Knotted Polymer Research Articles

The size of knots in polymers

Circular DNA in viruses and bacteria is often knotted. While mathematically problematic, the determination of the knot size is crucial for the study of the physical and biological behaviour of long macromolecules. Here, we review work on the size distribution of these knots under equilibrium conditions. We discuss knot localization in good and poor solvents, or in polymers that are adsorbed on a surface. We also discuss recent evidence that knot size is a crucial quantity in relaxation processes of knotted polymers.

A few notes about polymer knots

Basic theoretical results on knots in a single polymer chain are reviewed. The probability of a knot upon random chain closure and the effect of a preexisting knot on the physical characteristics of a molecule are considered. The probability of a knot in the linear chain is related to knot entropy and the minimum work required for a knot to become untied in a thermodynamically equilibrium environment. However, when chain topology is fixed by end closure, it controls, for example, the average radius of gyration of a coil: a coil appears to be more compact as compared with a phantom one when a knot is more complex than a certain threshold, and more swollen in the opposite case.

The elusiveness of polymer knots

Knots are a topological property of closed curves, and while not rigorously defined, are also expected to occur in sufficiently long open polymers.We argue, however, that energetic and entropic factors favor tight knots, with the entanglements localized on relatively small segments. Knots are statistically rare in swollen (coil) polymers, but abundant in compact (globular) configurations. Yet, our investigation of the Protein Data Bank reveals very few knotted structures in globular proteins. We shall discuss some particularly intriguing examples of this set, including the most complicated protein knot appearing in Human ubiquitin hydrolase.

Size of knots in ring polymers

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 modeled 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) approximately l(-c)f(l/N(D)) , with c approximately equal to 1.25 and D approximately equal to 1. Both D and c could be independent of knot type. As a consequence, the knot is weakly localized, i.e., <l> approximately N(t) , with t=2-c approximately equal to 0.75 . For a ring with fixed knot type, weak localization implies the existence of a peculiar characteristic length l(nu) approximately N(tnu) . In the scaling approximately N(nu) (nu approximately equal to 0.58) of the radius of gyration of the whole ring, this length determines a leading power law correction which is much stronger than that found in the case of unrestricted topology. The existence of this correction is confirmed by an analysis of extensive Monte Carlo data for the radius of gyration. The collapsed regime is studied by introducing in the model sufficiently strong attractive interactions for nearest neighbor sites visited by the self-avoiding polygon. In this regime knot length determinations can be based on the entropic competition between two knotted loops separated by a slip link. These measurements enable us to conclude that each knot is delocalized (t approximately equal to 1) .

A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales

In modeling many biological systems, it is important to take into account flexible structures which interact with a fluid. At the length scale of cells and cell organelles, thermal fluctuations of the aqueous environment become significant. In this work, it is shown how the immersed boundary method of [C.S. Peskin, The immersed boundary method, Acta Num. 11 (2002) 1–39.] for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic (τ−3/2) decay for long times [B.J. Alder, T.E. Wainwright, Decay of the Velocity Autocorrelation Function, Phys. Rev. A 1(1) (1970) 18–21]. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method. Specifically, we present simulations of osmotic effects of molecular dimers, worm-like chain polymer knots, and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force. The theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales.

Capturing knots in polymers

Capturing knots in polymers

Knots in polymers

Knots and topological entanglements play an important role in the statistical mechanics of polymers. While topological entanglement is a global property, it is possible to study the size of a knotted region both numerically and analytically. It can be shown that long-range repulsive interactions, as well as entropy favor small knots in dilute systems. However, in dense systems and at the Θ-point in two dimensions the uncontracted knot configuration is the most likely.

Topologically driven swelling of a polymer loop.

Numerical studies of the average size of trivially knotted polymer loops with no excluded volume were undertaken. Topology was 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 were generated for loops of up to N = 3,000 segments. Gyration radii of trivially knotted loops were found to follow a power law similar to that of self-avoiding walks consistent with earlier theoretical predictions.

Theory of chromatography of complex cyclic polymers: eight-shaped and daisy-like macromolecules

A theory of chromatography of eight-shaped, trefoil-shaped and daisy-like polymers is developed. For a model of an ideal chain in a slit-like pore exact equations and a number of approximate formulae for the distribution coefficient K of these polymers are derived. All modes of chromatography of complex macrocycles of arbitrary molar mass in both narrow and wide pores are covered by the theory. It is shown that complex macrocycles always elute after linear polymers and rings of the same contour length. The effective chromatographic radius of eight-shaped and daisy-like macromolecules, which determines retention in size-exclusion chromatography are calculated. The increase in the retention with molar mass is predicted for all types of macrocycles at the critical interaction condition. Non-monotonous molar mass dependences of K are found at pre-critical interaction. We simulate separation of complex cyclic polymers from linear and ring precursors, discuss possibilities to separate symmetric and asymmetric eights, and speculates on the use of chromatography for separating knotted and unknotted polymer rings. According to the theory, the chromatography under the critical and pre-critical interaction conditions is expected to be especially efficient in these and similar problems. Boundary conditions for the theory and its applicability to real systems are discussed.

Loose, Flat Knots in Collapsed Polymers

We consider single ring polymers which are confined on a plane but maintain a fixed three-dimensional knotted topology. The equilibrium statistics of such systems is studied on the basis of a model on square lattice in which the configurations are represented by N-step polygons with a number of self-intersections restricted to the minimum compatible with the topology. This allows to define the size, s, of the flat knots and to study their localization properties. Due to the presence of both excluded volume and attractive interactions, the model undergoes a theta transition. Accurate Monte Carlo results show that, while in the high temperature swollen regime both prime and composite knot components are localized (〈s〉 N ∼N t , with t=0), in the low temperature, compact phase they are fully delocalized (t=1). Right at the theta transition weak localization prevails (t=0.44±0.02). Part of the results can be interpreted by taking into account a dominance of figure eight shapes for the coarse grained knotted polymer configurations, and by applying the scaling theory of polymer networks of fixed topology. In particular t=3/7 can be conjectured as an exact exponent characterizing the weak knot localization at the theta point.

Non-Equilibrium Dynamics in Untying Knots

Flexible closed knots with strict topological constraint of no segment crossing are studied in our simulations using the bond-fluctuation model for polymers. A well-equilibrated polymer knot is cut at a randomly chosen segment and allow to relax to it new equilibrium state, which is the indistinguishable from the free linear chain. In the course of the relaxation, the topological information of the original knot is lost and the characteristic relaxation time measures its rate. The average relaxation time is the typical time scale needed to untie the cut knot by random Brownian motion. Some knot is found to have a longer relaxation time than other knots having more crossings. Remarkably, when all the knots are arranged into homologous groups with a common parametrization in terms of their Alexander polynomials, the relaxation times increase monotonically and linearly with C for all the four groups we studied. Our observation indicates that conventional labelling of knots can further be parametrized naturally into groups in a way that has a direct physical meaning in terms of the topological interactions in a knot. The linear stepwise increase of the relaxation time with the essential crossing suggests that the topological energy spectrum has equal spacing of for knots within a group.

Knots in charged polymers.

The interplay of topological constraints and Coulomb interactions in static and dynamic properties of charged polymers is investigated by numerical simulations and scaling arguments. In the absence of screening, the long-range interaction localizes irreducible topological constraints into tight molecular knots, while composite constraints are factored and separated. Even when the forces are screened, tight knots may survive as local (or even global) equilibria, as long as the overall rigidity of the polymer is dominated by the Coulomb interactions. As entanglements involving tight knots are not easy to eliminate, their presence greatly influences the relaxation times of the system. In particular, we find that tight knots in open polymers are removed by diffusion along the chain, rather than by opening up. The knot diffusion coefficient actually decreases with its charge density, and for highly charged polymers the knot's position appears frozen.

Variation in the coordination mode of arenedisulfonates: syntheses and structural characterization of mononuclear and dinuclear cadmium(II) arenedisulfonate complexes with two- to zero-dimensional architectures.

Seven cadmium(II) arenedisulfonate compounds, namely [Cd(2,2'-bpy)(2)(H(2)O)(peds)].4H(2)O (1), [Cd(2)(2,2'-bpy)(4)(H(2)O)(2)(1,5nds)](1,5nds).4H(2)O (2), [Cd(cyclam)(1,5nds)](2) (3), ([Cd(inia)(2)(H(2)O)(2)(2,6nds)].4H(2)O)(n)(4), ([Cd(inia)(2)(H(2)O)(2)(bpds)].4H(2)O)(n)(5), ([Cd(2)(inia)(4)(H(2)O)(3)(peds)(2)].2H(2)O)(n)(6), and [Cd(1,5nds)(H(2)O)(2)](n) (7), where 2,2'-bpy = 2,2'-bipyridyl, cyclam = 1,4,8,11-tetraazacyclotetradecane, inia = isonicotinamide, nds = naphthalenedisulfonate, bpds = 4,4'-biphenyldisulfonate, and peds = 4,4'-phenyletherdisulfonate, have been obtained from aqueous solution by using similar procedures and structurally characterized by X-ray single-crystal diffraction, IR spectroscopy, and thermal gravimetric analysis. In 1, the peds anion coordinates as a monodentate ligand, leading to a mononuclear unit. In 2 and 3, the 1,5nds anions coordinate as mu(2)-bridging ligands in different modes, producing charged or neutral dinuclear clusters. In 4 and 5, 2,6nds and bpds behave as mu(2)-spacers, resulting in 1-dimensional polymers. While in 6, the peds acts both as terminal and bridging ligands with the SO(3)(-) groups being either monodentate or mu(2)-bridging, creating a knotted 1-dimensional polymer with dinuclear clusters as the repeating units. In 7, 1,5nds acts as a bridging ligand with each SO(3)(-) coordinated as a mu(2)-bridging group to adjacent Cd(II) centers, leading to a 2-dimensional polymer. Together with the reported ([Cu(en)(2)(1,5nds)].2H(2)O)(n) (8), all of the six possible coordination modes adopted by organodisulfonate anions, on the assumption that each SO(3)(-) group could be monodentate or mu(2)-bridging, are realized by introducing nitrogen-containing organic ligands as auxiliaries.

Dynamics of polymer knots at equilibrium.

The relaxation and diffusion dynamics of knotted polymers at equilibrium under good solvent conditions are investigated by dynamic Monte Carlo simulations. Prime knots of chain lengths up to N=240 monomers and knots up to 20 essential crossings are studied. The relaxation dynamics of the prime knots at equilibrium do not display the classification into group as in the case of the nonequilibrium relaxation of cut knots [Phys. Rev. E 58, R1222 (1998); Phys. Rev. Lett. 87, 175503 (2001)]. Furthermore, the time autocorrelation functions for the radius of gyration of the nontrivial knots can be fitted by a sum of two exponential decays of long and short characteristic relaxation times. These two relaxation times decrease with the number of essential crossings C. The faster relaxation follows the Rouse behavior and scales as N1+2nu and its dependence on C is consistent with the scaling analysis using the blob picture. The mean-square displacement of the center of mass of the knots obeys the free diffusion behavior compatible with the Rouse dynamics. The diffusion coefficients of the knots, D approximately 1/N for large N, but D decreases for knots with increasing C. These results are analyzed using scaling theories and discussed in terms of topological interactions in the knots.

Polymer knot confined in a tube: statics and relaxation dynamics.

The statics and relaxation dynamics of a polymer knot confined in a cylindrical tube of diameter D are investigated by Monte Carlo simulations. The prime knots with crossings varying from 3(1) to 7(1) are considered. The equilibrium radius of gyration along the axial direction is scaled as R(parallel) approximately R(F)(R(F)/D)(m), where R(F) is the Flory radius of a knotted polymer. Our simulation results suggest that m=0.95, which is different from the value for a linear chain, m=2/3. The relaxation behavior of the knotted polymer is studied after the tubular constraint is removed. The relaxation time can be scaled as tau(parallel) approximately NR(1+1/m)(F)R(1-1/m)(parallel), which is confirmed by simulations. The effect of topological complexity on both properties is manifested through the Flory radius.

Nonequilibrium relaxation times in polymer knot groups.

Flexible polymer knots with strict topological constraint of no segment crossing are studied by Monte Carlo simulations. The nonequilibrium relaxation time of an equilibrated polymer knot cut at one point to relax to a linear chain is measured. Prime knots up to 20 essential crossings from the groups (3(1),5(1),7(1),...), (4(1),6(1),8(1),...) and (5(2),7(2),9(2),...), (6(2),8(2),10(2),...) are studied. The nonequilibrium relaxation time for knots within a group are found to increase stepwise linearly with the number of essential crossings of the original knot. Our results suggest an equally spaced topological interaction energy spectrum for knots in the same group and thus provide a quantitative description of topological interactions.

MECHANICAL MODELS FOR PREDICTING EXPANSIVE PRESSURE OF SELF EXPANDABLE METAL STENTS WITH COVER MEMBRANE

PURPOSE: Successful stenting is crucial for maintaining the patency of the organ with luminal obstruction. Three types of stents are commonly used: bare stent(wir only), coated stent(with polymer coating on wire), and covered stent(with polymer membrane on its peripheral surface). In the present study, we propose th mathematical models by which one can predict radial expansive pressure of coate and covered stents. METHODS: A coated stent exerts two types of forces agains radial compression: the spring restoring force exerted by metal wires and the forc derived from the moments by the knots of polymer coating. The helical sprin model[1] is employed to calculate wire spring force in bare stent. The polymer knot are considered as torsional springs and the spring moments are calculated. Given Imaginary longitudinal force from wire(Fwire) and knots(Fcoat), number of spring tums(c) diameter(D), length(L) and pitch angle(β) of compressed stent, one can derive th radial expansive pressure of coated stent by the following equation Pcoated stent = 2c (Fwire + Fcoat)/DL tanβ A covered stent exerts the force by the cover membrane itself in addition to two force exerted by coated stent. The membrane is regarded as tensile springs, and thus th covered stent is modeled as a combination of the following three: open-coiled helical springs, torsional springs, and tensile springs. Tensile test was performed and th resulting nonlinear stress-strain curve was employed to calculate the cover membran tensile force. The radial expansive pressure of coated stent is derived by th following equation Pcoated stent = 2c (Fwire + Fcoal + Fcover)/DL tanβ RESULTS: The proposed models may be instrumental in obtaining expansiv pressure properties of various stents. The radial expansive pressure values obtaine from mathematical models show excellent agreements with those from th experiments carried out according to the method by Moon et al.[2]. Most importantly the stents with cover membrane show rapid increase in radial expansive pressur during the initial phase of compression(0–10% compression) and slow increas beyond 10%, due to the nonlinear tensile characteristics of the cover material. Whil the cover membrane prevent tumor in-growth, it offers another advantage in that th membrane increases radial expansive pressure during the initial phase o compression, and hence helps to achieve full expansion. The mathematical model proposed in the present study could be useful in designing and developing new stent with optimal expansive behavior.

Crossings and writhe of flexible and ideal knots.

The data of ideal knots [Nature, 384, 142 (1996)] are reanalyzed and the average crossing number of the ideal knots <X>(ideal) shows a nonlinear behavior with the essential crossing number C. Supplemented with our Monte Carlo simulations using the bond fluctuation model on flexible knotted polymers, our analysis indicates that <X>(ideal) varies nonlinearly with both C and the corresponding average crossing number of the flexible knot, which is contrary to previous claims. Our extensive simulation data on the average crossing number of flexible knots suggest that it varies linearly with the square root of C. Furthermore, our data on the average writhe number <Wr> indicate that various knots are classified into holonomous groups, and <Wr> has a quantized linear increment with C in all four knot groups in our study.

Composite spring model in stretched polymer knots

The deformation of a knotted polymer under a stretching force is studied by modeling the deformed knot as a composite spring system. Our results predict that the elastic modulus of a knotted polymer is larger when compared to an equal-length linear chain. More complex knots are in general stiffer. The increase in stiffness of knots relative to the linear chain is also derived. Monte Carlo simulations are also performed to investigate the stetching of polymer knots. Chain lengths up to N = 82 and prime knots 0 1 , 3 1 , 4 1 , 5 1 , 6 1 and 8 1 are considered. Segregation of the crossings into a small tight region of the knot structure at strong forces is observed. The increase in stiffness predicted by the composite spring model agree well with the simulation data. Our simularion data show that the scaling laws proposed by de Gennes and Pincus for a single linear chain under traction force still hold for the knotted type polymers.

Structure and relaxation dynamics of polymer knots

Monte Carlo simulations are performed to study the equilibrium structure and nonequilibrium dynamic relaxation processes of knotted polymers. We find that topological complexity affects the static and dynamic behavior of knots in different ways to different extent. For statics, our results on the radii of gyration of knot polymers suggest that prime and two-factor composite knots belong to different groups, and we confirm that for knots in the same group, the average radius of gyration scales as R g ∼ N 3/5 p −4/15 in good solvents, where N is the number of monomers and p is the topological invariant representing the length-to-diameter ratio of the knot at its maximum inflated state. From the studies of nonequilibrium relaxation dynamics on prime knots cut at t=0, we find that even prime knots should be classified into different groups as (3 1,5 1,7 1,…), (4 1,6 1,8 1,…), (5 2,7 2,9 2,…) , etc., based on their topological similarity and their polynomial invariants such as Alexander polynomials. Our results suggest that the mathematical classification of knots can further be parametrized naturally into groups in a way that can have direct physical meaning in terms of structures and dynamics of knots. Furthermore, by scaling calculations, the nonequilibrium relaxation time is found to increase roughly as p 12/5. This prediction is further supported by our data.