Unexpected relaxation dynamics of a self-avoiding polymer in cylindrical confinement

Topological Friction and Relaxation Dynamics of Spatially Confined Catenated Polymers

We study catenated ring polymers confined inside channels and slits with Langevin dynamics simulations and address how the contour position and size of the interlocked or physically linked region evolve with time. We show that the catenation constraints generate a drag, or topological friction, that couples the contour motion of the interlocked regions. Notably, the coupling strength decreases as the interlocking is made tighter, but also shorter, by confinement. Though the coupling strength differs for channel and slit confinement, the data outline a single universal curve when plotted against the size of the linked region. Finally, we study how the relaxation kinetics changes after one of the rings is cut open and conclude that considering interlocked circular polymers is key for isolating the manifestations of topological friction. The results ought to be relevant for linked biomolecules in experimental or biological confining conditions.

Simulation of the Coronal Dynamics of Polymer-Grafted Nanoparticles

Polymer-grafted nanoparticles (PGNPs) are an important component of many advanced materials. The interplay between the nanoparticle surface curvature and spatial confinement by neighboring chains produces a complex set of structural and dynamical behaviors in the polymer corona surrounding the nanoparticle. For example, experiments have shown that the inner portion of the corona is more stretched and relaxes more slowly than the outer region. Here, we perform systematic core-modified dissipative particle dynamics (CM-DPD) simulations and analyze the relaxation dynamics using proper orthogonal decomposition (POD) of the monomer coordinates. We find that grafted chains relax more slowly than free chains and that the relaxation time of the grafted chains scales inversely with the confinement strength. For PGNPs in a polymer melt, the relaxation processes are always Rouse-like. However, we observe either Zimm-like or Rouse-like dynamics for PGNPs in solution depending on the confinement strength.

Adsorption of Semiflexible Polymers in Cylindrical Tubes

Conformations of wormlike chains in cylindrical pores with attractive walls are explored for varying pore radius and strength of the attractive wall potential by molecular dynamics simulations of a coarse-grained model. Local quantities such as the fraction of monomeric units bound to the surface and the bond-orientational order parameter as well as the radial density distribution are studied, as well as the global chain extensions parallel to the cylinder axis and perpendicular to the cylinder surface. A nonmonotonic convergence of these properties to their counterparts for adsorption on a planar substrate is observed due to the conflict between pore surface curvature and chain stiffness. Also the interpretation of partially adsorbed chains in terms of trains, loops, and tails is discussed.

The Longitudinal Superdiffusive Motion of Block Copolymer in a Tight Nanopore

The structure and dynamic properties of polymer chains in a confined environment were studied by means of the Monte Carlo method. The studied chains were represented by coarse-grained models and embedded into a simple 3D cubic lattice. The chains stood for two-block linear copolymers of different energy of bead-bead interactions. Their behavior was studied in a nanotube formed by four impenetrable surfaces. The long-time unidirectional motion of the chain in the tight nanopore was found to be correlated with the orientation of both parts of the copolymer along the length of the nanopore. A possible mechanism of the anomalous diffusion was proposed on the basis of thermodynamics of the system, more precisely on the free energy barrier of the swapping of positions of both parts of the chain and the impulse of temporary forces induced by variation of the chain conformation. The mean bead and the mass center autocorrelation functions were examined. While the former function behaves classically, the latter indicates the period of time of superdiffusive motion similar to the ballistic motion with the autocorrelation function scaling with the exponent t^5/3. A distribution of periods of time of chain diffusion between swapping events was found and discussed. The influence of the nanotube width and the chain length on the polymer diffusivity was studied.

Bacterial chromosome organization. II. Few special cross-links, cell confinement, and molecular crowders play the pivotal roles

Using a coarse-grained bead-spring model of bacterial chromosomes of Caulobacter crescentus and Escherichia coli, we show that just 33 and 38 effective cross-links in 4017 and 4642 monomer chains at special positions along the chain contour can lead to the large-scale organization of the DNA polymer, where confinement effects of the cell walls play a key role in the organization. The positions of the 33/38 cross-links along the chain contour are chosen from the Hi-C contact map of bacteria C. crescentus and E. coli. We represent 1000 base pairs as a coarse-grained monomer in our bead-spring flexible ring polymer model of the DNA polymer. Thus, 4017/4642 beads on a flexible ring polymer represent the C. crescentus/E. coli DNA polymer with 4017/4642 kilo-base pairs. Choosing suitable parameters from Paper I, we also incorporate the role of compaction of the polymer coil due to the presence of molecular crowders and the ability of the chain to release topological constraints. We validate our prediction of the organization of the bacterial chromosomes with available experimental data and also give a prediction of the approximate positions of different segments within the cell. In the absence of confinement, the minimal number of effective cross-links required to organize the DNA chains of 4017/4642 monomers was 60/82 [Agarwal et al., Europhys. Lett. 121, 18004 (2018) and Agarwal et al., J. Phys.: Condens. Matter 30, 034003 (2018)].

Relaxation of surface-tethered polymers under moderate confinement

We study the relaxation of surface-tethered polymers in microchannels under moderate confinement (i.e. h ∼ Rg, where h is the channel height and Rg is the radius of gyration of the polymer) by experiments with fluorescence-marked DNA molecules and coupled lattice-Boltzmann/molecular dynamics simulations. The determined scaling exponent suggests that the relaxation is dominated by Zimm-dynamics with significant intra-chain hydrodynamic interactions. The relaxation of the DNA molecules is slower in shallower channels, indicating a pronounced effect of confinement on the longest relaxation time. An experimental correlation is obtained for the longest relaxation time as a function of the molecular contour length and the channel height. Good agreement between the experimental and the simulation results is found.

A ring-polymer model shows how macromolecular crowding controls chromosome-arm organization in Escherichia coli

Macromolecular crowding influences various cellular processes such as macromolecular association and transcription, and is a key determinant of chromosome organization in bacteria. The entropy of crowders favors compaction of long chain molecules such as chromosomes. To what extent is the circular bacterial chromosome, often viewed as consisting of “two arms”, organized entropically by crowding? Using computer simulations, we examine how a ring polymer is organized in a crowded and cylindrically-confined space, as a coarse-grained bacterial chromosome. Our results suggest that in a wide parameter range of biological relevance crowding is essential for separating the two arms in the way observed with Escherichia coli chromosomes at fast-growth rates, in addition to maintaining the chromosome in an organized collapsed state. Under different conditions, however, the ring polymer is centrally condensed or adsorbed onto the cylindrical wall with the two arms laterally collapsed onto each other. We discuss the relevance of our results to chromosome-membrane interactions.

Effects of molecular crowding and confinement on the spatial organization of a biopolymer

A chain molecule can be entropically collapsed in a crowded medium in a free or confined space. Here, we present a unified view of how molecular crowding collapses a flexible polymer in three distinct spaces: free, cylindrical, and (two-dimensional) slit-like. Despite their seeming disparities, a few general features characterize all these cases, even though the ϕ_c-dependence of chain compaction differs between the two cases: a > a_c and a < a_c, where ϕ_c is the volume fraction of crowders, a is the monomer size, and a_c is the crowder size. For a > a_c (applicable to a coarse-grained model of bacterial chromosomes), chain size depends on the ratio aϕ_c/a_c, and "full" compaction occurs universally at aϕ_c/a_c ≈ 1; for a_c > a (relevant for protein folding), it is controlled by ϕ_c alone and crowding has a modest effect on chain size in a cellular environment (ϕ_c ≈ 0.3). Also for a typical parameter range of biological relevance, molecular crowding can be viewed as effectively reducing the solvent quality, independent of confinement.

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.

Polymers under confinement: single polymers, how they interact, and as model chromosomes

How confinement or a physical constraint modifies polymer chains is not only a classical problem in polymer physics but also relevant in a variety of contexts such as single-molecule manipulations, nanofabrication in narrow pores, and modelling of chromosome organization. Here, we review recent progress in our understanding of polymers in a confined (and crowded) space. To this end, we highlight converging views of these systems from computational, experimental, and theoretical approaches, and then clarify what remains to be clarified. In particular, we focus on exploring how cylindrical confinement reshapes individual chains and induces segregation forces between them - by pointing to the relationships between intra-chain organization and chain segregation. In the presence of crowders, chain molecules can be entropically phase-separated into a condensed state. We include a kernel of discussions on the nature of chain compaction by crowders, especially in a confined space. Finally, we discuss the relevance of confined polymers for the nucleoid, an intracellular space in which the bacterial chromosome is tightly packed, in part by cytoplasmic crowders.

Accuracy of the blob model for single flexible polymers inside nanoslits that are a few monomer sizes wide

The de Gennes' blob model is extensively used in different problems of polymer physics. This model is theoretically applicable when the number of monomers inside each blob is large enough. For confined flexible polymers, this requires the confining geometry to be much larger than the monomer size. In this paper, the opposite limit of polymer in nanoslits with one to several monomers width is studied, using molecular dynamics simulations. Extension of the polymer inside nanoslits, confinement force on the plates, and the effective spring constant of the confined polymer are investigated. Despite the theoretical limitations of the blob model, the simulation results are explained with the blob model very well. The agreement is observed for the static properties and the dynamic spring constant of the polymer. A theoretical description of the conditions under which the dynamic spring constant of the polymer is independent of the small number of monomers inside blobs is given. Our results on the limit of applicability of the blob model can be useful in the design of nanotechnology devices.

Induction of entropic segregation: the first step is the hardest

In confinement, overlapping polymers experience entropic segregating forces that tend to demix them. This plays a role during cell replication, where it facilitates the segregation of daughter chromosomes. It has been argued that these forces are strong enough to explain chromosome segregation in elongated bacteria such as E. coli without the need for additional active mechanisms [S. Jun and B. Mulder, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 12388]. However, entropic segregation can only set in after the initial symmetry has been broken. We demonstrate that the timescale for this induction phase is exponentially growing in the chain length, while the actual segregation time scales only quadratically in the chain length. Thus the induction quickly becomes the dominating, slow process, and makes entropic segregation much less efficient than previously thought. The slow induction might also explain the long delay in chromosome segregation observed in experiments on E. coli.

DNA confinement in nanochannels: physics and biological applications

DNA is the central storage molecule of genetic information in the cell, and reading that information is a central problem in biology. While sequencing technology has made enormous advances over the past decade, there is growing interest in platforms that can readout genetic information directly from long single DNA molecules, with the ultimate goal of single-cell, single-genome analysis. Such a capability would obviate the need for ensemble averaging over heterogeneous cellular populations and eliminate uncertainties introduced by cloning and molecular amplification steps (thus enabling direct assessment of the genome in its native state). In this review, we will discuss how the information contained in genomic-length single DNA molecules can be accessed via physical confinement in nanochannels. Due to self-avoidance interactions, DNA molecules will stretch out when confined in nanochannels, creating a linear unscrolling of the genome along the channel for analysis. We will first review the fundamental physics of DNA nanochannel confinement--including the effect of varying ionic strength--and then discuss recent applications of these systems to genomic mapping. Apart from the intense biological interest in extracting linear sequence information from elongated DNA molecules, from a physics view these systems are fascinating as they enable probing of single-molecule conformation in environments with dimensions that intersect key physical length-scales in the 1 nm to 100 µm range.

Fluctuation modes of nanoconfined DNA

We report an experimental investigation of the magnitude of length and density fluctuations in DNA that has been stretched in nanofluidic channels. We find that the experimental data can be described using a one-dimensional overdamped oscillator chain with nonzero equilibrium spring length and that a chain of discrete oscillators yields a better description than a continuous chain. We speculate that the scale of these discrete oscillators coincides with the scale at which the finite extensibility of the polymer manifests itself. We discuss how the measurement process influences the apparent measured dynamic properties, and outline requirements for the recovery of true physical quantities.

A model for segregation of chromatin after replication: segregation of identical flexible chains in solution

We study the segregation of two long chains from parallel but randomly twisted start conformations under good solvent conditions using Monte Carlo simulations to mimic chromatin segregation after replication in eukaryotic cells in the end of prophase. To measure the segregation process, we consider the center-of-mass separation between the two chains and the average square distance between the monomers which were connected before segregation starts. We argue that segregation is dominated by free diffusion of the chains, assuming that untwisting can be achieved by Rouse-like fluctuations on the length scale of a twisted loop. Using scaling analysis, we find that chain dynamics is in very good agreement with the free diffusion hypothesis, and segregation dynamics follows this scaling nearly. Long chains, however, show retardation effects that can be described by a new (to us) dynamical exponent, which is slightly larger than the dynamical exponent for Rouse-like diffusion. Our results indicate that nearly free diffusion of chains during a timescale of a few Rouse-times can lead to segregation of chains. A main obstacle during segregation by free diffusion is random twists between daughter strands. We have calculated the number of twists formed by the daughter strands in the start conformations, which turns out to be rather low and increases only with the square-root of the chain length.

Simulation of DNA Extension in Nanochannels

We have used a realistic model for double stranded DNA and Monte Carlo simulations to compute the extension (mean span) of a DNA molecule confined in a nanochannel over the full range of confinement in a high ionic strength buffer. The simulation data for square nanochannels resolve the apparent contradiction between prior simulation studies and the predictions from Flory theory, demonstrating the existence of two transition regimes between weak confinement (the de Gennes regime) and strong confinement (the Odijk regime). The simulation data for rectangular nanochannels support the use of the geometric mean for mapping data obtained in rectangular channels onto models developed for cylinders. The comparison of our results with experimental data illuminates the challenges in applying models for confined, neutral polymers to polyelectrolytes. Using a Flory-type approach, we also provide an improved scaling result for the relaxation time in the transition regime close to that found in experiments.

Density fluctuations dispersion relationship for a polymer confined to a nanotube

DNA confined to rigid nanotubes shows density fluctuations around its stretched equilibrium conformation. We report an experimental investigation of the length-scale dependent dynamics of these density fluctuations. We find that for highly elongated molecules a Rouse description is consistent with observations at sufficiently large length scales. We further find that for strongly fluctuating molecules, or short length scales, such Rouse modes cannot be detected due to strong mixing of fluctuation modes.

Single-polymer dynamics under constraints: scaling theory and computer experiment

The relaxation, diffusion and translocation dynamics of single linear polymer chains in confinement is briefly reviewed with emphasis on the comparison between theoretical scaling predictions and observations from experiment or, most frequently, from computer simulations. Besides cylindrical, spherical and slit-like constraints, related problems such as the chain dynamics in a random medium and the translocation dynamics through a nanopore are also considered. Another particular kind of confinement is imposed by polymer adsorption on attractive surfaces or selective interfaces--a short overview of single-chain dynamics is also contained in this survey. While both theory and numerical experiments consider predominantly coarse-grained models of self-avoiding linear chain molecules with typically Rouse dynamics, we also note some recent studies which examine the impact of hydrodynamic interactions on polymer dynamics in confinement. In all of the aforementioned cases we focus mainly on the consequences of imposed geometric restrictions on single-chain dynamics and try to check our degree of understanding by assessing the agreement between theoretical predictions and observations.

Self-avoiding polymer trapped inside a cylindrical pore: Flory free energy and unexpected dynamics

We study the elastic and dynamic behavior of a self-avoiding chain confined inside a cylindrical pore using a Flory-type approach and molecular-dynamics simulations. In the Hookean regime, we find that the effective spring constant of the chain is given by keff approximately N(-1)D(-gamma), where N is the number of monomers and D the diameter of the pore. While the Flory approach reproduces the earlier scaling result gamma=1/3, our simulations confirm a more recent numerical result gamma approximately 0.9 for the computationally accessible regimes. In the absence of hydrodynamic interactions, the relaxation dynamics of a stretched-and-released chain in this regime is characterized by a global relaxation time tauR approximately N2Dgamma with the same exponent gamma for keff. We also discuss how chain relaxation under confinement is influenced by hydrodynamic interactions. In the presence (or absence) of the hydrodynamic interaction, the finite-size effect observed in keff is shown to persist in chain relaxation, resulting in tauR markedly different from previous results.

Dynamics of a self-avoiding polymer chain in slit, tube, and cube confinements

Monte Carlo simulations are presented for the observation of the dynamics of a self-avoiding polymer in three types of confinement: slit, tube, and cube. We pay special attention to the parameter regime where the characteristic confinement dimension is smaller than the radius of gyration of the unconfined polymer. On the basis of the bond-fluctuation model, we measured the rotation time of the end-to-end vector of the polymer, the diffusion time for the center of the polymer to move a distance comparable to the root mean square end-to-end distance, and the looping time for the ends of the polymer to approach each other from an open position. As functions of the confinement width and polymer length, these three time scales are discussed in light of scaling theories.

Compression and stretching of a self-avoiding chain in cylindrical nanopores

Force-induced deformations of a self-avoiding chain confined inside a cylindrical cavity, with diameter D, are probed using molecular dynamics simulations, scaling analysis, and analytical calculations. We obtain and confirm a simple scaling relation -fD approximately R(-9/4) in the strong-compression regime, while for weak deformations, we find fD = -A(R/R0) + B(R/R0)(-2), where A and B are constants, f the external force, and R the chain extension (with R0 its unperturbed value). For a strong stretch, we present a universal, analytical force-extension relation. Our results can be used to analyze the behavior of biomolecules in confinement.

Shapes of semiflexible polymers in confined spaces

Biological macromolecules, living in the confines of a cell, often adopt conformations that are unlikely to occur in free space. In this paper, we investigate the effects of confinement on the shape of a semiflexible chain. Results of Monte Carlo simulations show the existence of a shape transition when the persistence length of the polymer becomes comparable to the dimensions of the box. An order parameter is introduced to quantify this behavior. A simple model is constructed to study the effect of the shape transition on the effective persistence length of the polymer.