Knotted Polymer Research Articles

A Simple and Efficient Algorithm to Identify the Chirality of Polymer Knots Based on the Alexander Polynomial

A Simple and Efficient Algorithm to Identify the Chirality of Polymer Knots Based on the Alexander Polynomial

Integrating Graph Convolutional Network to BiLSTM for Enhanced Polymer Knot Identification

Integrating Graph Convolutional Network to BiLSTM for Enhanced Polymer Knot Identification

Revisiting the second Vassiliev (In)variant for polymer knots

Knots in open strands such as ropes, fibers, and polymers, cannot typically be described in the language of knot theory, which characterizes only closed curves in space. Simulations of open knotted polymer chains, often parameterized to DNA, typically perform a closure operation and calculate the Alexander polynomial to assign a knot topology. This is limited in scenarios where the topology is less well-defined, for example when the chain is in the process of untying or is strongly confined. Here, we use a discretized version of the Second Vassiliev Invariant for open chains to analyze Langevin Dynamics simulations of untying and strongly confined polymer chains. We demonstrate that the Vassiliev parameter can accurately and efficiently characterize the knotted state of polymers, providing additional information not captured by a single-closure Alexander calculation. We discuss its relative strengths and weaknesses compared to standard techniques, and argue that it is a useful and powerful tool for analyzing polymer knot simulations.

Tube model for polymer knots: establishment and applications

纽结是链状物中的常见现象，既存在于耳机线等宏观物体，也存在于微观链状分子，包括DNA和蛋白质。近年来，实验观察到高分子纽结能显著改变力学、流变、催化、生物功能等性质，但是对高分子纽结的理论认识和解析推导较少，导致对纽结现象机理理解不足。理论研究的难点在于，纽结构象空间复杂，难以用统计物理理论进行严格解析推导。为解决该问题，我们发展纽结管子模型对纽结体系进行理论简化，将纽结问题转化成管子限制问题，使理论推导变得可行。基于该管子模型，结合蒙特卡洛模拟、分子动力学模拟、链生长法以及特定管子生成算法，我们将理论模型中的管子实物化、可视化和定量化，并得到管子形状和直径等参数。我们发现纽结管子是优美协调的心形。另外，基于该管子模型，结合受限链的标度行为，我们推导各种条件下高分子纽结的自由能、概率、纽结大小分布等性质，与模拟和实验结果吻合。在本专题论述中，我们将介绍本课题组近几年得到的高分子纽结研究结果。

Can Polymer Helicity Affect Topological Chirality of Polymer Knots?

We investigate the effect of helicity in isolated polymers on the topological chirality of their knots with computer simulations. Polymers are described by generic worm-like chains (WLC), where helical conformations are promoted by chiral coupling between segments that are neighbors along the chain contour. The sign and magnitude of the coupling coefficient u determine the sense and strength of helicity. The corrugation of the helix is adjusted via the radius R of a spherical, hard excluded volume around each WLC segment. Open and compact helices are, respectively, obtained for R that is either zero or smaller than the length of the WLC bond, and R that is a few times larger than the bond length. We use a Monte Carlo algorithm to sample polymer conformations for different values of u, spanning the range from achiral polymers to chains with well-developed helices. Monitoring the average helix torsion and fluctuations of chiral order as a function of u, for two very different chain lengths, demonstrates that the coil-helix transition in this model is not a phase transition but a crossover. Statistical analysis of conformations forming the simplest chiral knots, 31, 51, and 52, demonstrates that topological mirror symmetry is broken─knots formed by helices with a given sense prefer one handedness over the other. For the 31 and 51 knots, positive helical sense favors positive handedness. Intriguingly, an opposite trend is observed for 52 knots, where positive helical sense promotes negative handedness. We argue that this special coupling between helicity and topological chirality stems from a generic mechanism: conformations where some of the knot crossings are found in "braids" formed by two tightly interwoven sections of the polymer.

Quantifying the effects of slit confinement on polymer knots using the tube model.

Knots can spontaneously form in DNA, proteins, and other polymers and affect their properties. These knots often experience spatial confinement in biological systems and experiments. While confinement dramatically affects the knot behavior, the physical mechanisms underlying the confinement effects are not fully understood. In this work, we provide a simple physical picture of the polymer knots in slit confinement using the tube model. In the tube model, the polymer segments in the knot core are assumed to be confined in a virtual tube due to the topological restriction. We first perform Monte Carlo simulation of a flexible knotted chain confined in a slit. We find that with the decrease of the slit height from H=+∞ (the 3D case) to H=2a (the 2D case), the most probable knot size L_{knot}^{*} dramatically shrinks from (L_{knot}^{*})_{3D}≈140a to (L_{knot}^{*})_{2D}≈26a, where a is the monomer diameter of the flexible chain. Then we quantitatively explain the confinement-induced knot shrinking and knot deformation using the tube model. Our results for H=2a can be applied to a polymer knot on a surface, which resembles DNA knots measured by atomic force microscopy under the conditions that DNA molecules are weakly absorbed on the surface and reach equilibrium 2D conformations. This work demonstrates the effectiveness of the tube model in understanding polymer knots.

Survival Reassessment at Tumor Recurrence in Soft Matter

The paper reassesses a survival at tumor recurrence in soft matter. First, the stability of structural motifs under shear in clusters of dipolar spheres is characterized. Next, there are introduced transitions between polymer knots and rhythms of these transitions are obtained. The sensor is built for these rhythms. Treatment, with a tensile force protocol, is modeled, when the tumor in soft matter is observed by the above sensor. Survival probability, at tumor recurrence in soft matter, is defined for the treatment with a tensile force protocol. It is stated that the survival probability at a tensile force protocol treatment in soft matter confirms or specifies the prognostic survival of 32 patients with breast cancer.

Salt effects on knot dynamics in polyelectrolyte solutions

ABSTRACT We study the dynamics of linear polyelectrolyte (PE) knot in solution with doping salt by molecular dynamics simulations. The molecular mechanisms of PE knot dynamics are explained. The knot dynamics are the coupling results by the moving effect of chain entanglement and by the breathing effect of knot fluctuations. The moving effect is important as the whole chain is in the random coil state and becomes weak as shrinking to the close-to-globule (CTG) state. Interestingly, as the chain in the CTG state, the knot swells with occupying more segments. The breathing effect dominates the knot dynamics in the CTG state. This nontrivial behaviour (in comparison to neutral polymer knot) is prominent for the cases of doping salt with high concentration or of doping high-valent salt, which attributes to the enhanced electrostatic interactions and long-range fluctuation effects.

Channels with Helical Modulation Display Stereospecific Sensitivity for Chiral Superstructures.

By means of coarse-grained molecular dynamics simulations, we explore chiral sensitivity of confining spaces modelled as helical channels to chiral superstructures represented by polymer knots. The simulations show that helical channels exhibit stereosensitivity to chiral knots localized on linear chains by effect of external pulling force and also to knots embedded on circular chains. The magnitude of the stereoselective effect is stronger for torus knots, the effect is weaker in the case of twist knots, and amphichiral knots do exhibit no chiral effects. The magnitude of the effect can be tuned by the so-far investigated radius of the helix, the pitch of the helix and the strength of the pulling force. The model is aimed to simulate and address a range of practical situations that may occur in experimental settings such as designing of nanotechnological devices for the detection of topological state of molecules, preparation of new gels with tailor made stereoselective properties, or diffusion of knotted DNA in biological conditions.

Tube Model for Polymer Knots with Excluded Volume Interactions and Its Applications

Tube Model for Polymer Knots with Excluded Volume Interactions and Its Applications

Knots are Generic Stable Phases in Semiflexible Polymers

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 $r_b$ is the equilibrium bond length and $r_{\rm{min}}$ is the distance for the strongest nonbonded contacts. Our results provide evidence for both models that if the ratio $r_b/r_{\rm{min}}$ is outside a small window around unity then depending on the bending stiffness one always encounters stable knotted phases along with the usual frozen and bent-like structures at low temperatures. These findings prompt us to conclude that knots are generic stable phases in semiflexible polymers.

Revisiting the Non-monotonic Dependence of Polymer Knotting Probability on the Bending Stiffness

Knots can spontaneously form in polymers. How knotting affects polymer behavior depends on polymer knotting probability, pknot. An intriguing result about pknot in recent studies is that pknot exhibits a non-monotonic dependence on the bending stiffness and is maximized at Lp ≈ 8a, where Lp is the persistence length and a is the hardcore diameter of the monomer. In this work, we propose a new explanation for the non-monotonic behavior of pknot based on the fact that polymer knots are typically localized. We find that the non-monotonic behavior results from the competition of a special entropic effect arising from the variation in the sizes of localized knots and an effect arising from the variation in the free-energy cost of forming a localized knot on a fragment of a polymer. The first effect refers to the situation that shrinking the knot size for a polymer with a fixed length essentially increases the number of “slots” for knot formation and enhances pknot. Based on this explanation, we derive an approximate analytic equation that captures the non-monotonic behavior of pknot. Overall, this work provides new insights into pknot beyond previous studies, in particular, unifying the effect of the knot size on pknot and the effect of the polymer length on pknot. The results can be applied to understand DNA knotting, considering that the effective Lp/a for DNA can be widely varied by the ionic strength.

Thermophysical features of nanostructured polymer system based on polyurethane

Polymer systems containing nanoparticles are new class of polymer composite materials. Polymer matrix filling influences on network polymer structure forming mechanism. The filler influence appears in formation features of the network structure both near the filler surface and in polymer matrix volume. Polymer solid surface (filler, substrate, etc.) interaction results in polymer chains movement restriction, that in fact is equivalent of additional polymer network physical knots formation. The polymer – filler interface presence may lead to both increasing of physical polymer knots average amount and decreasing it because of decreasing the polymer –polymer type physical bonds amount. This in turn determines strength of the polymer composite value. Polyurethane composite filled with different type of gypsum has been studied. The semi-aquatic gypsum (water suspension and dry powder) and two-aquatic gypsum water suspension were used as the filler. The polyurethane oligomer was used as reactive polymer matrix. Thermal-and-physical properties of polymer composites were studied by differential scanning calorimetry. It is found that the semi-aquatic gypsum does not transform in two-aquatic gypsum within polymer matrix during observation time. Evidently, the first – the polymer matrix doesn’t have continues pores and the second – the polymer matrix forms dense surface layer that hinders the diffusion of water vapor to the filler. Obtained polymer compositions glass temperature shows weak dependence on gypsum concentration and method of gypsum filling, but shows strong dependence on heating multiplicity. Polymer composition change in heat capacity upon glass transition value ∆Cp decreases with filler concentration increases and strongly depends on filling process. To obtain stable material properties the annealing of internal strength is needed to carry out. Key words: polyurethane composite, gypsum, thermal-and-physical properties.

Separation of Geometrical and Topological Entanglement in Confined Polymers Driven out of Equilibrium.

We use Brownian dynamics simulations and advanced topological profiling methods to characterize the out-of-equilibrium evolution of self-entanglement in linear polymers confined into nanochannels and under periodic compression. By introducing suitable observables, we can distinguish two main forms of entanglement that we term geometrical and topological. The latter is measured by the number of (essential) crossings of the physical knot detected after a suitable bridging of the chain termini. The former is instead measured as the average number of times a linear chain appears to cross itself when viewed under all projections and is irrespective of the physical knotted state. The key discovery of our work is that these two forms of entanglement are uncoupled and evolve with distinct dynamics. While geometrical entanglement is typically in phase with the compression-elongation cycles and it is primarily sensitive to its force f, the topological measure is mildly sensitive to cyclic modulation but strongly depends on both compression force f and duration k. The findings could assist the interpretation of experiments using fluorescence molecular tracers to track physical knots in polymers. Furthermore, we identify optimal regions in the experimentally controllable parameter space in which to obtain more/less topological and geometrical entanglement; this may help designing polymers with targeted topology.

Developing the tube theory for polymer knots

This work presents an average of over a million knot-core conformations under equilibrium and obtains a universal, heart-shaped, trefoil knot. The author further studies the fluctuations around it

Polymer Knot in Solution near the θ Point.

We study knot effects on polymer dynamics in aqueous solution by dissipative particle dynamics. A methodology to identify the θ point is developed by combining simulation data and analytical methods. Polymer internal motions are investigated systematically by gradually changing solvent quality from a good to poor case. In a good solvent, the knot length grows with fluctuations (breathing stage) and then moves to chain ends (moving stage) to release. Nearby the θ point from the good solvent side, the breathing effect becomes stronger with a weak moving effect. As a result, it is easy for the knot to release because of strong fluctuations in chain conformation. In a poor solvent, both breathing and moving effects are weak. A knot is confined in the chain and suppresses polymer condensation because of the excluded volume effect. Our results are useful for the interpretation of relevant experiments and industrial applications of polymer knots.

Effects of Side Chains on Polymer Knots

As an intriguing phenomenon, knotting in DNA, proteins, and other molecules has been extensively investigated in recent years, not only because of fundamental interest in knotting but also because ...

Knot Energy, Complexity, and Mobility of Knotted Polymers

The Coulomb energy EC 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 EC 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 EC for all these knotted polymers using the program ZENO and find that the average Coulomb energy 〈EC〉 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.

Topological effects on the mechanical properties of polymer knots

The mechanical properties of knotted polymer rings under stretching in a bad or good solvent are investigated by applying a force F to a point of the knot while keeping another point fixed. The Monte Carlo sampling of the polymer conformations is performed on a simple cubic lattice using the Wang–Landau algorithm. The specific energy, specific heat capacity, gyration radius and the force–elongation curves are computed for several knot topologies with lengths up to 120 lattice units. The common features of the mechanical and thermal behavior of stretched short polymer rings forming knots of a given topological type are analyzed as well as the differences arising due to topology and size effects. It is found that these systems admit three different phases depending on the values of the tensile force F and the temperature T. The transitions from one phase to the other are well characterized by the peaks of the specific heat capacity and by the data of the gyration radius and specific energy. At very low temperatures the force–elongation curves show that the stretching of a knot is a stepwise process, which becomes smooth at higher temperatures. Criteria for distinguishing topological and size effects are provided. It turns out from our study that the behavior of short polymer rings is strongly influenced by topological effects. In particular, the swelling and the swelling rate of knots are severely limited by the topological constraints. Several other properties that are affected by topology, like the decay of the specific energy at high tensile forces, are discussed. The fading out of the influences of topological origin with increasing knot lengths has been verified. Some anomalies detected in the plots of the specific heat capacity of very short and complex knots have been explained by the limitations in the number of accessible energy states due to the topological constraints.

Knots in finite memory walks

We investigate the occurrence and size of knots in a continuum polymer model with finite memory via Monte Carlo simulations. Excluded volume interactions are local and extend only to a fixed number of successive beads along the chain, ensuring that at short length scales the excluded volume effect dominates, while at longer length scales the polymer behaves like a random walk. As such, this model may be useful for understanding the behavior of polymers in a melt or semi-dilute solution, where exactly the same crossover is believed to occur. In particular, finite memory walks allow us to investigate the role of local interactions in the transition from highly knotted ideal polymers to almost unknotted self-avoiding polymers. Even though knotting decreases substantially when a few next-nearest neighbor interactions are considered, we find that the knotting probability of a polymer chain of modest length of 500 steps only decays slowly as a function of the range of the excluded volume interaction. In this context, we also find evidence that for length scales up to the interaction length the knotting behavior of the finite memory walk resembles that of a self-avoiding walk (effectively suppressing small knots), while for larger length scales it resembles that of a random walk.