Inextensible Chain Research Articles

A novel constitutive model considering the role of elastic lamellae' structural heterogeneity in homogenizing transmural stress distribution in arteries.

Understanding how the homeostatic stress state can be reached in arterial tissues can provide new insights into vascular physiology. Even though the function of maintaining homeostasis is often linked to the concentric layers of medial elastic lamellae, how the lamellae are capable of evenly distributing the stress transmurally remains to be understood. The recent microstructural study by Yu et al. (2018 J. R. Soc. Interface 15, 20180492) revealed that, circumferentially, lamellar layers closer to the lumen are wavier than the ones further away from it and, thus, experience more unfolding when subjected to blood pressure. Motivated by this peculiar finding, the current study, for the first time, proposes a novel approach to model elastic lamellae and such structural heterogeneity using the extensible worm-like chain model. When implemented into the material description of the conventional two-layer artery model, in which adventitial collagen is modelled using the inextensible worm-like chain model, it is demonstrated that structural heterogeneity in elastic lamellae plays an important role in dictating transmural stress distribution and, therefore, the homeostasis of the arterial wall.

Plane-Parallel Sliding of a Flexible Inextensible Chain over the Rounded Edge of a Horizontal Table

Plane-Parallel Sliding of a Flexible Inextensible Chain over the Rounded Edge of a Horizontal Table

Dynamics of a long chain in turbulent flows: impact of vortices.

We show and explain how a long bead-spring chain, immersed in a homogeneous isotropic turbulent flow, preferentially samples vortical flow structures. We begin with an elastic, extensible chain which is stretched out by the flow, up to inertial-range scales. This filamentary object, which is known to preferentially sample the circular coherent vortices of two-dimensional (2D) turbulence, is shown here to also preferentially sample the intense, tubular, vortex filaments of three-dimensional (3D) turbulence. In the 2D case, the chain collapses into a tracer inside vortices. In the 3D case on the contrary, the chain is extended even in vortical regions, which suggests that the chain follows axially stretched tubular vortices by aligning with their axes. This physical picture is confirmed by examining the relative sampling behaviour of the individual beads, and by additional studies on an inextensible chain with adjustable bending-stiffness. A highly flexible, inextensible chain also shows preferential sampling in three dimensions, provided it is longer than the dissipation scale, but not much longer than the vortex tubes. This is true also for 2D turbulence, where a long inextensible chain can occupy vortices by coiling into them. When the chain is made inflexible, however, coiling is prevented and the extent of preferential sampling in two dimensions is considerably reduced. In three dimensions, on the contrary, bending stiffness has no effect, because the chain does not need to coil in order to thread a vortex tube and align with its axis. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Kinetics of rare events for non-Markovian stationary processes and application to polymer dynamics

How much time does it take for a fluctuating system, such as a polymer chain, to reach a target configuration that is rarely visited -- typically because of a high energy cost ? This question generally amounts to the determination of the first-passage time statistics to a target zone in phase space with lower occupation probability. Here, we present an analytical method to determine the mean first-passage time of a generic non-Markovian random walker to a rarely visited threshold, which goes beyond existing weak-noise theories. We apply our method to polymer systems, to determine (i) the first time for a flexible polymer to reach a large extension, and (ii) the first closure time of a stiff inextensible wormlike chain. Our results are in excellent agreement with numerical simulations and provide explicit asymptotic laws for the mean first-passage times to rarely visited configurations.

Inhomogeneous Forces in Semiflexible Biopolymers

Major developments in single-molecule experiments over the past few decades have permitted the manipulation of biomolecules and the study of their responses to force fields in unprecedented detail. These methods have led to the greater understanding of a wide range of biologically relevant processes, including the dynamics of cytoskeletal filaments, the stability of DNA double helix, or the formation of histone-DNA complexes. The wormlike chain model has been successfully applied to predict force-extension relations of semiflexible biopolymers. Despite these successes, the effects of inhomogeneous forces (for example effects of charge heterogeneity in a polyelectrolyte) are not well understood. In this work, we develop a new analytical model to understand the statistics of an inextensible polyelectrolyte chain under inhomogeneous tension. We show that an existing mean-field formalism that does not account for the inhomogeneity in the tension results in local over-stretching of the chain. We improve upon this approach by subdividing the chain into smaller segments over which the homogeneity assumption is only approximately valid. This technique can analytically impose inextensibility of the chain and help us study the statistical properties of the chain. We can also extend the theory to predict the protein-binding statistics in experiments that probe protein-DNA interactions under an external field, applicable to DNA wrapping around histone octamers. This proposed method may prove valuable to other biologically relevant problems involving inhomogeneities in actin bundles, semiflexible chains under shear flow or in the design of biomimetic materials.

The Catenary in History and Applications (La Catenaria nella Storia e nelle Applicazioni)

The catenary is one of the most common curves; it is, in fact, the shape assumed by a homogeneous and inextensible chain, fixed to the extremities, which is subject only to its own weight. The catenary has always fascinated not only mathematicians but also architects and engineers, who have often used it in their works due to its remarkable properties. In this note, the catenary is introduced by determining its equation and considering its history from Galileo Galilei to the present day, pointing out a historical mistake. Then, some of its applications to architecture and engineering are shown and the catenary is compared to the parabola, a curve that at first sight can look similar to catenary but must not be confused with it. Finally, some variants of the catenary are considered: the weighted catenary and the catenary of equal resistance, highlighting their properties and their practical applications.

Statistical model of a flexible inextensible polymer chain: The effect of kinetic energy.

Because of the holonomic constraints, the kinetic energy contribution in the partition function of an inextensible polymer chain is difficult to find, and it has been systematically ignored. We present the first thermodynamic calculation incorporating the kinetic energy of an inextensible polymer chain with the bending energy. To explore the effect of the translation-rotation degrees of freedom, we propose and solve a statistical model of a fully flexible chain of N+1 linked beads which, in the limit of smooth bending, is equivalent to the well-known wormlike chain model. The partition function with the kinetic and bending energies and correlations between orientations of any pair of links and velocities of any pair of beads are found. This solution is precise in the limits of small and large rigidity-to-temperature ratio b/T. The last exact solution is essential as even very "harmless" approximation results in loss of the important effects when the chain is very rigid. For very high b/T, the orientations of different links become fully correlated. Nevertheless, the chain does not go over into a hard rod even in the limit b/T→∞: While the velocity correlation length diverges, the correlations themselves remain weak and tend to the value ∝T/(N+1). The N dependence of the partition function is essentially determined by the kinetic energy contribution. We demonstrate that to obtain the correct energy and entropy in a constrained system, the T derivative of the partition function has to be applied before integration over the constraint-setting variable.

Stability of discoidal high-density lipoprotein particles

Motivated by experimental and numerical studies revealing that discoidal high-density lipoprotein (HDL) particles may adopt flat elliptical and nonplanar saddle-like configurations, it is hypothesized that these might represent stabilized configurations of initially unstable flat circular particles. A variational description is developed to explore the stability of a flat circular discoidal HDL particle. While the lipid bilayer is modeled as two-dimensional fluid film endowed with surface tension and bending elasticity, the apoA-I belt is modeled as one-dimensional inextensible twist-free chain endowed with bending elasticity. Stability is investigated using the second variation of the underlying energy functional. Various planar and nonplanar instability modes are predicted and corresponding nondimensional critical values of salient dimensionless parameters are obtained. The results predict that the first planar and nonplanar unstable modes occur due to in-plane elliptical and transverse saddle-like perturbations. Based on available data, detailed stability diagrams indicate the range of input parameters for which a flat circular discoidal HDL particle is linearly stable or unstable.

Harmonically Confined, Semiflexible Polymer in a Channel: Response to a Stretching Force and Spatial Distribution of the Endpoints

We consider an inextensible, semiflexible polymer or worm-like chain which is confined in the transverse direction by a parabolic potential and subject to a longitudinal force at the ends, so that the polymer is stretched out and backfolding is negligible. Simple analytic expressions for the partition function, valid in this regime, are obtained for chains of arbitrary length with a variety of boundary conditions at the ends. The spatial distribution of the end points or radial distribution function is also analyzed.

From constrained stochastic processes to the nonlinear sigma model. Two old problems revisited

In this work a method is presented to derive the generating functional in path integral form for a system with an arbitrary number of degrees of freedom and constrained by general conditions. The method is applied to the case of the dynamics of an inextensible chain subjected to external forces. Next, the generating functional of the inextensible chain is computed assuming that the interactions are switched off. Finally, the generating functional of a two-dimensional nonlinear sigma model with O ( 3 ) symmetry is derived exploiting its similarities with the model describing the dynamics of the inextensible chain.

Fluctuations of a long, semiflexible polymer in a narrow channel

We consider an inextensible, semiflexible polymer or wormlike chain, with persistence length P and contour length L, fluctuating in a cylindrical channel of diameter D. In the regime D<<P<<L , corresponding to a long, tightly confined polymer, the average length of the channel <R(∥)> occupied by the polymer and the mean-square deviation from the average vary as <R(∥)>=[1-α(∘)(D/P)(2/3)]L and <ΔR(∥)(2)>=β(∘)(D(2)P)L , respectively, where α(∘) and β(∘) are dimensionless amplitudes. In earlier work we determined α(∘) and the analogous amplitude α(square) for a channel with a rectangular cross section from simulations of very long chains. In this paper, we estimate β(∘) and β(square) from the simulations. The estimates are compared with exact analytical results for a semiflexible polymer confined in the transverse direction by a parabolic potential instead of a channel and with a recent experiment. For the parabolic confining potential we also obtain a simple analytic result for the distribution of R(∥) or radial distribution function, which is asymptotically exact for large L and has the skewed shape seen experimentally.

DNA force-extension curve under uniaxial stretching

Single-molecule experiments indicate that a double-stranded DNA (ds‐DNA) increases in length if put under tension greater than 10 pN; beyond this point, its conformation can no longer be described using an inextensible worm-like chain model. For this purpose, a general sequence-dependent elastic model for tensions greater than 10 pN and for both single-stranded (ss) and ds‐DNA is proposed, and the effective elastic bending and torsional rigidities are determined from experiments to characterise their deformation. The key to this progress is that the bending and torsional deformations of the DNA backbones, the base-stacking interactions and the hydrogen bond force between the complementary base pairs are quantitatively considered in this model. Moreover, this simple elastic model can be used to globally fit to the abrupt B–S experimental transition data over a wide range of DNA molecule extensions. Based on this robust model, further study may be warranted on the mechanical response of ss- and ds‐DNA molecules.

O(N) Generalized nonlinear sigma model and its applications

In this work the results of an interdisciplinary research between field theory and statistical mechanics will be presented. It is shown that the dynamics of an inextensible chain is described by a slight generalization of the O(d) nonlinear sigma model. It is checked, that in the large time limit the correct equilibrium distribution is reached. Our approach is based on path integrals, but it may be also connected to the usual description of the dynamics of a chain as a diffusion stochastic process. Some applications of our results will be discussed, like for instance the calculation of the average distance between two polymer segments.

On a path integral description of the dynamics of an inextensible chain and its connection to constrained stochastic dynamics

The dynamics of a freely jointed chain in the continuous limit is described by a field theory which closely resembles the nonlinear sigma model. The generating functional Ψ[J] of this field theory contains nonholonomic constraints, which are imposed by inserting in the path integral expressing Ψ[J] a suitable product of delta functions. The same procedure is commonly applied in statistical mechanics in order to enforce topological conditions on a system of linked polymers. The disadvantage of this method is that the contact with the stochastic process governing the diffusion of the chain is apparently lost. The main goal of this work is to re-establish this contact. For this purpose, it is shown here that the generating functional Ψ[J] coincides with the generating functional of the correlation functions of the solutions of a constrained Langevin equation. In the discrete case, this Langevin equation describes as expected the Brownian motion of beads connected together by links of fixed length.

Stretching of semiflexible polymers with elastic bonds.

A semiflexible harmonic chain model with extensible bonds is introduced and applied to the stretching of semiflexible polymers or filaments. The semiflexible harmonic chain model allows to study effects from bending rigidity, bond extension, discrete chain structure, and finite length of a semiflexible polymer in a unified manner. The interplay between bond extension and external force can be described by an effective inextensible chain with increased stretching force, which leads to apparently reduced persistence lengths in force-extension relations. We obtain force-extension relations for strong- and weak-stretching regimes which include the effects of extensible bonds, discrete chain structure, and finite polymer length. We discuss the associated characteristic force scales and calculate the crossover behaviour of the force-extension curves. Strong stretching is governed by the discrete chain structure and the bond extensibility. The linear response for weak stretching depends on the relative size of the contour length and the persistence length which affects the behaviour of very rigid filaments such as F-actin. The results for the force-extension relations are corroborated by transfer matrix and variational calculations.

Bending and twisting elasticity: A revised Marko-Siggia model on DNA chirality

A revised Marko-Siggia elastic model for the DNA double helix [Macromolecules 27, 981 (1994)] is proposed, which includes the inextensible wormlike chain bending energy and a new chiral twisting energy term. It is predicted that the mean helical repeat length (HRL) for short DNA rings increases with decreasing chain length, while for very long chains in solution, their mean HRL has the same value longer than that for rectilinear DNAs, independent both of the chain length and of whether the ends are closed. These theoretical results are in good agreement with recent experimental investigations, and it might be possible that the chirality in twisting will account for the long-standing linking number deficit puzzle observed in organelle DNAs.

Microscopic Elasticity of DNA from Torsionally-Constrained Stretching

AbstractWe investigate the statistical mechanics of a torsionally constrained polymer. The polymer is modelled as an inextensible chain with bend rigidity A, twist rigidity C, and twist-stretch coupling D. In such a model, thermal bend fluctuations couple geometrically to an applied torque through the relation Lk = Tw + Wr. We explore this coupling and find excellent agreement between the predictions of our model and the single λ-DNA molecule stretching experiments of Strick et al. [Science 271 (1996) 1835]. This analysis affords an experimental determination of the microscopic twist rigidity C. Quantitative agreement between theory and experiment is obtained using C = 120 nm and D = 50 nm. The theory further predicts a thermal reduction of the effective twist rigidity induced by bend fluctuations.

On the Dynamics of Chains

This paper classifies and constructs similarity solutions of the partial differential equations modelling the quasi-static motion of a heavy inextensible chain dragging over a rough plane as it is pulled at one or both ends. Coulomb friction is assumed to act between the chain and the supporting plane.. The solutions include translations, rotations, and travelling waves as well as more general motions. One solution involves propagation of a moving segment of chain into a stationary segment.

Oscillations of a suspended chain

The venerable problem is considered of finding the natural frequencies of small oscillations about the equilibrium configuration of a suspended chain, under the assumptions of a continuous, perfectly flexible, inextensible, and lossless chain, supported at two fixed endpoints in a uniform gravitational field. A variety of analytical, variational, and numerical methods gives results that are valid over the full range of chain span-to-length ratios, including the limiting cases of very slack and very taut chains. Some surprising predictions of the theory are confirmed by experimental data.

Negative thermal expansion of polymer crystals: Lattice model

AbstractThe lateral thermal motion of atoms in an inextensible polymer chain would cause a shortening along the chain direction. This effect could explain the observed negative thermal expansivity α along the chain direction of a polymer crystal. Using this idea, the expansivity α is calculated for a lattice of parallel linear chains. A value of −1.3 × 10−5 K is obtained for chains with a carbon backbone, in good agreement with data for polyethylene.