Properties of knotted ring polymers. I. Equilibrium dimensions

Conformational properties of hybrid star-shaped polymers comprised of linear and ring arms

We study the influence of arm architecture on the conformational properties of hybrid star-shaped macromolecules called rosette polymers containing linear and ring grafts connected to a central branching point in a good solvent regime. We utilize analytical methods and molecular dynamics simulations to determine the estimates for the relative size ratios of these polymers with respect to linear chains and starlike polymers composed of the same number of solely linear arms and equal molecular weights. The results of numerical simulations corroborate our theoretical prediction that rosette polymers undergo conformational compactification with increasing functionality of grafted rings. Our results quantitatively describe the impact of the complex architecture of the molecules with excluded volume on their effective size measures.

Topological and physical links in soft matter systems

Linking, or multicomponent topological entanglement, is ubiquitous in soft matter systems, from mixtures of polymers and DNA filaments packedin vivoto interlocked line defects in liquid crystals and intertwined synthetic molecules. Yet, it is only relatively recently that theoretical and experimental advancements have made it possible to probe such entanglements and elucidate their impact on the physical properties of the systems. Here, we review the state-of-the-art of this rapidly expanding subject and organize it as follows. First, we present the main concepts and notions, from topological linking to physical linking and then consider the salient manifestations of molecular linking, from synthetic to biological ones. We next cover the main physical models addressing mutual entanglements in mixtures of polymers, both linear and circular. Finally, we consider liquid crystals, fluids and other non-filamentous systems where topological or physical entanglements are observed in defect or flux lines. We conclude with a perspective on open challenges.

Interface Counterion Localization Induces a Switch between Tight and Loose Configurations of Knotted Weak Polyacid Rings despite Intermonomer Coulomb Repulsions

Stochastic simulations have been used to investigate the conformational behavior of knotted weak polyacid rings as a function of pH. Different from the commonly expected ionization-repulsion-expansion scheme upon increasing pH, theoretical results suggest a nonmonotonic behavior of the gyration radius R_g². Polyelectrolyte recontraction at high ionization is induced by the weakening of Coulomb repulsion due to counterions (CIs) localizing at the interphase between the polymer and solvent, and the more marked it appears, the more complex is the knot topology. Compared with strong polyelectrolytic species of identical ionization, weak polyacids present tighter knots due to their ability to localize neutral monomers inside the tangled part. Increasing the solvent Bjerrum length enhances CIs localization, lowering the pH at which polyacids start decreasing their average size. A similar effect is also obtained by increasing the amount of "localizable" cations by adding salts.

Entropy-induced Separation of Binary Semiflexible Ring Polymer Mixtures in Spherical Confinement

Coarse-grained molecular dynamics simulations are used to investigate the conformations of binary semiflexible ring polymers (SRPs) of two different lengths confined in a hard sphere. Segregated structures of SRPs in binary mixtures are strongly dependent upon the number density of system (ρ), the bending energy of long SRPs (Kb, long), and the chain length ratio of long to short SRPs (α). With a low ρ or a weak Kb, long at a small ratio α, long SRPs are immersed randomly in the matrix of short SRPs. As ρ and bending energy of long SRPs (Kb, long) are increased up to a certain value for a large ratio α, a nearly complete segregation between long and short SRPs is observed, which can be further characterized by the ratio of tangential and radial components of long SRPs velocity. These explicit segregated structures of the two components in spherical confinement are induced by a delicate competition between the entropic excluded volume (depletion) effects and bending contributions.

Trefoil Knot Hydrodynamic Delocalization on Sheared Ring Polymers

The behavior of unknotted and trefoil-knotted ring polymers under shear flow is here examined by means of mesoscopic simulations. In contrast to most polymers, ring polymers in a hydrodynamic solvent at high shear rates do not get shortened in the vorticity direction. This is a consequence of the backflow produced by the interaction of the sheared solvent with the end-free polymer topology. The extended structures of the ring in the vorticity-flow plane, when they are aligned in a constant velocity plane, favor ring contour fluctuations. This variety of conformations largely suppresses the tank-treading type of rotation with extended conformations in favor of the tumbling type of rotations, where stretched and collapsed conformations alternate. The extension of trefoil knots is also enhanced, so that the knots become delocalized. We anticipate that these effects, which disappear in the absence of hydrodynamic interactions, will have a crucial impact on the rheological properties of concentrated ring solutions, and will also influence the behavior of more complicated systems such as mixtures of polymers with different topologies.

Knot Energy, Complexity, and Mobility of Knotted Polymers

The Coulomb energy E C is defined by the energy required to charge a conductive object and scales inversely to the self-capacity C, a basic measure of object size and shape. It is known that C is minimized for a sphere for all objects having the same volume, and that C increases as the symmetry of an object is reduced at fixed volume. Mathematically similar energy functionals have been related to the average knot crossing number 〈m〉, a natural measure of knot complexity and, correspondingly, we find E C to be directly related to 〈m〉 of knotted DNA. To establish this relation, we employ molecular dynamics simulations to generate knotted polymeric configurations having different length and stiffness, and minimum knot crossing number values m for a wide class of knot types relevant to the real DNA. We then compute E C for all these knotted polymers using the program ZENO and find that the average Coulomb energy 〈E C〉 is directly proportional to 〈m〉. Finally, we calculate estimates of the ratio of the hydrodynamic radius, radius of gyration, and the intrinsic viscosity of semi-flexible knotted polymers in comparison to the linear polymeric chains since these ratios should be useful in characterizing knotted polymers experimentally.

To Tie or Not to Tie? That Is the Question

In this review, we provide an overview of entangled proteins. Around 6% of protein structures deposited in the PBD are entangled, forming knots, slipknots, lassos and links. We present theoretical methods and tools that enabled discovering and classifying such structures. We discuss the advantages and disadvantages of the non-trivial topology in proteins, based on available data about folding, stability, biological properties and evolutionary conservation. We also formulate intriguing and challenging questions on the border of biophysics, bioinformatics, biology and mathematics, which arise from the discovery of an entanglement in proteins. Finally, we discuss possible applications of entangled proteins in medicine and nanotechnology, such as the chance to design super stable proteins, whose stability could be controlled by chemical potential.

Minimal Surfaces on Unconcatenated Polymer Rings in Melt

In order to quantify the effect of mutual threading on conformations and dynamics of unconcatenated and unknotted rings in the melt we computationally examine their minimal surfaces. We found a linear scaling of the surface area with the ring length. Minimal surfaces allow for an unambiguous algorithmic definition of mutual threading between rings. Based on it, we found that, although ring threading is frequent, majority of cases correspond to short loops. These findings explain why approximate theories that neglect threading are so unexpectedly successful despite having no small parameter justification. We also examine threading dynamics and identify the threading order parameter that reflects the ring diffusivity.

Anisotropic effective interactions and stack formation in mixtures of semiflexible ring polymers

By means of extensive computer simulations, we investigate the formation of columnar structures (stacks) in concentrated solutions of semiflexible ring polymers. To characterize the stacks we employ an algorithm that identifies tube-like structures in the simulation cell. Stacks are found both in the real system and in the fluid of soft disks interacting through the effective anisotropic pair potential derived for the rings [P. Poier et al., Macromolecules, 2015, 48, 4983-4997]. Furthermore, we investigate binary mixtures of cluster-forming and non-cluster-forming rings. We find that monodispersity is not a requirement for stack formation. The latter is found for a broad range of mixture compositions, though the columns in the mixtures exhibit important differences to those observed in the monodisperse case. We extend the anisotropic effective model to mixtures. We show that it correctly predicts stack formation and constitutes a significant improvement with respect to the usual isotropic effective description based only on macromolecular centers-of-mass.

Knotting dynamics of DNA chains of different length confined in nanochannels

Langevin dynamics simulations are used to characterize the typical mechanisms governing the spontaneous tying, untying and the dynamical evolution of knots in coarse-grained models of DNA chains confined in nanochannels. In particular we focus on how these mechanisms depend on the chain contour length, Lc, at a fixed channel width D = 56 nm corresponding to the onset of the Odijk scaling regime where chain backfoldings and hence knots are disfavoured but not suppressed altogether. We find that the lifetime of knots grows significantly with Lc, while that of unknots varies to a lesser extent. The underlying kinetic mechanisms are clarified by analysing the evolution of the knot position along the chain. At the considered confinement, in fact, knots are typically tied by local backfoldings of the chain termini where they are eventually untied after a stochastic motion along the chain. Consequently, the lifetime of unknots is mostly controlled by backfoldings events at the chain ends, which is largely independent of Lc. The lifetime of knots, instead, increases significantly with Lc because knots can, on average, travel farther along the chain before being untied. The observed interplay of knots and unknots lifetimes underpins the growth of the equilibrium knotting probability of longer and longer chains at fixed channel confinement.

An Anisotropic Effective Model for the Simulation of Semiflexible Ring Polymers

We derive and introduce anisotropic effective pair potentials to coarse-grain solutions of semiflexible ring polymers of various lengths. The system has been recently investigated by means of full monomer-resolved computer simulations, revealing a host of unusual features and structure formation, which, however, cannot be captured by a rotationally averaged effective pair potential between the rings' centers of mass [Bernabei M.; Soft Matter2013, 9, 1287]. Our new coarse-graining strategy is to picture each ring as a soft, penetrable disk. We demonstrate that for the short- and intermediate-length rings the new model is quite capable of capturing the physics in a quantitative fashion, whereas for the largest rings, which resemble flexible ones, it fails at high densities. Our work opens the way for the physical justification of general, anisotropic penetrable interaction potentials.

Computational study on the progressive factorization of composite polymer knots into separated prime components

Using Monte Carlo simulations and advanced knot localization methods, we analyze the length and distribution of prime components in composite knots tied on freely jointed rings. For increasing contour length, we observe the progressive factorization of composite knots into separated prime components. However, we observe that a complete factorization, equivalent to the "decorated ring" picture, is not obtained even for rings of contour lengths N ≃ 3 N(0), about tens of times the most probable length of the prime knots tied on the rings. The decorated ring hypothesis has been used in the literature to justify the factorization of composite knot probabilities into the knotting probabilities of their prime components. Following our results, we suggest that such a hypothesis may not be necessary to explain the factorization of the knotting probabilities, at least when polymers excluding volume is not relevant. We rationalize the behavior of the system through a simple one-dimensional model in which prime knots are replaced by slip links randomly placed on a circle, with the only constraint being that the length of the loops has the same distribution as that of the length of the corresponding prime knots.

Effects of Knots on Ring Polymers in Solvents of Varying Quality

We employ extensive computer simulations to investigate the conformations and the interactions of ring polymers under conditions of worsening solvent quality, in comparison with those for linear polymers. We determine the dependence of the Θ-temperature on knotedness by considering ring polymers of different topologies. We establish a clear decrease of the former upon changing the topology of the polymer from linear to an unknotted ring and a further decrease of the same upon introducing trefoil- or 5-fold knots but we find no difference in the Θ-point between the two knotted molecules. Our results are based on two independent methods: one considering the scaling of the gyration radius with molecular weight and one based on the dependence of the effective interaction on solvent quality. In addition, we calculate several shape-parameters of the polymers to characterize linear, unknotted, and knotted topologies in good solvents and in the proximity of the Θ-point. The shape parameters of the knotted molecules show an interesting crossover at a degree of polymerization that depends on the degree of knottedness of the molecule.

Effect of Bending Rigidity on the Knotting of a Polymer under Tension

A coarse-grained computational model is used to investigate how the bending rigidity of a polymer under tension affects the formation of a trefoil knot. Thermodynamic integration techniques are applied to demonstrate that the free-energy cost of forming a knot has a minimum at nonzero bending rigidity. The position of the minimum exhibits a power-law dependence on the applied tension. For knotted polymers with nonuniform bending rigidity, the knots preferentially localize in the region with a bending rigidity that minimizes the free energy.

Multiscale entanglement in ring polymers under spherical confinement

The interplay of geometrical and topological entanglement in semiflexible knotted polymer rings confined inside a spherical cavity is investigated by using advanced numerical methods. By using stringent and robust algorithms for locating knots, we characterize how the knot length l(k) depends on the ring contour length L(c) and the radius of the confining sphere R(c). In the no- and strong-confinement cases, we observe weak knot localization and complete knot delocalization, respectively. We show that the complex interplay of l(k), L(c), and R(c) that seamlessly bridges these two limits can be encompassed by a simple scaling argument based on deflection theory. The same argument is used to rationalize the multiscale character of the entanglement that emerges with increasing confinement.

Maximum entropy principle applied to semiflexible ring polymers

Based on the success of the maximum entropy principle (MEP) in the study of semiflexible treelike polymers [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)], it is of much interest to establish MEP's potential for general semiflexible polymers which contain loops. Here, we embark on this endeavor by considering discrete semiflexible polymer rings in a Rouse-type scheme. Now, for treelike polymers a beads-and-bonds (i.e., a discrete) picture is essential for an easy inclusion of branching points. Moreover, one may envisage (similar to our former work [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)]) to impose for each angle between two bonds a distinct stiffness condition. Working in this way leads already for a polymer ring to a complicated problem. Hence, we follow a reduced variational approach as applied earlier to polymer chains, in which a single Lagrange multiplier is used for each set of identical conditions imposed on topologically equivalent bonds and bonds' orientations. In this way, we obtain for the discrete ring an analytically closed form which involves Chebyshev polynomials. This expression turns out to lead to a series of solutions: Apart from the regular solution, several other solutions appear. One may be tempted to discard the other solutions, since for them the potential energy matrix is not positive definite. A more careful analysis based on topological features suggests, however, that such solutions can be assigned to rings displaying knots. Monte Carlo simulations which take excluded volume interactions into account agree with our interpretation.

Properties of knotted ring polymers. II. Transport properties

We have calculated the hydrodynamic radius R(h) and intrinsic viscosity [eta] of both lattice self-avoiding rings and lattice theta-state rings that are confined to specific knot states by our path-integration technique. We observe that naive scaling arguments based on the equilibrium polymer size fail for both the hydrodynamic radius and the intrinsic viscosity, at least over accessible chain lengths. (However, we do conjecture that scaling laws will nevertheless prevail at sufficiently large N.) This failure is attributed to a "double" cross-over. One cross-over effect is the transition from delocalized to localized knotting: in short chains, the knot is distributed throughout the chain, while in long chains it becomes localized in only a portion of the chain. This transition occurs slowly with increasing N. The other cross-over, superimposed upon the first, is the so-called "draining" effect, in which transport properties maintain dependence on local structure out to very large N. The hydrodynamic mobility of knotted rings of the same length and backbone structure is correlated with the average crossing number X of the knots. The same correlation between mobility and knot complexity X has been observed for the gel-electrophoretic mobility of cyclic DNA molecules.