Coarse-Graining Reshapes Knot Dynamics in Polymers and Proteins

Abstract

Coarse-grained (CG) molecular dynamics (MD) simulations offer an efficient route to explore long-time-scale dynamics of knots in proteins and polymers. Despite extensive studies of knots using both atomistic (AA-) and CG-MD simulations, systematic comparisons of how coarse-graining alters knot relaxation mechanisms, intermediates, and time scalesparticularly in melts and proteinsremain scarce. Here, we employ AA- and CG-MD simulations to investigate the relaxation of tight knots in single polyethylene (PE) chains, PE-polymer melts, and a trefoil-knotted protein. Across all systems, CG models accelerate knot relaxation and enhance global mobility while preserving the dominant topological relaxation mechanism observed in AA-MD simulations. In single chains, CG-MD simulations produce smooth, monotonic unknotting trajectories, whereas AA-MD simulations reveal transient structural intermediates and intermittent shape rearrangements within the knot core. In melts, CG dynamics remain faster and more homogeneous, while AA polymers exhibit kinetically trapped intermediates and sliding-dominated relaxation driven by topological confinement. For the knotted protein, CG-MD simulations enable knot loosening and conformational rearrangements inaccessible on standard AA-MD time scales due to kinetic trapping. Collectively, these findings establish that coarse-graining systematically accelerates topological relaxation by smoothing the effective energy landscape and suppressing short-lived metastable knot conformations, while the dominant relaxation pathway is preserved.

1. Introduction

Knots are ubiquitous topological features in polymer systems, appearing in synthetic macromolecules, high-molecular-weight plastics, DNA, and proteins. − Unlike conventional conformational constraints, knots impose global topological restrictions while their dynamics are governed by local chain rearrangements, thereby exerting a profound influence on polymer relaxation, mechanical response, and transport properties. , As such, knotted polymers provide a unique platform for probing the interplay between topology, elasticity, and thermal motion in soft-matter systems.

The dynamics of polymer knotsencompassing knot tightening, diffusion along the backbone, trapping in metastable conformations, and eventual unknottinghave attracted sustained interest in both theoretical and computational studies. − These processes are inherently multiscale: local rearrangements at the monomeric level enable global topological simplification, often through rare events governed by subtle energetic and entropic balances. Understanding how knots relax is therefore essential not only for fundamental polymer physics but also for applications ranging from polymer processing and rheology to nanopore translocation and protein folding. −

Molecular dynamics (MD) simulations have played a central role in elucidating knot dynamics, particularly through AA models that explicitly resolve local interactions and chain stiffness. However, the long time scales associated with knot diffusion and unknottingespecially in long chains, dense melts, or biological macromoleculesoften exceed the reach of fully atomistic simulations. CG models, which offer substantial computational advantages by reducing degrees of freedom while preserving essential polymer physics such as chain connectivity, excluded volume, and flexibility, have therefore become indispensable tools for studying topologically constrained polymer dynamics. −

Despite their widespread use, a critical question remains largely unresolved: to what extent do CG models faithfully capture the dynamics of topological constraints resolved by chemically explicit AA models? While CG models are typically parametrized to reproduce equilibrium structural and thermodynamic properties, knot dynamics are governed by relaxation pathways that depend sensitively on local stress distributions, energetic roughness, and coupling between internal and global modes. As a result, it is not obvious whether the accelerated dynamics observed in CG simulations merely reflect a uniform rescaling of time, or whether coarse-graining fundamentally reshapes the free-energy landscape − governing knot relaxation. The specific consequences of this coarsening for complex topological processes, where local energy barriers and intermediate states may govern kinetics, have not been systematically characterized. ,

Previous studies have demonstrated that CG models often exhibit smoother effective potential energy surfaces and reduced local friction compared to their atomistic counterparts. ,,,− While CG simulations have been employed to study knot diffusion and unknotting in generic polymer models, these efforts have largely relied on bead–spring representations without systematic comparison to chemically explicit atomistic descriptions, leaving the mechanistic fidelity of CG knot dynamics uncharacterized. In particular, how the smoother CG energy landscape specifically influences rare, topology-driven events such as unknottingwhere local energy barriers and transient intermediates govern the relaxation pathwayremains poorly understood and has not been systematically examined across multiple levels of molecular complexity.

Here, we address these questions through a systematic, multiresolution investigation of knot dynamics spanning synthetic polymers and proteins. Using PE as a model polymer system, we directly compare AA simulations with CG representations across increasing levels of complexity: tightly knotted single chains, polymer melts containing knotted chains, and finally a biologically relevant knotted protein. This hierarchical approach enables us to disentangle universal features of topological relaxation from resolution-dependent effects introduced by coarse-graining. We demonstrate that CG models reproduce the global relaxation mechanism and length-dependent scaling laws of unknotting, while systematically accelerating the associated dynamics. This acceleration originates from a redistribution of local mechanical stress, consistent with reduced effective energetic roughness, which suppresses short-lived metastable knot conformations without altering the dominant topological pathway. These trends prove robust across systems of vastly different chemistry and connectivity. Our findings establish clear guidelines for deploying CG simulations in topologically constrained systemsfrom synthetic polymers to biological macromoleculesdelineating the conditions under which CG models provide reliable mechanistic insight and the regimes where atomistic resolution remains indispensable.

2. Simulation Models and Methods

2.1. Single-Chain PE Simulations

We investigated knot dynamics in flexible PE chains using both CG and AA representations. All CG simulations of PE were performed using the Shinoda–DeVane–Klein (SDK) CG model, ,− employing the condensed-phase parametrization for PE. In this representation, three PE monomer units are mapped onto a single CG bead (Figure ). Bonded, angular, and nonbonded interactions were described using the standard SDK functional forms, with all force-field parameters taken directly from the original parametrization by Hall et al. The polymer was simulated as a single fully flexible chain in an implicit condensed-phase (melt/crystal-like) environment without explicit solvent. Since polyethylene is electrically neutral, the model contains no partial charges and only bonded and short-range Lennard-Jones interactions were considered; consequently, no electrostatic interactions or long-range electrostatic treatments were included. AA simulations utilized the OPLS-AA force field. The benchmark system comprised a 50-bead CG PE chain with an initial trefoil (3₁) knot configuration, corresponding to 150 carbon atoms in the AA model. Additional chain lengths of 75 and 150 beads (225 and 450 carbons, respectively) were also studied to assess chain-length dependence of knot dynamics (Figure S1).

1.

Schematic overview of simulation systems and workflow. AA and CG representations of the systems studied. The first panel shows a PE chain containing a trefoil knot in its AA representation (top; hydrogen atoms shown in white) and its CG representation using the SDK potential (bottom). The corresponding polymer melt systems consisting of one knotted chain embedded within a melt of unknotted chains, shown for both the AA and CG models. For clarity, individual polymer chains in the melt are displayed using distinct colors. The other panel shows the shortest known trefoil-knotted protein, MJ0366 (PDB ID: 2EFV), represented using atomistic and MARTINI 3 CG models. Unless otherwise noted, polymer and protein chains are colored along the contour using a red–gray–blue color gradient to indicate position along the backbone.

All simulations followed a two-step protocol consisting of (i) knot tightening and (ii) unknotted dynamics. Initial configurations contained knots that were delocalized over the full polymer contour. These knots were first tightened by applying equal and opposite tensile forces to the chain termini, yielding compact, well-defined knot structures. The resulting tightened conformations were then used as initial conditions for equilibrium simulations aimed at probing the unknotting dynamics.

All MD simulations were performed in LAMMPS with a time step of 5 fs for CG and 0.5 fs for AA simulations. Simulations were performed at 300 K temperature. The simulations employed a Langevin thermostat to maintain the target temperature. Multiple replicas (at least six) were simulated to ensure statistical reproducibility.

2.2. Polymer Melt Simulations

To investigate the influence of intermolecular interactions, we simulated a melt system containing one knotted PE chain embedded in a melt of unknotted chains, considering both the CG and AA models (see Figure for the corresponding representative snapshots). The knotted chain had the same contour length as the single-chain benchmark system (50 beads in the CG model and 150 carbon atoms in the AA model). The melt system comprised a total of nine polymer chains, consistent with previously reported melt configurations of comparable chain lengths.

Initial melt configurations were arranged in an ordered crystalline-like packing with the knotted chain placed centrally among the unknotted chains. To preserve the knot topology during early-stage equilibration, temporary harmonic bond restraints were applied between the knot entry and exit points of the knotted chain, thereby maintaining the tightened knot configuration while the surrounding melt relaxed. The system was first equilibrated in the NPT ensemble at an elevated temperature (448 K) to melt the initial ordered structure, followed by further NPT equilibration at 300 K to establish a stable melt density. In both the AA and CG models, NPT equilibration allowed the systems to relax to the respective equilibrium melt densities prior to the production simulations. Subsequently, a series of short NVT simulations were performed, during which the restraints on the knotted chain were gradually released. This stepwise protocol ensured the preservation of the knot topology while allowing the surrounding melt to relax fully. After complete removal of restraints, production simulations were performed in the NVT ensemble at 300 K. All the simulations were performed using the LAMMPS tool, and each of the simulations was replicated at least six times to maintain statistical reproducibility.

2.3. Knotted Protein Simulations

Knotted protein simulations focused on MJ0366 (PDB ID: 2EFV), the shortest known trefoil-knotted protein (Figures and S1), modeled in both AA (CHARMM36) and CG (MARTINI 3) representations. − Initial protein structures were minimized and equilibrated prior to production simulations of 500 ns (AA) and equivalent scaled time for CG. The trefoil-knotted protein was solvated in explicit water and neutralized to maintain overall charge neutrality. The system was first energy-minimized, followed by NVT equilibration at 300 K and subsequent NPT production runs at 300 K and 1 bar. Atomistic simulations employed a 2 fs time step, whereas CG simulations used a 20 fs time step. All protein simulations were performed using GROMACS, with a minimum of three independent replicas for each model to ensure statistical reproducibility.

The knot size, knot position, and unknotting time (t _u) were determined using a geometric , sliding-window algorithm, conceptually inspired by Taylor’s method. , For each trajectory frame, contiguous subchains were estimated and the normalized radius of gyration of each segment was calculated relative to the full polymer. For a subchain containing s beads, the radius of gyration, R _g(s), was calculated and normalized by the corresponding value for the full polymer to account for chain-length dependence. The normalized compactness parameter was evaluated as

$$\chi (s)=\left[\left(\frac{R\mathrm{g}(s)}{R{\mathrm{g}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}}\right)\left(\frac{N}{s}\right)\right]$$ 1

where N is the total number of beads in the polymer. The shortest contiguous subchain with χ­(s) below a predefined threshold was identified as the knot core. The shortest segment involving the normalized R _g below a predefined threshold was indicated as the knot core, with the corresponding length (number of beads) identified as the knot size. The midpoint of this segment along the entire polymer contour was evaluated as the knot position, and the first frame is indicated as the unknotting time in which no subchain satisfied the criterion of the respective compactness. The threshold value was kept fixed throughout all simulations and was adopted from our previous work, where this geometric procedure was developed and validated. In that study, the method was benchmarked against the topology-based KymoKnot algorithm and shown to produce consistent knot-size evolution. The same validated protocol is employed here without modification. The utilization of local geometric measures, such as R _g , for identifying the compact, knot-like subchains, and the process of sliding-window scans has precedent in both protein and polymer studies to probe local knot properties. The approach for computing knot size has been previously validated and is adopted here without modification. Additional polymer metrics, including mean-square displacement (MSD), were calculated using MDAnalysis. ,

To characterize the distribution of local mechanical loading along the polymer contour, we analyzed the per-particle virial stress tensor obtained from the LAMMPS compute stress/atom command. , For each backbone atom in the atomistic model and each bead in the coarse-grained model, the six independent components of the per-particle stress tensor were extracted at every saved trajectory frame. An equivalent von Mises stress

$${\sigma }_{\mathrm{V}\mathrm{M}}=\sqrt{\frac{1}{2}[{({\sigma }_{xx}-{\sigma }_{yy})}^2+{({\sigma }_{yy}-{\sigma }_{zz})}^2+{({\sigma }_{zz}-{\sigma }_{xx})}^2]+3({\sigma }_{xy}^2+{\sigma }_{xz}^2+{\sigma }_{yz}^2)}$$ 2

was then computed for every backbone site. The stress profile was obtained by time averaging the von Mises stress over the entire knot-relaxation trajectory. To enable direct comparison between atomistic and coarse-grained representations, the averaged profiles were normalized using their respective minimum and maximum values. Consequently, the stress profile compares the relative spatial localization of mechanical stress along the polymer contour rather than absolute stress magnitudes, which are expected to differ because of the distinct interaction potentials and force-field parametrizations used in the two models.

To gain mechanistic insights, the temporal correlations of key observablesknot size, knot position, and polymer R _g were quantified via the normalized time autocorrelation function

$$C_x(\mathrm{\Delta }t)=\frac{⟨(x(t)-⟨x⟩)(x(t+\mathrm{\Delta }t)-⟨x⟩)⟩}{⟨{(x(t)-⟨x⟩)}^2⟩}$$ 3

$$\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}=\sum _{\mathrm{\Delta }t=0}^{\mathrm{\Delta }{t}^{*}}{C}_x(\mathrm{\Delta }t)\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{C}_x(\mathrm{\Delta }{t}^{*})<0$$ 4

In these expressions, x(t) denotes the time series of a given observable (knot size, knot position, or polymer R _g ), and ⟨·⟩ represents the time average. C _x (Δt), the normalized autocorrelation function, captures the temporal persistence of fluctuations in x(t) over a lag time Δt. The autocorrelation time is computed as the discrete integral (sum) of C _x up to the first zero-crossing at Δt = Δt*, beyond which correlations are no longer statistically meaningful.

To quantify the compactness of the localized knot, we define the local knot density as the number of beads contained within the identified knot core divided by the volume occupied by the knot. For each trajectory frame, the center of mass of the knot core was first determined. The radius of the knot

$$R_{\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}}={\max }_i|{\mathit{r}}_i-{\mathit{r}}_{\mathrm{C}\mathrm{O}\mathrm{M}}|$$ 5

was taken as the maximum distance of any bead in the knot core from its center of mass. Assuming an enclosing spherical volume

$$V_{\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}}=\frac{4}{3}\pi R_{\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}}^3$$ 6

the local knot density was calculated as

$${\rho }_{\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}}=\frac{N_{\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}}}{V_{\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}}}$$ 7

where N _knot is the number of beads comprising the knot core identified by the sliding-window algorithm. For comparison between AA and CG simulations, the density shown here is normalized by its initial value.

For detailed structural analysis, the shape anisotropy of the knot core was computed from the gyration tensor, G . To quantify the three-dimensional shape of the knot core, we computed G for all particles within the knot core in each trajectory frame

$$\mathit{G}=\frac{1}{N}\sum _{i=1}^N({\mathit{r}}_i-{\mathit{r}}_{\mathrm{c}\mathrm{o}\mathrm{m}}){({\mathit{r}}_i-{\mathit{r}}_{\mathrm{c}\mathrm{o}\mathrm{m}})}^T$$ 8

where r _i is the position of particle i, N is the number of particles in the knot core, and r _com is the center of mass of the core. The eigenvalues of G , λ₁ ≤ λ₂ ≤ λ₃, were then used to calculate the shape anisotropy

$${\kappa }^2=1-3\frac{{\lambda }_1{\lambda }_2+{\lambda }_2{\lambda }_3+{\lambda }_3{\lambda }_1}{{({\lambda }_1+{\lambda }_2+{\lambda }_3)}^2}$$ 9

This dimensionless measure κ² ranges from 0 for a perfectly spherical distribution to 1 for a highly elongated, anisotropic structure. The time-resolved κ²(t) profiles were analyzed to characterize temporal changes in knot shape and compared across CG and atomistic models.

To quantify local structural organization within the knotted region of polymers and proteins, we computed the three-dimensional (3D) isotropic structure factor, S(q), of the knot core. The 3D structure factor was computed as

$$S(q)=\frac{1}{N}{⟨\sum _{j,k=1}^N\exp [-i\mathbf{q}·({\mathbf{r}}_j-{\mathbf{r}}_k)]⟩}_{\mathbf{q}}$$ 10

where r _j and r _k denote the positions of particles within the knot core, N is the number of particles, and the average ⟨·⟩ _q is taken over 50 randomly oriented q-vectors for each magnitude q. A total of 100 q-values were sampled in the range of 0.1–10 Å^–1, providing a rotationally averaged, isotropic measure of local structural correlations within the knot.

The resulting S(q) profiles complement geometric measures such as the knot size, knot position, and shape anisotropy, providing a quantitative descriptor of the internal knot organization across simulation models and conditions.

3. Results and Discussion

3.1. CG Exhibits Accelerated Unknotting by Redistributing Local Stress of the Polymer

We begin by quantifying the characteristic time scales of knot relaxation in atomistic and CG representations of PE chains containing tight knots. During relaxation, the initially tightened knot progressively expands and slides along the polymer contour through local chain rearrangements until no localized compact knot core can be identified according to the criterion described in the Methods section, marking the unknotting time. For each model, the unknotting time was determined from ensemble-averaged trajectories under equilibrium conditions using consistent topological and structural criteria (see Methods). As shown in Figure A, CG simulations exhibit systematically shorter unknotting times than their atomistic counterparts. This acceleration is accompanied by a substantially larger effective diffusion coefficient of the polymer center of mass in the CG model, reflecting an enhanced overall mobility.

2.

Accelerated unknotting and stress redistribution in CG models. (A) Ensemble-averaged unknotting time ⟨t _u⟩ for AA (150C atoms) and CG (50 beads) PE single chains, together with the corresponding diffusion coefficient of the polymer atoms. The errors represent the standard deviation of the respective calculations averaged over multiple replica studies. (B) Representative contour-resolved stress profiles along the polymer backbone during knot relaxation for AA and CG models, normalized by the respective maximum force.

To elucidate the microscopic origin of this difference, we next examine the contour-resolved distribution of local mechanical stress along the polymer backbone during knot relaxation (see Methods). The profiles correspond to the time-averaged von Mises stress obtained from representative knot-relaxation trajectories. During these trajectories, the knot evolves naturally and is not constrained to a fixed contour position. Accordingly, the profiles are intended to illustrate the qualitative redistribution of local stress during knot relaxation rather than the instantaneous stress associated with a fixed knot location. Localized stress concentrations near knot entry and exit regions have long been recognized as a defining feature of polymer knot mechanics and were shown in pioneering atomistic simulations to govern the mechanical response and eventual failure of knotted polymer strands under tension. − Here, we extend this concept to equilibrium knot relaxation by comparing the time-averaged von Mises stress derived from the per-particle virial stress tensor in atomistic and coarse-grained representations. Figure B compares the stress profiles for AA and CG models, resolved along the polymer contour and normalized by the respective maximum atomic force. In the atomistic simulations, stress is highly localized at the knot entry and exit points, leading to pronounced force differentials between the knotted region and the polymer tails. These localized stress concentrations remain consistently pronounced and act as mechanical bottlenecks that hinder local rearrangements required for knot relaxation.

In contrast, the CG model displays a markedly more homogeneous stress distribution. The force differences between the knotted region and the tails are significantly reduced, and the stress is distributed more broadly along the polymer backbone near the knot. As a consequence, the knot entry and exit regions in the CG model experience lower peak forces that are shared among more interaction sites. This redistribution of stress facilitates smoother sliding and deformation of the knot, thereby enabling a more rapid relaxation.

While the earlier studies − established stress localization as a characteristic signature of knotted polymers under mechanical loading, our results demonstrate that coarse-graining substantially redistributes these stress concentrations during spontaneous knot relaxation. Although the locations of the stress maxima remain associated with the knotted region, their magnitude becomes significantly more homogeneous in the CG representation, providing a mechanistic explanation for the accelerated unknotting dynamics. The stress delocalization observed in the CG model can be directly attributed to its reduced spatial resolution, which leads to a smoother distribution of forces along the polymer contour. Similar smoothing of local interactions and force fluctuations has been widely reported in CG polymer models, ,,,− where the elimination of high-frequency intramolecular degrees of freedom leads to more homogeneous local dynamics. In this context, the reduced force contrasts and broader stress distribution observed here provide a clear mechanistic explanation for the accelerated knot relaxation in the CG model.

Taken together, these results demonstrate that the accelerated unknotting dynamics in CG simulations originate from a redistribution of local mechanical stress along the polymer backbonespecifically, the suppression of localized force concentrations at the knot entry and exit pointsrather than from enhanced global chain mobility alone. This mechanistic distinction is important because it indicates that the observed time scale differences cannot be explained solely by a uniform rescaling of dynamics but are also associated with differences in how local mechanical stress is represented in the atomistic and coarse-grained models. We note that the AA and CG simulations were performed using the same Langevin thermostat, which provides temperature control together with an implicit dissipative environment. Nevertheless, the absolute simulation times of the two representations are not expected to be directly equivalent because coarse-graining averages over several methylene units, yielding smoother effective interaction potentials, fewer internal degrees of freedom, and consequently faster effective dynamics. Thus, the larger center-of-mass diffusion coefficient and shorter unknotting times observed for the CG model reflect the combined effects of its intrinsically accelerated dynamics and the redistribution of local mechanical stress described above. Having established this microscopic origin of the time scale differences, we next examine whether coarse-graining preserves the length-dependent scaling laws of knot relaxation, which would indicate that the global relaxation behavior remains intact despite these local differences.

3.2. Length-Dependent Scaling Is Preserved across Resolutions

To assess whether coarse-graining alters the global topological behavior of knotted polymers, we next examined the scaling of knot relaxation with chain length (Figure A–D). Specifically, we analyzed the ensemble-averaged unknotting time (t _u ) as a function of the chain length outside the knot, M – < M _knot > for both AA and CG PE chains (Figure A,C), where M is the total polymer length and < M _knot > is the average contour length occupied by the knot. Despite the faster absolute dynamics observed in the CG model, the length dependence follows a clear power-law relationship in both representations. Fitting < t _u > ∼ ${(M-⟨M_{\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}}⟩)}^{\alpha }$ yields exponents α _AA = 1.92 ± 0.05 and α _CG = 1.86 ± 0.04, which are mutually consistent within uncertainty and close to the value of 2 expected for diffusion-controlled unknotting of flexible chains, ,,, indicating that coarse-graining preserves the fundamental scaling behavior of unknotting dynamics.

3.

Length scaling of knot relaxation in CG and AA PE single chains. (A,C) Scaling of the ensemble-averaged unknotting time ⟨t _u ⟩ with the effective contour length outside the knot, M – < knot > for CG (A) and AA (C) simulations. Power-law scaling exponents are indicated in the respective panels. (B,D) Log–log plots of the normalized average knot size, < M _knot >/M _tight, as a function of total chain length M for CG (B) and AA (D) simulations. Scaling exponents are provided in the corresponding plots.

The preservation of the scaling exponent despite faster absolute dynamics suggests that coarse-graining primarily affects local relaxation events, such as stress redistribution and segmental sliding, while the global diffusive or topologically constrained motion of the chain remains governed by the chain length. This separation of time scales allows CG models to accelerate unknotting without altering the fundamental polymer physics dictating how knot relaxation scales with chain length.

We further investigated the scaling of the knot size relative to the total chain length by computing the average knot contour length < M _knot > normalized by the tight-knot reference M _tight. Figure B,D shows a log–log plot of < M _knot > /M _tight versus M, revealing a power-law relationship < M _knot > /M _tight ∼ M ^β with exponents β _AA = 0.61 ± 0.03 and β _CG = 0.67 ± 0.03. This increase in knot size with chain length is consistent with prior studies of linear and knotted polymers, where larger chains allow the knot to explore a broader contour while retaining a tight core. The similar exponents observed for AA and CG models indicate that coarse-graining preserves this balance between local tightening and global chain accommodation.

Taken together, these results demonstrate that while coarse-graining accelerates the absolute time scales of knot relaxation, it preserves both the length-dependent scaling of unknotting times and the relative growth of knot size. These observations align with previous computational studies ,,, reporting power-law scaling of unknotting times with exponents near 2 for flexible chains under melt-like conditions and extend these findings by explicitly comparing CG and atomistic representations. The agreement in scaling exponents indicates that CG simulations provide a reliable framework for exploring knot dynamics in systems where atomistic simulations are computationally prohibitive, including long-chain polymers and biologically relevant knotted proteins.

3.3. Global Knot Relaxation Mechanism Is Preserved, Time Scales Differ

To examine whether coarse-graining preserves the overall mechanism of knot relaxation, we analyzed a set of time-resolved observables for both the AA and CG PE chains. These include the normalized knot size, maximum knot size, knot position along the polymer contour, and polymer radius of gyration (R _g ).

Figure A shows representative time profiles of the normalized knot sizes for AA and CG simulations. Despite the pronounced difference in absolute relaxation time scales, the temporal evolution of the knot size remains largely similar across the two representations. This indicates that the present coarse-grained representation preserves the overall topological extent and mode of knot relaxation while exhibiting shorter characteristic relaxation times. Consistent with this observation, both the time-averaged knot size and the maximum knot size are comparable between AA and CG models (Figure S2A,B), supporting the conclusion that coarse-graining does not significantly alter the average tightness or spatial extent of the knot.

4.

Comparison of knot relaxation mechanisms and time scales in AA and CG PE single chains. (A) Time evolution of the normalized knot size (actual knot size divided by the respective tight knot size) as a function of normalized time (t/⟨t _u⟩, where ⟨t _u⟩ is the ensemble-averaged unknotting time) for representative AA (150C atoms) and CG (50 beads) trajectories. (B) Autocorrelation times of knot size, knot position along the contour, and polymer R _g are shown in bar plots. The error bars represent the corresponding standard deviation computed from all replica simulations. Additional comparisons of time-averaged and maximum knot size are provided in the Supporting Information.

In addition, the time profiles of the normalized knot position along the polymer contour (Figure S3A) and the normalized polymer R _g (Figure S3B) follow qualitatively similar progressions in both AA and CG simulations. This overlap suggests that the relative progression of local knot motion and global chain conformational changes remains similar across the two modeling resolutions, further supporting the preservation of the overall relaxation mechanism.

To quantify the characteristic time scales associated with these dynamical processes, we computed autocorrelation functions for the knot size, knot position, and polymer R _g (Figure B). These quantities are used to compare the relative relaxation behaviors within each model. Because AA and CG representations possess different intrinsic dynamical time scales, the absolute autocorrelation times should not be interpreted as direct one-to-one dynamical comparisons between the two resolutions. In the CG simulations, all three observables exhibit comparable autocorrelation times, indicating a largely coherent relaxation of local knot degrees of freedom and global polymer conformations. In contrast, the AA simulations reveal a clear separation of time scales: while the autocorrelation times of knot size and knot position remain similar, the relaxation of the polymer R _g occurs on a noticeably longer time scale.

This distinction implies that in atomistic representations, local knot rearrangements can relax more rapidly than the global polymer conformation, revealing a partial decoupling between local topological rearrangements and global chain conformational relaxation. Such decoupling is significantly reduced in the CG model, where smoother local interactions and enhanced mobility promote more synchronized relaxation across length scales. Importantly, although the relative separation of the characteristic relaxation times differs between the AA and CG models, both representations characterize the evolution of the knot size, knot position, and global chain conformation during knot relaxation. The principal difference is that these processes become more strongly coupled in the CG model, leading to more synchronized relaxation across the observables.

Taken together, these results demonstrate that CG models reproduce the dominant features of knot relaxation, including the qualitative progression of knot size, knot position, and global chain conformational changes, while systematically accelerating the associated time scales. Although the relative separation between the characteristic relaxation times depends on the molecular resolution, both AA and CG models exhibit majorly similar evolution of the measured knot observables. The coarse-grained model exhibits a greater degree of coupling between local knot rearrangements and global chain conformational relaxation, whereas the atomistic model displays a more pronounced separation between these processes. In atomistic simulations, these processes exhibit a partial time scale separation that is compressed in CG models, consistent with the smoother effective interaction landscape and reduced effective friction of the coarse-grained models employed in the present study. This smoothing does not alter the dominant relaxation pathway but does suppress finer dynamical features, which we examine in the following section through the lens of the local knot structure and transient intermediates.

3.4. Local Knot Dynamics and Intermediates Are Resolution-Sensitive

While the preceding sections demonstrate that CG models reproduce the dominant features and qualitative progression of knot relaxation, local structural details within the knot may be more sensitive to the level of the molecular resolution. To assess this, we examined the coupled evolution of knot compactness and spatial extent using a parametric representation of the normalized local knot density (see Methods) as a function of the normalized knot radius of gyration R _g. Figure A shows these trajectories for AA and CG simulations, with color indicating normalized time along the unknotting pathway.

5.

Resolution-dependent local knot structure and intermediate conformations in AA and CG PE single chains. (A) Parametric plot of normalized knot local density versus normalized R _g (both normalized by their initial values), colored by normalized time, for AA (150 carbon atoms) and CG (50 beads) simulations of single PE chains. Time is normalized by the respective unknotting time. (B) Time evolution of knot shape anisotropy for AA and CG models. Time is normalized by the respective unknotting time. (C) Relative change in the knot structure factor during relaxation for AA and CG simulations, shown as bar plots.

Both AA and CG trajectories originate from a common region of high local density and small knot R _g , corresponding to the initially tight knot. As relaxation proceeds, the two models explore distinct regions of the density–R _g state space, revealing that coarse-graining not only accelerates relaxation but alters the structural pathway through which the knot loosens. In the CG simulations, the trajectory evolves smoothly and monotonically toward lower density and larger R _g , reflecting a gradual, continuous loosening in which knot compactness and spatial expansion decrease in concert. The color progression along the CG trajectory confirms that the system advances steadily toward the loosened state without pronounced reversals or trapping.

In contrast, AA simulations exhibit a more heterogeneous trajectory. Although the initial direction of relaxation is similar, the AA trajectory subsequently deviates from the monotonic CG pathway by revisiting regions of comparatively higher local density at similar knot R _g. These transient excursions reflect short-lived local structural rearrangements within the knot core rather than a reversal of the overall relaxation process (representative intermediate conformations are shown in Figure S4). Consistent with this interpretation, color-coded time progression indicates that AA trajectories access these states at intermediate and late stages of relaxation, suggesting the presence of transiently stabilized local configurations within the knot. These features are absent or strongly suppressed in the CG models. Overall, the AA simulations sample systematically higher normalized local knot densities than the CG model, reflecting finer-scale packing and localized structural rearrangements that are averaged out in the coarse-grained representation.

Further insight into these differences is provided by the time evolution (Figures B and S5) of the knot shape anisotropy (see eqs and and Methods for details). In CG simulations, the anisotropy remains relatively constant throughout relaxation, consistent with smooth deformation of the knot structure. By contrast, AA simulations show pronounced, recurring modulations in anisotropy, indicative of intermittent shape rearrangements and the transient emergence of metastable knot conformations driven by localized stress fluctuations that are smoothed out in the CG representation. These fluctuations correlate with the nonmonotonic features observed in the knot density–R _g trajectories and point to localized structural reorganization events within the knot core.

The enhanced structural variability in AA simulations is further supported by the relative change in the knot 3D structure factor (see eq and Methods for details) during relaxation (Figure C), which is markedly larger for AA than for CG, consistent with greater structural reorganization within the knot core at atomistic resolution. Together, these observations indicate that while coarse-graining captures the dominant topological relaxation pathway, it suppresses finer-scale structural intermediates that emerge at atomistic resolution and are smoothed out in the CG representation.

Overall, these results demonstrate that CG models reproduce the dominant topological relaxation pathway but systematically suppress local structural fluctuations, transient intermediates, and intermittent shape rearrangements within the knot core. This distinction is particularly relevant when interpreting CG-based predictions of knot dynamics in contexts in which transient intermediates or local structural heterogeneity may play a functional role. Having established these resolution-dependent differences for isolated chains, we next examined whether they persist under the more complex dynamical and topological constraints of a polymer melt environment.

3.5. Knot Relaxation in Polymer Melts across Resolutions

To assess whether the resolution-dependent trends observed for isolated polymer chains persist under more crowded conditions, we extended our analysis to PE chains embedded in a polymer melt where a single knotted chain is surrounded by unknotted chains. This configuration isolates the dynamics of a single topological defect under realistic intermolecular crowding, while the surrounding unknotted chains introduce the topological and dynamic constraints of a dense polymer environment. Together, these features provide a stringent test of the robustness of the resolution-dependent distinctions established for isolated chains.

Figure A shows the time evolution of the knot shape anisotropy for the AA and CG simulations in the melt. As in the single-chain case, the CG trajectories exhibit a smooth and weakly fluctuating anisotropy profile throughout the relaxation process. In contrast, the AA simulations display pronounced temporal modulations with recurring deviations from the mean anisotropy. These features indicate the emergence of transient knot conformations in the atomistic melt that are not prominently captured by the CG model. The persistence of these fluctuations under melt conditions suggests that local knot rearrangements remain sensitive to molecular resolution, even in the presence of intermolecular crowding.

6.

Knot dynamics in PE melts across models. (A) Time profiles of knot shape anisotropy for AA (150C atoms) and CG (50 beads) PE chains in a melt environment, where a single knotted chain is embedded among unknotted chains. (B) Autocorrelation times of knot size, knot position along the polymer contour, and polymer R _g for both AA and CG models in the melt. The inset shows the zoomed-in view.

Consistent trends are observed in the time profiles of the polymer R _g (Figure S6A). In AA simulations, R _g exhibits intermittent fluctuations during knot relaxation, suggestive of transient trapping or slowed global conformational rearrangements. In contrast, the CG simulations display a comparatively smooth evolution of R _g , reflecting more continuous chain relaxation in the melt environment. The more intermittent relaxation observed in the atomistic melt likely reflects the combined influence of stronger local steric constraints and more heterogeneous intermolecular interactions, which transiently impede knot sliding and global chain rearrangement. In contrast, the smoother effective interactions in the present CG representation reduce these local barriers, allowing for more continuous knot motion and conformational relaxation.

To quantify the associated time scales, we computed autocorrelation functions for the knot size, knot position, and polymer R _g in the melt (Figure B). For both AA and CG models, the autocorrelation time of the knot size exceeds that of the knot position, indicating that knot relaxation in the melt proceeds predominantly via sliding of the knot along the chain, rather than through rapid changes in knot tightness. Furthermore, the relative variance in the knot position along the contour is greater in the melt than in the single-chain case for both AA and CG models (Figure S6B), indicating enhanced knot sliding driven by topological confinement from surrounding chainsa trend consistently captured across both resolutions. This behavior contrasts with the single-chain case, where the relaxation times of knot size and position were more closely matched, highlighting the influence of surrounding chains on knot dynamics.

Despite this shift in the relaxation mode, the relative comparison between CG and AA remains consistent. Across all observables, CG simulations exhibit systematically shorter autocorrelation times than their atomistic counterparts, indicating faster overall dynamics in the CG representation. Importantly, however, both models identify knot sliding along the polymer contour as the dominant relaxation mechanism in the melt, although the relative separation between the characteristic relaxation times depends on the molecular resolution.

Taken together, these results establish that the resolution-dependent distinctions identified for isolated chainsaccelerated relaxation and systematic suppression of structural intermittency in CG modelspersist robustly under melt conditions. Beyond confirming these trends, the melt environment introduces a qualitatively distinct relaxation mode: both AA and CG models reveal that knot relaxation in the melt is dominated by sliding along the chain contour rather than by changes in knot tightness, driven by topological confinement from surrounding chains. This shift in relaxation mode is consistently captured across resolutions, further validating the use of CG models for probing topologically constrained dynamics in dense polymer systems, while underscoring that atomistic resolution remains necessary for resolving the local structural fluctuations and transient intermediates that accompany this unknotting process.

3.6. Knot Dynamics in a Knotted Protein: Atomistic versus CG Representations

To further test the generality of the trends observed in synthetic polymer systems, we extended our analysis to a biologically relevant systemMJ0366 (PDB ID: 2EFV), the shortest known trefoil-knotted protein, modeled in both AA (CHARMM36) and CG (MARTINI 3) representations. This system introduces additional structural heterogeneity compared to homopolymeric chains, providing a stringent benchmark for evaluating how coarse-graining influences knot dynamics in a chemically and conformationally complex biomolecular context.

Figure A shows the time evolution of the distance between the protein segments forming the knot crossing, along with the corresponding knot size and knot position profiles (Figure S7A,B). In the CG simulations using the MARTINI 3 force field, the distance between the crossing segments (Figure A) increases rapidly, leading to early knot loosening and eventual unknotting (Figure B). In contrast, the atomistic simulations employing the CHARMM36 force field show no complete unknotting event over the full simulation duration of 500 ns, with the knot remaining topologically stable despite thermal fluctuations.

7.

Knot dynamics in a trefoil-knotted protein across models. (A) Time evolution of the distance between protein segments forming the knot crossing for CG (MARTINI 3) and AA (CHARMM36) simulations. Corresponding knot size and knot position profiles are provided in the Supporting Information. (B) Representation of the unknotting event observed in the CG simulation, showing the initial knotted protein structure and the final unknotted structure. The protein chain is colored along the contour using a red–gray–blue gradient. (C) Mean-squared displacement (MSD) of the protein for CG and AA simulations. (D) Root-mean-square fluctuation (RMSF) of individual residues in the CG and AA models.

Building on these observations, the time profiles of knot size and knot position (Figure S7) provide further mechanistic insights into these dynamics. In the CG simulations, the knot size (Figure S7A) exhibits pronounced temporal fluctuations, reflecting continuous loosening and reorganization of the knot core during relaxation. Concomitantly, the knot core undergoes substantial displacement along the protein contour (Figure S7B), leading to significant variations in the knot position over time. These combined features indicate active knot sliding and reshaping, which facilitate rapid relaxation and unknottedness. By contrast, AA simulations display markedly reduced fluctuations in both knot size and knot position, with the knot remaining spatially localized and maintaining a nearly constant size throughout the trajectory, consistent with a kinetically stable knotted state.

Consistent with the accelerated knot dynamics, the mean-squared displacement (MSD) of the protein center of mass is significantly larger in the CG simulations than that in the atomistic ones (Figure C), indicating enhanced global mobility. Complementarily, the root-mean-square fluctuation (RMSF) of individual residues is systematically higher in the CG model (Figure D), reflecting an increased local flexibility along the protein backbone. Together, the enhanced global mobility and increased local backbone flexibility in the present CG simulations provide a consistent picture in which the coarse-grained representation promotes both local knot rearrangement and global conformational exploration, facilitating the accelerated unknotting observed in Figure B.

These results reinforce the resolution-dependent trends established across synthetic polymer chains and polymer melts. CG simulations consistently exhibit accelerated local and global dynamics, facilitating knot loosening and positional rearrangements that remain inaccessible on standard atomistic time scales. The atomistic protein, by contrast, remains kinetically trapped in its knotted conformation throughout the simulation, reflecting the higher topological persistence resolved at the full atomic resolution. We note, however, that the differing force fields, molecular representations, and effective dynamical properties of the AA and CG models preclude a direct one-to-one comparison of their absolute time scales. Consequently, the present comparison is intended to characterize the knot relaxation behavior observed within each model rather than to establish direct equivalence between their thermodynamic stability or free-energy landscapes. Accordingly, the observed unknotting event in the CG simulation should not be interpreted as implying that the atomistic system would necessarily follow an identical relaxation pathway under simple dynamical rescaling. Rather, these results illustrate how the present coarse-grained representation enables efficient exploration of knot relaxation events that remain inaccessible within the atomistic simulation window, while atomistic simulations remain appropriate for resolving the detailed stability and structural integrity of the knotted state.

4. Conclusion

This study provides a unified mechanistic framework for understanding knot relaxation dynamics across synthetic polymers and proteins by systematically comparing AA- and CG-molecular simulations. Through analysis of single PE chains, polymer melts, and a trefoil-knotted protein, we demonstrate that CG representations investigated here exhibit substantially accelerated topological relaxation and conformational rearrangements while retaining the dominant features of topological relaxation ,, resolved at an atomistic resolution.

For isolated PE chains, coarse-graining retains the scaling relationships between unknotting time, knot size, and chain length , that are characteristic of flexible polymer topology. The reduced energetic roughness inherent to CG force fieldswell-documented in prior studies of polymer dynamicspromotes smoother stress redistribution along the backbone, enabling more continuous knot sliding and deformation, and thereby accelerating relaxation. , Consequently, CG models quantitatively reproduce the global relaxation mechanism while suppressing the transient structural intermediates and intermittent shape rearrangements prominently resolved in AA trajectories. These intermediates, manifested as transient knot conformations and intermittent structural rearrangements within the knot core, are effectively averaged out in the CG representation, consistent with the smoother configurational landscapes previously reported for CG polymer models. −

In polymer melts, the dominant relaxation mode shifts from a balance of knot tightening and sliding in single chains toward sliding-dominated relaxation driven by topological confinement from surrounding chainsa trend consistently captured by both AA and CG models. Nonetheless, the key distinctions between resolutions persist: CG simulations exhibit systematically faster dynamics and reduced structural intermittency, while AA simulations continue to reveal kinetically trapped, localized knot rearrangements − that are suppressed by coarse-graining. The robustness of these trends under melt conditions underscores the suitability of CG approaches for probing topologically constrained dynamics in dense polymeric environments while reinforcing the important role of atomistic resolution in characterizing local structural fluctuations.

Extension to a trefoil-knotted protein further highlights the practical advantages of CG simulations in accessing knot relaxation events that remain kinetically inaccessible on standard atomistic time scales. CG simulations enable early-stage knot loosening, active sliding along the protein contour, and eventual unknotting, whereas AA simulations capture the kinetic persistence of the knotted state over the simulated time scalereflecting the high topological stability that atomistic resolution resolves with greater fidelity. These outcomes are best understood as complementary: CG models efficiently map the relaxation landscape and identify dominant unknotting pathways, while atomistic simulationsoptionally augmented with enhanced sampling strategiesremain the appropriate choice for resolving local knot-level structural features and residue-scale dynamic heterogeneity within the knotted region.

Taken together, this work provides a quantitative and physically transparent comparison of AA and CG descriptions of knot dynamics across chemically and topologically diverse systems. By identifying where coarse-graining faithfully reproduces global relaxation mechanisms and where it systematically departs from atomistic behavior, our findings offer practical guidance for the design of multiscale simulation strategies. We note that the conclusions drawn here pertain to the specific coarse-grained representations employed in this work. The observed acceleration of knot relaxation and redistribution of local stress should not be interpreted as an unavoidable consequence of coarse-graining itself. In principle, coarse-grained models incorporating many-body or density-dependent interactions, more accurate free-energy representations, or non-Markovian dynamical treatments may recover aspects of the atomistic behavior more faithfully. Investigating how such advanced coarse-grained methodologies influence knot relaxation dynamics represents an interesting direction for future work. The complementary strengths of AA and CG approachesmechanistic detail versus sampling efficiencycan be leveraged jointly to address open questions in polymer entanglement dynamics, topological polymer processing, and the biological function of knotted proteins, where both local structural fidelity and long-time scale accessibility are required.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcb.6c04142.

• Normalized average knot size and maximum knot size detected across the models; analysis of knot position and R _g of polymer in single-chain and melt systems; and comparison of knot-related metrics for the knotted protein system at different model resolutions (PDF)

The authors declare no competing financial interest.

Published as part of The Journal of Physical Chemistry B special issue “John E. Straub Festschrift”.