Persistence Length Of Polymers Research Articles

Dynamics of a semiflexible polymer or polymer ring in shear flow

Polymers exposed to shear flow exhibit a remarkably rich tumbling dynamics. While rigid rods rotate on Jeffery orbits, a flexible polymer stretches and coils up during tumbling. Theoretical results show that in both of these asymptotic regimes the corresponding tumbling frequency f(c) in a linear shear flow of strength γ scales as a power law Wi(2/3) in the Weissenberg number Wi = γτ, where τ is a characteristic time of the polymer's relaxational dynamics. For a flexible polymer these theoretical results are well confirmed by a large body of experimental single molecule studies. However, for the intermediate semiflexible regime, especially relevant for cytoskeletal filaments like F-actin and microtubules, the situation is less clear. While recent experiments on single F-actin filaments are still interpreted within the classical Wi(2/3) scaling law, theoretical results indicated deviations from it. Here we perform extensive Brownian dynamics simulations to explore the tumbling dynamics of semiflexible polymers over a broad range of shear strength and the polymer's persistence length l(p). We find that the Weissenberg number alone does not suffice to fully characterize the tumbling dynamics, and the classical scaling law breaks down. Instead, both the polymer's stiffness and the shear rate are relevant control parameters. Based on our Brownian dynamics simulations we postulate that in the parameter range most relevant for cytoskeletal filaments there is a distinct scaling behavior with f(c) τ* = Wi(3/4)f(c)(x) with Wi = γτ* and the scaling variable x = (l(p)/L)(Wi)(-1/3); here τ* is the time the polymer's center of mass requires to diffuse its own contour length L. Comparing these results with experimental data on F-actin we find that the Wi(3/4) scaling law agrees quantitatively significantly better with the data than the classical Wi(2/3) law. Finally, we extend our results to single ring polymers in shear flow, and find similar results as for linear polymers with slightly different power laws.

Estimation of persistence lengths of semiflexible polymers: Insight from simulations

The persistence length of macromolecules is one of their basic characteristics, describing their intrinsic local stiffness. However, it is difficult to extract this length from physical properties of the polymers, different recipes may give answers that disagree with each other. Monte Carlo simulations are used to elucidate this problem, giving a comparative discussion of two lattice models, the self-avoiding walk model extended by a bond bending energy, and bottle-brush polymers described by the bond fluctuation model. The conditions are discussed under which a description of such macromolecules by Kratky-Porod worm-like chains holds, and the question to what extent the persistence length depends on external conditions (such as solvent quality) is considered. The scattering function of semiflexible polymers is discussed in detail, a comparison to various analytic treatments is given, and an outlook to experimental work is presented.

Estimation of persistence lengths of semiflexible polymers: insight from simulations

The persistence length of macromolecules is one of their basic characteristics, describing their intrinsic local stiffness. However, it is difficult to extract this length from physical properties of the polymers, different recipes may give answers that disagree with each other. Monte Carlo simulations are used to elucidate this problem, giving a comparative discussion of two lattice models, the self-avoiding walk model extended by a bond bending energy, and bottle-brush polymers described by the bond fluctuation model. The conditions are discussed under which a description of such macromolecules by Kratky-Porod worm-like chains holds, and the question to what extent the persistence length depends on external conditions (such as solvent quality) is considered. The scattering function of semiflexible polymers is discussed in detail, a comparison to various analytic treatments is given, and an outlook to experimental work is presented.

Conformational and rheological properties of semiflexible polymers in shear flow

A theoretical description is provided for the nonequilibrium conformational and dynamical properties of a polymer in shear flow. Using a mean-field semiflexible chain model, which accounts for hydrodynamic interactions within the preaveraging approximation, analytical expressions are derived for the dependence of the deformation, orientation, and relaxation times on polymer persistence length and shear rate. Moreover, the rheological properties of a dilute polymer solution are discussed. The model yields shear thinning at large Weissenberg numbers. The analytical results are compared with fluorescence microscopy measurements of individual DNA molecules, which yield qualitative and partial quantitative agreement.

Transport of Flexible Molecules in Narrow Confinements

Dynamical characteristics of flexible polymer molecules in nanoscopic confinements are primarily dictated by the relative values of the confinement length scales with respect to the polymer persistence length. Depending on whether the channel height is larger [1] or smaller [2] than the polymer persistence length, altogether different polymer dynamics is ensued, as illustrated in the pioneering theoretical studies by de Gennes [1] and Odijk [2]. Rapid advances of nanofabrication and polymer handling, over the last few years, have been able to provide experimental validation to these studies and at the same time have been able to unravel different intriguing physical issues unique to nanoconfinement induced dynamics of polymer molecules. These studies have led to a plethora of new applications ranging from the estimation of structural and mechanical properties of polymer to fabrication of novel, portable diagnostic tool kits. In this review article, we shall revisit different physical and technological iss...

Tuning Polymer Thickness: Synthesis and Scaling Theory of Homologous Series of Dendronized Polymers

The thickness of dendronized polymers can be tuned by varying their generation g and the dendron functionality X. Systematic studies of this effect require (i) synthetic ability to produce large samples of high quality polymers with systematic variation of g, X and of the backbone polymerization degree N, (ii) a theoretical model relating the solvent swollen polymer diameter, r, and persistence length, lambda, to g and X. This article presents an optimized synthetic method and a simple theoretical model. Our theory approach, based on the Boris-Rubinstein model of dendrimers predicts r approximately n(1/4)g(1/2) and lambda approximately n(2) where n = [(X - 1)(g) - 1]/(X - 2) is the number of monomers in a dendron. The average monomer concentration in the branched side chains of a dendronized polymer increases with g in qualitative contrast to bottle brushes whose side chains are linear. The stepwise, attach-to, synthesis of X = 3 dendronized polymers yielded gram amounts of g = 1-4 polymers with N approximately = 1000 and N approximately = 7000 as compared to earlier maxima of 0.1 g amounts and of N approximately = 1000. The method can be modified to dendrons of different X. The conversion fraction at each attach-to step, as quantified by converting unreacted groups with UV labels, was 99.3% to 99.8%. Atomic force microscopy on mixed polymer samples allows to distinguish between chains of different g and suggests an apparent height difference of 0.85 nm per generation as well as an increase of persistence length with g. We suggest synthetic directions to allow confrontation with theory.

Effect of confinement on properties of stiff biological macromolecules

The behaviour of semiflexible chains, modelling biopolymers such as DNA and actin in confined spaces, was investigated by means of Monte Carlo simulations. Simulations, based on the coarse-grained worm-like chain (WLC) model, assumed confinement length-scales comparable to those used in micro- and nanofluidic devices. The end-to-end chain elongation R was determined as a function of the channel dimensions and chain bending rigidity. Three regions of chain elongation R, identified in simulations in a cylinder and a slit, were described by current theoretical concepts. In harmony with the measurements of confined DNA, an abrupt transition between the blob region at moderate confinement and the deflection region at strong cylindrical confinement was found. The conditions for hairpin formation were elucidated as a trade-off between confinement and chain stiffness. The intrinsic persistence length of unconfined polymers was calculated by four methods that provided practically identical results. However, in confined geometries only the rigorous and WLC methods predicted the dependence of apparent persistence length P on confinement in a qualitatively correct way. It was found that the simple exponential function, suitable for the description of orientation correlations in free chains is, in confined systems, limited only to short distances along the chain contour and, thus, the apparent persistence length determined by this method just reproduces the intrinsic value of P. The orientation correlations from simulations were compared with analytical predictions in the deflection regime under strong confinement and with the measurements of actin filaments.

X‐ray Diffraction Study of the Structure of Carboxymethylcellulose–Cationic Surfactant Complexes

We have investigated dilute aqueous solutions of an anionic polymer (carboxymethylcellulose) mixed with cationic surfactants of different chain lengths (dodecyl to octadecyl trimethylammonium bromides: DTAB, TTAB, CTAB and OTAB). The structures of the concentrated phases formed above the precipitation threshold were studied by X-ray diffraction. Different body-centred cubic structures with space groups Pm3n were observed in the presence of surfactant with a short aliphatic chain (DTAB), despite the fact that the polymer persistence length is comparable to the repeat distance of the structure (5 nm). For larger surfactant chain lengths (TTAB and CTAB), the structure of the precipitates can be either cubic (Pm3n) or 2D hexagonal depending on the initial surfactant and polymer concentrations. For still larger chain length (OTAB), the structure becomes lamellar. This structural evolution from micellar cubic towards 2D hexagonal and lamellar is attributed to the decrease of the local curvature of the surfactant aggregates, as observed for flexible synthetic polymers and short DNA fragments under similar conditions. Furthermore, the structure of the bulk complexes formed just below the precipitation threshold anticipates the structure seen in the precipitated phases.

Simple Model for the Binding of a Polyelectrolyte to an Oppositely Charged Curved Surface

We present a simple model for the binding of persistence-length segments of a polyelectrolyte to an oppositely charged spherical macroion with radius of the same order as the polymer persistence length. The polyelectrolyte is taken as a rod with elastic bending capacity. The macroion is abstracted as an absolutely rigid circular arc. In the bound state, the polyelectrolyte bends onto the macroion, and some of the opposite charges are mutually neutralized. The curvature of the macroion imposes stability limits on the bound state. The model predicts transitions to bound-state stability in the same conditions (polyelectrolyte charge density and bare persistence length, macroion radius, salt concentration) under which they are observed in experimental studies of the binding of polyelectrolytes to oppositely charged micelles. The calculations also suggest why macroscopic phase separation occurs in experiments below a critical salt concentration but not above.

Extremely Slow Counterion Dynamics in Xanthan Liquid Crystal through 23Na and 14N NMR

To describe the structure and dynamics in a lyotropic xanthan liquid crystal, we have done 23Na and 14N NMR experiments. The spectra are consistent with a powder average over randomly oriented liquid-crystalline domains. From 14N quadrupolar echo experiments it was concluded that the exchange among the several micron sized domains is slow with residence times in the millisecond range. Conventional spin-lock experiments (14N and 23Na) as well as spin-lock experiments filtered through the quadrupolar spin polarization order (23Na only) revealed the existence of an extremely low frequency, submegahertz dispersion in the spectral densities pertaining to both sodium and ammonium counterions. The corresponding correlation times are on the order of tens of microseconds and agree with a structural correlation length on the order of 500 nm. The latter correlation times reflect the rearrangement of counterions among polymer segments over the same spatial extent as the polymer persistence length.

Topological effects in ring polymers. II. Influence Of persistence length

The interplay of topological constraints and the persistence length of ring polymers in their own melt is investigated by means of dynamical Monte Carlo simulations of a three-dimensional lattice model. We ask if the results are consistent with an asymptotically regime where the rings behave like (compact) lattice animals in a self-consistent network of topological constraints imposed by neighboring rings. Tuning the persistence length provides an efficient route to increase the ring overlap required for this mean-field picture to hold: The effective Flory exponent for the ring size decreases down to nu less, similar1/3 with increasing persistence length. Evidence is provided for the emergence of one additional characteristic length scale d(t) approximately N0, only weakly dependent on the persistence length and much larger than the excluded volume screening length xi. At distances larger than d(t) the conformational properties of the rings are governed by the topological interactions; at smaller distances rings and their linear chain counterparts become similar. (At distances smaller than xi both architectures are identical.) However, the crossover between both limits is intricate and broad, as a detailed discussion of the local fractal dimension (e.g., obtained from the static structure factor) reveals. This is due to various crossover effects which we are unable to separate even for the largest ring size (N=1024) presented here. The increased topological interactions also influence the dynamical properties. Mean-square displacements and their distributions depend crucially on the ring overlap, and show evidence of the existence of additional size and time scales. The diffusion constant of the rings goes down from effectively D(N) approximately N-1.22 for flexible rings with low overlap to D(N) approximately N-1.68 for strongly overlapping semiflexible rings.

Knotted Fishing Line, Covalent Bonds, and Breaking Points

Every angler knows that a fishing line breaks easily at the place of a tight knot. In their report (1) “How strong is a covalent bond?” Michel Grandbois et al. present a “fly fishing method” for measuring the force needed to break long polysaccharide chains. In these experiments, a single molecule was attached with covalent bonds (Si–C, Si–O) to the tip of an atomic force microscope (AFM) at one end and to a substrate at the other. The polymers they used were very long (thousands of sugar rings) and were relaxed in a solution before attachment to the surfaces. Persistence length of such polymers roughly corresponds to the dimension of the composing monomers, as demonstrated in a report by S. B. Smithet al. (2), in which single-stranded DNA was used. Chains of this size (a thousandfold longer that their persistence length) are usually knotted and frequently will have more than one knot (3). This knotty property of long polymers was not discussed by Grandbois et al. (1), and it casts doubt on the interpretations of the data and the conclusions in the report.

Nature of PEVK-titin elasticity in skeletal muscle.

A unique sequence within the giant titin molecule, the PEVK domain, has been suggested to greatly contribute to passive force development of relaxed skeletal muscle during stretch. To explore the nature of PEVK elasticity, we used titin-specific antibodies to stain both ends of the PEVK region in rat psoas myofibrils and determined the region's force-extension relation by combining immunofluorescence and immunoelectron microscopy with isolated myofibril mechanics. We then tried to fit the results with recent models of polymer elasticity. The PEVK segment elongated substantially at sarcomere lengths above 2.4 micro(m) and reached its estimated contour length at approximately 3.5 micro(m). In immunofluorescently labeled sarcomeres stretched and released repeatedly above 3 micro(m), reversible PEVK lengthening could be readily visualized. At extensions near the contour length, the average force per titin molecule was calculated to be approximately 45 pN. Attempts to fit the force-extension curve of the PEVK segment with a standard wormlike chain model of entropic elasticity were successful only for low to moderate extensions. In contrast, the experimental data also could be correctly fitted at high extensions with a modified wormlike chain model that incorporates enthalpic elasticity. Enthalpic contributions are likely to arise from electrostatic stiffening, as evidenced by the ionic-strength dependency of titin-based myofibril stiffness; at high stretch, hydrophobic effects also might become relevant. Thus, at physiological muscle lengths, the PEVK region does not function as a pure entropic spring. Rather, PEVK elasticity may have both entropic and enthalpic origins characterizable by a polymer persistence length and a stretch modulus.

Trapping and Wiggling: Elastohydrodynamics of Driven Microfilaments

We present an analysis of the planar motion of single semiflexible filaments subject to viscous drag or point forcing. These are the relevant forces in dynamic experiments designed to measure biopolymer bending moduli. By analogy with the “Stokes problems” in hydrodynamics (motion of a viscous fluid induced by that of a wall bounding the fluid), we consider the motion of a polymer, one end of which is moved in an impulsive or oscillatory way. Analytical solutions for the time-dependent shapes of such moving polymers are obtained within an analysis applicable to small-amplitude deformations. In the case of oscillatory driving, particular attention is paid to a characteristic length determined by the frequency of oscillation, the polymer persistence length, and the viscous drag coefficient. Experiments on actin filaments manipulated with optical traps confirm the scaling law predicted by the analysis and provide a new technique for measuring the elastic bending modulus. Exploiting this model, we also present a reanalysis of several published experiments on microtubules.

Estimation of the Persistence Length of Polymers by MD Simulations on Small Fragments in Solution. Application to Cellulose

A procedure is described to estimate the persistence length and related properties of a persistent polymer by MD simulations of a small fragment, including solvent effects. The procedure is applied to cellulose in aqueous solution, which leads to an upper limit of the persistence length of 145 ± 10 Å. The occurrence of small amounts of folded conformations seems to be indicated by comparison of the theoretical with experimental data. The behavior of the cellulose chain is consistent with the Kratky−Porod wormlike model.

An efficient method of generating dense polymer model melts by computer simulation

A new method of generating equilibrium configurations of dense polymer model systems with freely jointed chains is presented. It uses the idea of self and mutually avoiding random walks combined with molecular dynamics simulation to generate freely jointed polymer chains from a densely packed simple liquid. A prior knowledge of polymer persistence length in the densely packed condition is not needed. The efficiency and the resultant equilibrium polymer configurations show that this method works better than the ones used previously in the literature.

Interacting rigid polyelectrolytes

The persistence length of charged linear polymers in semidilute solution must increase substantially upon addition of salt. For very long polymers the persistence length increases as the salt concentration to the 5/2 power provided this salt concentration is not too large. In deriving this scaling we correct our previous prediction in light of a recent paper by Barrat and Joanny, and in agreement with their qualitative prediction

Persistence length and related conformational properties of semiflexible polymers from Dirac propagator

A recently discovered connection between Dirac’s fermions and semiflexible polymers [Ann. Phys. (NY) 202, 186 (1990)] is developed further. The scattering form factor, diffusion and viscosity coefficients, and relaxation times are obtained for the arbitrary stiffness of the chain in the limit of infinite dilution. It is demonstrated explicitly how the excluded volume effects can be incorporated in calculations of the above observables. New methods of experimental determination of the persistence length in the presence of excluded volume effects are proposed along with a new method of experimental evaluation of the excluded volume parameter. Obtained results can be used for both homo and random semiflexible copolymers.

Correlation of polymer persistence length with Euler buckling fluctuations.

When a polymer is modeled by a long elastic rod of homogeneous and isotropic material, we find that its persistence length is proportional to the cube of the critical length for Euler buckling caused by the root-mean-square contraction force. The proportionality constant involves the radius of the circular cross section of the rod. For paramyosin, the persistence length is larger than the critical buckling length by a factor of about 3.

Polymer persistence length characterized as a critical length for instability caused by a fluctuating twist.

We establish the identity of the persistence length of an easily bendable polymer with the critical length for the onset of elastic instability toward bending induced by the root-mean-square twist. For a polymer of arbitrary bending stiffness the identity remains approximately valid; the ratio of persistence length to critical length is always within 20% of unity.