Knots “Choke Off” Polymers upon Stretching

Long polymer chains inevitably get tangled into knots. Like macroscopic ropes, polymer chains are substantially weakened by knots and the rupture point is always located at the “entry” or “exit” of the knot. However, these phenomena are only poorly understood at a molecular level. Here we show that when a knotted polyethylene chain is tightened, most of the stress energy is stored in torsions around the curved part of the chain. The torsions act as “work funnels” that effectively localize mechanical stress in the immediate vicinity of the knot. As a result, the knot “chokes” the chain at its entry or exit, thus leading to bond rupture at much lower forces than those needed to break a linear, unknotted chain. Our work not only explains the weakening of the polymer chain and the position of the rupture point, but more generally demonstrates that chemical bonds do not have to be extensively stretched to be broken.     

Langevin dynamics simulations of the diffusion of molecular knots in tensioned polymer chains

Motivated by recent experiments, in which knots have been tied in individual biopolymer molecules, we use Langevin dynamics simulations to study the diffusion of a knot along a tensioned polymer chain. We find that the dependence of the knot diffusion coefficient on the tension can be non-monotonic. This behavior can be explained by the model, in which the motion of the knot involves cooperative displacement of a local knot region. At low tension, the overall viscous drag force that acts on the knot region is proportional to the number N of monomers that participate in the knot, which decreases as the tension is increased, leading to faster diffusion. At high tension the knot becomes tight and its dynamics are dominated by the chain's internal friction, which increases with the increasing tension, thereby slowing down the knot diffusion. This model is further supported by the observation that the knot diffusion coefficient measured across a set of different knot types is inversely proportional to N. We propose that the lack of tension dependence of the knot diffusion coefficients measured in recent experiments is due to the fact that the experimental values of the tension are close to the turnover between the high- and low-force regimes.     

Coordination‐Driven Self‐Assembly of a Molecular Knot Comprising Sixteen Crossings

Molecular knots have become highly attractive to chemists because of their prospective properties in mimicking biomolecules and machines. Only a few examples of molecular knots from the billions tabulated by mathematicians have been realized and molecular knots with more than eight crossings have not been reported to date. We report here the coordination‐driven [8+8] self‐assembly of a higher‐generation molecular knot comprising as many as sixteen crossings. Its solid‐state X‐ray crystal structure and multinuclear 2D NMR findings confirmed its architecture and topology. The formation of this molecular knot appears to depend on the functionalities and geometries of donor and acceptor in terms of generating appropriate angles and strong π‐π interactions supported by hydrophobic effects. This study shows coordination‐driven self‐assembly offers a powerful potential means of synthesizing more and more complicated molecular knots and of understanding differences between the properties of knotted and unknotted structures.     

Langevin dynamics simulations of polymer translocation through nanopores

We investigate the dynamics of polymer translocation through a nanopore using two-dimensional Langevin dynamics simulations. In the absence of an external driving force, we consider a polymer which is initially placed in the middle of the pore and study the escape time tau(e) required for the polymer to completely exit the pore on either side. The distribution of the escape times is wide and has a long tail. We find that tau(e) scales with the chain length N as tau(e) approximately N(1+2nu), where nu is the Flory exponent. For driven translocation, we concentrate on the influence of the friction coefficient xi, the driving force E, and the length of the chain N on the translocation time tau, which is defined as the time duration between the first monomer entering the pore and the last monomer leaving the pore. For strong driving forces, the distribution of translocation times is symmetric and narrow without a long tail and tau approximately E(-1). The influence of xi depends on the ratio between the driving and frictional forces. For intermediate xi, we find a crossover scaling for tau with N from tau approximately N(2nu) for relatively short chains to tau approximately N(1+nu) for longer chains. However, for higher xi, only tau approximately N(1+nu) is observed even for short chains, and there is no crossover behavior. This result can be explained by the fact that increasing xi increases the Rouse relaxation time of the chain, in which case even relatively short chains have no time to relax during translocation. Our results are in good agreement with previous simulations based on the fluctuating bond lattice model of polymers at intermediate friction values, but reveal additional features of dependency on friction.     

Hydrocarbon links in an octet truss

We use the octet truss of R. Buckminster Fuller to develop a geometric placement method for synthesizing braid representations of knots and links of oligo (phenylene ethynylene)s using the 60° ortho, 120° meta or 180° para phenyl ring substitution angles and respecting the van der Waals repulsion constraints. We show that any knot or link can be realized by a phenylene ethynylene oligomer modeled on the octet truss. Use of this lattice is motivated by the structural constraints of these phenylene ethynylene units. Where in bio-organic chemistry, questions often involve identifying existing knots, for example in DNA strands, organic synthesis is concerned with assembling molecular structures that can be verified to exist in a desired knot topology. This physical realization of a knot as a construction of common organic molecular subunits then facilitates further study of the properties of knotted molecules in general.     

Dendronized molecular knots: selective synthesis of various generations,enantiomer separation,circular dichroism

For the first time, knot molecules (of the amide type) are synthesized, which bear one to three dendritic units of various generations at their periphery. They were obtained through two different routes: i) attachment of dendritic wedges to new mono-, di- and trihydroxy functionalized dodecaamide knots that have been obtained by selective debenzylation of oligobenzyloxy substituted knots, or ii) cyclization of already dendron substituted pyridine-2,6-dicarbonyl dichlorides with an "extended diamine" to directly yield the "tri-dendroknots". The derivatization of knot molecules by functional substituents and even large dendritic units is an important advance in the synthesis and property variation of molecular knots. This holds true in particular for substitution of the pyridine units of the knots, whereas the isophthalic acid units seem not to tolerate larger substitutents, as reflected in lower knot yields. These syntheses also demonstrate knots to be accessible indirectly by substitution of the corresponding mono-, di- and tri-functionalized knot skeleton. An advantage of dendritic "decoration" is the control of solubility and chromatographic behaviour of the molecular knots (knotanes). Suggestions are made about the threading mechanism by supramolecular template effects leading to the formation of amide-based molecular knots. The topological chirality of the new "dendroknots" is shown by efficient enantioseparations (separation factor alpha between 1.22 and 1.48). For this purpose (commercially unavailable) chiral column material of the Chiralpak type was used, in which the chiral component is covalently bonded to the silica gel support. The racemate splittings provide additional evidence for the knotted structure, as all other conceivable isomers such as macromonocyclic or catenated dodecaamides would not be chiral. The pure enantiomers obtained exhibit pronounced Cotton effects in their circular dichroism spectra. By comparison with the unsubstituted knot, the absolute configuration (Lambda, Delta) of all new knots is derived.     

Femtosecond/picosecond time-resolved fluorescence study of hydrophilic polymer fine particles

Femtosecond/picosecond time-resolved fluorescence study of hydrophilic polymer fine particles (polyacrylamide, PAAm) was reported. Ultrafast fluorescence dynamics of polymer/water solution was monitored using a fluorescent probe molecule (C153). In the femtosecond time-resolved fluorescence measurement at 480 nm, slowly decay components having lifetimes of tau(1) approximately 53 ps and tau(2) approximately 5 ns were observed in addition to rapid fluorescence decay. Picosecond time-resolved fluorescence spectra of C153/PAAm/H2O solution were also measured. In the time-resolved fluorescence spectra of C153/PAAm/H2O, a peak shift from 490 to 515 nm was measured, which can be assigned to the solvation dynamics of polymer fine particles. The fluorescence peak shift was related to the solvation response function and two time constants were determined (tau(3) approximately 50 ps and tau(4) approximately 467 ps). Therefore, the tau(1) component observed in the femtosecond time-resolved fluorescence measurement was assigned to the solvation dynamics that was observed only in the presence of polymer fine particles. Rotational diffusion measurements were also carried out on the basis of the picosecond time-resolved fluorescence spectra. In the C153/PAAm/H2O solution, anisotropy decay having two different time constants was also derived (tau(6) approximately 76 ps and tau(7) approximately 676 ps), indicating the presence of two different microscopic molecular environments around the polymer surface. Using the Stokes-Einstein-Debye (SED) equation, microscopic viscosity around the polymer surface was evaluated. For the area that gave a rotational diffusion time of tau(6) approximately 76 ps, the calculated viscosity is approximately 1.1 cP and for tau(7) approximately 676 ps, it is approximately 10 cP. The calculated viscosity values clearly revealed that there are two different molecular environments around the polyacrylamide fine particles.     

Sliding friction between polymer surfaces: a molecular interpretation

For two contacting rigid bodies, the friction force F is proportional to the normal load and independent of the macroscopic contact area and relative velocity V (Amonton's law). With two mutually sliding polymer samples, the surface irregularities transmit deformation to the underlying material. Energy loss along the deformation cycles is responsible for the friction force, which now appears to depend strongly on V [see, e.g., N. Maeda et al., Science 297, 379 (2002)]. We base our theoretical interpretation on the assumption that polymer chains are mainly subjected to oscillatory "reptation" along their "tubes." At high deformation frequencies-i.e., with a large sliding velocity V-the internal viscosity due to the rotational energy barriers around chain bonds hinders intramolecular mobility. As a result, energy dissipation and the correlated friction force strongly diminish at large V. Derived from a linear differential equation for chain dynamics, our results are basically consistent with the experimental data by Maeda et al. [Science 297, 379 (2002)] on modified polystyrene. Although the bulk polymer is below T(g), we regard the first few chain layers below the surface to be in the liquid state. In particular, the observed maximum of F vs V is consistent with physically reasonable values of the molecular parameters. As a general result, the ratio FV is a steadily decreasing function of V, tending to V(-2) for large velocities. We evaluate a much smaller friction for a cross-linked polymer under the assumption that the junctions are effectively immobile, also in agreement with the experimental results of Maeda et al. [Science 297, 379 (2002)].     

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 streching of polymer knots. Chain lengths up to N = 82 and prime knots 0₁ , 3₁ , 4₁ , 5₁ , 6₁ and 8₁ 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 simulation 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.     

Peptide chain dynamics in light and heavy water: zooming in on internal friction

Frictional effects due to the chain itself, rather than the solvent, may have a significant effect on protein dynamics. Experimentally, such "internal friction" has been investigated by studying folding or binding kinetics at varying solvent viscosity; however, the molecular origin of these effects is hard to pinpoint. We consider the kinetics of disordered glycine-serine and α-helix forming alanine peptides and a coarse-grained protein folding model in explicit-solvent molecular dynamics simulations. By varying the solvent mass over more than two orders of magnitude, we alter only the solvent viscosity and not the folding free energy. Folding dynamics at the near-vanishing solvent viscosities accessible by this approach suggests that solvent and internal friction effects are intrinsically entangled. This finding is rationalized by calculation of the polymer end-to-end distance dynamics from a Rouse model that includes internal friction. An analysis of the friction profile along different reaction coordinates, extracted from the simulation data, demonstrates that internal as well as solvent friction varies substantially along the folding pathways and furthermore suggests a connection between friction and the formation of hydrogen bonds upon folding.     

Unexpected relaxation dynamics of a self-avoiding polymer in cylindrical confinement

We report extensive simulations of the relaxation dynamics of a self-avoiding polymer confined inside a cylindrical pore. In particular, we concentrate on examining how confinement influences the scaling behavior of the global relaxation time of the chain, tau, with the chain length N and pore diameter D. An earlier scaling analysis based on the de Gennes blob picture led to tau approximately N(2)D(13). Our numerical effort that combines molecular dynamics and Monte Carlo simulations, however, consistently produces different tau results for N up to 2000. We argue that the previous scaling prediction is only asymptotically valid in the limit N"D(53)"1, which is currently inaccessible to computer simulations and, more interestingly, is also difficult to reach in experiments. Our results are thus relevant for the interpretation of recent experiments with DNA in nano- and microchannels.     

A Trefoil Knotted Polymer Produced through Ring Expansion

A synthetic strategy is reported for the production of a trefoil knotted polymer from a copper(I)‐templated helical knot precursor through ring expansion. The expected changes in the properties of the knotted polymer compared to a linear analogue, for example, reduced hydrodynamic radius and lower intrinsic viscosity, together with an atomic force microscopy (AFM) image of individual molecular knots, confirmed the formation of the resulting trefoil knotted polymer. The strategies employed here could be utilized to enrich the variety of available polymers with new architectures.     

Dissipative particle dynamics simulations of polymer melts. I. Building potential of mean force for polyethylene and cis-polybutadiene

We present the derivation of coarse-grained force fields for two types of polymers, polyethylene (PE), and cis-polybutadiene (cis-PB), using the concept of potential of mean force. Coarse-grained force fields were obtained from microscopic simulations for several coarse-graining levels, i.e., different number of monomers lambda per mesoscopic unit called "bead." These force fields are then used in dissipative particle dynamics (DPD) simulations to study structural and dynamical properties of polymer melts of PE and cis-PB. The radial distribution functions g(R), the end-to-end distance R0, the end-to-end vector relaxation time tau, and the chain center of mass self-diffusion D(CM), are computed for different chain lengths at different coarse-graining factor lambda. Scaling laws typical of the Rouse regime are obtained for both polymers for chain lengths ranging from 6 to 50 beads. It is found that the end-to-end distance R0 obtained from DPD simulations agree well with values obtained from both microscopic simulations and experiments. The dependence of the friction coefficient used in DPD simulations versus the coarse-graining level is discussed in view of the overall scaling of the dynamical properties.     

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.     

Single polymer dynamics of topologically complex DNA

Single molecule studies allow for the direct observation of polymer dynamics in dilute and concentrated solutions, thereby revealing polymer chain conformations and molecular sub-populations that may be obscured in ensemble-level measurements. Over the past two decades, researchers have used DNA as a model system to study polymer dynamics at the molecular level. The vast majority of studies have focused on linear DNA molecules; however, researchers have recently begun to study polymers with complex topologies and architectures at the single molecule level. Here, we explore recent work in single polymer dynamics focused on topologically complex DNA, including knots, ring polymers, and branched polymers. Experimental, computational, and theoretical advances have enabled in-depth studies of topologically complex DNA, with recent efforts focused on complex molecular conformations, intermolecular interactions, and topology-dependent dynamics. In this article, we highlight recent work aimed at understanding the interplay between molecular-scale behavior and the emergent properties of polymeric materials.     

Model carbyne knots vs ideal knots

The structure and stability of model carbyne knots built from 60 to 120 carbon atoms with 0, 3, 4,., 7 crossings have been estimated by semiempirical AM1 calculations. The calculations have shown an increase of the knot-cycle energy difference (deltaE) with an increasing number of knot crossings and a decrease of deltaE with an increasing number of atoms constituting the molecule. The deltaE changes nonlinearly with the characteristics of the corresponding ideal knots such as the average crossing number (ACN) and the length-to-diameter ratio (L/D). The molecular mechanic strain energy of carbyne knots correlates similarly with ACN and L/D of ideal knots. The calculated energy of the model carbyne knots correlates also with the electrophoretic mobility or sedimentation coefficient of DNA knots. Thus, similarly to characteristics of ideal knots, the energy of carbyne knots is a rather easily available parameter which can be used for further correlations with some characteristics of DNA knots.     

Microfluidics made of yarns and knots: from fundamental properties to simple networks and operations

We present and characterize cotton yarn and knots as building blocks for making microfluidic circuits from the bottom up. The yarn used is made up of 200-300 fibres, each with a lumen. Liquid applied at the extremity of the yarn spontaneously wets the yarn, and the wetted length increases linearly over time in untreated yarn, but progresses according to a square root relationship as described by Washburn's equation upon plasma activation of the yarn. Knots are proposed for combining, mixing and splitting streams of fluids. Interestingly, the topology of the knot controls the mixing ratio of two inlet streams into two outlet yarns, and thus the ratio can be adjusted by choosing a specific knot. The flow resistance of a knot is shown to depend on the force used to tighten it and the flow resistance rapidly increases for single-stranded knots, but remains low for double-stranded knots. Finally, a serial dilutor is made with a web made of yarns and double-stranded overhand knots. These results suggest that yarn and knots may be used to build low cost microfluidic circuits.     

Time-dependent Diffusion Coefficient and Conventional Diffusion Constant of Nanoparticles in Polymer Melts by Mode-coupling Theory (cited: 2)

Time-dependent diffusion coefficient and conventional diffusion constant are calculated and analyzed to study diffusion of nanoparticles in polymer melts. A generalized Langevin equa-tion is adopted to describe the diffusion dynamics. Mode-coupling theory is employed to calculate the memory kernel of friction. For simplicity, only microscopic terms arising from binary collision and coupling to the solvent density fluctuation are included in the formalism. The equilibrium structural information functions of the polymer nanocomposites required by mode-coupling theory are calculated on the basis of polymer reference interaction site modelwith Percus-Yevick closure. The effect of nanoparticle size and that of the polymer size are clarified explicitly. The structural functions, the friction kernel, as well as the diffusion coefficient show a rich variety with varying nanoparticle radius and polymer chain length. We find that for small nanoparticles or short chain polymers, the characteristic short time non-Markov diffusion dynamics becomes more prominent, and the diffusion coefficient takes longer time to approach asymptotically the conventional diffusion constant. This constant due to the microscopic contributions will decrease with the increase of nanoparticle size, while increase with polymer size. Furthermore, our result of diffusion constant from mode-coupling theory is compared with the value predicted from the Stokes-Einstein relation. It shows that the microscopic contributions to the diffusion constant are dominant for small nanoparticles or long chain polymers. Inversely, when nanonparticle is big, or polymer chain is short, the hydrodynamic contribution might play a significant role.     

The Self‐Sorting Behavior of Circular Helicates and Molecular Knots and Links

We report on multicomponent self‐sorting to form open circular helicates of different sizes from a primary monoamine, Fe^II ions, and dialdehyde ligand strands that differ in length and structure by only two oxygen atoms. The corresponding closed circular helicates that are formed from a diamine—a molecular Solomon link and a pentafoil knot—also self‐sort, but up to two of the Solomon‐link‐forming ligand strands can be accommodated within the pentafoil knot structure and are either incorporated or omitted depending on the stage that the components are mixed.