Knotted Ring Research Articles

A γ-Ray-emitting Collisional Ring Galaxy System in Our Galactic Neighborhood

The astrophysical γ-ray photons carry the signatures of the violent phenomena happening on various astronomical scales in our Universe. This includes supernova remnants, pulsars, and pulsar wind nebulae in the Galactic environment and extragalactic relativistic jets associated with active galactic nuclei (AGN). However, ∼30% of the γ-ray sources detected with the Fermi Large Area Telescope lack multiwavelength counterpart association, precluding us from characterizing their origin. Here we report, for the first time, the association of a collisional ring galaxy system in our Galactic neighborhood (distance ∼10 Mpc), formed as a consequence of a smaller “bullet” galaxy piercing through a larger galaxy, as the multifrequency counterpart of an unassociated γ-ray source 4FGL J1647.5−5724. The system, also known as “Kathryn’s Wheel,” contains two dwarf irregular galaxies and an edge-on, late-type, spiral galaxy surrounded by a ring of star-forming knots. We utilized observations taken from the Neil Gehrels Swift observatory, Rapid ASKAP Continuum Survey, SuperCOSMOS Hα Survey, Dark Energy Survey, and VIsible MultiObject Spectrograph at Very Large Telescope to ascertain the association with 4FGL J1647.5−5724 and to explore the connection between the star-forming activities and the observed γ-ray emission. We found that star formation alone cannot explain the observed γ-ray emission, and additional contribution likely from the pulsars/supernova remnants or buried AGN is required. We conclude that arcsecond/subarcsecond-scale observations of this extraordinary γ-ray-emitting galaxy collision will be needed to resolve the environment and explore the origin of cosmic rays.

Thermal properties of knotted block copolymer rings with charged monomers subjected to short-range interactions.

The thermal properties of coarse-grained knotted copolymer rings fluctuating in a highly screening solution are investigated on a simple cubic lattice using the Wang-Landau Monte Carlo algorithm. The rings contain two kinds of monomers A and B with opposite charges that are subjected to short-range interactions. In view of possible applications in medicine and the construction of intelligent materials, it is shown that the behavior of copolymer rings can be tuned by changing both their monomer configuration and topology. We find several phase transitions depending on the monomer distribution. They include the expansion and collapse of the knotted polymer as well as rearrangements leading to metastable states. The temperatures at which these phase transitions are occurring and other features can be tuned by changing the topology of the system. The processes underlying the observed transitions are identified. In knots formed by diblock copolymers, two different classes of behaviors are detected depending on whether there is an excess of monomers of one kind or not. Moreover, we find that the most stable compact states are formed by copolymers in which units of two A monomers are alternated by units of two B monomers. Remarkably, these compact states are in a lamellar phase. The transition from the lamellar to the expanded state produces in the specific heat capacity a narrow and high peak that is centered at temperatures that are much higher than those of the peaks observed in all other monomer distributions.

Machine Learning Understands Knotted Polymers

Simulated configurations of flexible knotted rings confined inside a spherical cavity are fed into long-short term memory neural networks (LSTM NNs) designed to distinguish knot types. The results show that they perform well in knot recognition even if tested against flexible, strongly confined and therefore highly geometrically entangled rings. In agreement with the expectation that knots are delocalized in dense polymers, a suitable coarse-graining procedure on configurations boosts the performance of the LSTMs when knot identification is applied to rings much longer than those used for training. Notably, when the NNs fail, usually the wrong prediction still belongs to the same topological family of the correct one. The fact that the LSTMs are able to grasp some basic properties of the ring's topology is corroborated by a test on knot types not used for training. We also show that the choice of the NN architecture is important: simpler convolutional NNs do not perform so well. Finally, all results depend on the features used for input: surprisingly, coordinates or bond directions of the configurations provide the best accuracy to the NNs, even if they are not invariant under rotations (while the knot type is invariant). Other rotational invariant features we tested are based on distances, angles, and dihedral angles.

Molecular dynamics simulations of the monomer density profiles of knotted ring polymer chains confined in a slit of two parallel walls with one attractive and another repulsive surface.

We have used Molecular Dynamics simulations to obtain the monomer density profiles for real linear and ring polymer chains of 360 monomers length with different topological structures such as simple knots: 31, 61, 91, 10124, complex knots 313151 and twisted knots with n = 10 and n = 20 in a slit geometry of two parallel walls with one attractive and another repulsive surface. We have used Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software to perform simulations with the Verlet integration algorithm. The interactions between monomers were simulated as Lennard-Jones 12-6 potential, for bonds we have used Finitely Extensible Nonlinear Elastic (FENE) potential and the interaction with the walls was taken into account via Lennard-Jones 9-3 potential. We observed that topologically complex polymers have lower monomer density profiles near the attractive wall, but at some distance in the direction to the repulsive wall this tendency changes to the opposite. We showed that most complex twisted knots have two maxima in narrow slits. In the wide slits we do not observe such relation for twisted knots at higher temperatures. These results are important for better understanding the nature of the depletion forces which arise in a slit geometry of two parallel walls with one attractive and one repulsive wall.

Penetration of linear chains into semiflexible knotted rings in linear-ring blends

Threading of ring chains by linear chains has a great effect on the long-time dynamics of the ring chains in linear-ring blends. In this paper, we performed coarse-grained molecular dynamics simulations on the mixtures of linear and ring chains to investigate the effects of chain rigidity on the penetration of knotted rings threaded by linear chains. As the chain rigidity Kb of rings increases, non-trivial rings exhibit a non-monotonous dependence of the chain-penetration on chain rigidity Kb, and the semiflexible knotted rings with Kb = 10 have the maximum chain-penetration ability. This anomalous result is due to the two competing effects: As the chain rigidity of rings increases, one is that the swollen of rings can create more free space in favour of chain penetration, and another is that the formation of ordered clusters for rings in which rings tend to get parallel to each other can strongly hinder linear chains to penetrate into rings. Meanwhile, the transition from disordered to ordered phase for the trivial rings in linear-ring blends is a quite sensitive dependence on the chain rigidity of rings, resulting in a sudden transition at Kb = 12.

The application of numerical topological invariants in simulations of knotted rings: A comprehensive Monte Carlo approach

In this work, a general Monte Carlo framework is proposed for applying numerical knot invariants in simulations of systems containing knotted one-dimensional ring-shaped objects like polymers and vortex lines in fluids, superfluids or other quantum liquids. A general prescription for smoothing the sharp corners appearing in discrete knots consisting of segments joined together is provided. Smoothing is very important for the correct evaluation of numerical knot invariants. A discrete version of framing is adopted in order to eliminate singularities that are possibly arising when computing the invariants. The presented algorithms for smoothing, eliminating potentially dangerous singularities and speeding up the calculations are quite general and can be applied to any discrete knot defined off- or on-lattice. This is one of the first attempts to use numerical knot invariants in order to avoid potential topology breakings during the sampling process taking place in computer simulations, in which millions of knot conformations are randomly generated. As an application, the energy domain of knotted polymer rings subjected to short-range interactions is studied using the so-called Vassiliev knot invariant of degree 2.

Autonomous topological time crystals and knotty molecular motors

We show that topology is a very effective tool, to construct classical Hamiltonian time crystals. For this we numerically analyze a general class of time crystalline Hamiltonians that are designed to model the dynamics of molecular closed strings. We demonstrate how the time crystalline qualities of a closed string are greatly enhanced when the string becomes knotted. The Hamiltonians that we investigate include a generalized Kratky–Porod wormlike chain model in combination with long range Coulomb and Lennard–Jones interactions. Such energy functions are commonplace in coarse grained molecular modeling. Thus we expect that physical realizations of Hamiltonian time crystals can be constructed in terms of knotted ring molecules.

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 Rg2. 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.

The free energy of compressed lattice knots

A compressed knotted ring polymer in a confining cavity is modelled by a knotted lattice polygon confined in a cube in . The GAS algorithm (Janse van Rensburg and Rechnitzer 2011 J. Knot Theol. Raman 20 1145–71) 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) Flory–Huggins theory (Flory 1942 J. Chem. Phys. 10 51–61; Huggins 1942 J. Phys. Chem. 46 151–8) is used to model the free energy as a function of monomer concentration. The Flory interaction parameter of knotted lattice polygons in is also estimated.

The free energy of compressed lattice knots

Abstract IOP Publishing has withdrawn this article upon the author's request due to several issues with the mathematical content. The osmotic pressure of monomers in a knotted ring polymer in a confining cavity is modelled by a lattice polygon confined in a cube in ${\mathbb Z}^3$. These polygons can be knotted and are called lattice knots. In this paper the GAS algorithm \cite{JvRR11} is used to estimate the free energy of lattice knots of knot types the unknot, the trefoil knot, and the figure eight knot, as a function 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) Flory-Huggins theory \cite{Flory42,H42} is used to model the free energy as a function of monomer concentration. The Flory interaction parameter of lattice polygons in ${\mathbb Z}^3$ is also estimated. At critical values of the concentration the osmotic pressure may vanish, and these critical concentrations, suitably rescaled, is dependent on knot type.

Hydrodynamics and Filtering of Knotted Ring Polymers in Nanochannels

We study the hydrodynamic transport of knotted ring polymers through modulated channels, establishing that the transport velocity is strongly dependent on the ring topology for Peclet numbers smaller than unity. As soon as convection dominates, transport properties become insensitive to the presence and type of knots. We identify two distinct modes of transport, corresponding to the motion being led by the knotted or unknotted portions of the ring, most surprisingly without impact on separation efficiency. The modes can be selected by the channel geometry, and this could be harnessed to design nanofluidic devices for the continuous topological sorting of entangled biopolymers.

Topological entanglement of interlocked knotted-unknotted polymer rings.

Topological entanglements in biopolymers could drive them to certain internal statics and dynamics with important implications for biological functions. In this study, by means of molecular dynamics simulations, we demonstrate that the minimal crossing pattern of a braid plays a major role in its structural and dynamical properties; the braid consists of a knotted ring and an interlocked entwined unknotted polymer ring. In particular, we show that depending on the bending rigidity of the chains, the conformational energy of the braid can be either lower or higher than the unlocked polymer rings. Additionally, we find that a non-identical crossing pattern in the braid could distinctly enforce concerted internal conformational fluctuations between the interlocked rings.

Communication: A comparison between the solution properties of knotted ring and star polymers.

We computationally investigate the good solvent solution properties of knotted ring and star polymers by combining molecular dynamics (MD) simulation and path-integral calculations. We consider knotted rings having a minimal crossing number m c in the range, 0 ≤ m c ≤ 9, and star polymers having a range of f star arms, 2 ≤ f ≤ 20, attached to a common core monomer particle. After generating configurational ensembles of these polymers by MD, we use the path-integration program ZENO to calculate basic configurational properties, i.e., radius of gyration, hydrodynamic radius, intrinsic viscosity, as well as fluctuations in these properties. Our simulations indicate that the configurational properties of knotted rings and star polymers in solution show a similar decrease with increasing m c and f. Moreover, fluctuations in these properties also decrease with increasing topological complexity. Our findings should be helpful in polymer characterization and more generally for understanding the role of polymer topology in polymer material properties.

Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots.

We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connectivity expressed by graphs “topological polymers”. Graphs with no loop have only trivial topology, while graphs with loops such as multiple-rings may have nontrivial topology of spatial graphs as embeddings in three dimensions, e.g., knots or links in some loops. We thus call also such polymers with nontrivial topology “topological polymers”, for simplicity. For various polymers with different structures in chemical connectivity, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of the renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer–Grest model) expressed with complex graphs. We then address topological properties of ring polymers in solution. We define the knotting probability of a knot K by the probability that a given random polygon or self-avoiding polygon of N vertices has the knot K. We show a formula for expressing it as a function of the number of segments N, which gives good fitted curves to the data of the knotting probability versus N. We show numerically that the average size of self-avoiding polygons with a fixed knot can be much larger than that of no topological constraint if the excluded volume is small. We call it “topological swelling”.

Pore translocation of knotted DNA rings

We use an accurate coarse-grained model for DNA and stochastic molecular dynamics simulations to study the pore translocation of 10-kbp-long DNA rings that are knotted. By monitoring various topological and physical observables we find that there is not one, as previously assumed, but rather two qualitatively different modes of knot translocation. For both modes the pore obstruction caused by knot passage has a brief duration and typically occurs at a late translocation stage. Both effects are well in agreement with experiments and can be rationalized with a transparent model based on the concurrent tensioning and sliding of the translocating knotted chains. We also observed that the duration of the pore obstruction event is more controlled by the knot translocation velocity than the knot size. These features should advance the interpretation and design of future experiments aimed at probing the spontaneous knotting of biopolymers.

The entropic pressure of lattice knots

.The entropic pressure in the vicinity of a cubic lattice knot is examined as a model of the entropic pressure near a knotted ring polymer in a good solvent. A model for the scaling of the pressure is developed and this is tested numerically by sampling lattice knots using a Monte Carlo algorithm. Good agreement is found between scaling predictions and numerical experiments.

Chain conformations of ring polymers under theta conditions studied by Monte Carlo simulation

We studied equilibrium conformations of trivial-, 31-knot, and 51-knot ring polymers with finite chain length at their θ-conditions using a Monte Carlo simulation. The polymer chains treated in this study were composed of beads and bonds on a face-centered-cubic lattice respecting the excluded volume. The Flory's critical exponent ν in Rg ~ N(ν) relationship was obtained from the dependence of the radius of gyration, Rg, on the segment number of polymers, N. In this study, the temperatures at which ν equal 1∕2 are defined as θ-temperatures of several ring molecules. The θ-temperatures for trivial-, 31-knot, and 51-knot ring polymers are lower than that for a linear polymer in N ≤ 4096, where their topologies are fixed by their excluded volumes. The radial distribution functions of the segments in each molecule are obtained at their θ-temperatures. The functions of linear- and trivial-ring polymers have been found to be expressed by those of Gaussian and closed-Gaussian chains, respectively. At the θ-conditions, the excluded volumes of chains and the topological-constraints of trivial-ring polymers can be apparently screened by the attractive force between segments, and the <Rg(2)> values for trivial ring polymers are larger than the half of those for linear polymers. In the finite N region the topological-constraints of 31- and 51-knot rings are stronger than that of trivial-ring, and trajectories of the knotted ring polymers cannot be described as a closed Gaussian even though they are under θ-conditions.

Effective interactions of knotted ring polymers

In the present article, we review recent computational investigations on the properties of ring polymers in solution. In particular, we focus on effective interactions obtained by means of coarse-graining techniques. We discuss the relative importance of the self-avoidance and the topological contributions in the qualitative features of the effective potential. We extend our previous results on identical rings and determine the effective potential between dissimilar ring polymers of distinct topology and size. The results obtained reveal the dramatic effects of the specific topology on the effective interactions, and hence in the structural correlations, of polymeric systems.

Topological constraint in ring polymers under theta conditions studied by Monte Carlo simulation

We studied equilibrium conformations of trivial-, 3(1)-, and 5(1)-knotted ring polymers together with a linear counterpart over the wide range of segment numbers, N, from 32 up through 2048 using a Monte Carlo simulation to obtain the dependence of the radius of gyration of these simulated polymer chains, R(g), on the number of segments, N. The polymer chains treated in this study are composed of beads and bonds placed on a face-centered-cubic lattice respecting the excluded volume. The Flory's critical exponent, ν, for a linear polymer is 1/2 at the θ-temperature, where the excluded volume is screened by the attractive force generated among polymer segments. The trajectories of linear polymers at the θ-condition were confirmed to be described by the Gaussian chain, while the ν values for trivial-, 3(1)-, and 5(1)-knots at the θ-temperature of a linear polymer are larger than that for a linear chain. This ν value increase is due to the constraint of preserving ring topology because the polymer chains dealt with in this study cannot cross themselves even though they are at the θ-condition. The expansion parameter, β, where N-dependence vanishes by the definition, for trivial-, 3(1)-, and 5(1)-knotted ring polymers is obtained at the condition of ν = 1/2. It has been found that β decreases with increasing the degree of the topological constraint in the order of trivial (0.526), 3(1) (0.422), and 5(1) knot (0.354). Since the reference β value for a random knot is 0.5, the trivial ring polymer is swollen at ν = 1/2 and the other knotted ring polymers are squeezed.

Lattice knots in a slab

In this paper the number and lengths of minimal length lattice knots confined to slabs of width L are determined. Our data on minimal length verify the recent results by Ishihara et al for the similar problem, except in a single case, where an improvement is found. From our data we construct two models of grafted knotted ring polymers squeezed between hard walls, or by an external force. In each model, we determine the entropic forces arising when the lattice polygon is squeezed by externally applied forces. The profile of forces and compressibility of several knot types are presented and compared, and in addition, the total work done on the lattice knots when they are squeezed to a minimal state is determined.