Bending Persistence Length Research Articles

Bent-shape effects of multi-walled carbon nanotube on the electrical conductivity and rheological properties of polycarbonate/multi-walled carbon nanotube nanocomposites

In this study, polycarbonate (PC)/multi-walled carbon nanotube (MWCNT) nancomposites have been prepared by pretreating MWCNT solutions with ultrasonication. We demonstrate that the electrical conductivity and rheological properties of PC/MWCNT nanocomposites strongly depend on the mesoscopic shape factor ( l sp/ d), which is represented by the ratio between the static bending persistence length ( l sp) and outer diameter ( d) of the MWCNT. The electrical conductivity of PC/MWCNT nanocomposites increases linearly with increasing ( l sp/ d) 2 and the percolation threshold of PC/MWCNT nanocomposites decreases linearly with increasing ( l sp/ d) 2 of MWCNTs. The storage modulus of PC/MWCNT nanocomposites increases linearly with increasing ( l sp/ d) 2 of MWCNTs at all frequency ranges.

Charge density and bending rigidity of a rodlike polyelectrolyte: Effects of multivalent counterions

We study line charge density and bending persistence length of a semiflexible polyelectrolyte rod typified by DNA in a z:1 electrolyte with or without pre-existing 1:1 salt. We use a wormlike chain model, where the effective Hamiltonian incorporates bending energy, as well as where the Coulomb interaction between charged segments is screened and mediated by the ions in the solution. By analytically evaluating the free energy associated with the contour undulation and charge density fluctuation affected by the adsorbed ions, we find renormalized mean line-charge-density and persistence length. Multivalent counterions bind to the polyelectrolyte more readily than monovalent counterions, due to electrostatic attraction dominant over the entropy. Also, our results show that at physiological conditions the electrostatic interaction gives no appreciable change on the persistence length from its bare value, against the implications of earlier studies. This is because the mean-field and charge fluctuation effects largely cancel each other in the rodlike conformation.

The effect of mesoscopic shape on thermal properties of multi-walled carbon nanotube mats

We highlight the effect that mesoscopic shape of individual multi-walled carbon nanotubes (MWCNTs) has on their collective thermal properties when they are lumped together into MWCNT mats. The thermal properties depend both on the mesoscopic shape of the MWCNTs and on the structure of the mat. The mesoscopic shape is represented by static bending persistence length ( l sp ) and the mat structure is represented by network length ( l e ). It is demonstrated that various thermal properties depend on l sp. The variable thermal properties are linear thermal expansivity, thermal diffusivity, specific heat capacity, and thermal conductivity. In the case of n∗ > 1, the MWCNT mats contract with increasing temperature. On the contrary, in the case of n∗ < 1, the mats expand with increasing temperature. The apparent thermal conductivity increases with increasing l sp .

Torsional Mechanics of DNA Are Regulated by Small-Molecule Intercalation

Whether the bend and twist mechanics of DNA molecules are coupled is unclear. Here, we report the direct measurement of the resistive torque of single DNA molecules to study the effect of ethidium bromide (EtBr) intercalation and pulling force on DNA twist mechanics. DNA molecules were overwound and unwound using recently developed magnetic tweezers where the molecular resistive torque was obtained from Brownian angular fluctuations. The effect of EtBr intercalation on the twist stiffness was found to be significantly different from the effect on the bend persistence length. The twist stiffness of DNA was dramatically reduced at low intercalator concentration (<10 nM); however, it did not decrease further when the intercalator concentration was increased by 3 orders of magnitude. We also determined the dependence of EtBr intercalation on the torque applied to DNA. We propose a model for the elasticity of DNA base pairs with intercalated EtBr molecules to explain the abrupt decrease of twist stiffness at low EtBr concentration. These results indicate that the bend and twist stiffnesses of DNA are independent and can be differently affected by small-molecule binding.

Anisotropic Elastic Network Modeling of Entire Microtubules

Microtubules are supramolecular structures that make up the cytoskeleton and strongly affect the mechanical properties of the cell. Within the cytoskeleton filaments, the microtubule (MT) exhibits by far the highest bending stiffness. Bending stiffness depends on the mechanical properties and intermolecular interactions of the tubulin dimers (the MT building blocks). Computational molecular modeling has the potential for obtaining quantitative insights into this area. However, to our knowledge, standard molecular modeling techniques, such as molecular dynamics (MD) and normal mode analysis (NMA), are not yet able to simulate large molecular structures like the MTs; in fact, their possibilities are normally limited to much smaller protein complexes. In this work, we developed a multiscale approach by merging the modeling contribution from MD and NMA. In particular, MD simulations were used to refine the molecular conformation and arrangement of the tubulin dimers inside the MT lattice. Subsequently, NMA was used to investigate the vibrational properties of MTs modeled as an elastic network. The coarse-grain model here developed can describe systems of hundreds of interacting tubulin monomers (corresponding to up to 1,000,000 atoms). In particular, we were able to simulate coarse-grain models of entire MTs, with lengths up to 350 nm. A quantitative mechanical investigation was performed; from the bending and stretching modes, we estimated MT macroscopic properties such as bending stiffness, Young modulus, and persistence length, thus allowing a direct comparison with experimental data.

Percolation of two-dimensional multiwall carbon nanotube networks

We highlight the significance of multiwall carbon nanotube (MWCNT) shape on the electrical percolation. For rodlike MWCNTs, percolation threshold depends on aspect ratio (L/d). For random coil-like MWCNTs, the percolation threshold does not depend on L/d but depends on the shape factor of MWCNTs represented by the ratio between static bending persistence length and diameter. Surface resistivities of various MWCNTs converge into single curve when we plot their dimensionless surface concentration versus surface resistivity. The surface resistivity of MWCNT films decrease with increasing temperature, particularly at low concentrations, indicating the films can be used as a negative temperature coefficient thermistor.

Structural properties of polymeric DNA from molecular dynamics simulations

Most of the reported DNA structural studies are based on oligonucleotide structures, which have artifacts due to unstable terminal base pairs (bps). We have carried out molecular dynamics simulation of DNA oligonucleotides in such a manner that gives rise to properties of polymeric DNA of infinite length. Molecular dynamics simulation studies of six homo- and heteropolymeric DNA sequences are reported here to understand structural features of all ten unique dinucleotide sequences. We observe that each of these dinucleotide sequences has unique features in agreement with Calladine's rule [C. R. Calladine, J. Mol. Biol. 161, 343 (1982)]. We noticed significant structural alternation between B(I) and B(II) forms for d(CA).d(TG) dinucleotide, where one of the strands showed frequent transitions between usual and unusual epsilon and zeta torsion angles associated with bp stacking geometry. In terms of the calculated bending rigidity and persistence length, pyrimidine-purine bp steps, namely, d(TA).d(TA), d(CA).d(TG), and d(CG).d(CG) are the most flexible dinucleotide bp steps. We estimated the major groove widths from our simulations. We did not observe much variation in major and minor groove widths depending on the base sequence. However, the distribution of water molecules in the minor groove shows sensitivity to the DNA sequence.

Kinetic and Thermodynamic DNA Elasticity at Micro- and Mesoscopic Scales

Recent theoretical and experimental studies have suggested that the elastic behavior of the small-length double-helical DNA does not correspond to the simple harmonic model. This article presents a thorough comparison of classical atom-level molecular dynamics (MD) and coarse-grained harmonic approximations. It is shown that the previously predicted duration of MD trajectories necessary for accurate assessment of DNA elasticity was significantly overestimated and that reliable estimates of elastic parameters can be obtained after a few tens of nanoseconds. The all-atom and coarse-grained ensembles were compared head-to-head, including the amplitudes and relaxation rates of internal fluctuations as well as translational diffusion. The computed diffusion rates were found to be similar, with good correspondence to experimental data. The torsional persistence length (PL) in MD agrees reasonably well with experiment, with the relaxation rate of twisting fluctuations corresponding well to the harmonic model. The bending PL also agrees reasonably well with experiment, but the corresponding relaxation rate is much higher than the harmonic approximation. For a tetradecamer DNA, the difference reaches 1 order of magnitude, with the bending dynamics faster than the twisting dynamics, in qualitative contrast to the coarse-grained model. The possible mechanisms of this anomalous behavior are discussed.

A NOTE ON THE ANALYSIS OF A HETEROGENEOUS ROD-LIKE CHAIN IN DNA MODELING

Two different results concerning the elastic behavior of the heterogeneous worm-like chain (WLC) [D. Bensimon et al., Europhys. Lett.42, 97 (1998)] and rod-like chain (RLC) [P. Nelson, Phys. Rev. Lett.80, 5810 (1998)] are compared. We argue that the RLC is a more suitable model for double-stranded (ds-) DNA. As the hetero-RLC is the basic model for studying sequence-dependent ds-DNA, a rigorous path integral analysis for the effective bending persistence length is performed in the weak disorder limit. The novelty of the paper is in analyzing a path integral on the Lie group SO(3) with random forcing, which supplies a rigorous basis for the analysis of RLC type models.

Multiscale modeling of nucleic acids: Insights into DNA flexibility

The elastic rod theory is used together with all-atom normal mode analysis in implicit solvent to characterize the mechanical flexibility of duplex DNA. The bending, twisting, stretching rigidities extracted from all-atom simulations (on linear duplexes from 60 to 150 base pairs in length and from 94-bp minicircles) are in reasonable agreement with experimental results. We focus on salt concentration and sequence effects on the overall flexibility. Bending persistence lengths are about 20% higher than most experimental estimates, but the transition from low-salt to high-salt behavior is reproduced well, as is the dependence of the stretching modulus on salt (which is opposite to that of bending). CTG and CGG trinucleotide repeats, responsible for several degenerative disorders, are found to be more flexible than random DNA, in agreement with several recent studies, whereas poly(dA).poly(dT) is the stiffest sequence we have encountered. The results suggest that current all-atom potentials, which were parameterized on small molecules and short oligonucleotides, also provide a useful description of duplex DNA at much longer length scales.

Statistics of Time-Limited Ensembles of Bent DNA Conformations

The paper considers statistical properties of ensembles of chain conformations obtained by short-time Brownian dynamics (BD) of a coarse-grained DNA model in order to find out if the conditions necessary for accurate evaluation of the polymer elasticity are attainable in atom-level molecular dynamics (MD) simulations. To measure the bending persistence length (PL) with a 10% error using data accumulated in a single trajectory of a double helix of 15 base pairs, dynamics should be continued for a few microseconds. However, these estimates should be scaled down by about 2 orders of magnitude because the bending dynamics of short double helices in MD features much smaller relaxation times. As a result, good qualitative agreement with the worm-like chain (WLC) theory is reached in MD after tens of nanoseconds. The presently accessible durations of MD trajectories provide reasonably accurate evaluation of DNA elasticity and allow modeling of its mesoscopic properties. The surprisingly fast bending dynamics of short double helices in MD suggests that the microscopic mechanisms of DNA flexibility differ from a simple harmonic model.

From rigid base pairs to semiflexible polymers: Coarse-graining DNA

The elasticity of double-helical DNA on a nm length scale is captured in detail by the rigid base-pair model, whose conformation variables are the relative positions and orientations of adjacent base pairs. Corresponding sequence-dependent elastic potentials have been obtained from all-atom MD simulation and from high-resolution structural data. On the scale of 100 nm, DNA is successfully described by a continuous wormlike chain model with homogeneous elastic properties, characterized by a set of four elastic constants which have been measured in single-molecule experiments. We present here a theory that links these experiments on different scales, by systematically coarse-graining the rigid base-pair model to an effective wormlike chain description. The average helical geometry of the molecule is accounted for exactly, and repetitive as well as random sequences are considered. Structural disorder is shown to produce a small, additive and short-range correction to thermal conformation fluctuations as well as to entropic elasticity. We also discuss the limits of applicability of the homogeneous wormlike chain on short scales, quantifying the anisotropy of bending stiffness, the non-Gaussian bend angle distribution and the variability of stiffness, all of which are noticeable below a helical turn. The coarse-grained elastic parameters show remarkable overall agreement with experimental wormlike chain stiffness. For the best-matching potential, bending persistence lengths of dinucleotide repeats span a range of 37-53 nm, with a random DNA value of 43 nm. While twist stiffness is somewhat underestimated and stretch stiffness is overestimated, the counterintuitive negative sign and the magnitude of the twist-stretch coupling agree with recent experimental findings.

Mechanism of RecA-mediated homologous recombination revisited by single molecule nanomanipulation

The mechanisms of RecA-mediated three-strand homologous recombination are investigated at the single-molecule level, using magnetic tweezers. Probing the mechanical response of DNA molecules and nucleoprotein filaments in tension and in torsion allows a monitoring of the progression of the exchange in real time, both from the point of view of the RecA-bound single-stranded DNA and from that of the naked double-stranded DNA (dsDNA). We show that strand exchange is able to generate torsion even along a molecule with freely rotating ends. RecA readily depolymerizes during the reaction, a process presenting numerous advantages for the cell's 'protein economy' and for the management of topological constraints. Invasion of an untwisted dsDNA by a nucleoprotein filament leads to an exchanged duplex that remains topologically linked to the exchanged single strand, suggesting multiple initiations of strand exchange on the same molecule. Overall, our results seem to support several important assumptions of the monomer redistribution model.

Anisotropy in Sickle Hemoglobin Fibers from Variations in Bending and Twist

We have studied the variations of twist and bend in sickle hemoglobin fibers. We find that these variations are consistent with an origin in equilibrium thermal fluctuations, which allows us to estimate the bending and torsional rigidities and effective corresponding material moduli. We measure bending by electron microscopy of frozen hydrated fibers and find that the bending persistence length, a measure of the length of fiber required before it starts to be significantly bent due to thermal fluctuations, is 130microm, somewhat shorter than that previously reported using light microscopy. The torsional persistence length, obtained by re-analysis of previously published experiments, is found to be only 2.5microm. Strikingly this means that the corresponding torsional rigidity of the fibers is only 6x10(-27)Jm, much less than their bending rigidity of 5x10(-25)Jm. For (normal) isotropic materials, one would instead expect these to be similar. Thus, we present the first quantitative evidence of a very significant material anisotropy in sickle hemoglobin fibers, as might arise from the difference between axial and lateral contacts within the fiber. We suggest that the relative softness of the fiber with respect to twist deformation contributes to the metastability of HbS fibers: HbS double strands are twisted in the fiber but not in the equilibrium crystalline state. Our measurements inform a theoretical model of the thermodynamic stability of fibers that takes account of both bending and extension/compression of hemoglobin (double) strands within the fiber.

Effects of DNA-distorting proteins on DNA elastic response.

Many DNA-binding proteins distort the double helix, and therefore can be studied using single-molecule experiments to investigate how they modify double-helix polymer elasticity. We study this problem theoretically using discrete wormlike-chain models to describe the mechanics of protein-DNA composites. We consider the cases of nonspecific and specific (sequence-targeted) binding. We find that, in general, proteins which bend DNA can be described in terms of a reduction of bending persistence length as long as the binding strength is relatively weak (well below the dissociation point). For strong binding, the force response depends strongly on the bending stiffness of the DNA-protein complex. Since most DNA-bending proteins will cause local DNA untwisting, we also show how the constraint of DNA linking number modifies the observed elastic response. We also show how essentially the same model may be used to describe the binding of proteins and drugs which stiffen and stretch the double helix.

Adaptive elastic properties of chromatin fiber

Chromatin is a complex of DNA and specific proteins forming an intermediary level of organization of eukaryotic genomes, between double-stranded DNA and chromosome. Within a generic modeling of chromatin assembly, we investigate the interplay between the mechanical properties of the chromatin fiber and its biological functions. A quantitative step is to relate the mechanics at the DNA level and the mechanics described at the chromatin fiber level. It allows to calculate the complete set of chromatin elastic constants (twist and bend persistence lengths, stretch modulus and twist–stretch coupling constant), in terms of DNA elastic properties and geometric features of the fiber. These elastic constants are strongly sensitive to the local architecture of the fiber and we argue that this tunable elasticity might be a key feature in chromatin functions, for instance in the initiation and regulation of transcription. Moreover, this analysis provides a framework to interpret micromanipulation studies of chromatin fiber and suggests further experiments involving intercalators to scan the tunable elasticity of the fiber.

Micromechanics of isolated sickle cell hemoglobin fibers: bending moduli and persistence lengths

Pathogenesis in sickle cell disease depends on polymerization of deoxyhemoglobin S into rod-like fibers, forming gels that rigidify red cells and obstruct the systemic microvasculature. Fiber structure, polymerization kinetics and equilibria are well characterized and intimately related to pathogenesis. However, data on gel rheology, the immediate cause of obstruction, are limited, and models for structure and rheology are lacking. The basis of gel rheology, micromechanics of individual fibers, has never been examined. Here, we isolate fibers by selective depolymerization of gels produced under photolytic deliganding of CO hemoglobin S. Using differential interference contrast (DIC) microscopy, we measure spontaneous, thermal fluctuations in fiber shape to obtain bending moduli (κ) and persistence lengths (λp). Some fibers being too stiff to decompose shape accurately into Fourier modes, we measure deviations of fiber midpoints from mean positions. Serial deviations, sufficiently separated to be independent, exhibit Gaussian distributions and provide mean-squared fluctuation amplitudes from which κ and λp can be calculated. λp ranges from 0.24 to 13 mm for the most flexible and stiffest fibers, respectively. This large range reflects formation of fiber bundles. If the most flexible are single fibers, then λp =13 mm represents a bundle of seven single fibers. Preliminary data on the bending variations of frozen, hydrated single fibers of HbS obtained by electron microscopy indicate that the value 0.24 mm is consistent with the persistence length of single fibers. Young’s modulus is 0.10 GPa, less than for structural proteins but much larger than for extensible proteins. We consider how these results, used with models for cross-linking, may apply to macroscopic rheology of hemoglobin S gels. This new technique, combining isolation of hemoglobin S fibers and measurement of micromechanical properties based on thermal fluctuations and midpoint deviations, can be used to study fibers of mutants, hemoglobin A/S, and mixtures and hybrids of hemoglobin S.

Chromatin: a tunable spring at work inside chromosomes.

This paper focuses on mechanical aspects of chromatin biological functioning. Within a basic geometric modeling of the chromatin assembly, we give a complete set of elastic constants (twist and bend persistence lengths, stretch modulus and twist-stretch coupling constant) of the so-called 30-nm chromatin fiber, in terms of DNA elastic properties and geometric properties of the fiber assembly. The computation naturally embeds the fiber within a current analytical model known as the "extensible wormlike rope," allowing a straightforward prediction of the force-extension curves. We show that these elastic constants are strongly sensitive to the linker length, up to 1 bp, or equivalently to its twist, and might locally reach very low values, yielding a highly flexible and extensible domain in the fiber. In particular, the twist-stretch coupling constant, reflecting the chirality of the chromatin fiber, exhibits steep variations, and sign changes when the linker length is varied. We argue that this tunable elasticity might be a key feature for chromatin function, for instance, in the initiation and regulation of transcription.

Flexibility of Duplex DNA on the Submicrosecond Timescale

Using a site-specific, Electron Paramagnetic Resonance (EPR)-active spin probe that is more rigidly locked to the DNA than any previously reported, the internal dynamics of duplex DNAs in solution were studied. EPR spectra of linear duplex DNAs containing 14–100 base pairs were acquired and simulated by the stochastic Liouville equation for anisotropic rotational diffusion using the diffusion tensor for a right circular cylinder. Internal motions have previously been assumed to be on a rapid enough time scale that they caused an averaging of the spin interactions. This assumption, however, was found to be inconsistent with the experimental data. The weakly bending rod model is modified to take into account the finite relaxation times of the internal modes and applied to analyze the EPR spectra. With this modification, the dependence of the oscillation amplitude of the probe on position along the DNA was in good agreement with the predictions of the weakly bending rod theory. From the length and position dependence of the internal flexibility of the DNA, a submicrosecond dynamic bending persistence length of around 1500 to 1700 Å was found. Schellman and Harvey (Biophys. Chem. 55:95–114, 1995) have estimated that, out of the total persistence length of duplex DNA, believed to be about 500 Å, approximately 1500 Å is accounted for by static bends and 750 Å by fluctuating bends. A measured dynamic persistence length of around 1500 Å leads to the suggestion that there are additional conformations of the DNA that relax on a longer time scale than that accessible by linear CW-EPR. These measurements are the first direct determination of the dynamic flexibility of duplex DNA in 0.1 M salt.

The mechanical properties of actin gels. Elastic modulus and filament motions.

To address large discrepancies reported in the literature, the viscoelastic properties of gels formed by purified actin filaments have been measured by five different techniques and five different instruments using actin preparations purified separately in four different laboratories. These measurements consistently showed that the elastic shear modulus of 2 mg/ml F-actin is on the order of several hundred pascals, and depends very strongly on the length of the filaments and on the history of the sample prior to measurement. Shortening of actin filaments with gelsolin and mechanical perturbations reduce the shear modulus to low values identical to some reported in the literature, indicating that such perturbations account for low shear moduli and poor responsiveness to filament modifying treatments reported previously. The structures of individual actin filaments within gels very similar or identical to those studied rheometrically were also examined by dynamic light scattering and fluorescence microscopy. Dynamic light scattering data were analyzed by a new method to confirm that actin filaments have no stable associations with each other and fluctuate in solution at a rate governed by the filament bending modulus or persistence length, determined to be approximately 10 microns. Fluorescence microscopy confirmed that applying even small shear stresses to F-actin can orient and rupture the filaments, and that in a minimally perturbed viscoelastic gel, long actin filaments are free to diffuse within a limit of constraints formed by their neighbors. These findings confirm that relatively isotropic F-actin networks are sufficiently strong to stabilize cells.