Simulation of Semiflexible Cyclic and Linear Chains Moderately and Strongly Confined in Nanochannels

Compression of a confined semiflexible polymer under direct and oscillating fields

The folding transition of biopolymers from the coil to compact structures has attracted wide research interest in the past and is well studied in polymer physics. Recent seminal works on DNA in confined devices have shown that these long biopolymers tend to collapse under an external field, which is contrary to the previously reported stretching of the chain. In this work, we capture the compression of a confined semiflexible polymer under direct and oscillating fields using a coarse-grained computer simulation model in the presence of long-range hydrodynamics. In the case of a semiflexible polymer chain, the inhomogeneous hydrodynamic drag from the center to the periphery of the coil couples with the chain bending to cause a swirling movement of the chain segments, leading to structural intertwining and compaction. Contrarily, a flexible chain of the same length lacks such structural deformation and forms a well-established tadpole structure. While bending rigidity profoundly influences the chain's folding favorability, we also found that subject to the direct field, chains in stronger confinements exhibit substantial compaction, contrary to the one in moderate confinements or bulk where such compaction is absent. However, an alternating field within an optimum frequency can effectuate this compression even in moderate or no confinement. This field-induced collapse is a quintessential hydrodynamic phenomenon, resulting in intertwined knotted structures even for shorter chains, unlike other spontaneous knotting experiments where it happens exclusively for longer chains.

Conformation of Flexible and Semiflexible Chains Confined in Nanoposts Array of Various Geometries

The conformation and distribution of a flexible and semiflexible chain confined in an array of nanoposts arranged in parallel way in a square-lattice projection of their cross-section was investigated using coarse-grained molecular dynamics simulations. The geometry of the nanopost array was varied at the constant post diameter dp and the ensuing modifications of the chain conformation were compared with the structural behavior of the chain in the series of nanopost arrays with the constant post separation Sp as well as with the constant distance between two adjacent post walls (passage width) wp. The free energy arguments based on an approximation of the array of nanopost to a composite of quasi-channels of diameter dc and quasi-slits of height wp provide semiqualitative explanations for the observed structural behavior of both chains. At constant post separation and passage width, the occupation number displays a monotonic decrease with the increasing geometry ratio dc/wp or volume fraction of posts, while a maximum is observed at constant post diameter. The latter finding is attributed to a relaxed conformation of the chains at small dc/wp ratio, which results from a combination of wide interstitial volumes and wide passage apertures. This maximum is approximately positioned at the same dc/wp value for both flexible and semiflexible chains. The chain expansion from a single interstitial volume into more interstitial volumes also starts at the same value of dc/wp ratio for both chains. The dependence of the axial chain extension on the dc/wp ratio turns out to be controlled by the diameter of the interstitial space and by the number of monomers in the individual interstitial volumes. If these two factors act in the same way on the axial extension of chain fragments in interstitial volumes the monotonic increase of the axial chain extension with the dc/wp in the nanopost arrays is observed. At constant wp, however, these two factors act in opposite way and the axial chain extension plotted against the dc/wp ratio exhibits a maximum. In the case of constant post diameter, the characteristic hump in the single chain structure factor whose position correlates with the post separation is found only in the structure factor of the flexible chain confined in the nanopost array of certain value of Sp. The structure factor of the flexible chain contains more information on the monomer organization and mutual correlations than the structure factor of the semiflexible chain. The stiffer chain confined in the nanopost array is composed of low number of statistical segments important for the presence of respective hierarchical regimes in the structure factor.

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 P_or 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.

Self-Avoiding Wormlike Chain Confined in a Cylindrical Tube: Scaling Behavior

Within a confining tube section, the multithreads of a strongly confined, backfolding polymer exert the excluded-volume repulsions on each other and produce physical properties that are very different from those of a confined ideal chain. The conformational properties of a such confined wormlike chain are of fundamental interest and are also practically useful in understanding the DNA confinement problems. Here, the excluded-volume effects are added to the standard wormlike-chain model by a self-consistent field theory. The numerical solutions are examined in light of their scaling properties.

Structural Behavior of a Semiflexible Polymer Chain in an Array of Nanoposts

The structural properties of a flexible and semiflexible circular chain confined in an array of parallel nanoposts with a square lattice cross-sectional projection were studied using coarse-grained molecular dynamics simulations. To address the effect of the circular topology, a comparison with linear analogs was also carried out. In the interpretation of the chain structural properties, the geometry of the post array is considered as a combination of a channel approximating the interstitial volume with the diameter dc and a slit approximating the passage aperture with the width wp. The number of interstitial volumes occupied by a chain monotonically increases with the decreasing ratio dc/wp regardless of the way the geometry of the post array is varied. However, depending on how the array geometry is modified, the chain span along the posts displays a monotonic (constant post separation) or a non-monotonic behavior (constant passage width) when plotted as a function of the post diameter. In the case of monotonic trend, the width of interstitial spaces increases with the increasing chain occupation number, while, in the case of non-monotonic trend, the width of interstitial spaces decreases with the increasing chain occupation number. In comparison with linear topology, for circular topology, the stiffness affects more significantly the relative chain extension along the posts and less significantly the occupation number. The geometrical parameters of the post arrays are stored in the single-chain structure factors. The characteristic humps are recognized in the structure factor which ensue from the local increase in the density of segments in the circular chains presented in an interstitial volume or from the correlation of parallel chain fragments separated by a row of posts. Although the orientation correlations provide qualitative information about the chain topology and the character of confinement within a single interstitial volume, information about the array periodicity is missing.

Conformational Properties of a Back-Folding Wormlike Chain Confined in a Cylindrical Tube

When a semiflexible chain is confined in a narrow cylindrical tube, the formation of a polymer hairpin is a geometrical conformation that accompanies an exponentially large local free energy and, hence, is a relatively rare event. Numerical solutions of the hairpin distribution functions for persistence-length-to-tube-radius ratios over a wide range are obtained in high precision, by using the Green's function approach for the wormlike-chain model. The crossover region between the narrow and moderately narrow tubes is critically investigated in terms of the hairpin free energy, global persistence length, mean hairpin-tip distance from the tube axis, and hairpin-plane orientational properties. Accurate representations of the solutions by simple interpolation formulae are suggested.

Comparison of a stripe and slab confinement for ring and linear macromolecules in nanochannel

The combined effects of the channel asymmetry and the closed chain topology on the chain extension, structure factor, and the orientation correlations were studied using coarse-grained molecular dynamics simulations for moderate chain lengths. These effects are related to applications in linearization experiments with a DNA molecule in nanofluidic devices. According to the aspect ratio, the channels are classified as a stripe or slabs. The chain segments do not have any freedom to move in the direction of the narrowest stripe size, being approximately the same size as the segment size. The chains of both ring and linear topologies are extended more in a stripe than in a slab; this effect is strengthened for a ring. For a ring in a stripe, the extension-confinement strength dependence leads to effective Flory exponents even larger than 3/4, which is characteristic for a self-avoiding two-dimensional chain. While the chain extension-confinement strength dependence for both topologies conforms to the de Gennes regime in a stripe, a linear chain undergoes gradual transition to the pseudoideal regime as the slab height increases in the slab-like confinement. For a confined circle, the onset of the pseudoideal regime is shifted to larger slab heights. The structure factor confirms the absence of the pseudoideal and extended de Gennes regime in a stripe and the transition from the extended to the pseudoideal regime of a circular and linear chain upon increasing the slab heights.

Modeling the relaxation of internal DNA segments during genome mapping in nanochannels

We have developed a multi-scale model describing the dynamics of internal segments of DNA in nanochannels used for genome mapping. In addition to the channel geometry, the model takes as its inputs the DNA properties in free solution (persistence length, effective width, molecular weight, and segmental hydrodynamic radius) and buffer properties (temperature and viscosity). Using pruned-enriched Rosenbluth simulations of a discrete wormlike chain model with circa 10 base pair resolution and a numerical solution for the hydrodynamic interactions in confinement, we convert these experimentally available inputs into the necessary parameters for a one-dimensional, Rouse-like model of the confined chain. The resulting coarse-grained model resolves the DNA at a length scale of approximately 6 kilobase pairs in the absence of any global hairpin folds, and is readily studied using a normal-mode analysis or Brownian dynamics simulations. The Rouse-like model successfully reproduces both the trends and order of magnitude of the relaxation time of the distance between labeled segments of DNA obtained in experiments. The model also provides insights that are not readily accessible from experiments, such as the role of the molecular weight of the DNA and location of the labeled segments that impact the statistical models used to construct genome maps from data acquired in nanochannels. The multi-scale approach used here, while focused towards a technologically relevant scenario, is readily adapted to other channel sizes and polymers.

The polymer physics of single DNA confined in nanochannels

In recent years, applications and experimental studies of DNA in nanochannels have stimulated the investigation of the polymer physics of DNA in confinement. Recent advances in the physics of confined polymers, using DNA as a model polymer, have moved beyond the classic Odijk theory for the strong confinement, and the classic blob theory for the weak confinement. In this review, we present the current understanding of the behaviors of confined polymers while briefly reviewing classic theories. Three aspects of confined DNA are presented: static, dynamic, and topological properties. The relevant simulation methods are also summarized. In addition, comparisons of confined DNA with DNA under tension and DNA in semidilute solution are made to emphasize universal behaviors. Finally, an outlook of the possible future research for confined DNA is given.

Knotting dynamics of DNA chains of different length confined in nanochannels

Langevin dynamics simulations are used to characterize the typical mechanisms governing the spontaneous tying, untying and the dynamical evolution of knots in coarse-grained models of DNA chains confined in nanochannels. In particular we focus on how these mechanisms depend on the chain contour length, Lc, at a fixed channel width D = 56 nm corresponding to the onset of the Odijk scaling regime where chain backfoldings and hence knots are disfavoured but not suppressed altogether. We find that the lifetime of knots grows significantly with Lc, while that of unknots varies to a lesser extent. The underlying kinetic mechanisms are clarified by analysing the evolution of the knot position along the chain. At the considered confinement, in fact, knots are typically tied by local backfoldings of the chain termini where they are eventually untied after a stochastic motion along the chain. Consequently, the lifetime of unknots is mostly controlled by backfoldings events at the chain ends, which is largely independent of Lc. The lifetime of knots, instead, increases significantly with Lc because knots can, on average, travel farther along the chain before being untied. The observed interplay of knots and unknots lifetimes underpins the growth of the equilibrium knotting probability of longer and longer chains at fixed channel confinement.

Segregated structures of ring polymer melts near the surface: a molecular dynamics simulation study

We study structural properties of a ring polymeric melt confined in a film in comparison to a linear counterpart using molecular dynamics simulations. Local structure orderings of ring and linear polymers in the vicinity of the surface are similar to each other because the length scale of surface-monomer excluded volume interactions is smaller than the size of an ideal blob of the ring. In a long length scale, while the Silberberg hypothesis can be used to provide the physical origin of the confined linear polymer results, it no longer holds for the ring polymer case. We also present different structural properties of ring and linear polymers in a melt, including the size of polymers, the adsorbed amount, and the coordination number of a polymer. Our observation reveals that a confined ring in a melt adopts a highly segregated conformation due to a topological excluded volume repulsion, which may provide a new perspective to understand the nature of biological processes, such as territorial segregation of chromosomes in eukaryotic nuclei.

Effects of excluded volume and hydrodynamic interaction on the deformation, orientation and motion of ring polymers in shear flow

A ring polymer is a classical model to explore the behaviors of biomacromolecules. Compared with its linear counterpart in shear flow, the ring polymer should be more sensitive to excluded volume and hydrodynamic interaction attributed to the absence of chain ends. We carried out multiparticle collision dynamics combined with molecular dynamics simulation to study the effects of excluded volume and hydrodynamic interaction on the behaviors of ring polymers in shear flow. The results show that in the absence of the strong excluded volume interaction, the ring polymer prefers a two-strand linear conformation with high deformation and orientation in the flow-gradient plane, and the tank-treading motion is nearly negligible. Ring polymers without excluded volume show no significant difference from linear polymers in the scaling exponents for the deformation, orientation and tumbling motion. We also observed that the hydrodynamic interaction could efficiently slow down the relaxation of ring polymers while the scaling exponents against the Weissenberg number have rarely been affected.

Force-Extension for DNA in a Nanoslit: Mapping between the 3D and 2D Limits

The force-extension relation for a semiflexible polymer confined in a nanoslit is investigated. Both the effective correlation length and force-extension relation change as the chain goes from 3D (large slit heights) to 2D (tight confinement). At low forces, correlations along the polymer give an effective dimensionality. The strong force limit can be interpolated with the weak force limit for two regimes: when confinement dominates over extensile force and vice versa. These interpolations give good agreement with simulations for all slit heights and forces. We thus generalize the Marko-Siggia force-extension relation for DNA and other semiflexible biopolymers in nanoconfinement.

Stripe to slab confinement for the linearization of macromolecules in nanochannels

We investigated the recently suggested advantageous analysis of chain linearization experiments with macromolecules confined in a stripe-like channel (Huang and Battacharya, EPL, 2014, 106, 18004) using Monte Carlo simulations. The enhanced chain extension in a stripe, which is due to the significant excluded volume interactions between the monomers in two dimensions, weakens considerably on transition to an experimentally feasible slit-like channel. Based on the chain extension-confinement strength dependence and the structure factor behavior for a chain in a stripe, we infer the excluded volume regime (de Gennes regime) typical for two-dimensional systems. On widening of the stripe in a direction perpendicular to the stripe plane, i.e. on the transition to the slab geometry, the advantageous chain extension decreases and a Gaussian regime is observed for not very long semiflexible chains. The evidence for pseudo-ideality in confined chains is based on four indicators: the extension curves, variation of the extension with the persistence length P, estimated limits for the regimes in the investigated systems, and the structure factor behavior. The slab behavior can be observed when the two-dimensional stripe (originally of a one-monomer thickness) reaches a reduced thickness D larger than approximately D/P ≈ 0.2 in the third dimension. This maximum height of a slab at which the advantage of a stripe is retained is very low and has implications for DNA linearization experiments.

Knotting and Unknotting Dynamics of DNA Strands in Nanochannels

The self-knotting dynamics of DNA strands confined in nanochannels is studied with Brownian simulations. The model DNA chains are several microns long and placed inside channels that are 50-300 nm wide. This width range covers the transition between different metric scaling regimes and the concomitant drop of DNA knotting probability for channel widths below ∼75 nm. We find that knots typically originate from deep looping and backfoldings of the chain ends. Upon lowering the channel width, backfoldings become shallower and rarer and the lifetime of knots decreases while that of unknots increases. This lifetimes interplay causes the dramatic reduction of knots incidence for increasing confinement. The results can aid the design of nanochannels capable of harnessing the self-knotting dynamics to quench or relax the DNA topological state as desired.

Free energy of a long semiflexible polymer confined in a spherical cavity

The free energy and conformational properties of a wormlike chain confined inside a spherical surface are investigated. We show that in the weak-confinement limit, the wormlike chain model exactly reproduces the confinement properties of a Gaussian chain; in such a case the confinement entropy dominates the free energy; in the strong-confinement limit, the free energy is dominated by the bending energy of the chain, which is forced to wrap around the confining surface. We also present a numerical solution within the crossover region between the two limits, solving the differential equation that the probability distribution function satisfies.

Gaussian semiflexible rings under angular and dihedral restrictions

Semiflexible polymer rings whose bonds obey both angular and dihedral restrictions [M. Dolgushev and A. Blumen, J. Chem. Phys. 138, 204902 (2013)], are treated under exact closure constraints. This allows us to obtain semianalytic results for their dynamics, based on sets of Langevin equations. The dihedral restrictions clearly manifest themselves in the behavior of the mean-square monomer displacement. The determination of the equilibrium ring conformations shows that the dihedral constraints influence the ring curvature, leading to compact folded structures. The method for imposing such constraints in Gaussian systems is very general and it allows to account for heterogeneous (site-dependent) restrictions. We show it by considering rings in which one site differs from the others.

Interplay between chain stiffness and excluded volume of semiflexible polymers confined in nanochannels

The properties of channel-confined semiflexible polymers are determined by a complicated interplay of chain stiffness and excluded volume effects. Using Pruned-Enriched Rosenbluth Method (PERM) simulations, we study the equilibrium properties of channel-confined polymers by systematically controlling chain stiffness and excluded volume. Our calculations of chain extension and confinement free energy for freely jointed chains with and without excluded volume show excellent agreement with theoretical predictions. For ideal wormlike chains, the extension is seen to crossover from Odijk behavior in strong confinement to zero-stretching, bulk-like behavior in weak confinement. In contrast, for self-avoiding wormlike chains, we always observe that the linear scaling of the extension with the contour length is valid in the long-chain limit irrespective of the regime of confinement, owing to the coexistence of stiffness and excluded volume effects. We further propose that the long-chain limit for the extension corresponds to chain lengths wherein the projection of the end-to-end distance along the axis of the channel is nearly equal to the mean span parallel to the axis. For DNA in nanochannels, this limit was identified using PERM simulations out to molecular weights of more than 1 megabase pairs; the molecular weight of λ-DNA is found to exhibit nearly asymptotic fractional extension for channels sizes used commonly in experiments.

Modeling the relaxation time of DNA confined in a nanochannel

Using a mapping between a Rouse dumbbell model and fine-grained Monte Carlo simulations, we have computed the relaxation time of λ-DNA in a high ionic strength buffer confined in a nanochannel. The relaxation time thus obtained agrees quantitatively with experimental data [Reisner et al., Phys. Rev. Lett. 94, 196101 (2005)] using only a single O(1) fitting parameter to account for the uncertainty in model parameters. In addition to validating our mapping, this agreement supports our previous estimates of the friction coefficient of DNA confined in a nanochannel [Tree et al., Phys. Rev. Lett. 108, 228105 (2012)], which have been difficult to validate due to the lack of direct experimental data. Furthermore, the model calculation shows that as the channel size passes below approximately 100 nm (or roughly the Kuhn length of DNA) there is a dramatic drop in the relaxation time. Inasmuch as the chain friction rises with decreasing channel size, the reduction in the relaxation time can be solely attributed to the sharp decline in the fluctuations of the chain extension. Practically, the low variance in the observed DNA extension in such small channels has important implications for genome mapping.

Comparison of linear and ring DNA macromolecules moderately and strongly confined in nanochannels

Understanding the mechanism of DNA extension in nanochannels is necessary for interpretation of experiments in nanofluidic channel devices that have been conducted recently with both linear and ring chains. The present article reviews the situation with linear chains and analyses the experimental results and simulations for channel-induced extension (linearization) of ring chains. Results for confined rings indicate a transition between moderate and strong confinement similar to that of linear chains. Owing to stronger self-avoidance in confined rings, the transition and chain extension is shifted relative to linear DNA. We suggest that a relationship similar to that used for the extension of linear chains may also be used for circular DNA.