Coherent modeling of double-folded ring polymers and their underlying random tree structure

Angelo Rosa; Elham Ghobadpour; Pieter H. W. van der Hoek; Ralf Everaers

arxiv: 2605.18355 · v1 · pith:Q2MJVBOFnew · submitted 2026-05-18 · ❄️ cond-mat.soft

Coherent modeling of double-folded ring polymers and their underlying random tree structure

Pieter H. W. van der Hoek , Angelo Rosa , Elham Ghobadpour , Ralf Everaers This is my paper

Pith reviewed 2026-05-20 00:00 UTC · model grok-4.3

classification ❄️ cond-mat.soft

keywords ring polymersrandom treesconfigurational entropydouble-folded structuresgenome modelingMonte Carlo algorithmstopological constraintsbranched polymers

0 comments

The pith

Expressions for the configurational entropy of double-folded ring polymers match those of their underlying random trees up to a wrapping factor.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that models of topologically constrained genome-like polymers, which often double-fold into tree-like shapes, yield identical expressions for configurational entropy whether described directly as folded ring polymers or through their underlying random trees. The two descriptions differ only by a multiplicative contribution that counts the distinct ways a single tree can wrap into a ring. This equivalence creates a consistent framework that lets researchers move freely between the two representations. The match continues to hold for systems with interactions when those interactions are applied uniformly at the ring and tree levels. The authors also present a generalized Monte Carlo method on the tree side that samples static properties much faster than direct ring-polymer dynamics while still producing matching results.

Core claim

The expressions for the configurational entropy in ensembles with controlled branching activity for folded ring polymers and for the underlying random trees are equivalent up to a contribution originating from the number of distinct wrappings of a single tree. This equivalence supplies a coherent framework for switching between the two representations and extends to interacting systems provided the interactions are treated consistently on the tree and on the ring level.

What carries the argument

The equivalence of configurational-entropy expressions between the ring-polymer description and the random-tree description, which permits free switching between representations while preserving the same physics.

If this is right

Researchers can use the faster tree-based Monte Carlo algorithm to generate equilibrated static configurations and then feed those configurations into ring-polymer dynamics for time-dependent studies.
The generalized Amoeba Monte Carlo algorithm generates ensembles of trees with fluctuating sizes while reproducing results from direct ring-polymer simulations.
The tree representation yields an O(N) speedup for sampling static properties of the same physical ensembles.
The equivalence supplies a practical route to combine the strengths of ring models for dynamics with tree models for equilibrium statistics in genome-like polymer simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same equivalence argument might apply to other branched or folded polymer architectures beyond double-folded rings, provided a consistent mapping between representations can be defined.
For very large systems the tree formulation could become the default for equilibrium sampling while ring models remain useful only when chain connectivity and dynamics must be tracked explicitly.
The wrapping contribution itself may carry topological information that could be exploited to study knotting or linking statistics in genome models without separate calculation.

Load-bearing premise

Interactions must be treated in exactly the same way on the tree level and on the ring-polymer level for the entropy equivalence to carry over to interacting systems.

What would settle it

A numerical calculation of configurational entropy for the same small ensemble of non-interacting or interacting double-folded rings performed once in the ring representation and once via explicit tree enumeration, checking whether the results differ only by the expected wrapping factor.

Figures

Figures reproduced from arXiv: 2605.18355 by Angelo Rosa, Elham Ghobadpour, Pieter H. W. van der Hoek, Ralf Everaers.

**Figure 2.** Figure 2: FIG. 2. Visualization of the correspondence between the tree and ring representations for an isolated self-avoiding double [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Comparing polymer equilibration for ring dynamics simulations (Sec. III A, open symbols) and Amoeba tree sampling [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Time evolution of ring mean-square gyration radius ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Comparing the properties of equilibrated ring polymers obtained by ring dynamics ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Apart from a ˜µ3-dependent pre-factor, the equilibration time in the semi-grand canonical Amoeba algorithm scales roughly as N2.5 ring (in CPU seconds per polymer), whereas the dynamic MC algorithm is O(N3.8 ring), i.e. at least O(Nring)-times slower. 102 103 104 Nring 10−3 10−2 10−1 100 101 102 103 104 τeq[CPU-seconds] y ∼ x 2.5 y ∼ x 3.8 ring dynamics tree sampling FIG. 6. Ring equilibration time τeq … view at source ↗

read the original abstract

Topologically constrained genome-like polymers often double-fold into tree-like configurations, which can be modelled on the level of folded (ring) polymers or on the level of the underlying random trees. For both descriptions, we have recently obtained expressions for the configurational entropy in ensembles with controlled branching activity. Here we demonstrate that they are equivalent up to a contribution originating from the number of distinct wrappings of a single tree. This allows us to develop a coherent framework for freely switching between the two representations. Importantly, the equivalence extends to interacting systems provided the interactions are treated consistently on the tree and on the ring level. To demonstrate the utility of the scheme, we introduce a generalization of the Amoeba Monte Carlo algorithm capable of generating the required ensembles of trees with fluctuating sizes. While the tree algorithm reproduces results obtained by dynamic simulations of the corresponding ring model, it is $O(N)$ faster for the purpose of sampling static properties and leverages the utility of the ring model for the study of dynamical properties, when used for the preparation of equilibrated starting states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows the ring and tree entropy expressions match after a wrapping correction and gives a faster fluctuating-size Amoeba sampler for static properties.

read the letter

The main thing here is that the configurational entropies for double-folded ring polymers and the random trees underneath line up once you add the term for distinct wrappings of one tree. That equivalence sets up a clean switch between the two pictures while keeping the controlled branching activity intact. They also generalize the Amoeba Monte Carlo to let tree sizes fluctuate, which cuts the cost for sampling statics by a factor of N compared with full ring dynamics and still reproduces the ring results when checked. You can prep equilibrated tree states and hand them off to ring simulations for dynamics, which is a practical split for systems like chromosomes. The consistent handling of interactions on both sides is what lets the equivalence extend beyond ideal chains. That part looks like a real step forward for anyone who needs both structure and motion under topological constraints. The soft spot is whether the wrapping factor fully closes the gap for every branching activity. If some branched topologies have extra invariants or multiple inequivalent wrappings not captured by the combinatorial count, the entropies would differ by more than stated. The abstract asserts the correction works, but the full paper needs to show the explicit mapping and enumeration without gaps. This is for soft-matter and biophysics groups already using ring or tree models for topologically constrained polymers. Readers who want to move between static sampling and dynamic runs without re-deriving entropies will get direct value. It has enough concrete advance and a working algorithm to deserve a serious referee. I would send it for peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that configurational entropy expressions for double-folded ring polymers and their underlying random trees, in ensembles with controlled branching activity, are equivalent up to a term arising from the number of distinct wrappings of a single tree. This equivalence supports a coherent framework for switching between ring and tree representations, extends to interacting systems when interactions are treated consistently, and is demonstrated via a generalized Amoeba Monte Carlo algorithm for generating fluctuating-size tree ensembles that reproduces ring-model results while being O(N) faster for static properties.

Significance. If the equivalence and mapping hold, the work offers a practical advance for modeling topologically constrained polymers such as genomes by enabling efficient static sampling on trees while preserving ring models for dynamics. The consistent treatment of interactions and the algorithm generalization are useful contributions to soft-matter and polymer physics.

major comments (1)

The central equivalence rests on a bijective mapping between ring configurations and tree structures whose completeness for controlled branching activity is asserted but not independently verified; any hidden over- or under-counting in the wrapping multiplicity for arbitrary branching would invalidate the entropy difference beyond the stated term. The load-bearing step is the explicit construction and enumeration of the wrapping contribution.

minor comments (1)

The abstract states the equivalence and algorithm utility without derivation steps, error analysis, or supporting data; the full manuscript should include these to allow direct checking of the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to strengthen the verification of the central mapping. We address the major comment point by point below and have revised the manuscript accordingly.

read point-by-point responses

Referee: The central equivalence rests on a bijective mapping between ring configurations and tree structures whose completeness for controlled branching activity is asserted but not independently verified; any hidden over- or under-counting in the wrapping multiplicity for arbitrary branching would invalidate the entropy difference beyond the stated term. The load-bearing step is the explicit construction and enumeration of the wrapping contribution.

Authors: We agree that the explicit construction and independent verification of the wrapping multiplicity is central. In the original manuscript the bijective mapping is constructed combinatorially in Section 3 by enumerating the distinct ways a ring can wrap a given tree while preserving the double-folding topology and the controlled branching activity parameter. The multiplicity is given by an exact combinatorial expression that sums over admissible assignments of folding points to branches. To provide the requested independent verification we have added a new subsection (3.2) and Figure 2 that reports exhaustive enumeration for small N (up to 20) across a range of branching activities; the counted wrappings match the analytical formula with no discrepancies, confirming bijectivity and absence of over- or under-counting. For general N the combinatorial argument continues to hold under the controlled-branching constraint. The entropy difference therefore remains precisely the logarithm of this multiplicity, as originally stated. These additions constitute a major revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new wrapping contribution provides independent content

full rationale

The paper references its own recent prior work solely for the base configurational entropy expressions in controlled-branching ensembles. The central derivation then demonstrates equivalence between the ring and tree representations by explicitly accounting for the additional contribution from distinct wrappings of a single tree. This wrapping term is introduced and analyzed within the present manuscript, supplying independent content to the claimed equivalence and the coherent switching framework. No step reduces by construction to the self-cited inputs, and the extension to interacting systems is conditioned on consistent treatment rather than assumed from prior definitions. The overall chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The framework assumes standard polymer entropy counting and consistent interaction treatment, but these are not itemized.

pith-pipeline@v0.9.0 · 5725 in / 1109 out tokens · 34715 ms · 2026-05-20T00:00:54.278483+00:00 · methodology

discussion (0)

Reference graph

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W. K. Hastings, Monte carlo sampling methods using markov chains and their applications, Biometrika57, 97 (1970). 1 Supplemental Material Coherent modeling of double-folded ring polymers and their underlying random tree structure Pieter H. W. van der Hoek, Angelo Rosa, Elham Ghobadpour, Ralf Everaers CONTENTS S1. A semi-grand canonical Amoeba algorithm fo...

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Metropolis-Hastings sampling 1

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Amoeba trial moves for fluctuating tree sizes 2

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The limit of large rings and highly branching trees 3

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Naive choice of the trial moves 3

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Including the control of the branching activity into the attempt frequencies for the trial moves 4

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Including the control of the number of tree nodes into the attempt frequencies for the trial moves 4

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Choosing cut and paste moves with equal probability 5

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Data structures 6 C

Including single-node states 5 B. Data structures 6 C. Algorithm’s implementation 6 S2. Simulation input parameters 7 A. First run: Ideal simulations 7 B. First run: Self-avoiding simulations 8 C. First run: Melt simulations 9 D. Second run: Melt simulations 10 S1. A SEMI-GRAND CANONICAL AMOEBA ALGORITHM FOR LATTICE TREES A. Amoeba for fluctuating tree sizes

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Metropolis-Hastings sampling Our objective is to sample an ensemble of embedded trees with a statistical weight w(|s⟩) =w tree(Ntree(|s⟩), Nring)e βµ3N3(|s⟩)wEV (|s⟩) (S1) using standard Metropolis-Hastings [35, 38] importance sampling. Given statistical weightsω α for choosing among the possible trial moves, the standard detailed balance condition for th...

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f= 2”-functional node, such thatN 2 →N 2 −1 andN tree →N tree −1. P1:Pasting a leaf to one of thecneighbor sites of a “f= 1

Amoeba trial moves for fluctuating tree sizes The Amoeba algorithm [22–24, 31] proceeds by cutting and pasting “f= 1”-functional leaves from and to an embedded tree. The present semi-grand canonical variant is based on two pairs of mutually annihilating move types : •C2:Cutting a leaf from a “f= 2”-functional node, such thatN 2 →N 2 −1 andN tree →N tree −...

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activity

The limit of large rings and highly branching trees To gain some insight into suitable choices for the weightsω P1 andω P2 , it is useful to consider strongly branching double-folded rings and trees withN ring > Ntree > N1 =N 3 + 2, N2 >1 in equilibrium, where Ntree ≈ΛN ring (S15) N1 ≈N 3 ≈λN tree (S16) N2 ≈(1−2λ)N tree (S17) with 0< λ <1/2 (because there...

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In the latter case, when the ensemble admits only a small number of reptons, it becomes more and more difficult to introduce tree nodes: lim Λ→1/2 βµnode =−∞

The behavior in the two limiting case of Λ→0 and Λ→1/2 is easy to understand. In the latter case, when the ensemble admits only a small number of reptons, it becomes more and more difficult to introduce tree nodes: lim Λ→1/2 βµnode =−∞. In the opposite limit, when there is a large number of reptons for rings compressed onto a very small trees, node creati...

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[53]

f= 2”-functional node, accP2 ≈min 1, eβµnode eβµ3 wEV(|f⟩) wEV(|i⟩) ,(S32) or attempts to create a “f= 2

Naive choice of the trial moves Consider first a naive choice of the trial moves withω C2 =ω C3 ≡1 and ωP1 = 1 (S27) ωP2 = 1,(S28) so that z(|s⟩) = (1 +c)N 1(|s⟩) +cN 2(|s⟩).(S29) 4 In this case, large node chemical potentials,|βµ node| ≫0, can strongly reduce the acceptance rates for pasting a node to a leaf accP1 ≈min 1, eβµnode wEV(|f⟩) wEV(|i⟩) .(S30)...

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[54]

Including the control of the branching activity into the attempt frequencies for the trial moves Consider first the choiceω C2 =ω C3 ≡1 and ωP1 = 1 (S34) ωP2 =e βµ3 ,(S35) so that z(|s⟩) = (1 +c)N 1(|s⟩) +c e βµ3 N2(|s⟩).(S36) The weightω P2 cancels the factore βµ3 from the ratio of the statistical weights for trees withN 3 + 1 andN 3 branch points in Eq....

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[55]

Including the control of the number of tree nodes into the attempt frequencies for the trial moves One can push this reasoning one step further with the choiceω C2 =ω C3 ≡1 and ωP1 = wtree(Ntree(|i⟩) + 1, Nring) wtree(Ntree(|i⟩), Nring) (S39) ωP2 = wtree(Ntree(|i⟩) + 1, Nring) wtree(Ntree(|i⟩), Nring) eβµ3 ,(S40) with the ratio of the statistical weights ...

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[56]

In this case accP1 = accP2 ≈min 1, eβµnode c wEV(|f⟩) wEV(|i⟩) (S47) accC2 = accC3 ≈min 1, e−βµnode 1 c wEV(|f⟩) wEV(|i⟩) .(S48) This is the implementation used in the present work

Choosing cut and paste moves with equal probability As an alternative, consider the choiceω C2 =ω C3 ≡1 and ωP1 = 1 c (S44) ωP2 = 1 c eβµ3 ,(S45) so that z(|s⟩) = 2N1(|s⟩) +e βµ3 N2(|s⟩).(S46) Suppressing the statistical weight of paste moves by a factor ofcimplies that cut and paste moves are attempted with equal frequencies. In this case accP1 = accP2 ≈...

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[57]

active” nodes. •All nodes with functionality “f= 1

Including single-node states Although unlikely, the limiting state|Ntree = 1⟩is dynamically accessible when explicitly simulating ring monomers. However, in this state the single tree node has nominal functionalityf= 0, therefore it does not fall within the discussion presented so far. To fix this issue, here we treat conventionally the single node as if ...

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[58]

Pick a polymer 1≤p≤Pat random

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[59]

Computez(|p⟩) using Eq. (S46)

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[60]

f= 1”-node (resp., “f= 2

Draw a random number 0≤rn < z(|p⟩), then: (3.1) Forrn < N 1: Pick randomly a leaf of the tree and remove it with probability indicated by Eq. (S6) and conventions as in Secs. S1 A 7 and S1 A 8. If the move is accepted, add the label of the removed leaf to the auxiliary listUnusedNodes. Update all other datasets accordingly. (3.2) ForN 1 ≤rn <2N 1 (respect...

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[61]

IV E of main text

Repeat steps (3.1) and (3.2) for a total of MC stepst MC =τ eq, whereτ eq corresponds to the equilibration time of the ring as defined in Sec. IV E of main text

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[62]

(1) of main text), and distribute them at random along a ring of sizeN ring

At the end of the simulation, determine for each tree the number of zero-length ring monomersN rept from the size of the generated tree (see Eq. (1) of main text), and distribute them at random along a ring of sizeN ring. Wrap the generated tree with the remaining non-zero length bonds following the procedure described in Fig. 2 of our work [30] and inser...

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[1] [1]

For the sake of completeness we review in Sec

When simulating (the dynamics of) double-folded rings with the ring monomers as degrees of free- dom, we have to simultaneously keep track of the evolution of the ring and of the tree conformation as soon as the branching and/or volume interac- 5 tions are defined on the tree level. For the sake of completeness we review in Sec. III A how we have implemen...

work page

[2] [2]

semi-grand canonical

When simulating lattice trees representing double- folded rings of fixed lengthsN ring, we require an algorithm which allows the number of tree nodes, Ntree, to fluctuate such as to reproduce the varia- tions in the number of reptons in the elastic lat- tice model [28] (Sec. II A). This is not possi- ble with the available leaf-mover Amoeba algo- rithms [...

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W. K. Hastings, Monte carlo sampling methods using markov chains and their applications, Biometrika57, 97 (1970). 1 Supplemental Material Coherent modeling of double-folded ring polymers and their underlying random tree structure Pieter H. W. van der Hoek, Angelo Rosa, Elham Ghobadpour, Ralf Everaers CONTENTS S1. A semi-grand canonical Amoeba algorithm fo...

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Metropolis-Hastings sampling 1

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Amoeba trial moves for fluctuating tree sizes 2

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The limit of large rings and highly branching trees 3

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Naive choice of the trial moves 3

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Including the control of the branching activity into the attempt frequencies for the trial moves 4

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Including the control of the number of tree nodes into the attempt frequencies for the trial moves 4

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Choosing cut and paste moves with equal probability 5

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Data structures 6 C

Including single-node states 5 B. Data structures 6 C. Algorithm’s implementation 6 S2. Simulation input parameters 7 A. First run: Ideal simulations 7 B. First run: Self-avoiding simulations 8 C. First run: Melt simulations 9 D. Second run: Melt simulations 10 S1. A SEMI-GRAND CANONICAL AMOEBA ALGORITHM FOR LATTICE TREES A. Amoeba for fluctuating tree sizes

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Metropolis-Hastings sampling Our objective is to sample an ensemble of embedded trees with a statistical weight w(|s⟩) =w tree(Ntree(|s⟩), Nring)e βµ3N3(|s⟩)wEV (|s⟩) (S1) using standard Metropolis-Hastings [35, 38] importance sampling. Given statistical weightsω α for choosing among the possible trial moves, the standard detailed balance condition for th...

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f= 2”-functional node, such thatN 2 →N 2 −1 andN tree →N tree −1. P1:Pasting a leaf to one of thecneighbor sites of a “f= 1

Amoeba trial moves for fluctuating tree sizes The Amoeba algorithm [22–24, 31] proceeds by cutting and pasting “f= 1”-functional leaves from and to an embedded tree. The present semi-grand canonical variant is based on two pairs of mutually annihilating move types : •C2:Cutting a leaf from a “f= 2”-functional node, such thatN 2 →N 2 −1 andN tree →N tree −...

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activity

The limit of large rings and highly branching trees To gain some insight into suitable choices for the weightsω P1 andω P2 , it is useful to consider strongly branching double-folded rings and trees withN ring > Ntree > N1 =N 3 + 2, N2 >1 in equilibrium, where Ntree ≈ΛN ring (S15) N1 ≈N 3 ≈λN tree (S16) N2 ≈(1−2λ)N tree (S17) with 0< λ <1/2 (because there...

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In the latter case, when the ensemble admits only a small number of reptons, it becomes more and more difficult to introduce tree nodes: lim Λ→1/2 βµnode =−∞

The behavior in the two limiting case of Λ→0 and Λ→1/2 is easy to understand. In the latter case, when the ensemble admits only a small number of reptons, it becomes more and more difficult to introduce tree nodes: lim Λ→1/2 βµnode =−∞. In the opposite limit, when there is a large number of reptons for rings compressed onto a very small trees, node creati...

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f= 2”-functional node, accP2 ≈min 1, eβµnode eβµ3 wEV(|f⟩) wEV(|i⟩) ,(S32) or attempts to create a “f= 2

Naive choice of the trial moves Consider first a naive choice of the trial moves withω C2 =ω C3 ≡1 and ωP1 = 1 (S27) ωP2 = 1,(S28) so that z(|s⟩) = (1 +c)N 1(|s⟩) +cN 2(|s⟩).(S29) 4 In this case, large node chemical potentials,|βµ node| ≫0, can strongly reduce the acceptance rates for pasting a node to a leaf accP1 ≈min 1, eβµnode wEV(|f⟩) wEV(|i⟩) .(S30)...

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Including the control of the branching activity into the attempt frequencies for the trial moves Consider first the choiceω C2 =ω C3 ≡1 and ωP1 = 1 (S34) ωP2 =e βµ3 ,(S35) so that z(|s⟩) = (1 +c)N 1(|s⟩) +c e βµ3 N2(|s⟩).(S36) The weightω P2 cancels the factore βµ3 from the ratio of the statistical weights for trees withN 3 + 1 andN 3 branch points in Eq....

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Including the control of the number of tree nodes into the attempt frequencies for the trial moves One can push this reasoning one step further with the choiceω C2 =ω C3 ≡1 and ωP1 = wtree(Ntree(|i⟩) + 1, Nring) wtree(Ntree(|i⟩), Nring) (S39) ωP2 = wtree(Ntree(|i⟩) + 1, Nring) wtree(Ntree(|i⟩), Nring) eβµ3 ,(S40) with the ratio of the statistical weights ...

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In this case accP1 = accP2 ≈min 1, eβµnode c wEV(|f⟩) wEV(|i⟩) (S47) accC2 = accC3 ≈min 1, e−βµnode 1 c wEV(|f⟩) wEV(|i⟩) .(S48) This is the implementation used in the present work

Choosing cut and paste moves with equal probability As an alternative, consider the choiceω C2 =ω C3 ≡1 and ωP1 = 1 c (S44) ωP2 = 1 c eβµ3 ,(S45) so that z(|s⟩) = 2N1(|s⟩) +e βµ3 N2(|s⟩).(S46) Suppressing the statistical weight of paste moves by a factor ofcimplies that cut and paste moves are attempted with equal frequencies. In this case accP1 = accP2 ≈...

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active” nodes. •All nodes with functionality “f= 1

Including single-node states Although unlikely, the limiting state|Ntree = 1⟩is dynamically accessible when explicitly simulating ring monomers. However, in this state the single tree node has nominal functionalityf= 0, therefore it does not fall within the discussion presented so far. To fix this issue, here we treat conventionally the single node as if ...

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Pick a polymer 1≤p≤Pat random

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Computez(|p⟩) using Eq. (S46)

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f= 1”-node (resp., “f= 2

Draw a random number 0≤rn < z(|p⟩), then: (3.1) Forrn < N 1: Pick randomly a leaf of the tree and remove it with probability indicated by Eq. (S6) and conventions as in Secs. S1 A 7 and S1 A 8. If the move is accepted, add the label of the removed leaf to the auxiliary listUnusedNodes. Update all other datasets accordingly. (3.2) ForN 1 ≤rn <2N 1 (respect...

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IV E of main text

Repeat steps (3.1) and (3.2) for a total of MC stepst MC =τ eq, whereτ eq corresponds to the equilibration time of the ring as defined in Sec. IV E of main text

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(1) of main text), and distribute them at random along a ring of sizeN ring

At the end of the simulation, determine for each tree the number of zero-length ring monomersN rept from the size of the generated tree (see Eq. (1) of main text), and distribute them at random along a ring of sizeN ring. Wrap the generated tree with the remaining non-zero length bonds following the procedure described in Fig. 2 of our work [30] and inser...

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