Stretching dynamics of semiflexible polymers

We analyze the nonequilibrium dynamics of single inextensible semiflexible biopolymers as stretching forces are applied at the ends. Based on different (contradicting) heuristic arguments, various scaling laws have been proposed for the propagation speed of the backbone tension which is induced in response to stretching. Here, we employ a newly developed unified theory to systematically substantiate, restrict, and extend these approaches. Introducing the practically relevant scenario of a chain equilibrated under some prestretching force f _pre that is suddenly exposed to a different external force f _ext at the ends, we give a concise physical explanation of the underlying relaxation processes by means of an intuitive blob picture. We discuss the corresponding intermediate asymptotics, derive results for experimentally relevant observables, and support our conclusions by numerical solutions of the coarse-grained equations of motion for the tension.     

Statistical mechanics of semiflexible polymers

We present the statistical-mechanical theory of semiflexible polymers based on the connection between the Kratky-Porod model and the quantum rigid rotator in an external homogeneous field, and treatment of the latter using the quantum mechanical propagator method. The expressions and relations existing for flexible polymers can be generalized to semiflexible ones, if one replaces the Fourier-Laplace transform of the end-to-end polymer distance, 1/(k ²/3 + p), through the matrix , where D and M are related to the spectrum of the quantum rigid rotator, and considers an appropriate matrix element of the expression under consideration. The present work provides also the framework to study polymers in external fields, and problems including the tangents of semiflexible polymers. We study the structure factor of the polymer, the transversal fluctuations of a free end of the polymer with fixed tangent of another end, and the localization of a semiflexible polymer onto an interface. We obtain the partition function of a semiflexible polymer in half space with Dirichlet boundary condition in terms of the end-to-end distribution function of the free semiflexible polymer, study the behaviour of a semiflexible polymer in the vicinity of a surface, and adsorption onto a surface.Received: 23 March 2004, Published online: 23 July 2004PACS: 36.20.-r Macromolecules and polymer molecules - 61.41. + e Polymers, elastomers, and plastics - 82.35.Gh Polymers on surfaces; adhesion     

Relaxation dynamics of semiflexible polymers

We study the relaxation dynamics of a semiflexible chain by introducing a time-dependent tension. The chain has one of its ends attached to a large bead, and the other end is fixed. We focus on the initial relaxation of the chain that is initially strongly stretched. Using a tension that is self-consistently determined, we obtain the evolution of the end-to-end distance with no free parameters. Our results are in good agreement with single molecule experiments on double stranded DNA.     

Anisotropic random networks of semiflexible polymers

Motivated by the organization of cross-linked cytoskeletal biopolymers, we present a semimicroscopic replica field theory for the formation of anisotropic random networks of semiflexible polymers. The networks are formed by introducing random permanent cross-links which fix the orientations of the corresponding polymer segments to align with one another. Upon increasing the cross-link density, we obtain a continuous gelation transition from a fluid phase to a gel where a finite fraction of the system gets localized at random positions. For sufficiently stiff polymers, this positional localization is accompanied by a continuous isotropic-to-nematic (IN) transition occurring at the same cross-link density. As the polymer stiffness decreases, the IN transition becomes first order, shifts to a higher cross-link density, and is preceded by an amorphous solid where the average polymer orientations freeze in random directions.     

Force-induced desorption and unzipping of semiflexible polymers

The thermally assisted force-induced desorption of semiflexible polymers from an adhesive surface or the unzipping of two bound semiflexible polymers by a localized force are investigated. The phase diagram in the force-temperature plane is calculated both analytically and by Monte Carlo simulations. Force-induced desorption and unzipping of semiflexible polymers are first order phase transitions. A characteristic energy barrier for desorption is predicted, which scales with the square root of the polymer bending rigidity and governs the initial separation process before a plateau of constant separation force is reached. This leads to activated desorption and unzipping kinetics accessible in single molecule experiments.     

Collapse transition of two-dimensional flexible and semiflexible polymers

The nature of the globule-coil transition of surface-confined polymers has been an issue of debate. Here this 2D collapse transition is studied through a partially directed lattice model. In the general case of polymers with positive bending stiffness (Delta>0), the collapse transition is first order; it becomes second order only in the limiting case of zero bending stiffness (Delta triple bond 0). These analytical results are confirmed by Monte Carlo simulations. We also suggest some possible future experiments.     

Triple points in solutions of polydisperse semiflexible polymers

When a mother solution of semiflexible polymers with differing molecular weights is forced to undergo phase transition, cloud and shadow curves emerge instead of a coexistence curve. For the first time, we calculate the cloud and shadow curves for an isotropic-nematic transition coupled to polydispersity and predict novel triple points. Because of the emergence of new triple points, polydispersity allows the occurrence of anisotropic phases at much lower polymer concentrations than for the monodisperse solutions.     

Activated dynamics of semiflexible polymers on structured substrates

We study the thermally activated motion of semiflexible polymers in double-well potentials using field-theoretic methods. Shape, energy, and effective diffusion constant of kink excitations are calculated, and their dependence on the bending rigidity of the semiflexible polymer is determined. For symmetric potentials, the kink motion is purely diffusive whereas kink motion becomes directed in the presence of a driving force. We determine the average velocity of the semiflexible polymer based on the kink dynamics. The Kramers escape over the potential barriers proceeds by nucleation and diffusive motion of kink-antikink pairs, the relaxation to the straight configuration by annihilation of kink-antikink pairs. We consider both uniform and point-like driving forces. For the case of point-like forces the polymer crosses the potential barrier only if the force exceeds a critical value. Our results apply to the activated motion of biopolymers such as DNA and actin filaments or of synthetic polyelectrolytes on structured substrates.     

Stretching instability of intrinsically curved semiflexible biopolymers: A lattice model approach

We apply a Monte Carlo simulation method to lattice systems to study the effect of an intrinsic curvature on the mechanical property of a semiflexible biopolymer.We find that when the intrinsic curvature is sufficiently large,the extension of a semiflexible biopolymer can undergo a first-order transition at finite temperature.The critical force increases with increasing intrinsic curvature.However,the relationship between the critical force and the bending rigidity is structuredependent.In a triangle lattice system,when the intrinsic curvature is smaller than a critical value,the critical force increases with the increasing bending rigidity first,and then decreases with the increasing bending rigidity.In a square lattice system,however,the critical force always decreases with the increasing bending rigidity.In contrast,when the intrinsic curvature is greater than the critical value,the larger bending rigidity always results in a larger critical force in both lattice systems.     

Triple minima in the free energy of semiflexible polymers

We study the free energy of the worm-like-chain model, in the constant-extension ensemble, as a function of the stiffness lambda for finite chains of length L. We find that the polymer properties obtained in this ensemble are qualitatively different from those obtained using constant-force ensembles. In particular, we find that as we change the stiffness parameter, t=L/lambda, the polymer makes a transition from the flexible to the rigid phase and there is an intermediate regime of parameter values where the free energy has three minima and both phases are stable. This leads to interesting features in the force-extension curves.     

Free energy of semiflexible polymers and structure of interfaces

The free energy of semiflexible polymers is calculated as a functional of the compositional scalar order parameter and the orientational order parameter of second-rank tensor S_ij on the basis of a microscopic model of wormlike chains with variable segment lengths. We use a density functional theory and a gradient expansion to evaluate the entropic part of the free energy, which is given in a power series of .The interaction term of the free energy is derived with a random phase approximation. For the rigid rod limit, the nematic-isotropic transition point is given by , N and w being the degree of polymerization and the anisotropic interaction parameter, respectively, and the degree of ordering at the transition point is 0.33448. We also find that the contour length of polymer chains becomes larger in a nematic phase than in an isotropic phase. Interface profiles are obtained numerically for some typical cases. In the neighborhood of isotropic-isotropic interfaces, polymer chains tend to align parallel to the interface on the polymer-rich side and perpendicular on the poor side. When an isotropic region and a nematic region coexist, orientational order parallel to the interface is preferred in the nematic region. Received: 28 May 1998 / Revised: 12 August 1998 / Accepted: 8 September 1998     

Elongation and fluctuations of semiflexible polymers in a nematic solvent

We directly visualize single polymers with persistence lengths l(p), ranging from 0.05 to 16 microm, dissolved in the nematic phase of rodlike fd virus. Polymers with a sufficiently large persistence length undergo a coil-rod transition at the isotropic-nematic transition of the background solvent. We quantitatively analyze the transverse fluctuations of the semiflexible polymers and show that at long wavelengths they are driven by the fluctuating nematic background. We extract the Odijk deflection length and the elastic constant of the background nematic phase from the data.     

Stretching of polymers in a random three-dimensional flow

Behavior of a dilute polymer solution in a random three-dimensional flow with an average shear is studied experimentally. Polymer contribution to the shear stress is found to be more than 2 orders of magnitude higher than in a laminar shear flow. The results indicate that the polymer molecules are strongly stretched by the random motion of the fluid.     

Potts model,Dirac propagator,and conformational statistics of semiflexible polymers

A new discretized version of the Dirac propagator ind space and one time dimensions is obtained with the help of the 2d-state, one-dimensional Potts model. The Euclidean version of this propagator describes all conformational properties of semiflexible polymers. It also describes all properties of fully directed self-avoiding walks. The case of semiflexible copolymers composed of a random sequence of fully flexible and semirigid monomer units is also considered. As a by-product, some new results for disordered one-dimensional Ising and Potts models are obtained. In the case of the Potts model the nontrivial extension of the results to higher dimensions is discussed briefly.     

On the origin of the heterogeneous orientation dynamics of semiflexible polymers in solutions

Collective rotational motion in nondilute isotropic solutions of semirigid chains is sensitively probed by depolarized light scattering over a broad time range. The bimodal shape and the peculiar dependence of the orientational relaxation function on the scattering angle might arise from the coupling between orientational and shear modes of molecular motion (P.G. de Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971)). The dynamic heterogeneity, i.e. the separation of the time scales and the shape of the relaxation functions appears to be system specific.     

Stretching of polymers in isotropic turbulence: a statistical closure

We present a new closure for the mean rate of stretching of a dissolved polymer by homogeneous isotropic turbulence. The polymer is modeled by a bead-spring-type model (e.g., Oldroyd B, FENE-P, Giesekus) and the analytical closure is obtained assuming the Lagrangian velocity gradient can be modeled as a Gaussian, white-noise stochastic process. The resulting closure for the mean stretching depends upon the ratio of the correlation time for strain and rotation. Additionally, we derived a second-order expression for circumstances when strain and rotation have a finite correlation time. Finally, the base level closure is shown to reproduce results from direct numerical simulations by simply modifying the coefficients.     

Single flexible and semiflexible polymers at high shear: Non-monotonic and non-universal stretching response

Using Brownian hydrodynamic simulation techniques, we study single polymers in shear. We investigate the effects of hydrodynamic interactions, excluded volume, chain extensibility, chain length and semiflexibility. The well-known stretching behavior with increasing shear rate [(g)\dot] \dot{{\gamma}} is only observed for low shear [(g)\dot] \dot{{\gamma}} < [(g)\dot]^max \dot{{\gamma}}^{{\max}}_{} , where [(g)\dot]^max \dot{{\gamma}}^{{\max}}_{} is the shear rate at maximum polymer extension. For intermediate shear rates [(g)\dot]^max \dot{{\gamma}}^{{\max}}_{} < [(g)\dot] \dot{{\gamma}} < [(g)\dot]^min \dot{{\gamma}}^{{\min}}_{} the radius of gyration decreases with increasing shear with minimum chain extension at [(g)\dot]^min \dot{{\gamma}}^{{\min}}_{} . For even higher shear [(g)\dot]^min \dot{{\gamma}}^{{\min}}_{} < [(g)\dot] \dot{{\gamma}} the chain exhibits again shear stretching. This non-monotonic stretching behavior is obtained in the presence of excluded-volume and hydrodynamic interactions for sufficiently long and inextensible flexible polymers, while it is completely absent for Gaussian extensible chains. We establish the heuristic scaling laws [(g)\dot]^max \dot{{\gamma}}^{{\max}}_{} ∼ N ^-1.4 and [(g)\dot]^min \dot{{\gamma}}^{{\min}}_{} ∼ N ^0.7 as a function of chain length N , which implies that the regime of shear-induced chain compression widens with increasing chain length. These scaling laws also imply that the chain response at high shear rates is not a universal function of the Weissenberg number Wi = [(g)\dot] \dot{{\gamma}} t \tau anymore, where t \tau is the equilibrium relaxation time. For semiflexible polymers a similar non-monotonic stretching response is obtained. By extrapolating the simulation results to lengths corresponding to experimentally studied DNA molecules, we find that the shear rate [(g)\dot]^max \dot{{\gamma}}^{{\max}}_{} to reach the compression regime is experimentally realizable.     

Shapes of semiflexible polymer rings

The shape of semiflexible polymer rings is studied over their whole range of flexibility. Investigating the joint distribution of asphericity and the nature of asphericity as well as their respective averages, we find two distinct shape regimes depending on the flexibility of the polymer. For a small perimeter to persistence length the fluctuating rings exhibit only planar, elliptical configurations. At higher flexibilities three-dimensional, crumpled structures arise. Analytic calculations confirm the qualitative behavior of the averaged shape parameters and the elliptical shape in the stiff regime.