Pressure of Linear and Ring Polymers Confined in a Cavity.

Nanoscale confinement of polymers in a cavity is central to a variety of biological and nanotechnology processes. Using the discrete WLC model we simulate the compression of flexible and semiflexible polymers of linear and ring topology in a closed cavity. Simulation reveals that polymer pressure inside the cavity increases with the chain stiffness but is practically unaffected by the chain topology. For flexible polymers, the computed dependence of pressure on the cavity size and polymer concentration is consistent with the scaling behavior expected for bulk polymers in a good solvent. However, the scaling behavior of semiflexible polymers is only in partial agreement with the theory prediction, with discrepancies arising from a continuous transition between regimes in chains of moderate lengths. The computed segment density profiles endorse the propensity of semiflexible polymers to concentrate beneath the cavity surface and thus elevate the pressure. The compaction of polymers by compression into the disordered globule or growing toroidal structure is documented.

Energy/entropy partition of force at DNA stretching.

We compute by molecular simulation the energy/entropic partition of the force in a stretched double-stranded (ds)DNA molecule that is not yet available from the single-molecule measurements. Simulation using the coarse-grained wormlike chain (WLC) model predicts a gradual decrease in the internal (bending) energy of DNA at stretching. The ensuing negative energy contribution to force fU is outweighed by the positive entropy contribution fS . The ratio fU /f, used to assess the polymer elasticity, is about -1 at the moderate extension of DNA. At the high extension, the extra energy expenses due to the contour length elongation make the ratio fU /f less negative. The simulation findings of the hybrid energy/entropy nature of DNA elasticity at weak and moderate forces are supported by computations using the thermoelastic method mimicking the polymer experiments in bulk. It is contended that the observation of the negative energy elasticity in DNA can be generalized to other semiflexible polymers described by the WLC model.

Compression and Stretching of Single DNA Molecules under Channel Confinement.

We study the compression and extension response of single dsDNA (double-stranded DNA) molecules confined in cylindrical channels by means of Monte Carlo simulations. The elastic response of micrometer-sized DNA to the external force acting through the chain ends or through the piston is markedly affected by the size of the channel. The interpretation of the force (f)-displacement (R) functions under quasi-one-dimensional confinement is facilitated by resolving the overall change of displacement ΔR into the confinement contribution ΔRD and the force contribution ΔRf. The external stretching of confined DNA results in a characteristic pattern of f-R functions involving their shift to the larger extensions due to the channel-induced pre-stretching ΔRD. A smooth end-chain compression into loop-like conformations observed in moderately confined DNA can be accounted for by the relationship valid for a Gaussian chain in bulk. In narrow channels, the considerably pre-stretched DNA molecules abruptly buckle on compression by the backfolding into hairpins. On the contrary, the piston compression of DNA is characterized by a gradual reduction of the chain span S and by smooth f-S functions in the whole spatial range from the 3d near to 1d limits. The observed discrepancy between the shape of the f-R and f-S functions from two compression methods can be important for designing nanopiston experiments of compaction and knotting of single DNA in nanochannels.

Force-displacement relations at compression of dsDNA macromolecules.

The elasticity of dsDNA molecules is investigated by Monte Carlo simulations based on a coarse-grained model of DNA. The force-displacement (f-r) curves are computed under the constraints of the constant force (Gibbs) or the constant length (Helmholtz) ensemble. Particular attention was paid to the compressional (negative) and weak tensile forces. It was confirmed that simulations using the vector Gibbs ensemble fail to represent the compression behavior of polymers. Simulations using the scalar Gibbs protocol resulted in a qualitatively correct compressional response of DNA provided that the quadratic averages of displacements were employed. Furthermore, a well-known shortcoming of the popular Marko-Siggia relation for DNA elasticity at weak tensile forces is elucidated. Conversely, the function f-r from the simulation at the constant length constraint, as well as the new closed-form expressions, provides a realistic depiction of the DNA elasticity over the wide range of negative and positive forces. Merely a qualitative resemblance of the compression functions f-r predicted by the employed approaches supports the notion that the elastic response of DNA molecules may be greatly affected by the specifics of the experimental setups and the kind of averaging of the measured variable.

Correlation anisotropy and stiffness of DNA molecules confined in nanochannels.

The anisotropy of orientational correlations in DNA molecules confined in cylindrical channels is explored by Monte Carlo simulations using a coarse-grained model of double-stranded (ds) DNA. We find that the correlation function ⟨C(s)⟩⊥ in the transverse (confined) dimension exhibits a region of negative values in the whole range of channel sizes. Such a clear-cut sign of the opposite orientation of chain segments represents a microscopic validation of the Odijk deflection mechanism in narrow channels. At moderate-to-weak confinement, the negative ⟨C(s)⟩⊥ correlations imply a preference of DNA segments for transverse looping. The inclination for looping can explain a reduction of stiffness as well as the enhanced knotting of confined DNA relative to that detected earlier in bulk at some channel sizes. Furthermore, it is shown that the orientational persistence length Por fails to convey the apparent stiffness of DNA molecules in channels. Instead, correlation lengths P∥ and P⊥ in the axial and transverse directions, respectively, encompass the channel-induced modifications of DNA stiffness.

Stretching and compression of DNA by external forces under nanochannel confinement.

Mechanical deformation of dsDNA molecules inside square nanochannels is investigated using simulations based on a coarse-grained model of DNA. The combined action of confinement and weak external forces is explored in a variety of confinement regimes, including the transition zone relevant to nanofluidic experiments. The computed free energy and force profiles are markedly affected by the channel size. Effective elastic softening of confined DNA molecules relative to the bulk DNA is observed in the channels of intermediate widths. The extension of DNA from its bulk equilibrium length in nanofluidic devices is resolved into contributions from the passive extension due to confinement and from the active stretching induced by force. Potential implications of the very different energy costs computed for the two extension modes (extension by confinement takes much more free energy than stretching by force) for behavior of DNA in nanofluidic chips are indicated.

Buckling transition in long α-helices.

The treatment of bending and buckling of stiff biopolymer filaments by the popular worm-like chain model does not provide adequate understanding of these processes at the microscopic level. Thus, we have used the atomistic molecular-dynamic simulations and the Amber03 force field to examine the compression buckling of α-helix (AH) filaments at room temperature. It was found that the buckling instability occurs in AHs at the critical force f(c) in the range of tens of pN depending on the AH length. The decrease of the force f(c) with the contour length follows the prediction of the classic thin rod theory. At the force f(c) the helical filament undergoes the swift and irreversible transition from the smoothly bent structure to the buckled one. A sharp kink in the AH contour arises at the transition, accompanied by the disruption of the hydrogen bonds in its vicinity. The kink defect brings in an effective softening of the AH molecule at buckling. Nonbonded interactions between helical branches drive the rearrangement of a kinked AH into the ultimate buckled structure of a compact helical hairpin described earlier in the literature.

A new two-state polymer folding model and its application to α-helical polyalanine.

A new two-state polymer folding model is proposed, in which the folding of a stiff helical polymer is enabled by allowing for short sequences of coils connecting shorter and separated helices. The folding is driven by short-range attraction energy among stacked helices and is opposed by the free-energy cost of forming coils from helical monomers. Principal outcomes of the model are equilibrium distribution of the number of helices and their length in helical polymers. The proposed model is applied to α-helical polyalanine. The distribution of the number of α-helices as a function of number of alanine residues is fitted to the corresponding result from molecular dynamics simulation employing an all-atom potential model with very good agreement. The influence and significance of the fitting parameters and possible use of the two-state folding model are discussed.

Molecular dynamics simulations of the folding of poly(alanine) peptides.

The secondary structures and the shapes of long-chain polyalanine (PA) molecules were investigated by all-atom molecular dynamics simulations using a modified Amber force field. Homopolymers of polyaminoacids such as PA are convenient models to study the mechanism of protein folding. It was found that the conformational structures of PA peptides are highly sensitive to the chain length. In the absence of solvent, straight α-helices dominate in short (n ∼ 20) peptides at room temperature. A shape transition occurs at a chain length n of 40-45; the compact helix-turn-helix structure (the double-leg hairpin) becomes favored over a straight α-helix. For n=60, double-leg and the triple-leg hairpins are the only structures present in PA molecules. An exploration of a chain organization in a cubic cavity revealed a clear predisposition of PA molecules for additional breaks in α-helices and the formation of multifolded hairpins. Furthermore, under confinement the hairpin structure becomes much looser, the antiparallel positions of helical stems are disturbed, and a sizeable proportion of the helical stems are transformed from α-helices into 3(10)-helices.

Persistence length of DNA molecules confined in nanochannels.

The influence of confinement on the persistence length of dsDNA molecules under a high ionic strength environment was explored by coarse-grained Monte Carlo simulations in channels of different profiles. It was found that under confinement three definitions of the persistence length of DNA molecules were not equivalent and represented different properties. In case of the global quantities, the projection and the WLC persistence lengths, the apparent values up to several hundred nanometres are observed for DNA confined in narrow channels. The orientational correlation function cos theta(s) of confined DNA shows a complex pattern, distinctive for semiflexible polymers. At weak and moderate confinements the function cos theta(s) suggests an unexpected increase in the apparent DNA flexibility. The orientational persistence length computed from the initial slope of the function cos theta(s) mirrors only short-scale correlations and gives the value close to the intrinsic persistence length of DNA. The simulation data of direct relevance to experimental studies of DNA in microfluidic devices are compared with analytical theories for stiff chains.

Chain extension of DNA confined in channels.

The mechanism of DNA elongation in nanochannels was explored by Monte Carlo simulations as a function of the channel dimension D, DNA length, and stiffness. Simulations were based on the bead-spring model, representing double-stranded DNA chains of moderate length at a high salt concentration. As a rule, the channel-induced elongation profiles of R( parallel) vs D from the simulations were in qualitative agreement with those from microfluidic measurements of DNA. The longitudinal chain elongation in narrow channels was found to be correctly predicted by the Odijk relation for the deflection regime. The scaling relation of R( parallel) vs D(-1), based on the statistics of ideal-chain blobs, was used to explain the simulation data at the intermediate channel widths. Contrary to the blob-theory presumption, the nonlinear dependence of DNA elongation R( parallel) on the chain length N was observed in simulations at moderate confinement. It was suggested that discrepancies found between the simulations and the blob theory arose from the formation of various DNA hairpin structures within channels.

Effect of confinement on properties of stiff biological macromolecules.

The behaviour of semiflexible chains, modelling biopolymers such as DNA and actin in confined spaces, was investigated by means of Monte Carlo simulations. Simulations, based on the coarse-grained worm-like chain (WLC) model, assumed confinement length-scales comparable to those used in micro- and nanofluidic devices. The end-to-end chain elongation R was determined as a function of the channel dimensions and chain bending rigidity. Three regions of chain elongation R, identified in simulations in a cylinder and a slit, were described by current theoretical concepts. In harmony with the measurements of confined DNA, an abrupt transition between the blob region at moderate confinement and the deflection region at strong cylindrical confinement was found. The conditions for hairpin formation were elucidated as a trade-off between confinement and chain stiffness. The intrinsic persistence length of unconfined polymers was calculated by four methods that provided practically identical results. However, in confined geometries only the rigorous and WLC methods predicted the dependence of apparent persistence length P on confinement in a qualitatively correct way. It was found that the simple exponential function, suitable for the description of orientation correlations in free chains is, in confined systems, limited only to short distances along the chain contour and, thus, the apparent persistence length determined by this method just reproduces the intrinsic value of P. The orientation correlations from simulations were compared with analytical predictions in the deflection regime under strong confinement and with the measurements of actin filaments.

Persistence lengths and structure factors of wormlike polymers under confinement.

The behavior of semiflexible chains modeling wormlike polymers such as DNA and actin in confined spaces was explored by coarse-grained Monte Carlo simulations. The persistence length P, mean end-to-end distance R2, mean radius of gyration Rg2, and the size ratio R2/Rg2 were computed for chains in slits, cylinders, and spheres. It was found that the intrinsic persistence length of a free chain undergoes on confinement substantial alteration into the apparent persistence length. The qualitative differences were found in trends of the apparent persistence lengths between slits and cylinders on one side and spheres on the other side. The quantities P, R2, Rg2, and R2/Rg2 display similar dependences upon squeezing the chains in nanopores. The above quantities change nonmonotonically with confinement in slits and cylinders, whereas they drop smoothly with decreasing radius of a sphere. For elongation of a chain in a cylinder, two regimes corresponding to strong and moderate confinements were found and compared to experiments and predictions of the blob and Odijk theories. In a spherical cavity, the toroidal chain structure with a hole in the center was detected under strong confinements. The scattering form factor S(q) computed for semiflexible confined chains revealed three regimes of behavior in a slit and a cylinder that matched up well with the scaling theory. The complex form of the function S(q) computed for a sphere was interpreted as a sign of the toroidal structure. A reasonable agreement was found between the simulations and measurements of DNA and actin filaments, confined in nano- and microfluidic channels and spherical droplets, pertaining to the changes of the persistence lengths, chain elongation, and toroidal structure formation.

Elastic properties of poly(hydroxybutyrate) molecules.

Elasticity of various poly(hydroxybutyrate) (PHB) molecules of regular and irregular conformational structure was examined by the molecular mechanics (MM) calculations. Force - distance functions and the Young's moduli E were computed by stretching of PHB molecules. Unwinding of the 2(1) helical conformation H is characterized at small deformations by the Young's modulus E = 1.8 GPa. The H form is transformed on stretching into the highly extended twisted form E, similar to the beta-structure observed earlier by X-ray fiber diffraction. The computations revealed that in contrast to paraffins, the planar all-trans structure of undeformed PHB is bent. Hence, a PHB molecule attains the maximum contour length in highly straightened, but slightly twisted conformations. A dependence of the single-chain moduli of regular and disordered conformations on the chain extension ratio x was found. The computed data were used to analyze elastic response of tie (bridging) molecules in the interlamellar (IL) region of a semi-crystalline PHB. A modification of the chain length distribution function of tie molecules tau(N) due to secondary crystallization of PHB was conjectured. The resulting narrow distribution tau(N) comprises the taut tie molecules of higher chain moduli prone to overstressing. The molecular model outlined is in line with the macroscopically observed increase in the modulus and brittleness of PHB with storage time.

Polymer-induced depletion interaction between weakly attractive plates.

Lattice Monte Carlo simulations have been employed to calculate depletion interaction of excluded volume chains in a weakly attractive slit, particularly in the region around the critical point of adsorption. The simulations were performed under full equilibrium conditions where a dilute solution in a slit was in contact with the reservoir. The free energy of confinement deltaA, the force f, and the relative pressurepI/pE on the slit walls were calculated as a function of slit width D and the attraction strength epsilon. The depletion region in the pressure profile pI/pE vs D is reduced by an increase in the attraction potential epsilon in a manner resembling the influence of polymer concentration. At the critical point of adsorption epsilonc the depletion interaction vanishes both in the pressure pI/pE and in the intraslit concentration profile phiI(x). The parameters used to assess the stability of colloidal dispersions such as the depletion potential W(D) (an integral of the net pressure deltap) reach a unique value at the critical condition. A monotonic repulsive profilepI vs D was found for chains trapped in the slit at restricted equilibrium. The mean dimensions (R2) of chains compressed in attractive slits feature a distinct minimum at intermediate slit widths.