: the Hairpin move. F: Forbidden move that does not preserve the double-folded structure.

Figure 2

Equilibration monitored using the mean-square radius of gyration as a function of time. Comparison between initially branched, compact double-folded chains (left column), and initially double-folded chains with no branches (right column) for three different systems: (a) ideal double-folded rings, (b) self-avoiding double-folded rings, and (c) double-folded rings in the melt state. The horizontal lines represent the average values after the chains have reached equilibrium. For better visualization, the time direction is reversed in the right column. In all three systems, at large times both initial states (from left and right) reach the same equilibrium values (reported in Table 1).

Figure 3

Equilibrated simulation snapshots of (randomly selected) configurations of the double-folded rings with $N=216$. Successive segments are represented with a HSV cyclic color map. (a) A single double-folded ideal ring; (b) a single double-folded self-avoiding ring; (c) a single double-folded ring in the melt state. The gray tubes show the longest paths of the trees. All the trees have the same bond scale. The size of the ring in the melt is larger than the ideal tree and smaller than the self-avoiding tree. (d) Sample configuration of the melt with 12 double-folded rings. Each ring is represented with a different color. The snapshots were produced using Blender 2.8 [60]. 3D views of these configurations are available in the Supplemental Materials, videos S1, S2, and S3 [59].

Figure 4

Conformational statistics of ideal double-folded rings for four different chain lengths (described in the legend). Data are shown for ring contour distances up to $N/2$. Column (a) are the average values of the tree contour distances between all possible pairs of monomers, $\langle L\left(n\right)\rangle \sim n^{\rho }$. Column (b) plots the squared internal distances as a function of $n, \langle R{\left(n\right)}^2\rangle \sim n^{2\nu }$. The exact exponents for the ideal case are $\rho =1/2$ and $\nu =1/4$. In panels $\left({\text{a}}_2\right)$ and $\left({\text{b}}_2\right)$ data are plotted as a function of $n_{\mathrm{eff}}$, which effectively reduces finite size effects. The straight dashed lines correspond to the expectation scaling exponents. $\left({\text{a}}_2\right)$ and $\left({\text{b}}_2\right)$ insets show the local slopes of the data in panels $\left({\text{a}}_2\right)$ and $\left({\text{b}}_2\right)$, respectively. These effective exponents appear to converge to the theoretical exponents (dashed horizontal lines). Error bars are the same size or smaller than the symbols.

Figure 5

Conformational statistics of self-avoiding double-folded rings. Column (a) plots the average value of the tree contour distances between all possible pairs of monomers. Flory theory predicts $\langle L\left(n\right)\rangle \sim n^{2/3}$. Column (b) plots the squared internal distance as a function of $n$. The exact scaling exponent is $\langle R{\left(n\right)}^2\rangle \sim n^1$. Notation and symbols are as in Fig. 4.

Figure 6

Conformational statistics of double-folded annealed trees in the melt state. Column (a) plots the average value of the tree contour distances between all possible pairs of monomers. Flory theory predicts, $\langle L\left(n\right)\rangle \sim n^{5/9}$. Column (b) plots the squared internal distance as a function of $n$. Flory theory predicts, $\langle R{\left(n\right)}^2\rangle \sim n^{2/3}$. Notations and symbols are as in Fig. 4.

Figure 7

Conformational statistics of double-folded rings. Left column: average tree contour distance $\langle L\rangle$ as a function of the chain length $N$. Straight lines correspond to the large-$N$ behavior, $\langle L\left(N\right)\rangle \sim N^{\rho }$. Right column: ring mean-square gyration radius $\langle R_g^2\rangle$ as a function of the chain length. Straight lines correspond to the large-$N$ behavior, $\langle R{\left(N\right)}^2\rangle \sim N^{2\nu }$. Insets show the local slopes of the data. These effective exponents appear to converge to the theoretical exponents (dashed horizontal lines). Error bars are the same size or smaller than the symbols.

Figure 8

MSDs for ideal double-folded rings. Panel (a) shows $g_1, g_2$, and $g_3$ for the ring with 216 monomers. The horizontal line corresponds to $2\times \langle R_g^2\rangle$. Panels (b), (c), and (d) plot $g_2\left(t\right), g_1\left(t\right)$, and $N\times g_3\left(t\right)$ vs time in the unit of MCs, respectively. In panel (b) the horizontal lines correspond to $2\times \langle R_g^2\rangle$. In panels (c) and (d) the dashed lines have slopes corresponding to the prediction of the theory, $g_1\left(t\right)\sim t^{\frac{2\nu }{(\rho +2)}}$ and $g_3\left(t\right)\sim t^{\frac{2\nu +1}{(\rho +2)}}$. (c, d) Insets show the local slopes of the data. The effective exponents appear to converge to the theoretical exponents (dashed horizontal lines). Panels $\left(\text{e}\right)$ and $\left(\text{f}\right)$ show rescaled $g_1\left(t\right)$ and $g_3\left(t\right)$ with the mean-square gyration radii vs the rescaled time with the diffusion relaxation times.

Figure 9

MSDs for self-avoiding double-folded rings. Notation and symbols are as in Fig. 8.

Figure 10

MSDs for double-folded rings in the melt state. Notation and symbols are as in Fig. 8.

Figure 11

Relaxation time in units of MCs vs $N$ (chain length). Diffusion relaxation time (triangles) is calculated where $g_3\left({\tau }_{\max }\right)$ and $\langle R_g^2\rangle$ are equal. Configurational relaxation time (circles) is calculated using ${\tau }_{\max }=\frac{T_{\mathrm{tot}}}{2N_{\mathrm{ind}}}$, where $N_{\mathrm{ind}}$ is number of independent samples. The black solid lines indicate the theoretically predicted slopes, ${\tau }_{\max }\sim N^{2+\rho }$, while the dashed lines are the best fit. Insets show the local slopes of the data. These effective exponents appear to converge to the theoretical exponents (dashed horizontal lines).

Figure 12

Tree contour distances between monomers $(i^{*},j^{*})$ flanking the longest path on the tree at an arbitrarily chosen time $t=0$ after equilibration. Left-hand side column: Rescaled probability distributions of the longest path length, $L_{\max }=L(i^{*},j^{*},t=0)$. Middle column: Rescaled time evolution of $\langle L(i^{*},j^{*},t)\rangle$. Right-hand side column: Rescaled probability distributions of the tree contour distance, $L(i^{*},j^{*},t=10{\tau }_{\max })$, between $(i^{*},j^{*})$ after all memory of the initial state at $t=0$ is lost. Top row: Self-avoiding double-folded rings. Bottom row: Double-folded rings in the melt state.

Figure 13

(a) An example of a tree with side branches. (b) Ring connectivity of the example tree (a) mapped on a circle together with bridge bonds (dashed lines) that are formed during the “burning” process. (c) Illustration of the distance between monomers 9 and 4 following the bridge bonds along the way (green: clockwise, and pink: counterclockwise).

Figure 14

(a) An example of the ideal double-folded ring. (b) Corresponding bridges. If all monomers on a site have bridges defined between them, it leads to an extra wrong bridge (red dashed line).

Figure 15

An example of branch tip detection in an ideal double-folded ring. (a) and $\left({\text{c}}_1\right)$: Examples of side branches in the ideal case. Ambiguous branch tips and their corresponding bonds are shown in blue. The tree structure is the same, but different interpretations of branch tips are possible. (b) Bridge bonds corresponding to interpretation (a). Removal of the first layer of tips in $\left({\text{c}}_1\right)$ results in the formation of bridge bonds $\left({\text{d}}_1\right)$ and the tree structure $\left({\text{c}}_2\right)$. The removal of the second layer results in the completion of the bridge bond detection displayed in $\left({\text{d}}_2\right)$.