Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers

Elastomer Mechanics of Cross-Linked Linear-Ring Polymer Blends

Cross-linking a blend of linear and ring polymers creates a new topology-based dual-network elastomer in which the two components differ significantly in their topology. We use molecular simulations and topological analysis to examine key mechanical properties as functions of ring polymer volume fraction ϕR. For ϕR < ϕR*, where the rings begin to overlap, the network shear modulus G and the maximum stretch ratio λp are weakly dependent on ϕR. For ϕR > ϕR*, entanglements trapped in the network are diluted as the rings overlap, leading to a significant decrease in G and an increase in λp with increasing ϕR. The peak tensile stress, σp, exhibits a maximum around ϕR*, indicating an enhancement of network strength due to the stronger cohesion from the entanglements between linear and ring polymers.

Chain Size and Knots of Ring Polymers in All-Crossing and Intra-Crossing Melts

Using dynamic Monte Carlo simulations based on the bond-fluctuation model, we systematically investigated the size and knots of ring polymers in all-crossing systems and intra-crossing systems. Our results demonstrate that the interchain constraint can increase the knotting probability, but does not alter the scaling relationship between knotting probability and chain length for ring polymers in melts. Having established that, we derived the interchain constraint contribution to the free energy of ring polymers in intra-crossing systems based on the knotting probability and obtained the scaling relationship between the size R and chain length N, i.e., R~N1/6. And, by calculating the mean-squared radius of gyration of ring polymers in intra-crossing systems, we validated these scaling results. Finally, we analyze the size of knotted ring polymers with different types and compare corresponding scaling exponents for size versus chain lengths of ring polymers with different knotting complexities. These results provide fundamental insights into the static properties of ring polymers in melts.

Dynamics of Nanoparticles in Solutions of Semiflexible Ring Polymers

We use hybrid molecular dynamics-multiparticle collision dynamics (MD-MPCD) simulations to investigate the influence of chain stiffness on the transport of nanoparticles (NPs) through solutions of semiflexible ring polymers. The NPs exhibit subdiffusive dynamics on short time scales before transitioning to normal diffusion at longer times. The terminal NP diffusivity decreases with increasing ring stiffness, similar to the behavior observed in solutions of semiflexible linear chains. The NP subdiffusive exponent is found to be strongly correlated with that of the polymer center of mass (COM) for the range of chain stiffnesses examined, which is at odds with the pronounced decoupling of the NP and polymer COM motions previously observed upon increasing the stiffness of linear chains. Our analysis indicates that these marked differences in the intermediate dynamics are rooted in distinct structural changes that emerge with increasing bending stiffness: Stiffer ring polymers adopt increasingly circular conformations and stack into transient tubes. The void space created near the ring centers is occupied by NPs and other polymers, resulting in strong dynamic coupling on short time scales.

Mapping deformation dynamics to composition of topologically-active DNA blends

Blends of circular and linear polymers have fascinated researchers for decades, and the role of topology on their stress response and dynamics remains fervently debated. While linear polymers adopt larger coil sizes and form stronger, more pervasive entanglements than their circular counterparts, threading of circular polymers by linear chains can introduce persistent constraints that dramatically decrease mobility, leading to emergent rheological properties in blends. However, the complex interplay between topology-dependent polymer overlap and threading propensity, along with the large amounts of material required to sample many compositions, has limited the ability to experimentally map stress response to composition with high resolution. Moreover, the role of supercoiling on the response of circular-linear blends remains poorly understood. Here, we leverage in situ enzymatic topological conversion to map the deformation dynamics of DNA blends with over 70 fractions of linear, ring and supercoiled molecules that span the phase space of possible topological compositions. We use OpTiDDM (optical tweezers integrating differential dynamic microscopy) to map strain-induced deformation dynamics to composition, revealing that strain-coupling, quantified by superdiffusive dynamics that are aligned with the strain, is maximized for blends with comparable fractions of ring and linear polymers. Increasing the supercoiled fraction dramatically reduces strain-coupling, while converting rings to linear chains offers more modest coupling reduction. We demonstrate that these results are a direct consequence of the interplay between increasing polymer overlap and decreasing threading probability as circular molecules are converted to linear chains, with a careful balance achieved for blends with ample ring fractions but devoid of supercoiled molecules.

Topological DNA blends exhibit resonant deformation fields and strain propagation dynamics tuned by steric constraints

Understanding how polymers deform in response to local stresses and strains, and how strains propagate from a local disturbance, are grand challenges in wide-ranging fields from materials manufacturing to cell mechanics. These dynamics are particularly complex for blends of polymers of distinct topologies, for which several different species-dependent mechanisms may contribute. Here, we use OpTiDDM (Optical Tweezers integrating Differential Dynamic Microscopy) to elucidate deformation fields and propagation dynamics of binary blends of linear, ring and supercoiled DNA of varying sizes. We reveal robust non-monotonic dependence of strain alignment and superdiffusive transport with strain rate. However, peak alignment and superdiffusivity are surprisingly decoupled, occurring at different strain rates resonant with the distinct relaxation rates of the different topologies. Despite this universal resonance, we find that strain propagation of ring-linear blends is dictated by entanglements while supercoiled-ring blends are governed by Rouse dynamics. Our results capture critical subtleties in propagation and deformation dynamics of topological blends, shedding new light on the governing physics and offering a route towards decoupled tuning of response features. We anticipate our approach to be broadly generalizable to mapping the deformation dynamics of polymer blends, with an eye towards bottom-up bespoke materials design. STATEMENT OF SIGNIFICANCE: In biology and in manufacturing, biomaterials are often subject to localized and spatially nonuniform strains and stresses. Yet, understanding the extent to which strains are absorbed, distributed, or propagated across different spatiotemporal scales remains a grand challenge. Here, we combine optical tweezers with differential dynamic microscopy to elucidate deformation fields and propagation dynamics of blends of linear, ring and supercoiled DNA, revealing robust non-monotonic trends and decoupling of strain alignment and superdiffusivity, and capturing critical subtleties in propagation and deformation dynamics. Our results, shedding important new physical insight to guide decoupled tuning of response features, may be leveraged to map the deformation dynamics of wide-ranging systems of biopolymers and other macromolecules, with an eye towards bottom-up bespoke biomaterials design.

Measuring Topological Constraint Relaxation in Ring-Linear Polymer Blends

Polymers are an effective test bed for studying topological constraints in condensed matter due to a wide array of synthetically available chain topologies. When linear and ring polymers are blended together, emergent rheological properties are observed as the blend can be more viscous than either of the individual components. This emergent behavior arises since ring-linear blends can form long-lived topological constraints as the linear polymers thread the ring polymers. Here, we demonstrate how the Gauss linking integral can be used to efficiently evaluate the relaxation of topological constraints in ring-linear polymer blends. For majority-linear blends, the relaxation rate of topological constraints depends primarily on reptation of the linear polymers, resulting in the diffusive time τ_{d,R} for rings of length N_{R} blended with linear chains of length N_{l} to scale as τ_{d,R}∼N_{R}^{2}N_{L}^{3.4}.

From Bioreactor to Bulk Rheology: Achieving Scalable Production of Highly Concentrated Circular DNA

DNA serves as a model system in polymer physics due to its ability to be obtained as a uniform polymer with controllable topology and nonequilibrium behavior. Currently, a major obstacle in the widespread adoption of DNA is obtaining it on a scale and cost basis that accommodates bulk rheology and high-throughput screening. To address this, recent advancements in bioreactor-based plasmid DNA production is coupled with anion exchange chromatography producing a unified approach to generating gram-scale quantities of monodisperse DNA. With this method, 1.1 grams of DNA is obtained per batch to generate solutions with concentrations up to 116 mg mL-1. This solution of uniform supercoiled and relaxed circular plasmid DNA, is roughly 69 times greater than the overlap concentration. The utility of this method is demonstrated by performing bulk rheology measurements at sample volumes up to 1 mL on DNA of different lengths, topologies, and concentrations. The measured elastic moduli are orders of magnitude larger than those previously reported for DNA and allowed for the construction of a time-concentration superposition curve that spans 12 decades of frequency. Ultimately, these results can provide important insights into the dynamics of ring polymers and the nature of highly condensed DNA dynamics.

Unified understanding of the impact of semiflexibility, concentration, and molecular weight on macromolecular-scale ring diffusion

Conformationally fluctuating, globally compact macromolecules such as polymeric rings, single-chain nanoparticles, microgels, and many-arm stars display complex dynamic behaviors due to their rich topological structure and intermolecular organization. Synthetic rings are hybrid objects with conformations that display both ideal random walk and compact globular features, which can serve as models of genomic DNA. To date, emphasis has been placed on the effect of ring molecular weight on their unusual behaviors. Here, we combine simulations and a microscopic force-level theory to build a unified understanding for how key aspects of ring dynamics depend on different tunable molecular properties including backbone rigidity, monomer concentration, degree of traditional entanglement, and molecular weight. Our large-scale molecular dynamics simulations of ring melts with very different backbone stiffnesses reveal unanticipated behaviors which agree well with our generalized theory. This includes a universal master curve for center-of-mass diffusion constants as a function of molecular weight scaled by a chemistry and thermodynamic state-dependent critical molecular weight that generalizes the concept of an entanglement cross-over for linear chains. The key physics is how backbone rigidity and monomer concentration induced changes of the entanglement length, interring packing, degree of interpenetration, and liquid compressibility slow down space-time dynamic-force correlations on macromolecular scales. A power law decay of the center-of-mass diffusion constant with inverse molecular weight squared is the first consequence, followed by an ultraslow activated hopping transport regime. Our results set the stage to address slow dynamics and kinetic arrest in different families of compact synthetic and biological polymeric systems.

Unified Understanding of the Structure, Thermodynamics, and Diffusion of Single-Chain Nanoparticle Fluids

Single-chain nanoparticles (SCNPs) are a fascinating class of soft nano-objects with promising properties and relevance to protein condensates, polymer nanocomposites, nanomedicine, bioimaging, catalysis, and drug delivery. We combine molecular dynamics simulations and equilibrium and time-dependent statistical mechanical theory to construct a unified understanding of how the internal conformational structure of SCNPs, of both a simple fractal globule-like form and more complex objects with multiple internal intermediate length scales, determines nm-scale intermolecular packing correlations, thermodynamic properties, and center-of-mass diffusion over a wide range of concentrations up to dense melts. The intermolecular pair correlations generically exhibit a distinctive deep correlation hole form due to SCNP internal connectivity structure and repulsive interparticle interactions associated with a globular-like conformation on the macromolecular scale, with concentration-dependent deviations at small separations. Unanticipated exponential-like dependences of the equation-of-state, osmotic compressibility, and center-of-mass diffusion constant on SCNP macromolecular packing fraction are theoretically predicted and confirmed via simulations. System-specific behaviors are found associated with SCNP internal structure, but overarching regularities are identified and understood based on a generalized effective globule conformation on macromolecular scales. Diffusivity slows down by 2-3 decades with increasing concentration and is understood as a consequence of a nonactivated excluded volume-driven weak-caging process associated with space-time correlated intermolecular forces experienced by the SCNP. Good agreement between the theory and simulations is established, testable predictions are made, and a quantitative comparison with viscosity measurements on a specific SCNP fluid is carried out. The basic theoretical approach can potentially be extended to treat the chemical and physical consequences of varying the structure of other classes of soft nanoparticles with distinctive internal nanoscale organization relevant in nanotechnology and nanomedicine, and the possible emergence of macromolecular kinetically arrested glasses.

Activity-driven chromatin organization during interphase: Compaction, segregation, and entanglement suppression

In mammalian cells, the cohesin protein complex is believed to translocate along chromatin during interphase to form dynamic loops through a process called active loop extrusion. Chromosome conformation capture and imaging experiments have suggested that chromatin adopts a compact structure with limited interpenetration between chromosomes and between chromosomal sections. We developed a theory demonstrating that active loop extrusion causes the apparent fractal dimension of chromatin to cross-over between two and four at contour lengths on the order of 30 kilo-base pairs. The anomalously high fractal dimension [Formula: see text] is due to the inability of extruded loops to fully relax during active extrusion. Compaction on longer contour length scales extends within topologically associated domains (TADs), facilitating gene regulation by distal elements. Extrusion-induced compaction segregates TADs such that overlaps between TADs are reduced to less than 35% and increases the entanglement strand of chromatin by up to a factor of 50 to several Mega-base pairs. Furthermore, active loop extrusion couples cohesin motion to chromatin conformations formed by previously extruding cohesins and causes the mean square displacement of chromatin loci during lag times ([Formula: see text]) longer than tens of minutes to be proportional to [Formula: see text]. We validate our results with hybrid molecular dynamics-Monte Carlo simulations and show that our theory is consistent with experimental data. This work provides a theoretical basis for the compact organization of interphase chromatin, explaining the physical reason for TAD segregation and suppression of chromatin entanglements which contribute to efficient gene regulation.

Polymer physics view of peripheral chromatin: de Gennes' self-similar carpet

Using scaling arguments to model peripheral chromatin localized near the inner surface of the nuclear envelope (NE) as a flexible polymer chain, we discuss the structural properties of the peripheral chromatin composed of alternating lamin-associated domains (LADs) and inter-LADs. Modeling the attraction of LADs to NE by de Gennes' self-similar carpet, which treats the chromatin layer as a polymer fractal, explains two major experimental observations. (i) The high density of chromatin close to the nuclear periphery decays to a constant density as the distance to the periphery increases. (ii) Due to the decreasing mesh size towards the nuclear periphery, the chromatin carpet inside NE excludes molecules (via nonspecific interactions) above a threshold size that depends on the distance from the nuclear periphery.

Supercoiled ring polymers under shear flow

We apply monomer-resolved computer simulations of supercoiled ring polymers under shear, taking full account of the hydrodynamic interactions, accompanied, in parallel, by simulations in which these are switched off. The combination of bending and torsional rigidities inherent in these polymers, in conjunction with hydrodynamics, has a profound impact on their flow properties. In contrast to their flexible counterparts, which dramatically deform and inflate under shear [Liebetreu et al., Commun. Mater. 2020, 1, 4], supercoiled rings undergo only weak changes in their overall shape and they display both a reduced propensity to tumbling (at fixed Weissenberg number) and a much stronger orientational resistance with respect to their flexible counterparts. In the presence of hydrodynamic interactions, the coupling of the polymer to solvent flow is capable of bringing about a topological transformation of writhe to twist at strong shear upon conservation of the overall linking number.

Mitotic chromosomes are self-entangled and disentangle through a topoisomerase-II-dependent two-stage exit from mitosis

The topological state of chromosomes determines their mechanical properties, dynamics, and function. Recent work indicated that interphase chromosomes are largely free of entanglements. Here, we use Hi-C, polymer simulations, and multi-contact 3C and find that, by contrast, mitotic chromosomes are self-entangled. We explore how a mitotic self-entangled state is converted into an unentangled interphase state during mitotic exit. Most mitotic entanglements are removed during anaphase/telophase, with remaining ones removed during early G1, in a topoisomerase-II-dependent process. Polymer models suggest a two-stage disentanglement pathway: first, decondensation of mitotic chromosomes with remaining condensin loops produces entropic forces that bias topoisomerase II activity toward decatenation. At the second stage, the loops are released, and the formation of new entanglements is prevented by lower topoisomerase II activity, allowing the establishment of unentangled and territorial G1 chromosomes. When mitotic entanglements are not removed in experiments and models, a normal interphase state cannot be acquired.

Enzymatic cleaving of entangled DNA rings drives scale-dependent rheological trajectories

DNA, which naturally occurs in linear, ring, and supercoiled topologies, frequently undergoes enzyme-driven topological conversion and fragmentation in vivo, enabling it to perform a variety of functions within the cell. In vitro, highly concentrated DNA polymers form entanglements that yield viscoelastic properties dependent on the topologies and lengths of the DNA. Enzyme-driven alterations of DNA size and shape therefore offer a means of designing active materials with programmable viscoelastic properties. Here, we incorporate multi-site restriction endonucleases into dense DNA solutions to linearize and fragment circular DNA molecules. We pair optical tweezers microrheology with differential dynamic microscopy and single-molecule tracking to measure the linear and nonlinear viscoelastic response and transport properties of entangled DNA solutions over a wide range of spatiotemporal scales throughout the course of enzymatic digestion. We show that, at short timescales, relative to the relaxation timescales of the polymers, digestion of these 'topologically-active' fluids initially causes an increase in elasticity and relaxation times followed by a gradual decrease. Conversely, for long timescales, linear viscoelastic moduli exhibit signatures of increasing elasticity. DNA diffusion, likewise, becomes increasingly slowed, in direct opposition to the short-time behavior. We hypothesize that this scale-dependent rheology arises from the population of small DNA fragments, which increases as digestion proceeds, driving self-association of larger fragments via depletion interactions, giving rise to slow relaxation modes of clusters of entangled chains, interspersed among shorter unentangled fragments. While these slow modes likely dominate at long times, they are presumably frozen out in the short-time limit, which instead probes the faster relaxation modes of the unentangled population.

Activity-driven chromatin organization during interphase: compaction, segregation, and entanglement suppression

In mammalian cells, the cohesin protein complex is believed to translocate along chromatin during interphase to form dynamic loops through a process called active loop extrusion. Chromosome conformation capture and imaging experiments have suggested that chromatin adopts a compact structure with limited interpenetration between chromosomes and between chromosomal sections. We developed a theory demonstrating that active loop extrusion causes the apparent fractal dimension of chromatin to cross over between two and four at contour lengths on the order of 30 kilo-base pairs (kbp). The anomalously high fractal dimension D = 4 is due to the inability of extruded loops to fully relax during active extrusion. Compaction on longer contour length scales extends within topologically associated domains (TADs), facilitating gene regulation by distal elements. Extrusion-induced compaction segregates TADs such that overlaps between TADs are reduced to less than 35% and increases the entanglement strand of chromatin by up to a factor of 50 to several Mega-base pairs. Furthermore, active loop extrusion couples cohesin motion to chromatin conformations formed by previously extruding cohesins and causes the mean square displacement of chromatin loci during lag times ( Δ t ) longer than tens of minutes to be proportional to Δ t 1 / 3 . We validate our results with hybrid molecular dynamics - Monte Carlo simulations and show that our theory is consistent with experimental data. This work provides a theoretical basis for the compact organization of interphase chromatin, explaining the physical reason for TAD segregation and suppression of chromatin entanglements which contribute to efficient gene regulation.

Topology-Based Detection and Tracking of Deadlocks Reveal Aging of Active Ring Melts

Connecting the viscoelastic behavior of stressed ring melts to the various forms of entanglement that can emerge in such systems is still an open challenge. Here, we consider active ring melts, where stress is generated internally, and introduce a topology-based method to detect and track consequential forms of ring entanglements, namely, deadlocks. We demonstrate that, as stress accumulates, more and more rings are co-opted in a growing web of deadlocks that entrap many other rings by threading, bringing the system to a standstill. The method ought to help the study of topological aging in more general polymer contexts.

Onset of glassiness in two-dimensional ring polymers: Interplay of stiffness and crowding

The effect of ring stiffness and pressure on the glassy dynamics of a thermal assembly of two-dimensional ring polymers is investigated using extensive coarse-grained molecular dynamics simulations. In all cases, dynamical slowing down is observed with increasing pressure, and thereby, a phase space for equilibrium dynamics is identified in the plane of the obtained monomer density and ring stiffness. When the rings are highly flexible, i.e., have low ring stiffness, glassiness sets in via the crowding of crumpled polymers, which take on a globular form. In contrast, at large ring stiffness, when the rings tend to have large asphericity under compaction, we observe the emergence of local domains having orientational ordering at high pressures. Therefore, our simulations highlight how varying the deformability of rings leads to contrasting mechanisms in driving the system toward the glassy regime.

Unraveling the Glass-like Dynamic Heterogeneity in Ring Polymer Melts: From Semiflexible to Stiff Chain

Ring polymers are an intriguing class of polymers with unique physical properties, and understanding their behavior is important for developing accurate theoretical models. In this study, we investigate the effect of chain stiffness and monomer density on the static and dynamic behaviors of ring polymer melts using molecular dynamics simulations. Our first focus is on the non-Gaussian parameter of center-of-mass displacement as a measure of dynamic heterogeneity, which is commonly observed in glass-forming liquids. We find that the non-Gaussianity in the displacement distribution increases with the monomer density and stiffness of the polymer chains, suggesting that excluded volume interactions between centers of mass have a strong effect on the dynamics of ring polymers. We then analyze the relationship between the radius of gyration and monomer density for semiflexible and stiff ring polymers. Our results indicate that the relationship between the two varies with chain stiffness, which can be attributed to the competition between repulsive forces inside the ring and from adjacent rings. Finally, we study the dynamics of bond-breakage virtually connected between the centers of mass of rings to analyze the exchanges of intermolecular networks of bonds. Our results demonstrate that the dynamic heterogeneity of bond-breakage is coupled with the non-Gaussianity in ring polymer melts, highlighting the importance of the bond-breaking method in determining the intermolecular dynamics of ring polymer melts. Overall, our study sheds light on the factors that govern the dynamic behaviors of ring polymers.

Cluster Formation in Solutions of Polyelectrolyte Rings

We use molecular dynamics simulations to explore concentrated solutions of semiflexible polyelectrolyte ring polymers, akin to the DNA mini-circles, with counterions of different valences. We find that the assembly of rings into nanoscopic cylindrical stacks is a generic feature of the systems, but the morphology and dynamics of such a cluster can be steered by the counterion conditions. In general, a small addition of trivalent ions can stabilize the emergence of clusters due to the counterion condensation, which mitigates the repulsion between the like-charged rings. Stoichiometric addition of trivalent ions can even lead to phase separation of the polyelectrolyte ring phase due to the ion-bridging effects promoting otherwise entropically driven clustering. On the other hand, monovalent counterions cause the formation of stacks to be re-entrant with density. The clusters are stable within a certain window of concentration, while above the window the polyelectrolytes undergo an osmotic collapse, disfavoring ordering. The cluster phase exhibits characteristic cluster glass dynamics with arrest of collective degrees of freedom but not the self-ones. On the other hand, the collapsed phase shows arrest on both the collective and single level, suggesting an incipient glass-to-glass transition, from a cluster glass of ring clusters to a simple glass of rings.

Cooperative Rheological State-Switching of Enzymatically-Driven Composites of Circular DNA And Dextran

Polymer topology, which plays a principal role in the rheology of polymeric fluids, and non-equilibrium materials, which exhibit time-varying rheological properties, are topics of intense investigation. Here, composites of circular DNA and dextran are pushed out-of-equilibrium via enzymatic digestion of DNA rings to linear fragments. These time-resolved rheology measurements reveal discrete state-switching, with composites undergoing abrupt transitions between dissipative and elastic-like states. The gating time and lifetime of the elastic-like states, and the magnitude and sharpness of the transitions, are surprisingly decorrelated from digestion rates and non-monotonically depend on the DNA fraction. These results are modeled using sigmoidal two-state functions to show that bulk state-switching can arise from continuous molecular-level activity due to the necessity for cooperative percolation of entanglements to support macroscopic stresses. This platform, coupling the tunability of topological composites with the power of enzymatic reactions, may be leveraged for diverse material applications from wound-healing to self-repairing infrastructure.

Mixing Linear Polymers with Rings and Catenanes: Bulk and Interfacial Behavior

We derive and parameterize effective interaction potentials between a multitude of different types of ring polymers and linear chains, varying the bending rigidity and solvent quality for the former species. We further develop and apply a density functional treatment for mixtures of both disconnected (chain-ring) and connected (chain-polycatenane) mixtures of the same, drawing coexistence binodals and exploring the ensuing response functions as well as the interface and wetting behavior of the mixtures. We show that worsening of the solvent quality for the rings brings about a stronger propensity for macroscopic phase separation in the linear-polycatenane mixtures, which is predominantly of the demixing type between phases of similar overall particle density. We formulate a simple criterion based on the effective interactions, allowing us to determine whether any specific linear-ring mixture will undergo a demixing phase separation.

Crumpled polymer with loops recapitulates key features of chromosome organization

Chromosomes are exceedingly long topologically-constrained polymers compacted in a cell nucleus. We recently suggested that chromosomes are organized into loops by an active process of loop extrusion. Yet loops remain elusive to direct observations in living cells; detection and characterization of myriads of such loops is a major challenge. The lack of a tractable physical model of a polymer folded into loops limits our ability to interpret experimental data and detect loops. Here, we introduce a new physical model - a polymer folded into a sequence of loops, and solve it analytically. Our model and a simple geometrical argument show how loops affect statistics of contacts in a polymer across different scales, explaining universally observed shapes of the contact probability. Moreover, we reveal that folding into loops reduces the density of topological entanglements, a novel phenomenon we refer as "the dilution of entanglements". Supported by simulations this finding suggests that up to ~ 1 - 2Mb chromosomes with loops are not topologically constrained, yet become crumpled at larger scales. Our theoretical framework allows inference of loop characteristics, draws a new picture of chromosome organization, and shows how folding into loops affects topological properties of crumpled polymers.

Unexpected Slow Relaxation Dynamics in Pure Ring Polymers Arise from Intermolecular Interactions

Ring polymers have fascinated scientists for decades, but experimental progress has been challenging due to the presence of linear chain contaminants that fundamentally alter dynamics. In this work, we report the unexpected slow stress relaxation behavior of concentrated ring polymers that arises due to ring-ring interactions and ring packing structure. Topologically pure, high molecular weight ring polymers are prepared without linear chain contaminants using cyclic poly(phthalaldehyde) (cPPA), a metastable polymer chemistry that rapidly depolymerizes from free ends at ambient temperatures. Linear viscoelastic measurements of highly concentrated cPPA show slow, non-power-law stress relaxation dynamics despite the lack of linear chain contaminants. Experiments are complemented by molecular dynamics (MD) simulations of unprecedentedly high molecular weight rings, which clearly show non-power-law stress relaxation in good agreement with experiments. MD simulations reveal substantial ring-ring interpenetrations upon increasing ring molecular weight or local backbone stiffness, despite the global collapsed nature of single ring conformation. A recently proposed microscopic theory for unconcatenated rings provides a qualitative physical mechanism associated with the emergence of strong inter-ring caging which slows down center-of-mass diffusion and long wavelength intramolecular relaxation modes originating from ring-ring interpenetrations, governed by the onset variable N/ND, where the crossover degree of polymerization ND is qualitatively predicted by theory. Our work overcomes challenges in achieving ring polymer purity and by characterizing dynamics for high molecular weight ring polymers. Overall, these results provide a new understanding of ring polymer physics.

Theory of chromatin organization maintained by active loop extrusion

The active loop extrusion hypothesis proposes that chromatin threads through the cohesin protein complex into progressively larger loops until reaching specific boundary elements. We build upon this hypothesis and develop an analytical theory for active loop extrusion which predicts that loop formation probability is a nonmonotonic function of loop length and describes chromatin contact probabilities. We validate our model with Monte Carlo and hybrid Molecular Dynamics-Monte Carlo simulations and demonstrate that our theory recapitulates experimental chromatin conformation capture data. Our results support active loop extrusion as a mechanism for chromatin organization and provide an analytical description of chromatin organization that may be used to specifically modify chromatin contact probabilities.

Rheological Role of Stiff Nanorings on Concurrently Strengthening and Toughening Polymer Nanocomposites

Nanorings, which are increasingly uncovered in natural systems and synthesized in man-made materials, exhibit dynamics distinct from those known for linear chains. We show in this study that, when immersed in a polymer melt matrix, segments of a stiff nanoring (SNR) have more facilitated subdiffusion, i.e., with a larger scaling exponent in the mean squared displacement, than those belonging to one flexible counterpart, while the whole SNR is more suppressed by its surroundings. It is revealed that adding SNRs contributes to achieving the long-anticipated rheological objective of sol- and gel-like characteristics at high and low shearing frequencies, respectively. This study suggests the promising prospect of exploiting SNRs to concurrently strengthen and toughen target polymer nanocomposites.

To thread or not to thread? Effective potentials and threading interactions between asymmetric ring polymers

We use computer simulations to study a system of two unlinked ring polymers, whose length and bending stiffness are systematically varied. We derive the effective potentials between the rings, calculate the areas of minimal surfaces of the same, and characterize the threading between them. When the two rings are of the same kind, threading of a one ring through the surface of the other is immanent for small ring-ring separations. Flexible rings pierce the surface of the other ring several times but only shallowly, as compared to the stiff rings which pierce less frequently but deeply. Typically, the ring that is being threaded swells and flattens up into an oblate-like conformation, while the ring that is threading the other takes a shape of an elongated prolate. The roles of the threader and the threaded ring are being dynamically exchanged. If, on the other hand, the rings are of different kinds, the symmetry is broken and the rings tend to take up roles of the threader and the threaded ring with unequal probabilities. We propose a method how to predict these probabilities based on the parameters of the individual rings. Ultimately, our work captures the interactions between ring polymers in a coarse-grained fashion, opening the way to large-scale modelling of materials such as kinetoplasts, catenanes or topological brushes.

Entanglement Length Scale Separates Threading from Branching of Unknotted and Non-concatenated Ring Polymers in Melts

Current theories on the conformation and dynamics of unknotted and non-concatenated ring polymers in melt conditions describe each ring as a tree-like double-folded object. While evidence from simulations supports this picture on a single ring level, other works show pairs of rings also thread each other, a feature overlooked in the tree theories. Here we reconcile this dichotomy using Monte Carlo simulations of the ring melts with different bending rigidities. We find that rings are double-folded (more strongly for stiffer rings) on and above the entanglement length scale, while the threadings are localized on smaller scales. The different theories disagree on the details of the tree structure, i.e., the fractal dimension of the backbone of the tree. In the stiffer melts we find an indication of a self-avoiding scaling of the backbone, while more flexible chains do not exhibit such a regime. Moreover, the theories commonly neglect threadings and assign different importance to the impact of the progressive constraint release (tube dilation) on single ring relaxation due to the motion of other rings. Despite that each threading creates only a small opening in the double-folded structure, the threading loops can be numerous and their length can exceed substantially the entanglement scale. We link the threading constraints to the divergence of the relaxation time of a ring, if the tube dilation is hindered by pinning a fraction of other rings in space. Current theories do not predict such divergence and predict faster than measured diffusion of rings, pointing at the relevance of the threading constraints in unpinned systems as well. Revision of the theories with explicit threading constraints might elucidate the validity of the conjectured existence of topological glass.

Entropic Mixing of Ring/Linear Polymer Blends

The topological constraints of nonconcatenated ring polymers force them to form compact loopy globular conformations with much lower entropy than unconstrained ideal rings. The closed-loop structure of ring polymers also enables them to be threaded by linear polymers in ring/linear blends, resulting in less compact ring conformations with higher entropy. This conformational entropy increase promotes mixing rings with linear polymers. Here, using molecular dynamics simulations for bead-spring chains, ring/linear blends are shown to be significantly more miscible than linear/linear blends and that there is an entropic mixing, negative χ, for ring/linear blends compared to linear/linear and ring/ring blends. In analogy with small angle neutron scattering, the static structure function S(q) is measured, and the resulting data are fit to the random phase approximation model to determine χ. In the limit that the two components are the same, χ = 0 for the linear/linear and ring/ring blends as expected, while χ < 0 for the ring/linear blends. With increasing chain stiffness, χ for the ring/linear blends becomes more negative, varying reciprocally with the number of monomers between entanglements. Ring/linear blends are also shown to be more miscible than either ring/ring or linear/linear blends and stay in single phase for a wider range of increasing repulsion between the two components.

Random Knotting in Fractal Ring Polymers

Many ring polymer systems of physical and biological interest exhibit both pronounced topological effects and nontrivial self-similarity, but the relationship between these two phenomena has not yet been clearly established. Here, we use theory and simulation to formulate such a connection by studying a fundamental topological property—the random knotting probability—for ring polymers with varying fractal dimension, d_f. Using straightforward scaling arguments, we generalize a classic mathematical result, showing that the probability of a trivial knot decays exponentially with chain size, N, for all fractal dimensions: P₀(N) ∝ exp(−N/N₀). However, no such simple considerations can account for the dependence of the knotting length, N₀, on d_f, necessitating a more involved analytical calculation. This analysis reveals a complicated double-exponential dependence, which is well supported by numerical data. By contrast, functional forms typical of simple scaling theories fail to adequately describe the observations. These findings are equally valid for two-dimensional ring polymer systems, where “knotting” is defined as the intersection of any two segments.

Geometric Predictors of Knotted and Linked Arcs

Inspired by how certain proteins “sense” knots and entanglements in DNA molecules, here, we ask if local geometric features that may be used as a readout of the underlying topology of generic polymers exist. We perform molecular simulations of knotted and linked semiflexible polymers and study four geometric measures to predict topological entanglements: local curvature, local density, local 1D writhe, and nonlocal 3D writhe. We discover that local curvature is a poor predictor of entanglements. In contrast, segments with maximum local density or writhe correlate as much as 90% of the time with the shortest knotted and linked arcs. We find that this accuracy is preserved across different knot types and also under significant spherical confinement, which is known to delocalize essential crossings in knotted polymers. We further discover that nonlocal 3D writhe is the best geometric readout of the knot location. Finally, we discuss how these geometric features may be used to computationally analyze entanglements in generic polymer melts and gels.

Superstretchable Elastomer from Cross-linked Ring Polymers

The stretchability of polymeric materials is critical to many applications such as flexible electronics and soft robotics, yet the stretchability of conventional cross-linked linear polymers is limited by the entanglements between polymer chains. We show using molecular dynamics simulations that cross-linked ring polymers are significantly more stretchable than cross-linked linear polymers. Compared to linear polymers, the entanglements between ring polymers do not act as effective cross-links. As a result, the stretchability of cross-linked ring polymers is determined by the maximum extension of polymer strands between cross-links, rather than between trapped entanglements as in cross-linked linear polymers. The more compact conformation of ring polymers before deformation also contributes to the increase in stretchability.

Formulation of Chromatin Mobility as a Function of Nuclear Size during C. elegans Embryogenesis Using Polymer Physics Theories

During early embryogenesis of the nematode, Caenorhabditis elegans, the chromatin motion markedly decreases. Despite its biological implications, the underlying mechanism for this transition was unclear. By combining theory and experiment, we analyze the mean-square displacement (MSD) of the chromatin loci, and demonstrate that MSD-vs-time relationships in various nuclei collapse into a single master curve by normalizing them with the mesh size and the corresponding time scale. This enables us to identify the onset of the entangled dynamics with the size of tube diameter of chromatin polymer in the C. elegans embryo. Our dynamical scaling analysis predicts the transition between unentangled and entangled dynamics of chromatin polymers, the quantitative formula for MSD as a function of nuclear size and timescale, and provides testable hypotheses on chromatin mobility in other cell types and species.

Effect of ring stiffness and ambient pressure on the dynamical slowdown in ring polymers

Using extensive molecular dynamics simulations, we investigate the slowdown of dynamics in a 3D system of ring polymers by varying the ambient pressure and the stiffness of the rings. Our study demonstrates that the stiffness of the rings determines the dynamics of the ring polymers, leading to glassiness at lower pressures for stiffer rings. The threading of the ring polymers, a unique feature that emerges only due to the topological nature of such polymers in three dimensions, is shown to be the determinant feature of dynamical slowdown, albeit only in a certain stiffness range. Our results suggest a possible framework for exploring the phase space spanned by ring stiffness and pressure to obtain spontaneously emerging topologically constrained polymer glasses.

Computer simulations of melts of ring polymers with nonconserved topology: A dynamic Monte Carlo lattice model

We present computer simulations of a dynamic Monte Carlo algorithm for polymer chains on a fcc lattice which explicitly takes into account the possibility to overcome topological constraints by controlling the rate at which nearby polymer strands may cross through each other. By applying the method to systems of interacting ring polymers at melt conditions, we characterize their structure and dynamics by measuring, in particular, the amounts of knots and links which are formed during the relaxation process. In comparison with standard melts of unknotted and unconcatenated rings, our simulations demonstrate that the mechanism of strand crossing makes polymer dynamics faster provided the characteristic timescale of the process is smaller than the typical timescale for chain relaxation in the unperturbed state, in agreement with recent experiments employing solutions of DNA rings in the presence of the type II topoisomerase enzyme. In the opposite case of slow rates the melt is shown to become slower, and this prediction may be easily validated experimentally.

Theory of Transient Localization, Activated Dynamics, and a Macromolecular Glass Transition in Ring Polymer Liquids

We construct a segmental scale force level theory for the center-of-mass diffusion constant and corresponding relaxation time for globally compact unconcatenated ring polymer solutions and melts (degree of polymerization N). The approach is based on slowly decaying macromolecular scale intermolecular force dynamic correlations as the origin of their unusual dynamics. Unentangled Rouse, weakly caged, and activated regimes are predicted. The barrier of the activated regime scales linearly with N and as a power law of concentration, which drives a kinetic glass transition on the radius-of-gyration scale. The values of N at the two dynamic crossovers (Rouse to weakly caged, weakly caged to activated) are proportional, with nonuniversality entering mainly via macromolecular volume fraction and dimensionless compressibility. Quantitative comparisons with simulation data reveal good agreement. Aspects of intermediate time dynamics are analyzed, and predictions are made for the conditions required to observe a macromolecular glass transition in the laboratory and on the computer.

Monte Carlo simulation of a lattice model for the dynamics of randomly branching double-folded ring polymers

Supercoiled DNA, crumpled interphase chromosomes, and topologically constrained ring polymers often adopt treelike, double-folded, randomly branching configurations. Here we study an elastic lattice model for tightly double-folded ring polymers, which allows for the spontaneous creation and deletion of side branches coupled to a diffusive mass transport, which is local both in space and on the connectivity graph of the tree. We use Monte Carlo simulations to study systems falling into three different universality classes: ideal double-folded rings without excluded volume interactions, self-avoiding double-folded rings, and double-folded rings in the melt state. The observed static properties are in good agreement with exact results, simulations, and predictions of Flory theory for randomly branching polymers. For example, in the melt state rings adopt compact configurations and exhibit territorial behavior. In particular, we show that the emergent dynamics is in excellent agreement with a recent scaling theory and illustrate the qualitative differences with the familiar reptation dynamics of linear chains.

Nonlinear rheometry of entangled polymeric rings and ring-linear blends

We present a comprehensive experimental rheological dataset for purified entangled ring polystyrenes and their blends with linear chains in nonlinear shear and elongation. In particular, data for shear stress growth coefficient, steady-state shear viscosity, and first and second normal stress differences are obtained and discussed as functions of shear rate as well as molecular parameters (molar mass, blend composition and decreasing molar mass of linear component in blend). Over the extended parameter range investigated, rings do not exhibit clear transient undershoot in shear, in contrast to their linear counterparts and ring-linear blends. For the latter, the size of the undershoot and respective strain appear to increase with shear rate. Universal scaling of strain at overshoot and fractional overshoot (ratio of maximum to steady-state shear stress growth coefficient) indicates subtle differences in the shear-rate dependence between rings and linear polymers or their blends. The shear thinning behaviour of pure rings yields a slope nearly identical to predictions (-4/7) of a recent shear slit model and molecular dynamics simulations. Data for the second normal stress difference are reported for rings and ring-linear blends. While N 2 is negative and its absolute value stays below that of N 1 , as for linear polymers, the ratio -N 2 /N 1 is unambiguously larger for rings compared to linear polymer solutions with the same number of entanglements (almost by factor of two), in agreement with recent non-equilibrium molecular dynamics simulations. Further, -N 2 exhibits slightly weaker shear rate dependence compared to N 1 at high rates, and the respective power-law exponents can be rationalized in view of the slit model (3/7) and simulations (0.6), although further work is needed to unravel the molecular original of the observed behaviour. The comparison of shear and elongational stress growth coefficients for blends reflects the effect of ring-linear threading which leads to significant viscosity enhancement in elongation. Along the same lines, the elongational stress is much larger than the first normal stress in shear, and their ratio is much larger for rings and ring-linear blends compared to linear polymers. This conforms the interlocking scenario of rings and their important role in mechanically reinforcing linear matrices.