Configurational entropy of randomly double-folding ring polymers

TL;DR

The work computes the configurational entropy of randomly double-folded ring polymers by encoding each configuration with a wrapping code that maps ring paths onto a randomly branching tree. Using Bertrand's ballot theorem, the authors obtain exact counts of admissible wrappers, leading to a closed form for the number of viable codes $\Omega_{\rm ring}$, and incorporate reptons to connect to an elastic lattice model via $Z_{\rm ring}$ and the distribution $p(N_{\rm tree},N_3|N_{\rm ring},\mu_3)$. They derive asymptotic scaling laws for the partition function, mean tree size, and node functionality distributions, and validate them against Monte Carlo simulations demonstrating excellent agreement. The framework is generalized to arbitrary node functionality, providing a robust, exact combinatorial foundation for the entropy of tightly double-folded rings and linking tree- and ring-level descriptions for potential applications to genome organization and polymer physics. All key results are expressed with explicit, calibrated formulas, enabling direct computation of observables in large systems. $$Z_{\rm ring}(N_{\rm tree},\mu_3) \sim \frac{\exp(-\beta\mu_3)}{\sqrt{\pi x_0}} \frac{\exp(N_{\rm tree} g(x_0)) c^{N_{\rm tree}-1}}{N_{\rm tree}^{3/2}},\; x_0=\frac{1}{2+e^{-\beta\mu_3/2}},\; \lambda(\mu_3)=\frac{1}{2+e^{-\beta\mu_3/2}}. $$

Abstract

Topologically constrained genome-like polymers often double-fold into tree-like configurations. Here we calculate the exact number of tightly double-folded configurations available to a ring polymer in ideal conditions. For this purpose, we introduce a scheme which allows us to define a ``code'' specifying how a ring wraps a randomly branching tree and calculate the number of admissible wrapping codes via a variant of Bertrand's ballot theorem. As a validation, we demonstrate that data from Monte Carlo simulations of an elastic lattice model of non-interacting tightly double-folded rings with controlled branching activity are in excellent agreement with exact expressions for branch-node and tree size statistics that can be derived from our expression for the ring entropy.

Figures (10)

• Figure 1: Illustration of the modeling steps and notation used in this work. In various physical and biological situations, (ring) polymers double-fold into conformations (a1) that can be characterised via acyclic trees (a2) and represented by (elastic) lattice models (a3) along the lines of Refs. RubinsteinPRL1994Ghobadpour2021Ghobadpour2025 where we have highlighted the monomer labeled “1” in red. (b1) Another conformation of the embedded ring for the same configuration of the tree and the same secondary structure of the double-folded ring. (b2) A cyclic permutation of the ring around the tree corresponds to a different configuration or secondary structure of the ring. (b3) A different tree configuration and hence also a different configuration of the double-folded ring.
• Figure 2: Rules to construct a tightly double-folded ring polymer (violet circles) wrapped around a tree and how to translate this into a corresponding wrapping code. At each step, the last placed ring monomer is colored in red. The possible directions of the wrapping procedure are indicated by the orange arrows. In every frame, the evolving construction of the wrapping code is shown in correspondence with the placed ring monomers. When a ring monomer is placed that does not correspond to a new entry in the wrapping code, this is indicated by the symbol "$||$".
• Figure 3: Density map representation of the $2d$ probability distributions (Eq. \ref{['eq:probability']}) of sampling a tuple ($N_{\rm tree}$, $N_3$) at a given $N_{\rm ring}$ and for different values (see legends) of the branching chemical potential $\beta {\mu}_3$. Yellow/blue regions are more/less likely to be sampled, as indicated by the color bar on the right. Tuples with a probability of sampling $<10^{-19}$ are white in the above plots. The red dots represent the sampled data points from computer simulations of the elastic polymer model, they all fall on the maxima of the predicted probability distributions with no systematic deviations observed.
• Figure 4: Expectation values of node functionalities normalized to the mean tree size $\langle N_{\rm tree}\rangle$ (main panel) and mean tree size normalized to the ring size $N_{\rm ring}$ (inset) as a function of the branching chemical potential $\beta\mu_3$. Results: ($\times$) sampled averages of the elastic lattice model; ($\square$) numerically evaluated sums Eqs. \ref{['eq:ElasticModel-Ntree']} and \ref{['eq:ElasticModel-N3']}; (lines) analytical asymptotic formulas Eqs. \ref{['eq:AsymptoticNtree']}-\ref{['eq:AsymptoticN2']}.
• Figure 5: Mean number of nodes of functionality $f$ for trees with maximal nodes' functionality $f_{\rm max}$ and branching chemical potentials $\{ {\mu}_f = 0 \}$. Symbols and lines are for sampled averages of the elastic lattice model and our theory.