Measuring the Confinement Free Energy and Effective Width of Single Polymer Chains via Single-Molecule Tetris

Equilibrium behaviour of two cavity-confined polymers: effects of polymer width and system asymmetries

Experiments using nanofluidic devices have proven effective in characterizing the physical properties of polymers confined to small cavities. Two recent studies using such methods examined the organization and dynamics of two DNA molecules in box-like cavities with strong confinement in one direction and with square and elliptical cross sections in the lateral plane. Motivated by these experiments, we employ Monte Carlo and Brownian dynamics simulations to study the physical behaviour of two polymers confined to small cavities with shapes comparable to those used in the experiments. We quantify the effects of varying the following polymer properties and confinement dimensions on the organization and dynamics of the polymers: the polymer width, the polymer contour length ratio, the cavity cross-sectional area, and the degree of cavity elongation for cavities with rectangular and elliptical cross sections. We find that the tendency for polymers to segregate is enhanced by increasing polymer width. For sufficiently small cavities, increasing cavity elongation promotes segregation and localization of identical polymers to opposite sides of the cavity along its long axis. A free-energy barrier controls the rate of polymers swapping positions, and the observed dynamics are roughly in accord with predictions of a simple theoretical model. Increasing the contour length difference between polymers significantly affects their organization in the cavity. In the case of a large linear polymer co-trapped with a small ring polymer in an elliptical cavity, the small polymer tends to lie near the lateral confining walls, and especially at the cavity poles for highly elongated ellipses.

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.

Confinement anisotropy drives polar organization of two DNA molecules interacting in a nanoscale cavity

There is growing appreciation for the role phase transition based phenomena play in biological systems. In particular, self-avoiding polymer chains are predicted to undergo a unique confinement dependent demixing transition as the anisotropy of the confined space is increased. This phenomenon may be relevant for understanding how interactions between multiple dsDNA molecules can induce self-organized structure in prokaryotes. While recent in vivo experiments and Monte Carlo simulations have delivered essential insights into this phenomenon and its relation to bacteria, there are fundamental questions remaining concerning how segregated polymer states arise, the role of confinement anisotropy and the nature of the dynamics in the segregated states. To address these questions, we introduce an artificial nanofluidic model to quantify the interactions of multiple dsDNA molecules in cavities with controlled anisotropy. We find that two dsDNA molecules of equal size confined in an elliptical cavity will spontaneously demix and orient along the cavity poles as cavity eccentricity is increased; the two chains will then swap pole positions with a frequency that decreases with increasing cavity eccentricity. In addition, we explore a system consisting of a large dsDNA molecule and a plasmid molecule. We find that the plasmid is excluded from the larger molecule and will exhibit a preference for the ellipse poles, giving rise to a non-uniform spatial distribution in the cavity that may help explain the non-uniform plasmid distribution observed during in vivo imaging of high-copy number plasmids in bacteria.

Boundaries of the Topologically Frustrated Dynamical State in Polymer Dynamics

Using fluorescence microscopy and single-particle tracking, we have directly observed the dynamics of λ-DNA trapped inside poly(acrylamide-co-acrylate) hydrogels under an externally applied electric field. Congruent with the recent discovery of the nondiffusive topologically frustrated dynamical state (TFDS) that emerges at intermediate confinements between the traditional entropic barrier and reptation regimes, we observe the immobility of λ-DNA in the absence of an electric field. The electrophoretic mobility of the molecule is triggered upon application of an electric field with strength above a threshold value E_c. The existence of the threshold value to elicit mobility is attributed to a large entropic barrier, arising from many entropic traps acting simultaneously on a single molecule. Using the measured E_c which depends on the extent of confinement, we have determined the net entropic barrier of up to 130 k_BT, which is responsible for the long-lived metastable TFDS. The net entropic barrier from multiple entropic traps is nonmonotonic with the extent of confinement and tends to vanish at the boundaries of the TFDS with the single-entropic barrier regime at lower confinements and the reptation regime at higher confinements. We present an estimate of the mesh size of the hydrogel that switches off the nondiffusive TFDS and releases chin diffusion in the heavily entangled state.

Entropic barrier of topologically immobilized DNA in hydrogels

The single most intrinsic property of nonrigid polymer chains is their ability to adopt enormous numbers of chain conformations, resulting in huge conformational entropy. When such macromolecules move in media with restrictive spatial constraints, their trajectories are subjected to reductions in their conformational entropy. The corresponding free energy landscapes are interrupted by entropic barriers separating consecutive spatial domains which function as entropic traps where macromolecules can adopt their conformations more favorably. Movement of macromolecules by negotiating a sequence of entropic barriers is a common paradigm for polymer dynamics in restrictive media. However, if a single chain is simultaneously trapped by many entropic traps, it has recently been suggested that the macromolecule does not undergo diffusion and is localized into a topologically frustrated dynamical state, in apparent violation of Einstein's theorem. Using fluorescently labeled λ-DNA as the guest macromolecule embedded inside a similarly charged hydrogel with more than 95% water content, we present direct evidence for this new state of polymer dynamics at intermediate confinements. Furthermore, using a combination of theory and experiments, we measure the entropic barrier for a single macromolecule as several tens of thermal energy, which is responsible for the extraordinarily long extreme metastability. The combined theory-experiment protocol presented here is a determination of single-molecule entropic barriers in polymer dynamics. Furthermore, this method offers a convenient general procedure to quantify the underlying free energy landscapes behind the ubiquitous phenomenon of movement of single charged macromolecules in crowded environments.

Equilibrium organization, conformation, and dynamics of two polymers under box-like confinement

Motivated by recent nanofluidics experiments, we use Brownian dynamics and Monte Carlo simulations to study the conformation, organization and dynamics of two polymer chains confined to a single box-like cavity. The polymers are modeled as flexible bead-spring chains, and the box has a square cross-section of side length L and a height that is small enough to compress the polymers in that dimension. For sufficiently large L, the system behaviour approaches that of an isolated polymer in a slit. However, the combined effects of crowding and confinement on the polymer organization, conformation and equilibrium dynamics become significant when where is the transverse radius of gyration for a slit geometry. In this regime, the centre-of-mass probability distribution in the transverse plane exhibits a depletion zone near the centre of the cavity (except at very small L) and a 4-fold symmetry with quasi-discrete positions. Reduction in polymer size with decreasing L arises principally from confinement rather than inter-polymer crowding. By contrast, polymer diffusion and internal motion are strongly affected by inter-polymer crowding. The two polymers tend to occupy opposite positions relative to the box centre, about which they diffuse relatively freely. Qualitatively, this static and dynamical behaviour differs significantly from that previously observed for confinement of two polymers to a narrow channel. The simulation results for a suitably chosen box width are qualitatively consistent with results from a recent experimental study of two λ-DNA chains confined to a nanofluidic cavity.

Polymer translocation into cavities: Effects of confinement geometry, crowding, and bending rigidity on the free energy

Monte Carlo simulations are used to study the translocation of a polymer into a cavity. Modeling the polymer as a hard-sphere chain with a length up to N=601 monomers, we use a multiple-histogram method to measure the variation of the conformational free energy of the polymer with respect to the number of translocated monomers. The resulting free-energy functions are then used to obtain the confinement free energy for the translocated portion of the polymer. We characterize the confinement free energy for a flexible polymer in cavities with constant cross-sectional area A for various cavity shapes (cylindrical, rectangular, and triangular) as well as for tapered cavities with pyramidal and conical shape. The scaling of the free energy with cavity volume and translocated polymer subchain length is generally consistent with predictions from simple scaling arguments, with small deviations in the scaling exponents likely due to finite-size effects. The confinement free energy depends strongly on cavity shape anisometry and is a minimum for an isometric cavity shape with a length-to-width ratio of unity. Entropic depletion at the edges or vertices of the confining cavity are evident in the results for constant-A and pyramidal cavities. For translocation into infinitely long cones, the scaling of the free energy with taper angle is consistent with a theoretical prediction employing the blob model. We also examine the effects of polymer bending rigidity on the translocation free energy for cylindrical cavities. For isometric cavities, the observed scaling behavior is in partial agreement with theoretical predictions, with discrepancies arising from finite-size effects that prevent the emergence of well-defined scaling regimes. In addition, translocation into highly anisometric cylindrical cavities leads to a multistage folding process for stiff polymers. Finally, we examine the effects of crowding agents inside the cavity. We find that the confinement free energy increases with crowder density. At constant packing fraction the magnitude of this effect lessens with increasing crowder size for a crowder-to-monomer size ratio ≥1.

Block Copolymer of Flexible and Semi-Flexible Block Confined in Nanopost Array

Coarse-grained molecular dynamics simulations of a diblock copolymer consisting of a flexible and semi-flexible block in a dense array of parallel nanoposts with a square lattice packing were performed. The mutual interactions between the two blocks of the confined diblock chain were investigated through a comparison of their size, structure, and penetration among nanoposts with the corresponding separate chains. The geometry of a nanopost array was varied at constant post separation or at constant width of the passage between nanoposts. The size of a single interstitial volume was comparable to or smaller than the size of the diblock chain. A comparison of the blocks with their separate analogous chains revealed that the mutual interactions between the blocks were shielded by the nanoposts and, thus, the blocks behaved independently. At constant passage width, competitive effects of the axial chain extension in interstitial volumes and the lateral chain expansion among interstitial volumes led to a nonmonotonic behavior of the axial span. The position of the maximum in the span plotted against the filling fraction for a diblock chain was dictated by the semi-flexible block. The semi-flexible block penetrates among the nanoposts more readily and the expansion of the whole diblock copolymer is governed by the semiflexible block. The main findings were explained using the free energy arguments when an interstitial volume was approximated by a channel geometry and a passage aperture by a slit geometry. Detail knowledge of controlled conformational behavior in a compartmentalized environment can contribute to new processes in the storage and retrieval of information.

Anomalous packing and dynamics of a polymer chain confined in a static porous environment

Polymers in confined porous environments are ubiquitous throughout biology, physics, materials science, and engineering. Several experiments have suggested that in some porous environments, chain dynamics can become extremely slow. While phenomenological explanations exist, the exact mechanisms for these slow dynamics have not been fully characterized. In this work, we initiate a joint simulation-theory study to investigate chain packing and dynamics in a static porous environment. The main theoretical concept is the free energy of the chain partitioning into several chambers of the porous environment. Both the theoretical results and Langevin dynamics simulations show that chain packing in each of the chambers is predominantly independent of chain length; it is determined by the maximal packing of segments in each chamber. Dynamically, short chains (compared to the chamber size) become trapped in a single chamber and dynamics become extremely slow, characteristic of an Ogston sieving-like behavior. For longer chains, on the other hand, a hierarchy of slow dynamics is observed due to entropic trapping, characterized by sub-diffusive behavior and a temporary plateau in the mean square displacement. Due to the slow nature of the dynamics, the inevitable long-time diffusive behavior of the chains is not captured by our simulations. Theoretically, the slow dynamics are understood in terms of a free energy barrier required to thread the chain from one chamber to the next. There is overall qualitative and quantitative agreement between simulations and theory. This work provides foundations for a better understanding of how chain dynamics are affected by porous environments.

Probing the organization and dynamics of two DNA chains trapped in a nanofluidic cavity

Here we present a pneumatically-actuated nanofluidic platform that has the capability of dynamically controlling the confinement environment of macromolecules in solution. Using a principle familiar from classic devices based on soft-lithography, the system uses pneumatic pressure to deflect a thin nitride lid into a nanoslit, confining molecules in an array of cavities embedded in the slit. We use this system to quantify the interactions of multiple confined DNA chains, a key problem in polymer physics with important implications for nanofluidic device performance and DNA partitioning/organization in bacteria and the eukaryotes. In particular, we focus on the problem of two-chain confinement, using differential staining of the chains to independently assess the chain conformation, determine the degree of partitioning/mixing in the cavities and assess coupled diffusion of the chain center-of-mass positions. We find that confinement of more than one chain in the cavity can have a drastic impact on the polymer dynamics and conformation.

Compressing a confined DNA: from nano-channel to nano-cavity

We analyze the behavior of a semiflexible polymer confined in nanochannel under compression in axial direction. Key to our discussion is the identification of two length scales; the correlation length ξ of concentration fluctuation and what we call the segregation length [Formula: see text]. These length scales, while degenerate in uncompressed state in nanochannel, generally split as [Formula: see text] upon compression, and the way they compete with the system size during the compression determines the crossover from quasi-1D nanochannel to quasi-0D nanocavity behaviors. For a flexible polymer, the story becomes very simple, which corresponds to a special limit of our description, but a much richer behavior is expected for a semiflexible polymer relevant to DNA in confined spaces. We also briefly discuss the dynamical properties of the compressed polymer.

Motion of Knots in DNA Stretched by Elongational Fields

Knots in DNA occur in biological systems, serve as a model system for polymer entanglement, and affect the efficacy of modern genomics technologies. We study the motion of complex knots in DNA by stretching molecules with a divergent electric field that provides an elongational force. We demonstrate that the motion of knots is nonisotropic and driven towards the closest end of the molecule. We show for the first time experimentally that knots can go from a mobile to a jammed state by varying an applied strain rate, and that this jamming is reversible. We measure the mobility of knots as a function of strain rate, demonstrating the conditions under which knots can be driven towards the ends of the molecule and untied.

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.

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.

Compressive response and helix formation of a semiflexible polymer confined in a nanochannel

Configurations of a single semiflexible polymer is studied when it is pushed into a nanochannel in the case where the polymer persistence length l_{p} is much longer than the channel diameter D:l_{p}/D≫1. Using numerical simulations, we show that the polymer undergoes a sequence of recurring structural transitions upon longitudinal compression: random deflection along the channel, a helix going around the channel wall, double-fold random deflection, double-fold helix, etc. We find that the helix transition can be understood as buckling of deflection segments, and the initial helix formation takes place at very small compression with no appreciable weak compression regime of the random deflection polymer.

Wall depletion length of a channel-confined polymer

Numerous experiments have taken advantage of DNA as a model system to test theories for a channel-confined polymer. A tacit assumption in analyzing these data is the existence of a well-defined depletion length characterizing DNA-wall interactions such that the experimental system (a polyelectrolyte in a channel with charged walls) can be mapped to the theoretical model (a neutral polymer with hard walls). We test this assumption using pruned-enriched Rosenbluth method (PERM) simulations of a DNA-like semiflexible polymer confined in a tube. The polymer-wall interactions are modeled by augmenting a hard wall interaction with an exponentially decaying, repulsive soft potential. The free energy, mean span, and variance in the mean span obtained in the presence of a soft wall potential are compared to equivalent simulations in the absence of the soft wall potential to determine the depletion length. We find that the mean span and variance about the mean span have the same depletion length for all soft potentials we tested. In contrast, the depletion length for the confinement free energy approaches that for the mean span only when depletion length no longer depends on channel size. The results have implications for the interpretation of DNA confinement experiments under low ionic strengths.

Waves of DNA: Propagating excitations in extended nanoconfined polymers

We use a nanofluidic system to investigate the emergence of thermally driven collective phenomena along a single polymer chain. In our approach, a single DNA molecule is confined in a nanofluidic slit etched with arrays of embedded nanocavities; the cavity lattice is designed so that a single chain occupies multiple cavities. Fluorescent video-microscopy data shows fluctuations in intensity between cavities, including waves of excess fluorescence that propagate across the cavity-straddling molecule, corresponding to propagating fluctuations of contour overdensity in the cavities. The transfer of DNA between neighboring pits is quantified by examining the correlation in intensity fluctuations between neighboring cavities. Correlations grow from an anticorrelated minimum to a correlated maximum before decaying, corresponding to a transfer of contour between neighboring cavities at a fixed transfer time scale. The observed dynamics can be modeled using Langevin dynamics simulations and a minimal lattice model of coupled diffusion. This study shows how confinement-based sculpting of the polymer equilibrium configuration, by renormalizing the physical system into a series of discrete cavity states, can lead to new types of dynamic collective phenomena.

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.

Transition state theory demonstrated at the micron scale with out-of-equilibrium transport in a confined environment

Transition state theory (TST) provides a simple interpretation of many thermally activated processes. It applies successfully on timescales and length scales that differ several orders of magnitude: to chemical reactions, breaking of chemical bonds, unfolding of proteins and RNA structures and polymers crossing entropic barriers. Here we apply TST to out-of-equilibrium transport through confined environments: the thermally activated translocation of single DNA molecules over an entropic barrier helped by an external force field. Reaction pathways are effectively one dimensional and so long that they are observable in a microscope. Reaction rates are so slow that transitions are recorded on video. We find sharp transition states that are independent of the applied force, similar to chemical bond rupture, as well as transition states that change location on the reaction pathway with the strength of the applied force. The states of equilibrium and transition are separated by micrometres as compared with angstroms/nanometres for chemical bonds.