Wormlike Chain Research Articles

Physicochemical behaviour of semi-rigid biopolymers in aqueous medium

In this paper, the objective is to point out some characteristics of stereoregular biopolymers, especially concerning their behaviour in aqueous solutions. Firstly, stereoregular polymers such as DNA, polypeptides and some polysaccharides adopt helical conformations at given thermodynamic conditions. This conformation is semi-rigid forming liquid cristalline phases and/or cooperative interactions stabilizing physical gels. Secondly, they are often charged and have positive or more frequently negative sites distributed regularly along the main chain, becoming polyelectrolytes. An unique viscometric behaviour is observed, but in the presence of slight amount of external salt (preferably ≥ 0.05M NaCl), their behaviour reaches that of a polymer having the same chemical structure and the same contour length but no ionic sites allowing their macromolecular characterization. Thirdly, the semi-rigid character gives an unique hydrodynamic behaviour: a model of worm like chain must be adopted, giving a quite different behaviour compared with the flexible chain model. Especially, even in specific thermodynamic conditions (the θ-conditions), the chains are not gaussian but they are expanded due to the local stiffness (expressed by the persistence length) with free draining behaviour and a Mark Houwink exponent larger than 0.5.

Scattering and Gaussian Fluctuation Theory for Semiflexible Polymers.

The worm-like chain is one of the best theoretical models of the semiflexible polymer. The structure factor, which can be obtained by scattering experiment, characterizes the density correlation in different length scales. In the present review, the numerical method to compute the static structure factor of the worm-like chain model and its general properties are demonstrated. Especially, the chain length and persistence length involved multi-scale nature of the worm-like chain model are well discussed. Using the numerical structure factor, Gaussian fluctuation theory of the worm-like chain model can be developed, which is a powerful tool to analyze the structure stability and to predict the spinodal line of the system. The microphase separation of the worm-like diblock copolymer is considered as an example to demonstrate the usage of Gaussian fluctuation theory.

Stretching a Semiflexible Polymer in a Tube.

How the statistical behavior of semiflexible polymer chains may be affected by force stretching and tube confinement is a classical unsolved problem in polymer physics. Based on the Odijk deflection theory and normal mode decomposition in terms of Fourier expansion, we have derived a new compact formula for the extension of a wormlike chain of finite length strongly confined in a tube and simultaneously stretched by an external force. We have also suggested a new deflection length, which together with the force-extension relation is valid for a very extended range of the tube-diameter/persistence-length ratio comparing to the classic Odijk theory. The newly derived formula has no adjustable fitting parameters for the whole deflection regime; in contrast, the classic Odijk length needs different prefactors to fit the free energy and average extension, respectively. Brownian dynamics simulations based on the Generalized Bead-Rod (GBR) model were extensively performed, which justified the theoretical predictions.

The influence of fiber on the rheological properties, microstructure and suspension behavior of the supramolecular viscoelastic fracturing fluid

An innovative fiber-based fracturing fluid technology has been developed to enhance the productivity of low-permeability and ultra-low-permeability reservoirs. However, few studies have focused on the influence of fiber on the rheological properties, microstructure and suspension behavior of a supramolecular fracturing fluid. In this article, the rheological properties, microstructure and proppant suspension behavior of the supramolecular fluid affected by the fibers were discussed. The rheological measurement showed the addition of fibers facilitated the formation of the fiber network and strengthened the mechanical network structure which improved the rheological properties of the fluid. The addition of fibers did not change the reversible shear thinning behavior of this fluid. Meanwhile, it was proven that the power-law model could accurately predict the rheological parameters of this fiber-laden fracturing fluid. Both temperature and shear rate could disassemble the complex structure of this fiber-based fracturing fluid resulting in the decrease of viscosity. The scanning electron microscopes (SEM) micrographs demonstrated the supramolecular network honeycomb structure of the fluid, which was made up of the wormlike micelles and polymer chains with extremely efficient physical crosslinking. The proppant suspension data indicated the elasticity factor did not represent the proppant suspension capacity of the liquid. The results of the rheological measurement, SEM and the static-column test showed that the fiber network could improve the proppant suspension capacity through strengthening the ability of load bearing and consolidation of the fluid.

Modeling the relaxation of internal DNA segments during genome mapping in nanochannels.

We have developed a multi-scale model describing the dynamics of internal segments of DNA in nanochannels used for genome mapping. In addition to the channel geometry, the model takes as its inputs the DNA properties in free solution (persistence length, effective width, molecular weight, and segmental hydrodynamic radius) and buffer properties (temperature and viscosity). Using pruned-enriched Rosenbluth simulations of a discrete wormlike chain model with circa 10 base pair resolution and a numerical solution for the hydrodynamic interactions in confinement, we convert these experimentally available inputs into the necessary parameters for a one-dimensional, Rouse-like model of the confined chain. The resulting coarse-grained model resolves the DNA at a length scale of approximately 6 kilobase pairs in the absence of any global hairpin folds, and is readily studied using a normal-mode analysis or Brownian dynamics simulations. The Rouse-like model successfully reproduces both the trends and order of magnitude of the relaxation time of the distance between labeled segments of DNA obtained in experiments. The model also provides insights that are not readily accessible from experiments, such as the role of the molecular weight of the DNA and location of the labeled segments that impact the statistical models used to construct genome maps from data acquired in nanochannels. The multi-scale approach used here, while focused towards a technologically relevant scenario, is readily adapted to other channel sizes and polymers.

Semiflexible Chains at Surfaces: Worm-Like Chains and beyond.

We give an extended review of recent numerical and analytical studies on semiflexible chains near surfaces undertaken at Institut Charles Sadron (sometimes in collaboration) with a focus on static properties. The statistical physics of thin confined layers, strict two-dimensional (2D) layers and adsorption layers (both at equilibrium with the dilute bath and from irreversible chemisorption) are discussed for the well-known worm-like-chain (WLC) model. There is mounting evidence that biofilaments (except stable d-DNA) are not fully described by the WLC model. A number of augmented models, like the (super) helical WLC model, the polymorphic model of microtubules (MT) and a model with (strongly) nonlinear flexural elasticity are presented, and some aspects of their surface behavior are analyzed. In many cases, we use approaches different from those in our previous work, give additional results and try to adopt a more general point of view with the hope to shed some light on this complex field.

Theory for nonlinear dynamic force spectroscopy

Dynamic force spectroscopy (DFS) is an experimental technique that is commonly used to assess information on the strength, energy landscape, and lifetime of noncovalent bio-molecular interactions. DFS traditionally requires an applied force that increases linearly with time so that the bio-complex under investigation is exposed to a constant loading rate. However, tethers or polymers can modulate the applied force in a nonlinear manner. For example, bacterial adhesion pili and polymers with worm-like chain properties are structures that show nonlinear force responses. In these situations, the theory for traditional DFS cannot be readily applied. In this work, we expand the theory for DFS to also include nonlinear external forces while still maintaining compatibility with the linear DFS theory. To validate the theory, we modeled a bio-complex expressed on a stiff, an elastic, and a worm-like chain polymer, using Monte Carlo methods, and assessed the corresponding rupture force spectra. It was found that the nonlinear DFS (NLDFS) theory correctly predicted the numerical results. We also present a protocol suggesting an experimental approach and analysis method of the data to estimate the bond length and the thermal off-rate.

A new force–extension formula for stretched macromolecules and polymers based on the Ising model

Abstract In this paper, we derive a new force–extension formula for stretched macromolecules and homogeneous polymer matrices. The Ising model arising from paramagnetism is employed, where the magnetic force is replaced by the external force, and the resistance energy is addressed in this model instead of the usual persistent length arising from the freely jointed chain and worm-like chain models. While the force–extension formula reveals the distinctive stretching features for stretched polymers, the resistance energy is found to increase almost linearly with the external force for our two polysaccharides stretching examples with and without ring conformational changes. In particular, a jump in the resistance energy which is caused by a conformational transition is investigated, and the gap between the jump determines the energy barrier between two conformational configurations. Our theoretical model matches well with experimental results undergoing no and single conformational transitions, and a Monte Carlo simulation has also been performed to ensure the correctness of the resistance energy. This technique might also be employed to determine the binding energy from other causes during molecular stretching and provide vital information for further theoretical investigations.

Studying PMMA films on silica surfaces with generic microscopic and mesoscale models

Polymer films on solid substrates present significant interest for fundamental polymer physics and industrial applications. For their mesoscale study, we develop a hybrid particle-based representation where polymers are modeled as worm-like chains and non-bonded interactions are introduced through a simple density functional. The mesoscale description is parameterized to match a generic microscopic model, which nevertheless can represent real materials. Choosing poly (methyl methacrylate) adsorbed on silica as a case study, the consistency of both models in describing conformational and structural properties in polymer films is investigated. We compare selected quantifiers of chain-shape, the structure of the adsorbed layer, as well as the statistics of loops, tails, and trains. Overall, the models are found to be consistent with each other. Some deviations in conformations and structure of adsorbed layer can be attributed to the simplified description of polymer/surface interactions and local liquid packing in the mesoscale model. These results are encouraging for a future development of pseudo-dynamical schemes, parameterizing the kinetics in the hybrid model via the dynamics of the generic microscopic model.

Understanding the stiffness of macromolecules: From linear chains to bottle-brushes

The intrinsic local stiffness of a polymer is characterized by its persistence length. However, its traditional definition in terms of the exponential decay of bond orientational correlations along the chain backbone is accurate only for Gaussian phantom-chain-like polymers. Also care is needed to clarify the conditions when the Kratky-Porod wormlike chain model is applicable. These problems are elucidated by Monte Carlo simulations of simple lattice models for polymers in both d = 2 and d = 3 dimensions. While the asymptotic decay of the bond orientational correlations for real polymers always is of power law form, the Kratky-Porod model is found to be applicable for rather stiff (but not too long) thin polymers in d = 3 (but not in d = 2). However, it does not describe thick chains, e.g., bottle-brush polymers, where stiffness is due to grafted flexible side-chains, and the persistence length grows proportional to the effective thickness of the bottle-brush. A scaling description of bottle-brushes is validated by simulations using the bond fluctuation model.

ICCM2015(Paper ID:1148): DPD Simulation of the Movement and Deformation of Bioconcave Cells

This paper presents a dissipative particle dynamics (DPDs) method for investigating the movement and deformation of biconcave shape red blood cells (RBCs) with the worm-like chain (WLC) bead spring. First, the stretching of a RBC is modeled and the obtained shape evolution of the cell agrees well with experimental results. Second, the movement and deformation of a RBC in shear flows are investigated and three typical modes (tumbling, intermittent and tank-treading) are observed. Lastly, an illustrating example of multi-RBCs in Poiseuille flow in a tube is simulated. We conclude that the presented DPD method with WLC spring can effectively model the movement and deformation of bioconcave cells.

Dilute solution properties of poly(di-tert-butyl fumarate)

The mean-square radius of gyration 〈S 2〉 is determined for poly(di-tert-butyl fumarate) (PDtBF) for a range of weight-average number of repeat units n w from 382 to 1520 in tetrahydrofuran at 30.0 °C. The data are analyzed on the basis of the Kratky–Porod (KP) wormlike chain with excluded volume (EV). It is shown that the average chain dimension of PDtBF is significantly larger than that of poly(diisopropyl fumarate) (PDiPF) because of a remarkable difference in chain stiffness between the two polymers.

On the properties of a bundle of flexible actin filaments in an optical trap.

We establish the statistical mechanics framework for a bundle of Nf living and uncrosslinked actin filaments in a supercritical solution of free monomers pressing against a mobile wall. The filaments are anchored normally to a fixed planar surface at one of their ends and, because of their limited flexibility, they grow almost parallel to each other. Their growing ends hit a moving obstacle, depicted as a second planar wall, parallel to the previous one and subjected to a harmonic compressive force. The force constant is denoted as the trap strength while the distance between the two walls as the trap length to make contact with the experimental optical trap apparatus. For an ideal solution of reactive filaments and free monomers at fixed free monomer chemical potential μ1, we obtain the general expression for the grand potential from which we derive averages and distributions of relevant physical quantities, namely, the obstacle position, the bundle polymerization force, and the number of filaments in direct contact with the wall. The grafted living filaments are modeled as discrete Wormlike chains, with F-actin persistence length ℓp, subject to discrete contour length variations ±d (the monomer size) to model single monomer (de)polymerization steps. Rigid filaments (ℓp = ∞), either isolated or in bundles, all provide average values of the stalling force in agreement with Hill's predictions Fs (H)=NfkBTln(ρ1/ρ1c)/d, independent of the average trap length. Here ρ1 is the density of free monomers in the solution and ρ1c its critical value at which the filament does not grow nor shrink in the absence of external forces. Flexible filaments (ℓp < ∞) instead, for values of the trap strength suitable to prevent their lateral escape, provide an average bundle force and an average trap length slightly larger than the corresponding rigid cases (few percents). Still the stalling force remains nearly independent on the average trap length, but results from the product of two strongly L-dependent contributions: the fraction of touching filaments ∝〈L〉(O.T.) (2) and the single filament buckling force ∝〈L〉(O.T.) (-2).

Effects of solution conditions on characteristics and size exclusion chromatography of pneumococcal polysaccharides and conjugate vaccines

Molecular properties of bacterial polysaccharides and protein-polysaccharide conjugates play an important role in the efficiency and immunogenicity of the final vaccine product. Size exclusion chromatography (SEC) is commonly used to analyze and characterize biopolymers, including capsular polysaccharides. The objective of this work was to determine the effects of solution ionic strength and pH on the SEC retention of several capsular polysaccharides from S. pneumoniae bacteria in their native and conjugated forms. The retention time of the charged polysaccharides increased with increasing ionic strength and decreasing pH due to compaction of the polysaccharides associated with a reduction in the intramolecular electrostatic interactions. The calculated radius of gyration was in good agreement with model calculations based on the worm-like chain model accounting for the increase in chain stiffness and excluded volume of the charged polysaccharide at low ionic strength. These results provide important insights into the effects of solution ionic strength on physical properties and SEC behavior of capsular polysaccharides and their corresponding conjugates.

Modeling the Abrupt Buckling Transition in dsDNA During Supercoiling

When deoxyribonucleic (DNA), held at a fixed tension, is subjected to torsional deformations, it responds by forming plectonemic supercoils accompanied by a reduction in its end-to-end extension. This transition from the extended state to the supercoiled state is marked by an abrupt buckling of the DNA accompanied by a rapid “hopping” of the DNA between the extended and supercoiled states. This transition is studied by means of Brownian dynamics simulations using a discrete wormlike-chain (dWLC) model of DNA. The simulations reveal, among other things, the distinct regimes that occur during DNA supercoiling and the probabilities of states within the buckling transition regime.

DNA Twist Stability Changes with Magnesium(2+) Concentration.

To understand DNA elasticity at high forces (F>30 pN), its helical nature must be taken into account, as a coupling between twist and stretch. The prevailing model, the wormlike chain, was previously extended to include this twist-stretch coupling. Motivated by DNA's charged nature, and the known effects of ionic charges on its elasticity, we set out to systematically measure the impact of buffer ionic conditions on twist-stretch coupling. After developing a robust fitting approach, we show, using our new data set, that DNA's helical twist is stabilized at high concentrations of the magnesium divalent cation. DNA's persistence length and stretch modulus are, on the other hand, relatively insensitive to the applied range of ionic strengths.

Impact of Conformational and Chemical Correlations on Microphase Segregation in Random Copolymers

Random copolymers play an important role in a range of soft materials applications and biological phenomena. An individual monomer is typically a single chemical unit whose length is comparable to or less than a Kuhn length, resulting in a monomer segment that is structurally rigid at length scales of a segregated domain. Previous work on random copolymer phase segregation addresses the impact of correlations between the chemical identities along the chains for flexible polymers. In these works, a single monomer unit is effectively a large polymer block that behaves as a random walk without conformational correlation associated with semiflexibility. In our work, we develop a model of semiflexible random copolymers using the wormlike chain model to capture conformational correlation of the polymer chains. To address the thermodynamics of microphase segregation and the structure of the segregated domains, we develop a random phase approximation up to quartic order in density fluctuations that leverages our ...

Elasticity of a semiflexible filament with a discontinuous tension due to a cross-link or a molecular motor.

We analyze the stretching elasticity of a wormlike chain with a tension discontinuity resulting from a Hookean spring connecting its backbone to a fixed point. The elasticity of isolated semiflexible filaments has been the subject in a significant body of literature, primarily because of its relevance to the mechanics of biological matter. In real systems, however, these filaments are usually part of supramolecular structures involving cross-linkers or molecular motors, which cause tension discontinuities. Our model is intended as a minimal structural element incorporating such a discontinuity. We obtain analytical results in the weakly bending limit of the filament, concerning its force-extension relation and the response of the two parts in which the filament is divided by the spring. For a small tension discontinuity, the linear response of the filament extension to this discontinuity strongly depends on the external tension. For large external tension f, the spring force contributes a subdominant correction ∼1/f^{3/2} to the well-known ∼1/sqrt[f]-dependence of the end-to-end extension.

When Computer Simulation Excels Experiment

For many biophysical problems we have, in general, two approaches. A problem can be studied experimentally, or the corresponding system can be simulated. So far, for the great majority of problems, simulation cannot compete with experiment. Still, simulation seems very attractive because if we solve a problem this way, we can obtain much more interesting additional information about the system. This can be seen in the field of large-scale conformational properties of DNA where simulation and experiment compete successfully and, of course, complement each other.

Simulation of DNA Supercoil Relaxation

Several recent single-molecule experiments observe the response of supercoiled DNA to nicking endonucleases and topoisomerases. Typically in these experiments, indirect measurements of supercoil relaxation are obtained by observing the motion of a large micron-sized bead. The bead, which also serves to manipulate DNA, experiences significant drag and thereby obscures supercoil dynamics. Here we employ our discrete wormlike chain model to bypass experimental limitations and simulate the dynamic response of supercoiled DNA to a single strand nick. From our simulations, we make three major observations. First, extension is a poor dynamic measure of supercoil relaxation; in fact, the linking number relaxes so fast that it cannot have much impact on extension. Second, the rate of linking number relaxation depends upon its initial partitioning into twist and writhe as determined by tension. Third, the extensional response strongly depends upon the initial position of plectonemes.