Wormlike Chain Research Articles

Force-extension and longitudinal response of wormlike chains with single cross-link

Certain important biopolymers, such as actin filaments, are known to have cross-links at their interfaces, which significantly influence their mechanical properties. To explore these effects, the force-extension and longitudinal response of wormlike chains (WLCs) with a single cross-link under tension in two-dimension are examined using both analytical methods and Brownian dynamics simulations. The cross-link is modeled as a spring in the analytical method, and mode analysis is used to calculate the path integrals associated with the partition function. These theoretical results are then validated through Brownian dynamics simulations. Final results indicate that the simulation results are consistent with the theoretical predictions, particularly for situations involving large tensile force and short chain, which are prerequisites for the application of the weak bending approximation.

Exploring protein-mediated compaction of DNA by coarse-grained simulations and unsupervised learning

Protein-DNA interactions and protein-mediated DNA compaction play key roles in a range of biological processes. The length scales typically involved in DNA bending, bridging, looping, and compaction (≥1 kbp) are challenging to address experimentally or by all-atom molecular dynamics simulations, making coarse-grained simulations a natural approach. Here, we present a simple and generic coarse-grained model for DNA-protein and protein-protein interactions and investigate the role of the latter in the protein-induced compaction of DNA. Our approach models the DNA as a discrete worm-like chain. The proteins are treated in the grand canonical ensemble, and the protein-DNA binding strength is taken from experimental measurements. Protein-DNA interactions are modeled as an isotropic binding potential with an imposed binding valency without specific assumptions about the binding geometry. To systematically and quantitatively classify DNA-protein complexes, we present an unsupervised machine learning pipeline that receives a large set of structural order parameters as input, reduces the dimensionality via principal-component analysis, and groups the results using a Gaussian mixture model. We apply our method to recent data on the compaction of viral genome-length DNA by HIV integrase and find that protein-protein interactions are critical to the formation of looped intermediate structures seen experimentally. Our methodology is broadly applicable to DNA-binding proteins and protein-induced DNA compaction and provides a systematic and semi-quantitative approach for analyzing their mesoscale complexes.

Contrast dependence of scattering profiles for poly(ethylene glycol) in water: Investigation by small-angle neutron scattering with 3He spin filter and small-angle x-ray scattering.

The small-angle neutron scattering (SANS) and small-angle x-ray scattering (SAXS) measurements were performed for deuterated and non-deuterated poly(ethylene glycol) (d-PEG and h-PEG, respectively) in D2O and a D2O/H2O mixed solvent (Mix) to compare the scattering profiles. To determine the coherent scattering intensity of SANS, a 3He spin filter was utilized. The scattering profiles determined by the SANS measurements were analyzed in terms of the wormlike chain model with touched beads along the contour of the chain. However, the SAXS profiles were not explained by the same model with uniform beads but with beads each consisting of a core and a shell having different electron densities. To explore the chain thickness determined from the SANS profile, the scattering intensities for different combinations of d-PEG/D2O, d-PEG/Mix, h-PEG/D2O, and h-PEG/Mix were also examined.

Forced extension of a wormlike chain in the Gibbs and Helmholtz ensembles.

A semiflexible polymer can be stretched by either applying a force to it or by fixing the positions of its endpoints. The two approaches generally yield different results and correspond to experiments performed in either the Gibbs or Helmholtz statistical ensembles. Here, we derive the Helmholtz force-extension relationship for the commonly used wormlike-chain model in the strongly stretched regime. By analyzing it in comparison with the Gibbs ensemble result, we show that equivalence between the two relationships is achieved only in the long-chain thermodynamic limit.

Dimension and Flexibility of Polystyrenesulfonate Chains in Methanol-Water.

The influence of the relative permittivity of the solvent medium on the single-chain dimension and flexibility of sodium polystyrenesulfonate chains has been investigated in mixed solvent media of methanol and water using viscosity experiments. Particular attention has been paid to explore the effect of the added low-molar-mass electrolyte. The root-mean-square (rms) radii of gyration of the chains in the unperturbed state have been calculated by applying the Flory model, while the intrinsic persistence lengths by the Benoit-Doty equation on the basis of the Kratky-Porod worm-like chain model. Estimation of the expansion factors for the rms radius of gyration, and the electrostatic persistence length helps evaluate the rms radii of gyration and the total persistence length of polystyrenesulfonate chains in the presence of varying amount of the supporting electrolyte. The polyion chains are highly extended at low ionic strengths but exhibit coil-like behavior with small persistence lengths when an excess of the supporting electrolyte is added in all the methanol-water mixtures investigated. Specifically, in the investigated solvent media, the polystyrenesulfonate chains have been found to shrink by ∼63-65% in the θ-state from their expanded conformation in the presence of 0.0001 mol L-1 NaCl. The chain dimensions pass through a maximum as the medium becomes richer in methanol, which could be explained by the formation and breakup of internal rings involving the polyion chain and water and/or methanol molecules. The intrinsic persistence length of sodium polystyrenesulfonate in a methanol-water mixture containing 0.1 mole fraction of methanol is ca. 1.3 times that in a medium with 0.3 mole fraction of methanol, indicating that flexibility of the polyion depends appreciably on the relative permittivity of the medium.

Microstructure-based nuclear lamina constitutive model.

The nuclear lamina is widely recognized as the most crucial component in providing mechanical stability to the nucleus. However, it is still a significant challenge to model the mechanics of this multilayered protein network. We developed a constitutive model of the nuclear lamina network based on its microstructure, which accounts for the deformation phases at the dimer level, as well as the orientational arrangement and density of lamin filaments. Instead of relying on homology modeling in the previous studies, we conducted molecular simulations to predict the force-extension response of a highly accurate lamin dimer structure obtained through X-ray diffraction crystallography experimentation. Furthermore, we devised a semiflexible worm-like chain extension-force model of lamin dimer as a substitute, incorporating phases of initial stretching, uncoiling of the dimer coiled-coil, and transition of secondary structures. Subsequently, we developed a 2D network continuum model for the nuclear lamina by using our extension-force lamin dimer model and derived stress resultants. By comparing with experimentally measured lamina modulus, we found that the lamina network has sharp initial strain-hardening behavior. This also enabled us to carry out finite element simulations of the entire nucleus with an accurate microstructure-based nuclear lamina model. Finally, we conducted simulations of transendothelial transmigration of a nucleus and investigated the impact of varying network density and uncoiling constants on the critical force required for successful transmigration. The model allows us to incorporate the microstructure characteristics of the nuclear lamina into the nucleus model, thereby gaining insights into how laminopathies and mutations affect nuclear mechanics.

Coexisting superhydrophobicity and superadhesion features of Ziziphus mauritiana abaxial leaf surface with possibility of biomimicking using electrospun microfibers

The coexistence of superhydrophobicity and superadhesion features is being reported for the abaxial leaf of Indian jujube (Ziziphus mauritiana) possessing hairy, matted surface texture with fiber dia ranging ∼ 5.6–7.1 μm. Very high-water contact angle (WCA > 143°) and high contact angle hysteresis (CAH ∼ 30°–46°) were observed and compared for the tender, mature, and senescent leaf states. The tender leaf exhibits a maximal superhydrophobicity with WCA as high as ∼151° and an increased roll-off angle from ∼21° to 33°. Conversely, next two leaf states are characterized by excellent adhesion even up to a base tilting of 90° without any tendency to roll-off. An attempt has also been made for biomimicking the leaf's hairy fiber microstructure with polyvinylidene fluoride (PVDF) by employing an electrospinning setup, with adjustable control parameters. The fabricated ∼4.3-μm-dia PVDF-based nonwoven fibers were seen to be replicated at par with excellent superhydrophobicity and high adhesion features. The WCA and CAH of artificially grown fibers were estimated to be ∼145.7° and 49.4°. The nonwoven, yarn-like surface construct of microfibers fits well in worm-like chain model, which considers a normal distribution of segments described through discrete jointed length, persistent length, and bending angle between successive segments.

Scaling regimes for wormlike chains confined to cylindrical surfaces under tension.

We compute the free energy of confinement [Formula: see text] for a wormlike chain (WLC), with persistence length [Formula: see text], that is confined to the surface of a cylinder of radius R under an external tension f using a mean field variational approach. For long chains, we analytically determine the behavior of the chain in a variety of regimes, which are demarcated by the interplay of [Formula: see text], the Odijk deflection length ([Formula: see text]), and the Pincus length ([Formula: see text], with [Formula: see text] being the thermal energy). The theory accurately reproduces the Odijk scaling for strongly confined chains at [Formula: see text], with [Formula: see text]. For moderate values of f, the Odijk scaling is discernible only when [Formula: see text] for strongly confined chains. Confinement does not significantly alter the scaling of the mean extension for sufficiently high tension. The theory is used to estimate unwrapping forces for DNA from nucleosomes.

Correlation functions for confined wormlike chains

Polymer models describing the statistics of biomolecules under confinement have applications to a wide range of single-molecule experimental techniques and give insight into biologically relevant processes in vivo. In this paper, we determine the transverse position and bending correlation functions for a wormlike chain confined within slits and cylinders (with one and two confined dimensions, respectively) using a mean-field approach that enforces rigid constraints on average. We show the theoretical predictions accurately capture the statistics of a wormlike chain from Monte Carlo simulations in both confining geometries for both weak and strong confinement. We also show that the longitudinal correlation function is accurately computed for a chain confined to a slit and leverages the accuracy of the model to suggest an experimental technique to infer the (often unobservable) transverse statistics from the (directly observable) longitudinal end-to-end distance.

MODELING A RED BLOOD CELL CYTOSKELETON: INSIGHTS AND TIPS

The aim of this work is to provide a clear and ready-to-implement mathematical model to describe the red blood cell (RBC) cytoskeleton. The motivation for this methodo- logy work lies in the lack of available sources facilitating the implementation of computational (in silico) modelling of RBC mechanical properties. Our approach adopts a worm-like chain force model to mimic an individual spectrin deformation and the virtual work principle to extrapolate the spectrin network dynamics. We report modelling tips using coarse-grained scaling techniques and optical tweezers experiments. We validate the model in virtual expe- riments gaining insights on single spectrin equilibrium, the impact of cytoskeletal symmetry defects on system behaviour, and stress transmission and distribution within the network. By performing these virtual experiments, we elucidate the role that the cytoskeleton plays in de- termining the remarkable RBC mechanical behaviour. These findings emphasise the need to consider the cytoskeleton component in the latest and most advanced single-cell mathematical models.

Molecular Adhesion of a Pilus-Derived Peptide Involved in Pseudomonas aeruginosa Biofilm Formation on Non-Polar ZnO-Surfaces.

Bacterial colonization and biofilm formation on abiotic surfaces are initiated by the adhesion of peptides and proteins. Understanding the adhesion of such peptides and proteins at a molecular level thus represents an important step toward controlling and suppressing biofilm formation on technological and medical materials. This study investigates the molecular adhesion of a pilus-derived peptide that facilitates biofilm formation of Pseudomonas aeruginosa, a multidrug-resistant opportunistic pathogen frequently encountered in healthcare settings. Single-molecule force spectroscopy (SMFS) was performed on chemically etched ZnO surfaces to gather insights about peptide adsorption force and its kinetics. Metal-free click chemistry for the fabrication of peptide-terminated SMFS cantilevers was performed on amine-terminated gold cantilevers and verified by X-ray photoelectron spectroscopy (XPS) and polarization-modulated infrared reflection absorption spectroscopy (PM-IRRAS). Atomic force microscopy (AFM) and XPS analyses reveal stable topographies and surface chemistries of the substrates that are not affected by SMFS. Rupture events described by the worm-like chain model (WLC) up to 600 pN were detected for the non-polar ZnO surfaces. The dissociation barrier energy at zero force ΔG(0), the transition state distance xb and bound-unbound dissociation rate at zero force koff (0) for the single crystalline substrate indicate that coordination and hydrogen bonds dominate the peptide/surface interaction.

Asymmetric Periodic Boundary Conditions for All-Atom Molecular Dynamics and Coarse-Grained Simulations of Nucleic Acids.

Periodic boundary conditions are commonly applied in molecular dynamics simulations in the microcanonical (NVE), canonical (NVT), and isothermal-isobaric (NpT) ensembles. In their simplest application, a biological system of interest is placed in the middle of a solvation box, which is chosen 'sufficiently large' to minimize any numerical artifacts associated with the periodic boundary conditions. This practical approach brings limitations to the size of biological systems that can be simulated. Here, we study simulations of effectively infinitely long nucleic acids, which are solvated in the directions perpendicular to the polymer chain, while periodic boundary conditions are also applied along the polymer chain. We study the effects of these asymmetric periodic boundary conditions (APBC) on the simulated results, including the mechanical properties of biopolymers and the properties of the surrounding solvent. To get some further insights into the advantages of using the APBC, a coarse-grained worm-like chain model is first studied, illustrating how the persistence length can be extracted from the local properties of the polymer chain, which are less affected by the APBC than some global averages. This is followed by all-atom molecular dynamics simulations of DNA in ionic solutions, where we use the APBC to investigate sequence-dependent properties of DNA molecules and properties of the surrounding solvent.

Marginal Distributions at the Tip of a Grafted Stiff Polymer: Analytical and Monte Carlo Investigations

AbstractThis study investigates the marginal distributions of grafted stiff polymer tips in a 2D embedding space using both analytical methods and Monte Carlo simulations. By mapping active Brownian particle (ABP) trajectories in the short‐time regime, analytical expressions for the elongation of the free end of the polymer under horizontal and vertical forces are derived and these expressions are validated using Monte Carlo simulations. These results indicate that the theoretical predictions match well with the simulation results when the chain length is short or the force is large. However, a slight discrepancy is observed between the theoretical and simulation results when the chain is extremely long, although the qualitative asymptotic results remain valid. Additionally, expressions are provided for the horizontal and vertical force versus displacement for the wormlike chain under the weakly bending approximation. This research provides insights into how the trajectories of an ABP correspond to the equilibrium configuration of a semiflexible polymer. These findings have potential applications in various fields, including biophysics and materials science, where understanding the behavior of grafted polymers is crucial.

Structure of Cellulose Ether Affected by Ionic Surfactant and Solvent: A Small-Angle Neutron Scattering Investigation.

Polysaccharides and their derivatives are commonly used in pharmaceutical and agricultural formulations as rheology modifiers. Their performance is related to their conformation in solution, which in turn is affected by other ingredients present in the formulation. This study focuses on modulating the conformation of relatively rigid cellulose chains in aqueous solutions. In particular, we have investigated the nonionic cellulose derivative ethyl(hydroxyethyl)cellulose (EHEC) in water in the presence of the ionic surfactant sodium dodecyl sulfate (SDS) and/or ethanol acting as modulating agents. We have used small angle neutron scattering (SANS) with contrast variation to determine the EHEC chain conformation in the presence of (but not masked by) ethanol and SDS. In dilute and semidilute aqueous solutions, EHEC exhibits worm-like chain conformation due to the rigid cellulose backbone. Addition of ethanol does not impact the polymer conformation to a great extent. Addition of SDS alters the EHEC chain conformation, resulting in polyelectrolyte-like scattering behavior due to repulsive interactions between bound charged micelles which show similar structure as the free SDS micelles in solution (in the absence of polymers). Ethanol affects the polymer + surfactant system primarily by acting on the surfactant (bound on polymer) which, in turn, affects the polymer conformation. At higher ethanol concentrations (20 wt %), EHEC regains the worm-like chain conformation because of the detachment of the bound SDS micelles. To the best of our knowledge, this is the only study providing details on chain conformation of the rigid polymer EHEC in dilute or semidilute aqueous solutions in the presence of surfactant and alcohol and one of very few papers utilizing SANS for the characterization of polymer + surfactant + water + alcohol interactions. Such fundamental understanding of interactions and structure in multicomponent mixtures supports the design of industrial formulations.

Arabinoxylan in Water through SANS: Single-Chain Conformation, Chain Overlap, and Clustering.

Using small-angle neutron scattering (SANS), we examine the structure and conformational behavior of wheat arabinoxylan (AX) prepared at various concentrations in a sodium phosphate aqueous buffer. As for another major hemicellulose, xyloglucan, we observe a small number of large clusters surrounded by AX chains that behave exactly as a polymer in good solvent with a Flory exponent ν = 0.588. The fit of the data at high q-values to a standard worm-like chain model gives the persistence length lp = 45 Å and cross section of the chains 2Rc = 11-12 Å. In addition, using a dedicated modeling approach, we extract from the SANS data at the intermediate q-range the correlation length ξ of the solutions in the semidilute regime. The decay of ξ with concentration follows a scaling law that further confirms the self-avoiding statistical behavior of the AX chains. This first comprehensive study about the properties of water-soluble AX at different length scales may help in the development of products and processes involving AX as a substitute for fossil carbon molecules.

Hierarchical Model of Weak Polyelectrolytes with Ionization and Conformation Consistency

Conventional models of weak polyelectrolytes are formulated in terms of either an average degree of ionization or a rigid polymer conformation. While the decoupling of segment-level ionization and polymer structure simplifies the theoretical procedure and is beneficial from a computational perspective, it misses intrachain correlations (i.e., interactions beyond adjacent monomers) that are important for systems at a low polymer concentration with strong electrostatic interactions. In this work, we propose a hierarchical method that incorporates the liquid-state theories with the site-binding model to account for the effects of the local ionic environment on both inter- and intrachain correlations. A wormlike-chain model is introduced to describe the coupling of polymer conformation and monomeric ionization with the long-range electrostatic interactions accounted for by using the Gibbs–Bogoliubov variational principle. Through an extensive comparison of the theoretical predictions with experimental titration curves for poly(acrylic acid) in different alkali chloride solutions, we demonstrate that the hierarchical model is able to quantify the charging behavior of weak polyelectrolytes over a broad range of solution conditions.

Effect of flexibility of arms on the nematic ordering of V-shaped molecules

The nematic ordering of a melt of V-shaped molecules composed of two semiflexible arms connected by one end at an external angle α is inspected within the Landau theory of phase transitions. The conformational statistics of arms is described by a discrete worm-like chain model. It is found that for a sufficiently high stiffness of the arms the phase diagram contains the regions of stability of isotropic (I), prolate (NU+) and oblate (NU−) uniaxial, and biaxial (NB) nematic phases. The isotropic–uniaxial nematic and uniaxial nematic – biaxial nematic phase transitions are of the first and second order, respectively. At lowering the stiffness of arms the stability area of NB phase decreases; the re-entrant NB phase becomes possible in the certain range of angles α and the sequence of phase transitions I−NU−−NB−NU+−NB occurs. At a sufficiently low stiffness, the stability region of the nematic biaxial phase disappears completely, and phase diagrams display only the first order transitions between I−NU+, I−NU−, and NU+−NU− phases. A further decrease in the stiffness of arms leaves only the first order I−NU+ transition in the phase diagrams. The tricritical points appear at the curves of NU−−NB transition in the phase diagrams in certain intervals of values of the stiffness parameter.

Characterization of the effect of chromium salts on tropocollagen molecules and molecular aggregates

Though the capability of chromium treatment to improve the stability and mechanical properties of collagen fibrils is well-known, the influence of different chromium salts on collagen molecules (tropocollagen) is not well characterized. In this study, the effect of Cr3+ treatment on the conformation and hydrodynamic properties of collagen was studied using atomic force microscopy (AFM) and dynamic light scattering (DLS). Statistical analysis of contours of adsorbed tropocollagen molecules using the two-dimensional worm-like chain model revealed a reduction of the persistence length (i.e., the increase of flexibility) from ≈72 nm in water to ≈56–57 nm in chromium (III) salt solutions. DLS studies demonstrated an increase of the hydrodynamic radius from ≈140 nm in water to ≈190 nm in chromium (III) salt solutions, which is associated with protein aggregation. The kinetics of collagen aggregation was shown to be ionic strength dependent. Collagen molecules treated with three different chromium (III) salts demonstrated similar properties such as flexibility, aggregation kinetics, and susceptibility to enzymatic cleavage. The observed effects are explained by a model that considers the formation of chromium-associated intra- and intermolecular crosslinks. The obtained results provide novel insights into the effect of chromium salts on the conformation and properties of tropocollagen molecules.

On the kratky-porod model for semi-flexible polymers in an external force field

On the kratky-porod model for semi-flexible polymers in an external force field

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.