The statistical mechanics of a long knotted collapsed polymer is determined by a free-energy with a knot-dependent subleading term, which is linked to the length of the shortest polymer that can hold such knot. The only other parameter depending on the knot kind is an amplitude such that relative probabilities of knots do not vary with the temperature $T$, in the limit of long chains. We arrive at this conclusion by simulating interacting self-avoiding rings at low $T$ on the cubic lattice, both with unrestricted topology and with setups where the globule is divided by a slip link in two loops (preserving their topology) which compete for the chain length, either in contact or separated by a wall as for translocation through a membrane pore. These findings suggest that in macromolecular environments there may be entropic forces with a purely topological origin, whence portions of polymers holding complex knots should tend to expand at the expense of significantly shrinking other topologically simpler portions.
We compare Monte Carlo simulations of knotted and unknotted polymers whose ends are connected to two parallel walls. The force $f$ exerted on the polymer is measured as a function of the separation $R$ between the walls. For unknotted polymers of several monomer numbers $N$, the product $fN^ν$ is a simple function of $R/N^ν$, where $ν\simeq 0.59$. By contrast, knotted polymers exhibit strong finite size effects which can be interpreted in terms of a new length scale related to the size of the knot. Based on this interpretation, we conclude that the number of monomers forming the knot scales as…
We investigate the Rouse dynamics of a flexible ring polymer with a prime knot. Within a Monte Carlo approach, we locate the knot, follow its diffusion, and observe the fluctuations of its length. We characterise a topological time scale, and show that it is related to a self-reptation of the knotted region. The associated dynamical exponent, $z_T=2.32\pm.1$, can be related to that of the equilibrium knot length distribution and determines the behaviour of several dynamical quantities.
We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, $R_G$, and that of the region containing the knot, $R_{G,k}$, are small compared to the distance b between the obstacles, the knot is weakly localised and $R_G$ scales as in a good solvent with an amplitude that depends on knot type. In an intermediate regime where $R_G > b > R_{G,k}$, the geometry of the polymer becomes branched. When $R_{G,k}…
Two types of average structures of a single knotted ring polymer are studied by Brownian dynamics simulations. For a ring polymer with N segments, its structure is represented by a 3N -dimensional conformation vector consisting of the Cartesian coordinates of the segment positions relative to the center of mass of the ring polymer. The average structure is given by the average conformation vector, which is self-consistently defined as the average of the conformation vectors obtained from a simulation each of which is rotated to minimize its distance from the average conformation vector. From e…
Inspired by how certain proteins "sense" knots and entanglements in DNA molecules, here we ask if there exist local geometric features that may be used as a read-out of the underlying topology of generic polymers. We perform molecular simulations of knotted and linked semiflexbile polymers and study four geometric measures to predict topological entanglements: local curvature, local density, local 1D writhe and non-local 3D writhe. We discover that local curvature is a poor predictor of entanglements. In contrast, segments with maximum local density or writhe correlate as much as 90% of the ti…
By performing Monte Carlo sampling of $N$-steps self-avoiding polygons embedded on different Bravais lattices we explore the robustness of universality in the entropic, metric and geometrical properties of knotted polymer rings. In particular, by simulating polygons with $N$ up to $10^5$ we furnish a sharp estimate of the asymptotic values of the knot probability ratios and show their independence on the lattice type. This universal feature was previously suggested although with different estimates of the asymptotic values. In addition we show that the scaling behavior of the mean squared radi…
We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modeled by a closed polygon drawn on the square diagonal lattice, with possible crossings describing pairs of strands of polymer passing on top of each other. Each polygon configuration can be viewed as the two- dimensional projection of a particular knot. We study numerically the statistics of large polygons with a fixed knot type, using a generalization of the BFACF algorithm for self-avoiding walks. This new algorithm incorporates both the displacement of crossings and the t…
The effects of entanglement in solutions and melts of unknotted ring polymers have been addressed by several theoretical and numerical studies. The system properties have been typically profiled as a function of ring contour length at fixed solution density. Here, we use a different approach to investigate numerically the equilibrium and kinetic properties of solutions of model ring polymers. Specifically, the ring contour length is maintained fixed, while the interplay of inter- and intra-chain entanglement is modulated by varying both solution density (from infinite dilution up to \approx 40…
Several nontrivial properties are shown for the mean square radius of gyration $R_K^2$ of ring polymers with a fixed knot type K. Through computer simulation, we discuss both finite-size and asymptotic behaviors of the gyration radius under the topological constraint for self-avoiding polygons consisting of N cylindrical segments with radius r. We find that the average size of ring polymers with a knot K can be much larger than that of no topological constraint. The effective expansion due to the topological constraint depends strongly on the parameter r which is related to the excluded volume…
The thermal properties of coarse grained knotted polymers containing two kinds of monomers $A$ and $B$ fluctuating in a solution are investigated on a simple cubic lattice using the Wang-Landau MC algorithm. These knots have a more complex phase diagram than knots formed by homopolymers, including the possible presence of metastable states. Two different setups are considered: i) charged block copolymers in a ion solution and ii) neutral copolymers with the $A$ monomers above and the $B$ monomers below the theta point. A precise interpretation of the peaks observed in the plots of the specific…
Numerical studies of the average size of trivially knotted polymer loops with no excluded volume are undertaken. Topology is identified by Alexander and Vassiliev degree 2 invariants. Probability of a trivial knot, average gyration radius, and probability density distributions as functions of gyration radius are generated for loops of up to N=3000 segments. Gyration radii of trivially knotted loops are found to follow a power law similar to that of self avoiding walks consistent with earlier theoretical predictions.
Bridging algorithms are global Monte Carlo moves which allow for an efficient sampling of single polymer chains. In this manuscript we discuss the adaptation of three bridging algorithms from lattice to continuum models, and give details on the corrections to the acceptance rules which are required to fulfill detailed balance. For the first time we are able to compare the efficiency of the moves by analyzing the occurrence of knots in globular states. For a flexible homopolymer chain of length N=1000, independent configurations can be generated up to two orders of magnitude faster than with sl…
An analysis of extensive simulations of interacting self-avoiding polygons on cubic lattice shows that the frequencies of different knots realized in a random, collapsed polymer ring decrease as a negative power of the ranking order, and suggests that the total number of different knots realized grows exponentially with the chain length. Relative frequencies of specific knots converge to definite values because the free energy per monomer, and its leading finite size corrections, do not depend on the ring topology, while a subleading correction only depends on the crossing number of the knots.
Semiflexible polymer models are widely used as a paradigm to understand structural phases in biomolecules including folding of proteins. Since stable knots are not so common in real proteins, the existence of stable knots in semiflexible polymers has not been explored much. Here, via extensive replica exchange Monte Carlo simulation we investigate the same for a bead-stick and a bead-spring homopolymer model that covers the whole range from flexible to stiff. We establish the fact that the presence of stable knotted phases in the phase diagram is dependent on the ratio $r_b/r_{\rm{min}}$ where…
We introduce and implement a Monte Carlo scheme to study the equilibrium statistics of polymers in the globular phase. It is based on a model of "interacting elastic lattice polymers" and allows a sufficiently good sampling of long and compact configurations, an essential prerequisite to study the scaling behaviour of free energies. By simulating interacting self-avoiding rings at several temperatures in the collapsed phase, we estimate both the bulk and the surface free energy. Moreover from the corresponding estimate of the entropic exponent $α-2$ we provide evidence that, unlike for swollen…
Using a combination of the replica-exchange Monte Carlo algorithm and the multicanonical method, we investigate the influence of bending stiffness on the conformational phases of a bead-stick homopolymer model and present the pseudo-phase diagram for the complete range of semi-flexible polymers, from flexible to stiff. Although a simple model, we observe a rich variety of conformational phases, reminiscent of conformations observed for synthetic polymers or biopolymers. Changing the internal bending stiffness, the model exhibits different pseudo phases like bent, hairpin or toroidal. In partic…
A compressed knotted ring polymer in a confining cavity is modelled by a knotted lattice polygon confined in a cube in ${\mathbb Z}^3$. The GAS algorithm [17] is used to sample lattice polygons of fixed knot type in a confining cube and to estimate the free energy of confined lattice knots. Lattice polygons of knot types the unknot, the trefoil knot, and the figure eight knot, are sampled and the free energies are estimated as functions of the concentration of monomers in the confining cube. The data show that the free energy is a function of knot type at low concentrations, and (mean-field) F…
Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either "local" or "non-local" and, for several ring polymer models, we provide both theoretical and numerical evidence that, at equilibrium, the non-local configurations are more likely than the local ones. These characterisations are based on a standard approach for measuring the "size" of a knot within a knotted polymer chain. The method involves associating knot-types to subarcs of the chain, and then identifying a knotted subarc with minimal arclength; this arclength i…
A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of $N$ segments follows a decaying exponential form, $ \sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to loo…
We study numerically the tightness of prime flat knots in a model of self-attracting polymers with excluded volume. We find that these knots are localised in the high temperature swollen regime, but become delocalised in the low temperature globular phase. Precisely at the collapse transition, the knots are weakly localised. Some of our results can be interpreted in terms of the theory of polymer networks, which allows to conjecture exact exponents for the knot length probability distributions.
We show that the collapsed globular phase of a polymer accommodates a scale-free incompatibility graph of its contacts. The degree distribution of this network is found to decay with the exponent $γ= 1/(2-c)$ up to a cut-off degree $d_c \propto L^{2-c}$, where $c$ is the loop exponent for dense polymers ($c=11/8$ in two dimensions) and $L$ is the length of the polymer. Our results exemplify how a scale-free network (SFN) can emerge from standard criticality.
A self-consistent-field theory (SCFT) in the grand canonical ensemble formulation is used to study transitions in a helix-coil multiblock copolymer globule. The helices are modeled as stiff rods. In addition to the established coil-globule transition we show for the first time that, even without explicit rod-rod alignment interaction, the system undergoes a transition to a nematic liquid-crystalline (LC) globular state. The LC-globule formation is driven by the hydrophobic helical segment attraction and the anisotropy of the globule surface energy. The full phase diagram of the copolymer was c…
The interplay of geometrical and topological entanglement in semiflexible knotted polymer rings confined inside a spherical cavity is investigated using advanced numerical methods. By using stringent and robust algorithms for locating knots, we characterize how the knot length lk depends on the ring contour length, Lc and the radius of the confining sphere, Rc . In the no- and strong- confinement cases we observe weak knot localization and complete knot delocalization, respectively. We show that the complex interplay of lk, Lc and Rc that seamlessly bridges these two limits can be encompassed …
The elastic interactions between objects embedded in a nematic liquid crystal are usually caused by the average distorsion-rather than by the fluctuations-of the nematic orientational field. We argue that for sufficiently small particles, the nematic-mediated interaction originates purely from the fluctuations of the nematic director. This Casimir interaction decays as d^(-6), d being the distance between the particles, and it dominates van der Waals interactions close to the isotropic-to-nematic transition. Considering the nematic as a polymer solvent, we show that the onset of this Casimir i…
We study the behavior of a flexible polymer chain in the presence of a low-molecular weight solvent in the vicinity of a liquid-gas critical point within the framework of a self-consistent field theory. The total free energy of the dilute polymer solution is expressed as a function of the radius of gyration of the polymer and the average solvent number density within the gyration volume at the level of the mean-field approximation. Varying the strength of attraction between polymer and solvent we show that two qualitatively different regimes occur at the liquid-gas critical point. In case of w…
We construct a tractable model to describe the rate at which a knotted polymer is ejected from a spherical capsid via a small pore. Knots are too large to fit through the pore and must reptate to the end of the polymer for ejection to occur. The reptation of knots is described by symmetric exclusion on the line, with the internal capsid pressure represented by an additional biased particle that drives knots to the end of the chain. We compute the exact ejection speed for a finite number of knots L and find that it scales as 1/L. We also construct a continuum theory for many knots that matches …
In this work a new strategy is proposed in order to build analytic and microscopic models of fluctuating polymer rings subjected to topological constraints. The topological invariants used to fix these constraints belong to a wide class of the so-called numerical topological invariants. For each invariant it is possible to derive a field theory that describes the statistical behavior of knotted and linked polymer rings following a straightforward algorithm. The treatment is not limited to the partition function of the system, but it allows also to express the expectation values of general obse…
We have used Langevin dynamics to simulate the forced translocation of linked polymer rings through a narrow pore. For fixed size (i.e. fixed number of monomers) the translocation time depends on the link type and on whether the rings are knotted or unknotted. For links with two unknotted rings, the crossings between the rings can slow down the translocation and are responsible for a delay as the crossings pass through the pore. The results fall on a set of relatively smooth curves for different link families with the translocation time not always increasing with crossings number within the sa…
We propose a potential for wormlike polymer chains which can be used to model the low-temperature conformational structures. We successfully reproduced helix ground states up to 6.5 helical loops, using the multicanonical Monte Carlo simulation method. We demonstrate that the coil-helix transition involves four distinct phases: coil(gaslike), collapsed globular(liquidlike), and two helical phases I and II (both solidlike). The helix I phase is characterized by a helical structure with dangling loose ends, and the helix II phase corresponds to a near perfect helix ordering in the entire crystal…
The conformational complexity of linear polymers far exceeds that of point-like atoms and molecules. Polymers can bend, twist, even become knotted. Thus they may also display a much richer phase structure than point particles. But it is not very easy to characterize the phase of a polymer. Essentially, the only attribute is the radius of gyration. The way how it changes when the degree of polymerization becomes different, and how it evolves when the ambient temperature and solvent properties change, discloses the phase of the polymer. Moreover, in any finite length chain there are corrections …
We give two different, statistically consistent definitions of the length l of a prime knot tied into a polymer ring. In the good solvent regime the polymer is modelled by a self avoiding polygon of N steps on cubic lattice and l is the number of steps over which the knot ``spreads'' in a given configuration. An analysis of extensive Monte Carlo data in equilibrium shows that the probability distribution of l as a function of N obeys a scaling of the form p(l,N) ~ l^(-c) f(l/N^D), with c ~ 1.25 and D ~ 1. Both D and c could be independent of knot type. As a consequence, the knot is weakly loca…
A long flexible neutral polymer chain immersed in a poor solvent and interacting with an impenetrable attractive surface exhibits a phase known as surface attached globule ({\bf SAG}) in addition to other adsorbed and desorbed phases. In the thermodynamic limit, the {\bf SAG} phase has the same free energy per monomer as the globular phase, and the transition between them is a surface transition. We have investigated the phase diagrams of such a chain in both two- and three- dimensions and calculated the distribution of monomers in different domains of the phase diagram.
A self-consistent variational theory for globules based on the uniform expansion method is presented. This method, first introduced by Edwards and Singh to estimate the size of a self-avoiding chain, is restricted to a good solvent regime, where two-body repulsion leads to chain swelling. We extend the variational method to a poor solvent regime where the balance between the two-body attractive and the three-body repulsive interactions leads to contraction of the chain to form a globule. By employing the Ginzburg criterion, we recover the correct scaling for the $θ$-temperature. The introducti…
