Effect of molecular architecture on ring polymer dynamics in semidilute linear polymer solutions

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.

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.

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.

Adherent Moving of Polymers in Spherical Confined Binary Semiflexible Ring Polymer Mixtures

Based on the coarse-grained model, we used molecular dynamics methods to calculate and simulate a semiflexible long ring–semiflexible short ring blended polymer system confined in a hard sphere. We systematically studied the distribution and motion characteristics of the long ring chain. The results show that when the short ring is short enough (Lshort < 20), the long ring (Llong = 50) is separated from the blend system and then distributed against the inner wall. As the length of the short ring increases (Lshort ≥ 20), the long ring can no longer be separated from the blending system. Moreover, we found that the long ring demonstrates a random direction of adherent walking behavior on the inner surface of the hard sphere. The velocity of the long ring decreases with the increase in the short ring length Lshort. Specifically for Lshort ≥ 20, the system does not undergo phase separation and the speed of the long ring decreases sharply along with the long ring distributed inside the confined bulk. This is related to the inner wall layer moving faster than the inside bulk of the restricted system. Our simulation results can help us to understand the distribution of macromolecules in biological systems in confined systems, including the restricted chromosome partitioning distribution and packing structure of circular DNA molecules.

Conformation Dynamics of Single Polymer Strands in Solution

Conformational changes in macromolecules significantly affect their functions and assembly into high-level structures. Despite advances in theoretical and experimental studies, investigations into the intrinsic conformational variations and dynamic motions of single macromolecules remain challenging. Here, liquid-phase transmission electron microscopy enables the real-time tracking of single-chain polymers. Imaging linear polymers, synthetically dendronized with conjugated aromatic groups, in organic solvent confined within graphene liquid cells, directly exhibits chain-resolved conformational dynamics of individual semiflexible polymers. These experimental and theoretical analyses reveal that the dynamic conformational transitions of the single-chain polymer originate from the degree of intrachain interactions. In situ observations also show that such dynamics of the single-chain polymer are significantly affected by environmental factors, including surfaces and interfaces.

Topological digestion drives time-varying rheology of entangled DNA fluids

Understanding and controlling the rheology of polymeric complex fluids that are pushed out-of-equilibrium is a fundamental problem in both industry and biology. For example, to package, repair, and replicate DNA, cells use enzymes to constantly manipulate DNA topology, length, and structure. Inspired by this feat, here we engineer and study DNA-based complex fluids that undergo enzymatically-driven topological and architectural alterations via restriction endonuclease (RE) reactions. We show that these systems display time-dependent rheological properties that depend on the concentrations and properties of the comprising DNA and REs. Through time-resolved microrheology experiments and Brownian Dynamics simulations, we show that conversion of supercoiled to linear DNA topology leads to a monotonic increase in viscosity. On the other hand, the viscosity of entangled linear DNA undergoing fragmentation displays a universal decrease that we rationalise using living polymer theory. Finally, to showcase the tunability of these behaviours, we design a DNA fluid that exhibits a time-dependent increase, followed by a temporally-gated decrease, of its viscosity. Our results present a class of polymeric fluids that leverage naturally occurring enzymes to drive diverse time-varying rheology by performing architectural alterations to the constituents.

Understanding and controlling the rheology of polymeric complex fluids is of fundamental importance in both industry and biology. Here, Michieletto et al. show how to achieve time-dependent rheology of DNA solutions via enzymatically-driven architectural alterations by restriction endonucleases.

Crooks Fluctuation Theorem for Single Polymer Dynamics in Time-Dependent Flows: Understanding Viscoelastic Hysteresis

Nonequilibrium work relations have fundamentally advanced our understanding of molecular processes. In recent years, fluctuation theorems have been extensively applied to understand transitions between equilibrium steady-states, commonly described by simple control parameters such as molecular extension of a protein or polymer chain stretched by an external force in a quiescent fluid. Despite recent progress, far less is understood regarding the application of fluctuation theorems to processes involving nonequilibrium steady-states such as those described by polymer stretching dynamics in nonequilibrium fluid flows. In this work, we apply the Crooks fluctuation theorem to understand the nonequilibrium thermodynamics of dilute polymer solutions in flow. We directly determine the nonequilibrium free energy for single polymer molecules in flow using a combination of single molecule experiments and Brownian dynamics simulations. We further develop a time-dependent extensional flow protocol that allows for probing viscoelastic hysteresis over a wide range of flow strengths. Using this framework, we define quantities that uniquely characterize the coil-stretch transition for polymer chains in flow. Overall, generalized fluctuation theorems provide a powerful framework to understand polymer dynamics under far-from-equilibrium conditions.

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.

Threading-Unthreading Transition of Linear-Ring Polymer Blends in Extensional Flow

Adding small amounts of ring polymers to a matrix of their linear counterparts is known to increase the zero-shear-rate viscosity because of linear-ring threading. Uniaxial extensional rheology measurements show that, unlike its pure linear and ring constituents, the blend exhibits an overshoot in the stress growth coefficient. By combining these measurements with ex-situ small-angle neutron scattering and nonequilibrium molecular dynamics simulations, this overshoot is shown to be driven by a transient threading-unthreading transition of rings embedded within the linear entanglement network. Prior to unthreading, embedded rings deform affinely with the linear entanglement network and produce a measurably stronger elongation of the linear chains in the blend compared to the pure linear melt. Thus, rings uniquely alter the mechanisms of transient elongation in linear polymers.

Anomalous and heterogeneous DNA transport in biomimetic cytoskeleton networks

The cytoskeleton, a complex network of protein filaments and crosslinking proteins, dictates diverse cellular processes ranging from division to cargo transport. Yet, the role the cytoskeleton plays in the intracellular transport of DNA and other macromolecules remains poorly understood. Here, using single-molecule conformational tracking, we measure the transport and conformational dynamics of linear and relaxed circular (ring) DNA in composite networks of actin and microtubules with variable types of crosslinking. While both linear and ring DNA undergo anomalous, non-Gaussian, and non-ergodic subdiffusion, the detailed dynamics are controlled by both DNA topology (linear vs. ring) and crosslinking motif. Ring DNA swells, exhibiting heterogeneous subdiffusion controlled via threading by cytoskeleton filaments, while linear DNA compacts, exhibiting transport via caging and hopping. Importantly, while the crosslinking motif has little effect on ring DNA, linear DNA in networks with actin-microtubule crosslinking is significantly less ergodic and shows more heterogeneous transport than with actin-actin or microtubule-microtubule crosslinking.

Dynamics of poly[n]catenane melts

Inspired by advances in the chemical synthesis of interlocking polymer architectures, extensive molecular dynamics simulations have been conducted to study the dynamical properties of poly[n]catenanes-polymers composed entirely of interlocking rings-in the melt state. Both the degree of polymerization (number of links) and the number of beads per ring are systematically varied, and the results are compared to linear and ring polymers. A simple Rouse-like model is presented, and its analytical solution suggests a decomposition of the dynamics into "ring-like" and "linear-like" regimes at short and long times, respectively. In agreement with this picture, multiple sub-diffusive regimes are observed in the monomer mean-squared-displacements even though interchain entanglement is not prevalent in the system. However, the Rouse-type model does not account for the topological effects of the mechanical bonds, which significantly alter the dynamics at intermediate length scales both within the rings and at the chain segment scales. The stress relaxation in the system is extremely rapid and may be conveniently separated into ring-like and linear-like contributions, again in agreement with the Rouse picture. However, the viscosity has a non-monotonic dependence on the ring size for long chains, which disagrees strongly with theoretical predictions. This unexpected observation cannot be explained in terms of chain disentanglement and is inconsistent with other measures of polymer relaxation. Possible mechanisms for this behavior are proposed and implications for materials design are discussed.

Threading-Induced Dynamical Transition in Tadpole-Shaped Polymers

The relationship between polymer topology and bulk rheology remains a key question in soft matter physics. Architecture-specific constraints (or threadings) are thought to control the dynamics of ring polymers in ring-linear blends, which thus affects the viscosity to range between that of the pure rings and a value larger, but still comparable to, that of the pure linear melt. Here we consider qualitatively different systems of linear and ring polymers, fused together in "chimeric" architectures. The simplest example of this family is a "tadpole"-shaped polymer, a single ring fused to the end of a single linear chain. We show that polymers with this architecture display a threading-induced dynamical transition that substantially slows chain relaxation. Our findings shed light on how threadings control dynamics and may inform design principles for chimeric polymers with topologically tunable bulk rheological properties.

Stretching DNA to twice the normal length with single-molecule hydrodynamic trapping

Single-molecule force spectroscopy has brought many new insights into nanoscale biology, from the functioning of molecular motors to the mechanical response of soft materials within the cell. To expand the single-molecule toolbox, we have developed a surface-free force spectroscopy assay based on a high-speed hydrodynamic trap capable of applying extremely high tensions for long periods of time. High-speed single-molecule trapping is enabled by a rigid and gas-impermeable microfluidic chip, rapidly and inexpensively fabricated out of glass, double-sided tape and UV-curable adhesive. Our approach does not require difficult covalent attachment chemistries, and enables simultaneous force application and single-molecule fluorescence. Using this approach, we have induced a highly extended state with twice the contour length of B-DNA in regions of partially intercalated double-stranded (dsDNA) by applying forces up to 250 pN. This highly extended state resembles the hyperstretched state of dsDNA, which was initially discovered as a structure fully intercalated by dyes under high tension. It has been hypothesized that hyperstretched DNA could also be induced without the aid of intercalators if high-enough forces were applied, which matches our observation. Combining force application with single-molecule fluorescence imaging is critical for distinguishing hyperstretched DNA from single-stranded DNA that can result from peeling. High-speed hydrodynamic trapping is a powerful yet accessible force spectroscopy method that enables the mechanics of biomolecules to be probed in previously difficult to access regimes.

Dynamic Heterogeneity in Ring-Linear Polymer Blends

We present results from a direct statistical analysis of long molecular dynamics (MD) trajectories for the orientational relaxation of individual ring molecules in blends with equivalent linear chains. Our analysis reveals a very broad distribution of ring relaxation times whose width increases with increasing ring/linear molecular length and increasing concentration of the blend in linear chains. Dynamic heterogeneity is also observed in the pure ring melts but to a lesser extent. The enhanced degree of dynamic heterogeneity in the blends arises from the substantial increase in the intrinsic timescales of a large subpopulation of ring molecules due to their involvement in strong threading events with a certain population of the linear chains present in the blend. Our analysis suggests that the relaxation dynamics of the rings are controlled by the different states of their threading by linear chains. Unthreaded or singly-threaded rings exhibit terminal relaxation very similar to that in their own melt, but multiply-threaded rings relax much slower due to the long lifetimes of the corresponding topological interactions. By further analyzing the MD data for ring molecule terminal relaxation in terms of the sum of simple exponential functions we have been able to quantify the characteristic relaxation times of the corresponding mechanisms contributing to ring relaxation both in their pure melts and in the blends, and their relative importance. The extra contribution due to ring-linear threadings in the blends becomes immediately apparent through such an analysis.

Stress relaxation in symmetric ring-linear polymer blends at low ring fractions

We combine linear viscoelastic measurements and modelling in order to explore the dynamics of blends of the same-molecular-weight ring and linear polymers in the regime of the low volume fraction (0.3 or lower) of the ring component. The stress relaxation modulus is affected by the constraint release (CR) of both rings and linear components due to the motion of linear chains. We develop a CR-based model of ring-linear blends that predicts the stress relaxation function in the low fraction regime of ring component in excellent agreement with experiments. Rings trapped by their entanglements with linear chains can only relax by linear-chain-induced constraint release, resulting in much slower relaxation of rings than of linear chains. The relative viscosity η ( ϕ R * ) / η L of the blend with respect to the linear melt viscosity η _L at ring overlap volume fraction ϕ R * increases proportionally to the square root of ring molecular weight M w , R . Our experimental results clearly demonstrate that it is possible to enhance the viscosity and simultaneously the structural relaxation time of linear polymer melts by adding a small fraction of ring polymers. These results not only provide fundamental insights into the physics of the CR process but also suggest ways to fine-tune the flow properties of linear polymers by means of adding rings.

Topological Linking Drives Anomalous Thickening of Ring Polymers in Weak Extensional Flows

Molecular dynamics simulations confirm recent extensional flow experiments showing ring polymer melts exhibit strong extension-rate thickening of the viscosity at Weissenberg numbers Wi≪1. Thickening coincides with the extreme elongation of a minority population of rings that grows with Wi. The large susceptibility of some rings to extend is due to a flow-driven formation of topological links that connect multiple rings into supramolecular chains. Links form spontaneously with a longer delay at lower Wi and are pulled tight and stabilized by the flow. Once linked, these composite objects experience larger drag forces than individual rings, driving their strong elongation. The fraction of linked rings depends nonmonotonically on Wi, increasing to a maximum when Wi∼1 before rapidly decreasing when the strain rate approaches 1/τ_{e}.

Unexpected entanglement dynamics in semidilute blends of supercoiled and ring DNA

Blends of polymers of different topologies, such as ring and supercoiled, naturally occur in biology and often exhibit emergent viscoelastic properties coveted in industry. However, due to their complexity, along with the difficulty of producing polymers of different topologies, the dynamics of topological polymer blends remains poorly understood. We address this void by using both passive and active microrheology to characterize the linear and nonlinear rheological properties of blends of relaxed circular and supercoiled DNA. We characterize the dynamics as we vary the concentration from below the overlap concentration c* to above (0.5c* to 2c*). Surprisingly, despite working at the dilute-semidilute crossover, entanglement dynamics, such as elastic plateaus and multiple relaxation modes, emerge. Finally, blends exhibit an unexpected sustained elastic response to nonlinear strains not previously observed even in well-entangled linear polymer solutions.

Equilibrium structure and deformation response of 2D kinetoplast sheets

The considerable interest in two-dimensional (2D) materials and complex molecular topologies calls for a robust experimental system for single-molecule studies. In this work, we study the equilibrium properties and deformation response of a complex DNA structure called a kinetoplast, a 2D network of thousands of linked rings akin to molecular chainmail. Examined in good solvent conditions, kinetoplasts appear as a wrinkled hemispherical sheet. The conformation of each kinetoplast is dictated by its network topology, giving it a unique shape, which undergoes small-amplitude thermal fluctuations at subsecond timescales, with a wide separation between fluctuation and diffusion timescales. They deform elastically when weakly confined and swell to their equilibrium dimensions when the confinement is released. We hope that, in the same way that linear DNA became a canonical model system on the first investigations of its polymer-like behavior, kinetoplasts can serve that role for 2D and catenated polymer systems.

Topology-dependent anomalous dynamics of ring and linear DNA are sensitive to cytoskeleton crosslinking

Cytoskeletal crowding plays a key role in the diffusion of DNA molecules through the cell, acting as a barrier to effective intracellular transport and conformational stability required for processes such as transfection, viral infection, and gene therapy. Here, we elucidate the transport properties and conformational dynamics of linear and ring DNA molecules diffusing through entangled and crosslinked composite networks of actin and microtubules. We couple single-molecule conformational tracking with differential dynamic microscopy to reveal that ring and linear DNA exhibit unexpectedly distinct transport properties that are influenced differently by cytoskeleton crosslinking. Ring DNA coils are swollen and undergo heterogeneous and biphasic subdiffusion that is hindered by crosslinking. Conversely, crosslinking actually facilitates the single-mode subdiffusion that compacted linear chains exhibit. Our collective results demonstrate that transient threading by cytoskeleton filaments plays a key role in the dynamics of ring DNA, whereas the mobility of the cytoskeleton dictates transport of linear DNA.

Simulation of semidilute polymer solutions in planar extensional flow via conformationally averaged Brownian noise

The dynamics and rheology of semidilute polymer solutions in strong flows are of great practical relevance. Processing applications can in principle be designed utilizing the relationship between nonequilibrium polymer conformations and the material properties of the solution. However, the interplay between concentration, flow, hydrodynamic interactions (HIs), and topological interactions which govern semidilute polymer dynamics is challenging to characterize. Brownian dynamics (BD) simulations are particularly valuable as a way to directly visualize how molecular interactions arise in these systems and are quantitatively comparable to single-molecule experiments. However, such simulations are often computationally intractable and are limited by the need to calculate the correlated Brownian noise via decomposition of the diffusion tensor. Previously, we have introduced an iterative conformational averaging (CA) method for BD simulations which bypasses these limitations by preaveraging the HI and Brownian noise in an iterative procedure. In this work, we generalize the CA method to flowing semidilute solutions by introducing a conformation dependent diffusion tensor and a strain dependent approximation to the conformationally averaged Brownian noise. We find that this approach nearly quantitatively reproduces both transient and steady state polymer dynamics and rheology while achieving an order of magnitude computational acceleration. We then utilize the CA method to investigate the concentration and flow rate dependence of polymer dynamics in planar extensional flows. Our results are consistent with previous experimental and simulation studies and provide a detailed view of broad conformational distributions in the semidilute regime. We observe interconversion between stretched and coiled states at steady state, which we conjecture occur due to the effect of concentration on the conformation dependent polymer drag. Additionally, we observe transient flow-induced intermolecular hooks in the startup of flow which lead to diverse and unique stretching pathways.