Semiflexible Polymer Chains Research Articles

Conformational transition of single semiflexible polymer chain as a coarsegrained model of DNA

Conformational transition of single semiflexible polymer chain as a coarsegrained model of DNA

Ejection Dynamics of a Semiflexible DNA Polymer from a Capsid

We present Langevin dynamics simulations to study the ejection of a semiflexible DNA polymer chain from a spherical capsid. We explore the ejection process, forces, and velocity, depending on the bending rigidity of semiflexible polymers. We find that there are three stages in the ejection process as a function of time and the number of ejected beads. The ejection speed of semiflexible polymers has a maximum at an intermediate stage of the ejection. The bending penalty of a semiflexible DNA polymer inside a capsid holds the key to determining the maximum speed.

Stretching semiflexible polymer chains: Evidence for the importance of excluded volume effects from Monte Carlo simulation

Semiflexible macromolecules in dilute solution under very good solvent conditions are modeled by self-avoiding walks on the simple cubic lattice (d = 3 dimensions) and square lattice (d = 2 dimensions), varying chain stiffness by an energy penalty ε(b) for chain bending. In the absence of excluded volume interactions, the persistence length l(p) of the polymers would then simply be l(p) = l(b)(2d - 2)(-1)q(b) (-1) with q(b) = exp(-ε(b)/k(B)T), the bond length l(b) being the lattice spacing, and k(B)T is the thermal energy. Using Monte Carlo simulations applying the pruned-enriched Rosenbluth method (PERM), both q(b) and the chain length N are varied over a wide range (0.005 ≤ q(b) ≤ 1, N ≤ 50,000), and also a stretching force f is applied to one chain end (fixing the other end at the origin). In the absence of this force, in d = 2 a single crossover from rod-like behavior (for contour lengths less than l(p)) to swollen coils occurs, invalidating the Kratky-Porod model, while in d = 3 a double crossover occurs, from rods to Gaussian coils (as implied by the Kratky-Porod model) and then to coils that are swollen due to the excluded volume interaction. If the stretching force is applied, excluded volume interactions matter for the force versus extension relation irrespective of chain stiffness in d = 2, while theories based on the Kratky-Porod model are found to work in d = 3 for stiff chains in an intermediate regime of chain extensions. While for q(b) ≪ 1 in this model a persistence length can be estimated from the initial decay of bond-orientational correlations, it is argued that this is not possible for more complex wormlike chains (e.g., bottle-brush polymers). Consequences for the proper interpretation of experiments are briefly discussed.

Harmonically Confined, Semiflexible Polymer in a Channel: Response to a Stretching Force and Spatial Distribution of the Endpoints

We consider an inextensible, semiflexible polymer or worm-like chain which is confined in the transverse direction by a parabolic potential and subject to a longitudinal force at the ends, so that the polymer is stretched out and backfolding is negligible. Simple analytic expressions for the partition function, valid in this regime, are obtained for chains of arbitrary length with a variety of boundary conditions at the ends. The spatial distribution of the end points or radial distribution function is also analyzed.

The convex hull for a random acceleration process in two dimensions

We compute exactly the mean perimeter 〈L(T)〉 and the mean area 〈A(T)〉 of the convex hull of a random acceleration process of duration T in two dimensions. We use an exact mapping that relates, via Cauchy's formulae, the computation of the perimeter and the area of the convex hull of an arbitrary two-dimensional stochastic process [x(t), y(t)] to the computation of the extreme value statistics of the associated one-dimensional component process x(t). The latter can be computed exactly for the one-dimensional random acceleration process even though the process is non-Markovian. Physically, our results are relevant in describing the average shape of a semi-flexible ideal polymer chain in two dimensions.

Gelation in semiflexible polymers

Molecular dynamics simulations have been employed to study the formation of a physical gel by semiflexible polymer chains. The formation of a geometrically connected network of these chains is investigated as a function of temperature and rate of cooling. The stiffness of the molecules is controlled via a potential between beads separated by two bonds. As the temperature is lowered, a percolated homogeneous solution phase separates to form a high-density, non-percolated nematic fluid and a low-density gas phase. On further decreasing the temperature, the chains are dynamically arrested preventing the completion of the vapor-liquid (VL) phase separation. As a result, the chains are stuck in a three-dimensional network of nematic bundles forming a percolated gel. Apart from temperature, the rate of cooling also plays an important role in the formation of the gel. Cooling the system at a faster rate yields gel while slower rates result in complete VL phase separation.

Effect of geometrical constraint on conformational properties of a polymer chain

In this article, we analyse the effect of geometrical constraint on the conformational properties of an infinitely long linear semiflexible polymer chain confined in-between two constraints under good solvent condition in two dimensions. The constraints are two impenetrable stair shaped surface and for two-dimensional space, the surface is a one-dimensional line. The semiflexibility of the chain is accounted by introducing a Boltzmann weight of bending energy required to produce each turn in the chain and good solvent condition was accounted by using self-avoiding walk model of the chain. We have calculated exact critical value of step fugacity required for polymerization of an infinitely long polymer chain confined in-between the constraints for different values of separation between the constraints for directed version of the model. We have also calculated possible maximum, minimum values of the persistent length for such chains and the maximum value of bending energy required for each turn in the chain for few values of separation between the constraints.

Local coil–helix transition of semiflexible polymers confined in spheres

An off-lattice Monte Carlo method is used to study the conformations of semiflexible polymer chains confined in spheres. The local coil–helix transition is observed for the chains with moderate rigidity confined in small spheres, and the helical structures can be identified by the combination of the dihedral angles and the correlation functions between the tangent vectors. Through calculating the average number of monomers per helix turn, the average number of helix turns per chain, and the ratio of average helix pitch to average helix radius, we find that the radius of the confining sphere and the bending energy are the key factors in the formation of helical structures. This study can help us understand how DNA chains fold in confined spaces, such as viruses and cells.

Chromosome dynamics, molecular crowding, and diffusion in the interphase cell nucleus: a Monte Carlo lattice simulation study

Using Monte Carlo simulations, we have investigated the decondensation of chromosomes during interphase and the diffusive transport of spherical probe particles in the chromatin network. The chromatin fibers are modeled as semiflexible polymer chains on a fixed three-dimensional grid, taking into account their flexibility and eventual chain crossing by the aid of topoisomerases. The network thus created will obstruct the diffusion of macromolecules. A result of our simulations is that crowding of diffusing molecules leaves the dynamics of the chromosomes and the behavior of other diffusing molecules qualitatively unaffected. Furthermore, the capability of the simulated chromatin network to trap diffusing molecules over long times is lower than that measured in microrheological experiments. Microrheology is a technique that allows to determine the viscoelastic properties of a material by the motion of embedded tracer particles. Long-time trapping requires a stiff network, as only such a network quickly responds to the diffusive fluctuations of tracers and prevents them from squeezing through meshes. A high degree of crosslinking amplifies this effect. The presence of a flexible and uncrosslinked polymer simply increases the effective viscosity sensed by tracer particles. The diffusion of tracers in our simulations reveals rather viscous than elastic chromatin networks, suggesting that chromatin alone cannot account for the high elasticity of the cell nucleus.

Brownian dynamics simulations of charged semiflexible polymers confined to curved surfaces

As an extension of the generalized bead–rod model developed earlier by the authors, this paper proposes a method for Brownian dynamics simulations of charged semiflexible polymers confined to various curved surfaces such as spherical, cylindrical, ellipsoidal and toroidal. We model charged semiflexible polymers as discrete wormlike chains consisting of virtual beads connected by inextensible rods with length varying according to the characteristic radius of curvature of the confining surface. The long-range electrostatic interactions are incorporated via the Debye–Hueckel potential. The geometrical constraints associated with the inextensible rods are realized by the so-called linear constraint solver. For a semiflexible polymer chain confined to a spherical surface, an analytical expression for the winding number is obtained by using an existing exact closed-form solution of the mean-square end-to-end distance. The proposed simulation method is then validated against theoretical predictions for both charged and uncharged polymer chains under surface confinements.

Polymer escape from a metastable Kramers potential: Path integral hyperdynamics study

We study the dynamics of flexible, semiflexible, and self-avoiding polymer chains moving under a Kramers metastable potential. Due to thermal noise, the polymers, initially placed in the metastable well, can cross the potential barrier, but these events are extremely rare if the barrier is much larger than thermal energy. To speed up the slow rate processes in computer simulations, we extend the recently proposed path integral hyperdynamics method to the cases of polymers. We consider the cases where the polymers' radii of gyration are comparable to the distance between the well bottom and the barrier top. We find that, for a flexible polymers, the crossing rate (R) monotonically decreases with chain contour length (L), but with the magnitude much larger than the Kramers rate in the globular limit. For a semiflexible polymer, the crossing rate decreases with L but becomes nearly constant for large L. For a fixed L, the crossing rate becomes maximum at an intermediate bending stiffness. For the self-avoiding chain, the rate is a nonmonotonic function of L, first decreasing with L, and then, above a certain length, increasing with L. These findings can be instrumental for efficient separation of biopolymers.

Fluctuations of a long, semiflexible polymer in a narrow channel

We consider an inextensible, semiflexible polymer or wormlike chain, with persistence length P and contour length L, fluctuating in a cylindrical channel of diameter D. In the regime D<<P<<L , corresponding to a long, tightly confined polymer, the average length of the channel <R(∥)> occupied by the polymer and the mean-square deviation from the average vary as <R(∥)>=[1-α(∘)(D/P)(2/3)]L and <ΔR(∥)(2)>=β(∘)(D(2)P)L , respectively, where α(∘) and β(∘) are dimensionless amplitudes. In earlier work we determined α(∘) and the analogous amplitude α(square) for a channel with a rectangular cross section from simulations of very long chains. In this paper, we estimate β(∘) and β(square) from the simulations. The estimates are compared with exact analytical results for a semiflexible polymer confined in the transverse direction by a parabolic potential instead of a channel and with a recent experiment. For the parabolic confining potential we also obtain a simple analytic result for the distribution of R(∥) or radial distribution function, which is asymptotically exact for large L and has the skewed shape seen experimentally.

Flexibility and local structure of a worm-like cylinder of self-assembled discotic triazine triamide

Small- and wide-angle X-ray scattering from an organogel system made from discotic triazine triamide (TRI) in octanol was carried out using a synchrotron X-ray source to provide scattering profiles for a wide range of scattering vectors, of range 0.050 nm−1 <q <20 nm−1, where q is the magnitude of the scattering vector. The analyses of the profiles revealed the following features: (1) the amorphous nature of the molecular stacking at a distance of 0.32 nm; (2) the presence of a semiflexible rod-like scattering object with a radius of 1.3 nm and a Kuhn length of 55 nm; and (3) branching of the cylinder formed in the gel state. These findings lead to the conclusion that the aggregated TRI functions in the manner of a semiflexible polymer chain in octanol and that the polymer-chain-like behavior of TRI cylinders is the origin of the gel nature of this system. X-ray scattering from an organogel made from discotic triazine triamide was measured at SPring-8 at a quite wide q range of 0.050 nm−1 <q <20 nm−1. Analyses revealed (1) an amorphous nature of the molecular stacking at a distance of 0.32 nm, (2) the presence of a semiflexible rod-like scattering object with a radius of 1.3 nm and a Kuhn length of 55 nm and (3) branching of the cylinder formed at the gel state.

Stretching of a semiflexible chain composed of elastic bonds

We study the extension of semiflexible (persistent) polymer chains composed of elastic bonds under the action of forces applied to their ends. For a given discrete model of a chain, the effective potential energy includes three components: the energy of bonds in the external dipole field, the energy of elastic deformation of bonds, and the energy of bending, which depends on the angles between neighboring bonds. The extension/contraction modulus of bonds is high but finite. To calculate the relative extension and its variance, the variational method for finding the maximum eigenvalue of the transfer operator in the space of orientations of bonds is used. For chains composed of more than ten bonds, the results appear to approach the data of simulation of chain extension by the collisional molecular dynamics method. Two proposed extensionforce dependences are compared with the computer experiment, and this comparison makes it possible to define the limits of their applicability.

Large scale behavior of semiflexible heteropolymers

On considère un modèle discret pour un polymère semi-flexible et hétérogène. Le bruit thermique et le caractère hétérogène du polymère (le désordre) sont modélisés en termes de rotations aléatoires. Nous nous concentrons sur le régime de désordre gélé, c’est-à-dire, l’analyse est effectuée pour une réalisation fixée du désordre. Les modèles semi-flexibles diffèrent sensiblement des marches aléatoires à petite échelle, mais à grande échelle un comportement brownien apparaît. En exploitant des techniques de calcul tensoriel et d’analyse de Fourier non-commutative, nous établissons le caractère brownien du modèle à grande échelle et nous obtenons une expression pour la constante de diffusion. Nous donnons aussi des conditions qui entraînent des propriétés quantitatives de mélange.

Stiffness dependence of critical exponents of semiflexible polymer chains situated on two-dimensional compact fractals

We present an exact and Monte Carlo renormalization group (MCRG) study of semiflexible polymer chains on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension df is equal to 2 for all members of the fractal family enumerated by the odd integer b(3<or=b<infinity). For various values of stiffness parameter s of the chain, on the PF fractals (for 3<or=b<or=9 ), we calculate exactly the critical exponents nu (associated with the mean squared end-to-end distances of polymer chain) and gamma (associated with the total number of different polymer chains). In addition, we calculate nu and gamma through the MCRG approach for b up to 201. Our results show that for each particular b, critical exponents are stiffness dependent functions, in such a way that the stiffer polymer chains (with smaller values of s) display enlarged values of nu, and diminished values of gamma. On the other hand, for any specific s, the critical exponent nu monotonically decreases, whereas the critical exponent gamma monotonically increases, with the scaling parameter b. We reflect on a possible relevance of the criticality of semiflexible polymer chains on the PF family of fractals to the same problem on the regular Euclidean lattices.

Dynamics of loop formation in a semiflexible polymer

The dynamics of loop formation by linear polymer chains has been a topic of several theoretical and experimental studies. Formation of loops and their opening are key processes in many important biological processes. Loop formation in flexible chains has been extensively studied by many groups. However, in the more realistic case of semiflexible polymers, not much results are available. In a recent study [K. P. Santo and K. L. Sebastian, Phys. Rev. E 73, 031923 (2006)], we investigated opening dynamics of semiflexible loops in the short chain limit and presented results for opening rates as a function of the length of the chain. We presented an approximate model for a semiflexible polymer in the rod limit based on a semiclassical expansion of the bending energy of the chain. The model provided an easy way to describe the dynamics. In this paper, using this model, we investigate the reverse process, i.e., the loop formation dynamics of a semiflexible polymer chain by describing the process as a diffusion-controlled reaction. We make use of the "closure approximation" of Wilemski and Fixman [G. Wilemski and M. Fixman, J. Chem. Phys. 60, 878 (1974)], in which a sink function is used to represent the reaction. We perform a detailed multidimensional analysis of the problem and calculate closing times for a semiflexible chain. We show that for short chains, the loop formation time tau decreases with the contour length of the polymer. But for longer chains, it increases with length obeying a power law and so it has a minimum at an intermediate length. In terms of dimensionless variables, the closing time is found to be given by tau approximately Ln exp(const/L), where n=4.5-6. The minimum loop formation time occurs at a length Lm of about 2.2-2.4. These are, indeed, the results that are physically expected, but a multidimensional analysis leading to these results does not seem to exist in the literature so far.

Viscoelasticity of a Single Semiflexible Polymer Chain

We study the dynamic properties of a single semiflexible polymer chain based on the theory for the wormlike-chain model developed by Hallatschek et al. The linear viscoelastic response is investigated under oscillatory forces acting at the two chain ends. The complex compliance and complex modulus are obtained analytically as a function of the oscillation frequency ω. The real part of the complex compliance in the low frequency limit ω → 0 is consistent with the static result of Marko and Siggia whereas the imaginary part in the low frequency regime exhibits the power-law dependence ω1/2. On the other hand, these compliances decrease as ω−7/8 in the high frequency limit ω → ∞. This is consistent with the theoretical result obtained by Everaers et al. in a slightly different condition. A scaling argument is developed to get understanding of these novel results.

Prestress-dependent Rheology of Semiflexible Polymers of the Cytoskeleton

Rheological properties of living cells are essential for their physiological functions. Microrheological measurements have shown that cytoskeletal contractile stress (or prestress) and weak power-law viscoelasticity are governing principles of cell rheology, and that these two properties are closely associated in living cells for reasons that are largely unknown. In this study, we develop a stochastic model of a semiflexible polymer of the cytoskeleton that links the power-law rheology to the prestress. We describe a semiflexible polymer chain as a three-dimensional elastically-jointed chain composed of nonlinearly elastic bonds jointed by linearly elastic torsional springs. Assuming that the chain dynamics is thermally driven, we use a Monte-Carlo-based algorithm to obtain numerical simulations of the chain's creep behavior during uniaxial stretching. We obtain that the creep curves follow a power-law and that this behavior changes with prestress in a manner that is consistent with previously reported data from living cells and reconstituted crosslinked actin gels. We show that the power-law creep results from a finite-speed propagation of free energy from the chain's end points towards the center of the chain in response to externally applied stretching force. We also show that the power-law dependence on the prestress results from the chain's nonlinear, stiffening behavior that originates from both entropic and enthalpic contributions. Based on qualitative similarities between model simulations and experimental data from living cells and actin gels, it is conceivable that the mechanisms embodied in our model may also be key determinants of the overall viscoelastic properties of living cells and actin gels.

Effect of hydrodynamic interaction on partially stretched polymers

We compute the effect of hydrodynamic interaction and stretching on the fluctuation properties of a polymer, with its end points held fixed. Computing the preaveraged hydrodynamic tensor exactly for this geometry, we study both flexible and semiflexible polymer chains, such as Zimm, freely jointed chain, and wormlike chain (WLC) models. We compare the spectra of relaxation-time scales for the effective normal modes of these models. The spectra differ across models with respect to the degree of stretch, but their power-law scaling with low mode numbers turns out to be the same. The characteristics of the transverse modes of WLC agree very well with the experimental data on DNA. The crossover scaling function for (1/r), the inverse of the distance along the polymer contour, yields a modified formula for the size of a "Pincus blob," appropriate for the fixed-end boundary condition.