The Odijk Regime in Slits

Multi-feature clustering of CTCF binding creates robustness for loop extrusion blocking and Topologically Associating Domain boundaries

Topologically Associating Domains (TADs) separate vertebrate genomes into insulated regulatory neighborhoods that focus genome-associated processes. TADs are formed by Cohesin-mediated loop extrusion, with many TAD boundaries consisting of clustered binding sites of the CTCF insulator protein. Here we determine how this clustering of CTCF binding contributes to the blocking of loop extrusion and the insulation between TADs. We identify enrichment of three features of CTCF binding at strong TAD boundaries, consisting of strongly bound and closely spaced CTCF binding peaks, with a further enrichment of DNA-binding motifs within these peaks. Using multi-contact Nano-C analysis in cells with normal and perturbed CTCF binding, we establish that individual CTCF binding sites contribute to the blocking of loop extrusion, but in an incomplete manner. When clustered, individual CTCF binding sites thus create a stepwise insulation between neighboring TADs. Based on these results, we propose a model whereby multiple instances of temporal loop extrusion blocking create strong insulation between TADs.

Entropic force of cone-tethered polymers interacting with a planar surface

Computer simulations are used to characterize the entropic force of one or more polymers tethered to the tip of a hard conical object that interact with a nearby hard flat surface. Pruned-enriched Rosenbluth method Monte Carlo simulations are used to calculate the variation of the conformational free energy F of a hard-sphere polymer with respect to the cone-tip-to-surface distance h from which the variation of the entropic force f≡|dF/dh| with h is determined. We consider the following cases: a single freely jointed tethered chain, a single semiflexible tethered chain, and several freely jointed chains of equal length each tethered to the cone tip. The simulation results are used to test the validity of a prediction by Maghrebi et al. [Maghrebi et al., Europhys. Lett. 96, 66002 (2011)0295-507510.1209/0295-5075/96/66002; Phys. Rev. E 86, 061801 (2012)1539-375510.1103/PhysRevE.86.061801] that f∝(γ_{∞}-γ_{0})h^{-1}, where γ_{0} and γ_{∞} are universal scaling exponents for the partition function of the tethered polymer for h=0 and h=∞, respectively. The measured functions f(h) are generally consistent with the predictions, with small quantitative discrepancies arising from the approximations employed in the theory. In the case of multiple tethered polymers, the entropic force per polymer is roughly constant, which is qualitatively inconsistent with the predictions.

Compression and Stretching of Confined Linear and Ring Polymers by Applying Force

We use Langevin dynamics to study the deformations of linear and ring polymers in different confinements by applying compression and stretching forces on their two sides. Our results show that the compression deformations are the results of an interplay among of polymer rigidity, degree of confinement, and force applied. When the applied force is beyond the threshold required for the buckling transition, the semiflexible chain under the strong confinement firstly buckles; then comes helical deformation. However, under the same force loading, the semiflexible chain under the weaker confinement exhibits buckling instability and shrinks from the folded ends/sides until it becomes three-folded structures. This happens because the strong confinement not only strongly reduces the buckling wavelength, but also increases the critical buckling force threshold. For the weakly confined polymers, in compression process, the flexible linear polymer collapses into condensed states under a small external force, whereas the ring polymer only shows slight shrinkage, due to the excluded volume interactions of two strands in the crowded states. These results are essential for understanding the deformations of the ring biomacromolecules and polymer chains in mechanical compression or driven transport.

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.

Statistical Behaviors of Semiflexible Polymer Chains Stretched in Rectangular Tubes

We investigated the statistical behaviors of semiflexible polymer chains, which were simultaneously subjected to force stretching and rectangular tube confinement. Based on the wormlike chain model and Odijk deflection theory, we derived a new deflection length, by using which new compact formulas were obtained for the confinement free energy and force⁻confinement⁻extension relations. These newly derived formulas were justified by numerical solutions of the eigenvalue problem associated with the Fokker⁻Planck governing equation and extensive Brownian dynamics simulations based on the so-called generalized bead-rod (GBR) model. We found that, compared to classical deflection theory, these new formulas were valid for a much more extended range of the confinement size/persistence length ratio and had no adjustable fitting parameters for sufficiently long semiflexible chains in the whole deflection regime.

Investigation of nematic to smectic phase transition and dynamical properties of strongly confined semiflexible polymers using Langevin dynamics

We investigated the nematic to smectic phase transition for strongly confined semiflexible polymer solutions in slit-like confinements using GPU-accelerated Langevin dynamics. We characterized the phase transitions from the nematic to smectic phases for semi-flexible polymer solutions as the polymer density increased. The dependence for the lyotropic nematic to smectic transition can be collapsed by scaling exponents between 0.2 and 0.3. The smectic C phase is found for all the cases with the polymer orientation director tilted with respect to smectic layer lateral alignment. As the chain rigidity increases, the transition density decreases for systems in which the polymer persistence length (P) to slit height (H) ratios are 1.25, 2.5, 3.75, 5 and 25. We also characterized the polymer dynamics for the isotropic-nematic-smectic transitions. The overall polymer diffusivity decreased steadily as the polymer density increased. We observed anomalous polymer diffusion along the nematic director near the isotropic-nematic transition, similar to previously reported behavior for nematic-forming ellipsoids. Polymer diffusivity decreased sharply by two orders of magnitude upon the nematic-smectic transition.

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.

Evidence for the extended de Gennes regime of a semiflexible polymer in slit confinement

We use off-lattice, pruned-enriched Rosenbluth method (PERM) simulations to compute the confinement free energy of a real wormlike chain of effective width w and persistence length lp in a slit of height H. For slit heights much larger than the persistence length of the polymer and much smaller than the thermal blob size, the excess free energy of the confined chain is consistent with a modified version of the scaling theory for the extended de Gennes regime in a channel that reflects the blob statistics in slit confinement. Explicitly, for channel sizes [Formula: see text], the difference between the confinement free energy of the real chain and that of an ideal chain scales like w/H.

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.

Simulations of knotting of DNA during genome mapping

Genome mapping involves the confinement of long DNA molecules, in excess of 150 kilobase pairs, in nanochannels near the circa 50 nm persistence length of DNA. The fidelity of the map relies on the assumption that the DNA is linearized by channel confinement, which assumes the absence of knots. We have computed the probability of forming different knot types and the size of these knots for long chains (approximately 164 kilobase pairs) via pruned-enriched Rosenbluth method simulations of a discrete wormlike chain model of DNA in channel sizes ranging from 35 nm to 60 nm. Compared to prior simulations of short DNA in similar confinement, these long molecules exhibit both complex knots, with up to seven crossings, and multiple knots per chain. The knotting probability is a very strong function of channel size, ranging from 0.3% to 60%, and rationalized in the context of Odijk's theory for confined semiflexible chains. Overall, the knotting probability and knot size obtained from these equilibrium measurements are not consistent with experimental measurements of the properties of anomalously bright regions along the DNA backbone during genome mapping experiments. This result suggests that these events in experiments are either knots formed during the processing of the DNA prior to injection into the nanochannel or regions of locally high DNA concentration without a topological constraint. If so, knots during genome mapping are not an intrinsic problem for genome mapping technology.

Stretching of Tethered DNA in Nanoslits

External forces and confinement are two fundamental and complementary approaches for biopolymer stretching. By employing micro- and nanofluidics, we study the force-extension dynamics by simultaneously applying external forces and confinement to single-DNA molecules. In particular, we apply external electric fields to stretch single DNA molecules that are attached to microspheres anchored at a nanoslit entrance. Using this method, we measure the force-extension relation of tethered DNA and describe this relation with modified wormlike chain models. This allowed experimental validations of several theoretical predictions, including the increase in the global persistence length of confined DNA with increasing degree of confinement and the "confined Pincus" regime in slit confinement.

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.

Effect of excluded volume on the force-extension of wormlike chains in slit confinement

We use pruned-enriched Rosenbluth method simulations to develop a quantitative phase diagram for the stretching of a real wormlike chain confined in a slit. Our simulations confirm the existence of a "confined Pincus" regime in slit confinement, analogous to the Pincus regime in free solution, where excluded volume effects are sensible. The lower bound for the confined Pincus regime in the force-molecular weight plane, as well as the scaling of the extension with force and slit size, agree with an existing scaling theory for this regime. The upper bound of the confined Pincus regime depends on the strength of the confinement. For strong confinement, the confined Pincus regime ends when the contour length in the Pincus blob is too short to have intrablob excluded volume. As a result, the chain statistics become ideal and the confined Pincus regime at low forces is connected directly to ideal chain stretching at large forces. In contrast, for weak confinement, the confined Pincus regime ends when the Pincus blobs no longer fit inside the slit, even though there is sufficient contour length to have excluded volume inside the Pincus blob. As a result, weak confinement leads to a free-solution Pincus regime intervening between the confined Pincus regime for weak forces and ideal chain stretching at strong forces. Our results highlight shortcomings in existing models for the stretching of wormlike chains in slits.

The Backfolded Odijk Regime for Wormlike Chains Confined in Rectangular Nanochannels

We confirm Odijk's scaling laws for (i) the average chain extension; (ii) the variance about the average extension; and (iii) the confinement free energy of a wormlike chain confined in a rectangular nanochannel smaller than its chain persistence length through pruned-enriched Rosenbluth method (PERM) simulations of asymptotically long, discrete wormlike chains. In the course of this analysis, we also computed the global persistence length of ideal wormlike chains for the modestly rectangular channels that are used in many experimental systems. The results are relevant to genomic mapping systems that confine DNA in channel sizes around 50 nm, since fabrication constraints generally lead to rectangular cross-sections.

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.

Rigorous study of molecular dynamics of a single dsDNA confined in a nanochannel: Introduction of a critical mobility behaviour

The essential and effective characteristics of a double-stranded DNA (dsDNA) confined in a nanochannel is revisited by employing the rigorous full numerical approach of Molecular Dynamics (MD). The deformation of dsDNA and wall-biomolecule interaction which is critical in highly confined regime has been precisely imposed in numerical simulations. The numerical approach has been justified against available theoretical outcomes. A new and general expression for DNA electrophoretic mobility versus DNA length is extracted from numerical simulation which is out of reach of experimental methods due to practical shortcomings. The newly derived expression suggests an essential correction in the previously proposed expression for the critical case of small DNA molecules and reveals an astonishingly unbeknown trend of small DNA's mobility. Sub-molecular phenomenon of dsDNA melting under the condition of large external force is also studied. Assuming strong electric field exertion, the MD approach aptly demonstrates the elaborate melting phenomenon for dsDNA in sub-molecular scale.

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.

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.

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.

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.

Nanoslit Confined DNA at Low Ionic Strengths

We present nanoslit confined DNA conformations at very low ionic strengths and a theory to explain most measurements for single DNA molecule size under strong nanoslit confinement. Very low ionic strength conditions not only increase the DNA persistence length dramatically, but also cause DNA molecules to swell to the extent that the effective diameter of DNA becomes larger than the nanoslit height. By accounting for these effects, our results and theory provide a reasonable clue for a current controversy regarding the dependence of the DNA conformation on slit height (h), persistence length (p), and effective diameter (w).