Wormlike Chain Research Articles

The hierarchical emergence of worm-like chain behaviour from globular domain polymer chains.

Biological organisms make use of hierarchically organised structures to modulate mechanical behaviour across multiple lengthscales, allowing microscopic objects to generate macroscopic effects. Within these structural hierarchies, the resultant physical behaviour of the entire system is determined not only by the intrinsic mechanical properties of constituent subunits, but also by their organisation in three-dimensional space. When these subunits are polyproteins, colloidal chains or other globular domain polymers, the Kratky-Porod model is often assumed for the individual subunits. Hence, it is implicitly asserted that the polymeric object has an intrinsic parameter, the persistence length, that defines its flexibility. However, the persistence lengths extracted from experiment vary, and are often relatively small. Through a series of simulations on polymer chains formed of globular subunits, we show that the persistence length itself is a hierarchical structural property, related not only to the intrinsic mechanical properties of the underlying monomeric subunits, but emerging due to the organisation of inhomogenous geometry along the polymer contour.

The molecular basis of interaction domains of full-length PrP with lipid membranes.

PrP-lipid membrane interactions are critical to PrP structural conversion and neurotoxicity, but its molecular mechanism remains unclear. A two-dimensional histogram of force-distance curves and a worm-like chain model revealed three binding regions at the PrP N-terminal, providing the molecular basis for understanding the interactions between full-length PrP and lipid membranes.

Effects of particle stiffness on the extensional rheology of model rod-like nanoparticle suspensions.

The linear and nonlinear rheological behavior of two rod-like particle suspensions as a function of concentration is studied using small amplitude oscillatory shear, steady shear and capillary breakup extensional rheometry. The rod-like suspensions are composed of fd virus and its mutant fdY21M, which are perfectly monodisperse, with a length on the order of 900 nm. The particles are semiflexible yet differ in their persistence length. The effect of stiffness on the rheological behavior in both, shear and extensional flow, is investigated experimentally. The linear viscoelastic shear data is compared in detail with theoretical predictions for worm-like chains. The extensional properties are compared to Batchelor's theory, generalized for the shear thinning nature of the suspensions. Theoretical predictions agree well with the measured complex moduli at low concentrations as well as the nonlinear shear and elongational viscosities at high flow rates. The results in this work provide guidelines for enhancing the elongational viscosity based on purely frictional effects in the absence of strong normal forces which are characteristic for high molecular weight polymers.

Comparative study of worm‐like chain models for collagen molecules

AbstractCollagen is the main structural protein in various biological tissues and of high interest for microstructural multi‐scale models due to its load bearing functionality. The worm‐like chain (WLC) model [4] with its approximation [6] is a widely used approach for the modeling of DNA, which has a helical structure similar to tropocollagen. However, the classical WLC is suitable only for the entropic deformation regime of collagen. By modeling the extension behavior, it fails to give accurate predictions when the helical structure of the molecule starts twisting under higher loads.The twistable worm‐like chain [3] and its special case referred to as the extensible worm‐like chain [7] aim to include the mechanical coupling between bond stretching and twist of the helix using a twist‐stretch coupling term, leading to more accurate predictions in the energetic regime. This contribution discusses different variations and coupling terms describing the molecular behavior behind the transition state between the entropic and energetic regime. Experimental data of the literature [1] are used to make a quantitative comparison of the discussed models and their applicability.

How Necklace Pearls Evolve in Hydrophobic Polyelectrolyte Chains under Good Solvent Addition: A SANS Study of the Conformation

The chain conformation in sulfonated polystyrene PSSNa of a degree of sulfonation 0.34 ≤ f ≤ 1, i.e., of various hydrophobicity, is followed in mixtures of water and increasing content of tetrahydrofuran (THF), a good solvent of the hydrophobic polystyrene moieties (which improves the solvent quality of the mixture). This is achieved by measuring the chain form factor by small-angle neutron scattering using the zero average contrast method (ZAC). Polymer concentrations 0.17 and 0.34 M correspond in our case to the semidilute regime or its limit with dilute regime depending on the chain conformation. The main result is the monitoring with added THF of the pearl necklace conformation. This heterogeneous structure, made of wormlike chain parts (strings) and pearls, was observed formerly in water: when f decreases, the string contribution decreases, and the pearls size, characterized by a maximum in Kratky q2S1(q) representation, slightly increases. Here we see that in the presence of increasing content of ad...

Nematic order in solutions of semiflexible polymers: Hairpins, elastic constants, and the nematic-smectic transition.

Coarse-grained models of lyotropic solutions of semiflexible polymers are studied by both molecular dynamics simulations and density functional theory calculations, using an implicit solvent bead-spring model with a bond-angle potential. We systematically vary the monomer density, persistence length, and contour length over a wide range and explore the full range from the isotropic-nematic transition to the nematic-smectic transition. In the nematic regime, we span the entire regime from rigid-rod like polymers to thin wormlike chains, confined in effective straight tubes caused by the collective nematic effective ordering field. We show that the distribution of bond angles relative to the director is well described by a Gaussian, irrespective of whether the chains are rod-like or rather flexible. However, the related concept of "deflection length" is shown to make sense only in the latter case for rather dilute solutions since otherwise the deflection length is of the order of about two bond lengths only. When the solution is semi-dilute, a substantial renormalization of the persistence length occurs, while this effect is absent in the isotropic phase even at rather high monomer densities. The effective radii of the "tubes" confining the chains in the related description of orientational ordering are significantly larger than the distances between neighboring chains, providing evidence for a pronounced collective character of orientational fluctuations. Hairpins can be identified close to the isotropic-nematic transition, and their probability of occurrence agrees qualitatively with the Vroege-Odijk theory. The corresponding theoretical predictions for the elastic constants, however, are not in good agreement with the simulations. We attribute the shortcomings of the theories to their neglect of the coupling between local density and orientational fluctuations. Finally, we detected for this model a transition to a smectic phase for reduced monomer densities near 0.7.

Kinky DNA in solution: Small-angle-scattering study of a nucleosome positioning sequence

DNA is a flexible molecule, but the degree of its flexibility is subject to debate. The commonly-accepted persistence length of $l_p \approx 500\,$\AA\ is inconsistent with recent studies on short-chain DNA that show much greater flexibility but do not probe its origin. We have performed X-ray and neutron small-angle scattering on a short DNA sequence containing a strong nucleosome positioning element, and analyzed the results using a modified Kratky-Porod model to determine possible conformations. Our results support a hypothesis from Crick and Klug in 1975 that some DNA sequences in solution can have sharp kinks, potentially resolving the discrepancy. Our conclusions are supported by measurements on a radiation-damaged sample, where single-strand breaks lead to increased flexibility and by an analysis of data from another sequence, which does not have kinks, but where our method can detect a locally enhanced flexibility due to an $AT$-domain.

Polymer-Based Accurate Positioning: An Exact Worm-like-Chain Study.

Precise positioning of molecular objects from one location to another is important for nanomanipulation and is also involved in molecular motors. Here, we study single-polymer-based positioning on the basis of the exact solution to the realistic three-dimensional worm-like-chain (WLC) model. The results suggest the possibility of a surprisingly accurate flyfishing-like positioning in which tilting one end of a flexible short polymer enables positioning of the other diffusing end to a distant location within an error of ∼1 nm. This offers a new mechanism for designing molecular positioning devices. The flyfishing effect (and reverse process) likely plays a role in biological molecular motors and may be used to improve speed of artificial counterparts. To facilitate these applications, a new force–extension formula is obtained from the exact WLC solution. This formula has an improved accuracy over the widely used Marko–Siggia formula for stretched polymers and is valid for compressed polymers too. The new formula is useful in analysis of single-molecule stretching experiments and in estimating intramolecular forces of molecular motors, especially those involving both stretched and compressed polymer components.

Morphological Diversity, Protein Adsorption, and Cellular Uptake of Polydopamine-Coated Gold Nanoparticles.

Polydopamine (PDA)-coated nanoparticles are adhesive bionanomaterials widely utilized in intracellular applications, yet how their adhesiveness affects their colloidal stability and their interactions with serum proteins and mammalian cells remain unclear. In this work, we systematically investigate the combined effects of dopamine (DA) concentration and polymerization time (both reaction parameters spanning 2 orders of magnitude) on the morphological diversity of PDA-coated nanoparticles by coating PDA onto gold nanoparticle cores. Independent of the DA concentration, Au@PDA NPs remain largely aggregated upon several hours of limited polymerization; interestingly, extended polymerization for 2 days or longer yield randomly aggregated NPs, nearly monodisperse NPs, or worm-like NP chains in the ascending order of DA concentration. Upon exposure to serum proteins, the specific type of proteins adsorbed to the Au@PDA NPs strongly depends upon the DA concentration. As DA concentration increases, less albumin and more hemoglobin subunits adhere. Moreover, cellular uptake is a strong function of polymerization time. Serum-stabilized Au@PDA NPs prepared by limited polymerization enter Neuro-2a and HeLa cancer cells more abundantly than those prepared by extended polymerization. Our data underscore the importance of DA concentration and polymerization time for tuning the morphology and degree of intracellular delivery of PDA-coated nanostructures.

Advancing the computational methodology of rigid rod and semiflexible polymer systems: A new solution to the wormlike chain model with rod‐coil copolymer calculations

ABSTRACTA wormlike chain model for rod type blocks in a rod‐coil diblock copolymer is implemented in the self‐consistent field theory (SCFT) formalism. A pseudo‐spectral method is used to solve for the single‐chain partition function of this copolymer system. Orientation degrees of freedom are discretized using Lebedev sphere rules such that orientation integrations are carried out through a Lebedev quadrature, an approach not used previously in tandem with the pseudo‐spectral method. Phase behavior in the rigid‐rod limit as a function of rod segment volume fraction, Flory–Huggins interaction parameter χ, degree of polymerization N, and rod contour length ratio β are examined in detail in one and two dimensions. Examples extending to three dimensions are included. Semiflexible behavior via the rod bending rigidity κ is explored. An approximation is used for rigid‐rods that do not need spherical harmonics leading to increased speed in finding equilibrium morphologies. The results show that standing vertical structures may be more easily produced with rigid‐rod blocks compared to coil‐coil lamellae, an important feature in nanolithographic applications. Suggestions are made for using the model in future molecular orientation studies where the model can be used with inverse search methods to measure the values of the model parameters for the real systems. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019, 57, 29–39

Environmentally Controlled Curvature of Single Collagen Proteins

The predominant structural protein in vertebrates is collagen, which plays a key role in extracellular matrix and connective tissue mechanics. Despite its prevalence and physical importance in biology, the mechanical properties of molecular collagen are far from established. The flexibility of its triple helix is unresolved, with descriptions from different experimental techniques ranging from flexible to semirigid. Furthermore, it is unknown how collagen type (homo- versus heterotrimeric) and source (tissue derived versus recombinant) influence flexibility. Using SmarTrace, a chain-tracing algorithm we devised, we performed statistical analysis of collagen conformations collected with atomic force microscopy to determine the protein’s mechanical properties. Our results show that types I, II, and III collagens—the key fibrillar varieties—exhibit similar molecular flexibilities. However, collagen conformations are strongly modulated by salt, transitioning from compact to extended as KCl concentration increases in both neutral and acidic pH. Although analysis with a standard worm-like chain model suggests that the persistence length of collagen can attain a wide range of values within the literature range, closer inspection reveals that this modulation of collagen’s conformational behavior is not due to changes in flexibility but rather arises from the induction of curvature (either intrinsic or induced by interactions with the mica surface). By modifying standard polymer theory to include innate curvature, we show that collagen behaves as an equilibrated curved worm-like chain in two dimensions. Analysis within the curved worm-like chain model shows that collagen’s curvature depends strongly on pH and salt, whereas its persistence length does not. Thus, we find that triple-helical collagen is well described as semiflexible irrespective of source, type, pH, and salt environment. These results demonstrate that collagen is more flexible than its conventional description as a rigid rod, which may have implications for its cellular processing and secretion.

The gold nanoparticle springs’ spectrum modulation

The nanoparticles’ localized surface plasma resonance (LSPR) reveals an excellent light manipulation potentiality for the nano scale optical device development. Due to the coupling effect, the LSPR is very sensitive to the distance between the nanoparticles. Using the surface modification one nanoparticle could be grafted onto another one. But the LSPR peak shifting of such twin-nanoparticles architecture was stationary. In this study, the gold nanoparticles springs (80 nm AuNp (gold nanoparticle)—PEG75 (Polyethylene glycol) chain—45 nm AuNp) were on-situ self-assembly anchored on the glass substrate inside the ITO (indium-tin-oxide) electrodes pair trenches. Since the 80 nm AuNPs were grafted on the glass substrate by the Au-S bonds. Utilizing the large angle laser irradiation dark field microscopic imaging technology, the scattering spectral shifting of the deformed gold nanoparticles springs could be easily traced. With the electric field changing and reversing, the scattering spectra of the parallel gold nanoparticle springs showed an accredited linear (r2 = 0.8) modulation from 570 nm to 620 nm. Although the linked PEG75 molecular is a long flexible transparent polymer chain. Deducting from the semi-empirical formula of LSPR, zeta potential, and the PEG worm like chain (WLC) deformation model, the wreathed length (5–18 nm) of the parallel gold nanoparticles springs under the dynamic electrostatic field (−100–100 KV m−1) could be speculated. The FDTD (Finite Difference Time Domain) simulation also provided the similar spectral shifting numerical results. With the development of the gold nanoparticles springs precisely controlling, many nano optical spectral modulation applications will be promoted.

Measuring the wall depletion length of nanoconfined DNA.

Efforts to study the polymer physics of DNA confined in nanochannels have been stymied by a lack of consensus regarding its wall depletion length. We have measured this quantity in 38 nm wide, square silicon dioxide nanochannels for five different ionic strengths between 15 mM and 75 mM. Experiments used the Bionano Genomics Irys platform for massively parallel data acquisition, attenuating the effect of the sequence-dependent persistence length and finite-length effects by using nick-labeled E. coli genomic DNA with contour length separations of at least 30 µm (88 325 base pairs) between nick pairs. Over 5 × 106 measurements of the fractional extension were obtained from 39 291 labeled DNA molecules. Analyzing the stretching via Odijk's theory for a strongly confined wormlike chain yielded a linear relationship between the depletion length and the Debye length. This simple linear fit to the experimental data exhibits the same qualitative trend as previously defined analytical models for the depletion length but now quantitatively captures the experimental data.

DNA Confined in a Nanodroplet: A Molecular Dynamics Study.

As a major genetic material, the configuration and the mechanical properties of a double-stranded DNA (dsDNA) molecule in confinement are crucial for the application of nanotechnology and biological engineering. In the present paper, molecular dynamics simulation is utilized to study the configuration of dsDNA in a nanodroplet on a graphene substrate. The results show that the semiflexible dsDNA molecule changes its configuration with radius of gyration ( Rg) of a few nanometers because of the confined space, that is, the Rg of the dsDNA molecule decreases with the reduction of the nanodroplet size. In comparison, the dsDNA in the bulk usually has a persistent length of tens of nanometers. Especially, if the nanodroplet is small enough, the dsDNA molecule might form a loop structure inside. The dsDNA molecule affects the wetting properties of the graphene substrate. It is found that the graphene becomes more hydrophilic in smaller systems containing the dsDNA molecule, whereas for larger droplets, the changes of the contact angles are not significant with the presence of dsDNA. Moreover, the results indicate that for larger droplets, the line tension of the droplet containing DNA is positive and greater than that without DNA; for smaller droplets, the line tension becomes negative because the dsDNA is compressed and bent in the confinement, and has the potential to expand outwards. The worm-like chain model is used to study the bending energy of a dsDNA molecule in a droplet. The results address that the bending energy of the non-loop-structured dsDNA decreases as the droplet becomes larger, and it is larger than that of loop-structured dsDNA, as the loop structure efficiently prevents the DNA from bending in the vertical direction.

Worm-like chain model extensions for highly stretched tropocollagen molecules

Tropocollagen plays a very important role in the load bearing functionality of soft tissues. In the context of multi-scale modeling the response of tropocollagen molecules to stretch should be very carefully predicted in order to describe the mechanical behavior of soft tissues. To this end, the worm-like chain (WLC) model is often applied, although it is restricted to the entropic force regime which is essential at moderate deformations. To describe molecular forces under larger stretches several extensions of the WLC have been proposed for deoxyribonucleic acid (DNA). This contribution aims to investigate the applicability of these models in the context of tropocollagen and discusses the feasibility of their application. Finally, the models are validated in comparison to experimental data available in the literature.

Systematic bottom-up theory for the viscoelastic response of worm-like chain networks

Using the frequency-dependent force-extension response functions of single worm-like chains as the only input, the linear complex viscoelastic modulus of a polymeric network with given connectivity is derived from a systematic bottom-up theory via iterative coarse-graining. Choosing a cubic connectivity and accounting for random network orientation, we find excellent agreement with experimental data for actin networks under shear over the entire frequency range with the strength of the osmotic pressure that acts within the polymeric network as the only fitting parameter. In particular, we obtain a viscoelastic plateau regime at low frequencies and a crossover to an intermediate-frequency regime characterized by a power law behavior and an inhomogeneous shear deformation field.

Strong deformations of DNA: Effect on the persistence length

.Extreme deformations of the DNA double helix attracted a lot of attention during the past decades. Particularly, the determination of the persistence length of DNA with extreme local disruptions, or kinks, has become a crucial problem in the studies of many important biological processes. In this paper we review an approach to calculate the persistence length of the double helix by taking into account the formation of kinks of arbitrary configuration. The reviewed approach improves the Kratky-Porod model to determine the type and nature of kinks that occur in the double helix, by measuring a reduction of the persistence length of the kinkable DNA.Graphical abstract

FEATHER: Automated Analysis of Force Spectroscopy Unbinding and Unfolding Data via a Bayesian Algorithm

Single-molecule force spectroscopy (SMFS) provides a powerful tool to explore the dynamics and energetics of individual proteins, protein-ligand interactions, and nucleic acid structures. In the canonical assay, a force probe is retracted at constant velocity to induce a mechanical unfolding/unbinding event. Next, two energy landscape parameters, the zero-force dissociation rate constant (ko) and the distance to the transition state (Δx‡), are deduced by analyzing the most probable rupture force as a function of the loading rate, the rate of change in force. Analyzing the shape of the rupture force distribution reveals additional biophysical information, such as the height of the energy barrier (ΔG‡). Accurately quantifying such distributions requires high-precision characterization of the unfolding events and significantly larger data sets. Yet, identifying events in SMFS data is often done in a manual or semiautomated manner and is obscured by the presence of noise. Here, we introduce, to our knowledge, a new algorithm, FEATHER (force extension analysis using a testable hypothesis for event recognition), to automatically identify the locations of unfolding/unbinding events in SMFS records and thereby deduce the corresponding rupture force and loading rate. FEATHER requires no knowledge of the system under study, does not bias data interpretation toward the dominant behavior of the data, and has two easy-to-interpret, user-defined parameters. Moreover, it is a linear algorithm, so it scales well for large data sets. When analyzing a data set from a polyprotein containing both mechanically labile and robust domains, FEATHER featured a 30-fold improvement in event location precision, an eightfold improvement in a measure of the accuracy of the loading rate and rupture force distributions, and a threefold reduction of false positives in comparison to two representative reference algorithms. We anticipate FEATHER being leveraged in more complex analysis schemes, such as the segmentation of complex force-extension curves for fitting to worm-like chain models and extended in future work to data sets containing both unfolding and refolding transitions.

The effect of chain stiffness and salt on the elastic response of a polyelectrolyte.

We present simulations of the force-extension curves of strong polyelectrolytes with varying intrinsic stiffness as well as specifically treating hyaluronic acid, a polyelectrolyte of intermediate stiffness. Whereas fully flexible polyelectrolytes show a high-force regime where extension increases nearly logarithmically with force, we find that the addition of even a small amount of stiffness alters the short-range structure and removes this logarithmic elastic regime. This further confirms that the logarithmic regime is a consequence of the short-ranged "wrinkles" in the flexible chain. As the stiffness increases, the force-extension curves tend toward and reach the wormlike chain behavior. Using the screened Coulomb potential and a simple bead-spring model, the simulations are able to reproduce the hyaluronic acid experimental force-extension curves for salt concentrations ranging from 1 to 500 mM. Furthermore, the simulation data can be scaled to a universal curve like the experimental data. The scaling analysis is consistent with the interpretation that, in the low-salt limit, the hyaluronic acid chain stiffness scales with salt with an exponent of -0.7, rather than either of the two main theoretical predictions of -0.5 and -1. Furthermore, given the conditions of the simulation, we conclude that this exponent value is not due to counterion condensation effects, as had previously been suggested.

Sphere sedimentation in wormlike micelles: Effect of micellar relaxation spectrum and gradients in micellar extensions

We report experiments on the flow of wormlike micellar solutions based on cetylpyridinium chloride/sodium salicylate past a falling sphere via a combination of particle tracking velocimetry, particle image velocimetry, rheology, and flow induced birefringence (FIB). Recent studies have shown that beyond a critical extensional Deborah number, a falling sphere in wormlike micelles never reaches a constant terminal velocity; instead, it settles with an unsteady velocity. This behavior is linked to the wormlike micellar chain scission in the wake of the sphere. Similar instabilities in viscoelastic polymer solutions, where polymer chain scission is highly unlikely, are thought to be the results of a single-mode relaxation spectrum of the polymer chains or the asymmetry in the polymer chain extensions on the flanks of the falling sphere. In this paper, we examine the effect of micellar relaxation spectrum and gradients in micellar extensions on sphere instability in wormlike micelles over a wide range of flow parameters (10−2 < DeE < 40.7 and 10−6 < Re < 10). In wormlike micelles with a single-mode relaxation spectrum, the sphere instability occurs for DeE ≥ 2.6. However, for similar conditions (2.5 ≤ DeE ≤ 40.7 and 10−2 < Re < 10), spheres never exhibit unsteady motion in the wormlike micelles with a broad spectrum of relaxation times. This indicates the importance of the micellar relaxation spectrum on dynamics of sphere sedimentation in wormlike micelles. We show that a criterion based on the ratio of dissipated energy to the stored elastic energy of micelles can successfully describe the effect of micellar relaxation spectrum on sphere sedimentation dynamics. In addition, for conditions that give rise to sphere instability, FIB indicates that micellar extensions on sphere sideways are perfectly symmetric.