Wormlike Chain Research Articles

Single-molecule force spectroscopy of membrane proteins from membranes freely spanning across nanoscopic pores.

Single-molecule force spectroscopy (SMFS) provides detailed insight into the mechanical (un)folding pathways and structural stability of membrane proteins. So far, SMFS could only be applied to membrane proteins embedded in native or synthetic membranes adsorbed to solid supports. This adsorption causes experimental limitations and raises the question to what extent the support influences the results obtained by SMFS. Therefore, we introduce here SMFS from native purple membrane freely spanning across nanopores. We show that correct analysis of the SMFS data requires extending the worm-like chain model, which describes the mechanical stretching of a polypeptide, by the cubic extension model, which describes the bending of a purple membrane exposed to mechanical stress. This new experimental and theoretical approach allows to characterize the stepwise (un)folding of the membrane protein bacteriorhodopsin and to assign the stability of single and grouped secondary structures. The (un)folding and stability of bacteriorhodopsin shows no significant difference between freely spanning and directly supported purple membranes. Importantly, the novel experimental SMFS setup opens an avenue to characterize any protein from freely spanning cellular or synthetic membranes.

How a short double-stranded DNA bends.

A recent experiment using fluorescence microscopy showed that double-stranded DNA fragments shorter than 100 base pairs loop with the probabilities higher by the factor of 10(2)-10(6) than predicted by the worm-like chain (WLC) model [R. Vafabakhsh and T. Ha, Science 337, 1101(2012)]. Furthermore, the looping probabilities were found to be nearly independent of the loop size. The results signify a breakdown of the WLC model for DNA mechanics which works well on long length scales and calls for fundamental understanding for stressed DNA on shorter length scales. We develop an analytical, statistical mechanical model to investigate what emerges to the short DNA under a tight bending. A bending above a critical level initiates nucleation of a thermally induced bubble, which could be trapped for a long time, in contrast to the bubbles in both free and uniformly bent DNAs, which are either transient or unstable. The trapped bubble is none other than the previously hypothesized kink, which releases the bending energy more easily as the contour length decreases. It leads to tremendous enhancement of the cyclization probabilities, in a reasonable agreement with experiment.

The study of the structure factor of a wormlike chain in an orientational external field.

A precise representation of the structure factor of a wormlike chain for the arbitrary chain flexibility in an orientational external field is obtained by virtue of the numerical solution to the modified diffusion equation satisfied by the Green's function. The model is built from a standard wormlike chain formalism in a continuous version which crossovers from the rigid-rod limit to the flexible chain limit and the Maier-Saupe interaction which describes the orientational effects from the nematic field. The behaviors of the structure factor in the distinct wavevector k regimes are numerically investigated as functions of chain flexibility and tilt angle between the directors of the nematic field and k. The radius of gyration extracted from the structure factor in small-k regime is also carefully analysed in both the directions along and perpendicular to the nematic axis. Our calculations exactly recover the prediction of the structure factor undergoing an orientational field in the rigid rod limit.

Kirkwood diffusivity of long semiflexible chains in nanochannel confinement.

We compute the axial diffusivity of asymptotically long semiflexible polymers confined in square channels. Our calculations employ the Kirkwood approximation of the mobility tensor by combining computational fluid dynamics (CFD) calculations of the hydrodynamic tensor in channel confinement with pruned-enriched Rosenbluth method (PERM) simulations of a discrete wormlike chain model. Three key results emerge from our study. First, for the classic de Gennes regime, we confirm that Brochard and de Gennes' blob theory correctly predicts the scaling of the axial diffusivity, contrary to the conclusions of previous analyses. Second, for the extended de Gennes regime, we show that a modified blob theory, which has been used to incorporate the effect of local stiffness on DNA diffusion in nanoslits, explains the deviation from the prediction of classic blob theory for diffusion in nanochannels. Third, we provide a calculation similar to the modified blob theory to explain the relative insensitivity of the diffusivity to channel size for channels between the extended de Gennes regime and the Odijk regime, which is the most relevant regime for experiments and technological applications of DNA confinement in nanochannels. Our results are not only relevant to the dynamics of confined semiflexible polymers such as DNA, but also reveal interesting analogies between confinement in channels and slits.

Flexibility of short DNA helices with finite-length effect: From base pairs to tens of base pairs.

Flexibility of short DNA helices is important for the biological functions such as nucleosome formation and DNA-protein recognition. Recent experiments suggest that short DNAs of tens of base pairs (bps) may have apparently higher flexibility than those of kilo bps, while there is still the debate on such high flexibility. In the present work, we have studied the flexibility of short DNAs with finite-length of 5-50 bps by the all-atomistic molecular dynamics simulations and Monte Carlo simulations with the worm-like chain model. Our microscopic analyses reveal that short DNAs have apparently high flexibility which is attributed to the significantly strong bending and stretching flexibilities of ∼6 bps at each helix end. Correspondingly, the apparent persistence length lp of short DNAs increases gradually from ∼29 nm to ∼45 nm as DNA length increases from 10 to 50 bps, in accordance with the available experimental data. Our further analyses show that the short DNAs with excluding ∼6 bps at each helix end have the similar flexibility with those of kilo bps and can be described by the worm-like chain model with lp ∼ 50 nm.

The elastic free energy of a tandem modular protein under force

Recent studies have provided a theoretical framework for including entropic elasticity in the free energy landscape of proteins under mechanical force. Accounting for entropic elasticity using polymer physics models has helped explain the hopping behavior seen in single molecule experiments in the low force regime. Here, we expand on the construction of the free energy of a single protein domain under force proposed by Berkovich et al. to provide a free energy landscape for N tandem domains along a continuous polypeptide. Calculation of the free energy of individual domains followed by their concatenation provides a continuous free energy landscape whose curvature is dominated by the worm-like chain at forces below 20 pN. We have validated our free energy model using Brownian dynamics and reproduce key features of protein folding. This free energy model can predict the effects of changes in the elastic properties of a multidomain protein as a consequence of biological modifications such as phosphorylation or the formation of disulfide bonds. This work lays the foundations for the modeling of tissue elasticity, which is largely determined by the properties of tandem polyproteins.

Probing a label-free local bend in DNA by single molecule tethered particle motion.

Being capable of characterizing DNA local bending is essential to understand thoroughly many biological processes because they involve a local bending of the double helix axis, either intrinsic to the sequence or induced by the binding of proteins. Developing a method to measure DNA bend angles that does not perturb the conformation of the DNA itself or the DNA-protein complex is a challenging task. Here, we propose a joint theory-experiment high-throughput approach to rigorously measure such bend angles using the Tethered Particle Motion (TPM) technique. By carefully modeling the TPM geometry, we propose a simple formula based on a kinked Worm-Like Chain model to extract the bend angle from TPM measurements. Using constructs made of 575 base-pair DNAs with in-phase assemblies of one to seven 6A-tracts, we find that the sequence CA6CGG induces a bend angle of 19° ± 4°. Our method is successfully compared to more theoretically complex or experimentally invasive ones such as cyclization, NMR, FRET or AFM. We further apply our procedure to TPM measurements from the literature and demonstrate that the angles of bends induced by proteins, such as Integration Host Factor (IHF) can be reliably evaluated as well.

Intermolecular interaction and the extended wormlike chain conformation of chitin in NaOH/urea aqueous solution.

The intra- and intermolecular interactions of chitin in NaOH/urea aqueous system were studied by a combination of NMR measurements (including (13)C NMR, (23)Na NMR, and (15)N NMR) and differential scanning calorimetry. The results revealed that the NaOH and chitin formed a hydrogen-bonded complex that was surrounded by the urea hydrates to form a sheath-like structure, leading to the good dissolution. The optimal concentration range, in which chitin was molecularly dispersed in NaOH/urea aqueous, was found to investigate the chain conformation in the dilute solution with a combination of static and dynamic light scattering. The weight-average molecular weight (Mw), radii of gyration (⟨Rg⟩z), and hydrodynamic radii (⟨Rh⟩z) values of chitin were determined, and the structure-sensitive parameter (ρ) and persistent length (Lp) were calculated to be >2.0 and ∼30 nm, respectively, suggesting an extended wormlike chain conformation. The visualized images from TEM, cryo-TEM, and AFM indicated that, chitin nanofibers were fabricated from the parallel aggregation of chitin chains in NaOH/urea system. This work would provide a theoretical guidance for constructing novel chitin-based nanomaterials via "bottom-up" method at the molecular level.

The influence of DNA shape fluctuations on fluorescence resonance energy transfer efficiency measurements in nucleosomes

Fluorescence resonance energy transfer (FRET) measurements allow one to observe site exposure in nucleosomes, i.e. the transient unwrapping of a part of the wrapped DNA from the histone octamer. In such experiments one can typically distinguish between a closed state and an open state but in principle one might hope to detect several states, each corresponding to a certain number of open binding sites. Here we show that even in an ideal FRET setup it would be hard to detect unwrapping states with intermediate levels of FRET efficiencies. As the unwrapped DNA molecule, modelled here as a wormlike chain, has a finite stiffness, shape fluctuations smear out FRET signals completely from such intermediate states.

DNA-Binding Properties of Peptide-Functionalized Graphene Quantum Dots

We present here the synthesis and characterization of nanoscale materials with DNA binding properties. We have functionalized graphene quantum dots (GQDs) with graphite-binding peptides to obtain graphene/peptide conjugates. The conjugates form stable 2D crystalline beta-sheet domains on the graphene surface. Upon incubation with DNA, we observe that DNA binding to the domain is directional, and the DNA preferentially aligns to the peptide direction. Upon binding the peptide-functionalized GQDs to DNA we observe an architectural role of GQDs in global and local flexibility, looping, and compacting of DNA. We use worm like chain (WLC) model to determine the persistence length of DNA in the absence and in the presence of GQDs, confirming a decreased persistence length upon GQD binding. Further, we find that the presence of peptide augments the number of DNA binding sites, as compared with exposure of bare GQDs to DNA. Our results show that peptide-modified GQDs can be a potentially ideal nanomaterial mimic of DNA-binding proteins similar to nucleosomal structures and other DNA-metabolic enzymes.

Bringing Force Probe Molecular Dynamics Simulations Closer to Experiments

The interaction between Streptavidin and Biotin is one of the strongest known non covalent bonds in nature. For this reason, many experimental and computational studies have been done on this complex, which revealed many details on binding and unbinding of biotin, which turned out to be an astonishingly complex process.However, taken all these studies together and despite considerable technical advances over the past years, there is still a time scale gap remaining, i.e., the loading rates possible in force spectroscopy experiments and those achievable in molecular dynamics (MD) simulations hardly overlap.Recent high speed force spectroscopy (HS-FS) experiments by Scheuring offer the possibility to reach pulling velocities up to 5 mm/s.Here we present extensive MD simulations covering pulling velocities from 0.5 mm/s to 50 m/s, thus providing an overlap of one order of magnitude with experimental time scales. Further, to resemble the AFM experiment as closely as possible in the simulations, we modeled the PEG-linker between Biotin and the cantilever as a wormlike chain and chose a spring constant of the pulling potential to match the stiffness of the cantilever in HS-FS experiments done by Scheuring.Over the whole interval, agreement of the probability distributions of unbinding forces is seen. Taken together, these data cover unbinding time scales over 10 orders of magnitude, which offers a unique opportunity to compare against several proposed transition state treatments of this process.

Brownian Dynamics Study of DNA Supercoil Relaxation

The stresses induced in DNA during cell processes such as replication and transcription lead to the formation of plectonemic supercoils. Supercoiling in turn affects these processes. As a result of this relationship between supercoiling and these processes, the degree of supercoiling in DNA needs to be controlled closely in order to optimize these cell processes. This control is facilitated by enzymes such as type 1b topoisomerases and nicking endonucleases which relax supercoiling in DNA.The dynamics of DNA supercoil relaxation have been studied in recent experiments by means of single-molecule magnetic and/or optical tweezer experiments. Novel as these experiments are, they do not permit a direct observation of the structural changes that occur in DNA during supercoil relaxation. They depend on components (e.g. a paramagnetic bead) attached to the DNA to indirectly obtain information about these dynamics.We studied the dynamics of supercoil relaxation by means of Brownian Dynamics simulations of a discrete wormlike-chain (dWLC) model of DNA. These simulations parallel the single-molecule experiments in which a single DNA molecule is held under constant tension so that its end-to-end extension increases as supercoils are relaxed by a nicking endonuclease. The dWLC model accounts for elasticity, electrostatics and entropic forces as well as for hydrodynamic interactions.From the simulation results, we more directly extract the rate at which supercoiling in DNA is relaxed by nicking endonucleases. We also determine the dependence of the relaxation timescales on the tension applied to the DNA molecule.

Supercoil Diffusion along Stretched DNA by Brownian Dynamics

The dynamics of DNA supercoils are intimately connected to several fundamental cellular processes. Transcription is a prime example. While the rate of transcription is sensitive to the level of supercoiling, transcription produces positive (and negative) supercoils ahead of (and behind) RNA polymerase. Subject to thermal fluctuations and DNA-protein interactions, supercoils are dynamic structures which accumulate, rearrange, translocate, and dissipate. Recent single molecule studies have observed the dynamic formation, diffusion, hopping, and dissipation of supercoils. Here, we employ Brownian dynamics simulations of a discrete worm-like chain to build understanding beyond that provided by recent experimental efforts. Our computational model accounts for hydrodynamic interactions, thermal fluctuations, bending, torsion, extension, and electrostatics in stretched DNA. We perform many simulations including trajectories representing 21 kilobasepairs of supercoiled DNA over the course of about 500 ms. We observe several metrics describing the dynamics of supercoils, including: the average number of supercoils, their lifetime, first juxtaposition time, and diffusion constants. In addition, we explore the sensitivity of these quantities to DNA extension as well as sequence dependence (through the introduction of sites with elevated DNA flexibility).

Modeling the stretching of wormlike chains in the presence of excluded volume.

We propose an interpolation formula (the EV-WLC relation) for the force-extension behavior of wormlike chains in the presence of hard-core excluded volume interactions, analogous to the classic interpolation formula from Marko and Siggia for ideal wormlike chains. Using pruned-enriched Rosenbluth method (PERM) simulations of asymptotically long, discrete wormlike chains in an external force, we show that the error in the EV-WLC interpolation formula to describe discrete wormlike chains is systematically smaller than the error in the Marko-Siggia interpolation formula, except for the saturation region in which both formulas have the same limiting behavior. We anticipate that the EV-WLC interpolation formula will prove useful in the coarse-graining of wormlike chain models for dynamic simulations. Related results for the excess free energy due to excluded volume provide strong support for the physical basis of the Pincus regime.

Statistical analysis of molecular nanotemplate driven DNA adsorption on graphite.

In this work, we have studied the conformation of DNA molecules aligned on the nanotemplates of octadecylamine, stearyl alcohol, and stearic acid on highly oriented pyrolytic graphite (HOPG). For this purpose, fluctuations of contours of adsorbed biopolymers obtained from atomic force microscopy (AFM) images were analyzed using the wormlike chain model. Moreover, the conformations of adsorbed biopolymer molecules were characterized by the analysis of the scaling exponent ν, which relates the mean squared end-to-end distance and contour length of the polymer. During adsorption on octadecylamine and stearyl alcohol nanotemplates, DNA forms straight segments, which order along crystallographic axes of graphite. In this case, the conformation of DNA molecules can be described using two different length scales. On a large length scale (at contour lengths l > 200-400 nm), aligned DNA molecules have either 2D compact globule or partially relaxed 2D conformation, whereas on a short length scale (at l ≤ 200-400 nm) their conformation is close to that of rigid rods. The latter type of conformation can be also assigned to DNA adsorbed on a stearic acid nanotemplate. The different conformation of DNA molecules observed on the studied monolayers is connected with the different DNA-nanotemplate interactions associated with the nature of the functional group of the alkane derivative in the nanotemplate (amine, alcohol, or acid). The persistence length of λ-DNA adsorbed on octadecylamine nanotemplates is 31 ± 2 nm indicating the loss of DNA rigidity in comparison with its native state. Similar values of the persistence length (34 ± 2 nm) obtained for 24-times shorter DNA molecules adsorbed on an octadecylamine nanotemplate demonstrate that this rigidity change does not depend on biopolymer length. Possible reasons for the reduction of DNA persistence length are discussed in view of the internal DNA structure and DNA-surface interaction.

Persistence-length renormalization of polymers in a crowded environment of hard disks.

The most conspicuous property of a semiflexible polymer is its persistence length, defined as the decay length of tangent correlations along its contour. Using an efficient stochastic growth algorithm to sample polymers embedded in a quenched hard-disk fluid, we find apparent wormlike chain statistics with a renormalized persistence length. We identify a universal form of the disorder renormalization that suggests itself as a quantitative measure of molecular crowding.

Hydrodynamics of DNA confined in nanoslits and nanochannels.

Modeling the dynamics of a confined, semi exible polymer is a challenging problem, owing to the complicated interplay between the configurations of the chain, which are strongly affected by the length scale for the confinement relative to the persistence length of the chain, and the polymer-wall hydrodynamic interactions. At the same time, understanding these dynamics are crucial to the advancement of emerging genomic technologies that use confinement to stretch out DNA and "read" a genomic signature. In this mini-review, we begin by considering what is known experimentally and theoretically about the friction of a wormlike chain such as DNA confined in a slit or a channel. We then discuss how to estimate the friction coefficient of such a chain, either with dynamic simulations or via Monte Carlo sampling and the Kirk-wood pre-averaging approximation. We then review our recent work on computing the diffusivity of DNA in nanoslits and nanochannels, and conclude with some promising avenues for future work and caveats about our approach.

Backfolding of Wormlike Chains Confined in Nanochannels

Using pruned-enriched Rosenbluth method (PERM) simulations of a discrete wormlike chain model, we provide compelling evidence in support of Odijk’s prediction of two distinct Odijk regimes for a long wormlike chain confined in a nanochannel. In both cases, the chain of persistence length lp is renormalized into a series of deflection segments of characteristic length D2/3lp1/3, where D is the channel size. In the first (classic) Odijk regime, these deflection segments are linearly ordered. In the second Odijk regime, thin, long wormlike chains can backfold at a length scale quantified by the global persistence length. We have measured this quantity by simulations and modified Odijk’s global persistence length theory to account for thermal fluctuations. The global persistence length, which is defined to be independent of the effect of excluded volume, provides the requisite closure to Odijk’s scaling theory for the second regime and thus allows us to resolve much of the confusion surrounding the so-called “transition” regime for DNA confined in a nanochannel. We show that Odijk’s theory for the backfolded regime correctly describes both the average chain extension and the variance about this extension for wormlike chains in channel sizes between the classic Odijk regime and the de Gennes blob regimes, with our data spanning several decades in terms of Odijk’s scaling parameter ξ. Although the backfolded Odijk regime occupies a very narrow range of D/lp, it is indeed a regime when viewed in terms of ξ and grows in size with increasing monomer anisotropy.

Large amplitude oscillatory shear studies on the strain-stiffening behavior of gelatin gels

Linear and nonlinear viscoelasticity of gelatin solutions was investigated by rheology. The dynamic mechanical properties during the sol-gel transition of gelatin followed the time-cure superposition. The fractal dimension df of the critical gel was estimated as 1.76, which indicated a loose network. A high sol fraction ws = 0.61 was evaluated from the plateau modulus by semi-empirical models. Strain-stiffening behavior was observed under large amplitude oscillatory shear (LAOS) for the gelatin gel. The strain and frequency dependence of the minimum strain modulus GM, energy dissipation Ed, and nonlinear viscoelastic parameter NE was illustrated in Pipkin diagrams and explained by the strain induced helix formation reported previously by others. The BST model described the strain-stiffening behavior of gelatin gel quite well, whereas the Gent and worm-like chain network models overestimated the strain-stiffening at large strains.

Atomic force microscopy study of DNA flexibility on short length scales: smooth bending versus kinking.

The apparently anomalous flexibility of DNA on short length scales has attracted a lot of attention in recent years. We use atomic force microscopy (AFM) in solution to directly study the DNA bending statistics for small lengths down to one helical turn. The accuracy of experimental estimates could be improved due to a large data volume and a refined algorithm for image processing and measuring bend angles. It is found that, at length scales beyond two helical turns (7 nm), DNA is well described by the harmonic worm-like chain (WLC) model with the bending persistence length of 56 nm. Below this threshold, the AFM data are also described by the WLC model assuming that the accuracy of measured bend angles is limited by the physical width of the double helix. We conclude that the double helical DNA behaves as a uniform elastic rod even at very short length scales. Strong bends due to kinks, melting bubbles and other deviations from the WLC model are statistically negligible.