Effects of Intrachain Interactions on the Knot Size of a Polymer

Effect of Counterion Size on Knotted Polyelectrolyte Conformations

Using Langevin dynamics simulations and a coarse-grained primitive model of electrolytes, we show that the behavior of knotted circular strong polyelectrolytes (PEs) in diluted aqueous solution is largely affected by the diameter of the counterions (CIs), σCI. Indeed, we observe that both gyration radius and knot length vary nonmonotonically with σCI, with both small and bulky CIs favoring knot localization, while medium-sized ones promote delocalized knots. We also show that the conformational change from delocalized to tight knots occurs via the progressive coalescence of the knot's essential crossings. The emerging conformers correspond to the minima of the free energy landscape profiled as a function of the knot length or PE size. We demonstrate that different conformational states can coexist, the transition between them appearing first-order-like and controlled by the enthalpic and entropic trade-off of the amount of CIs condensed on the PE. Such balance can be further altered by varying CI concentrations, thus providing an additional and more convenient tuning parameter for the system properties. Our results lay the foundation for achieving broader and more precise external adjustability of knotted PE size and shape by choosing the nature of its CIs. Thus, they offer new intriguing possibilities for designing novel PE-based materials that are capable of responding to changes in ionic solution properties.

Quantifying the effects of slit confinement on polymer knots using the tube model

Knots can spontaneously form in DNA, proteins, and other polymers and affect their properties. These knots often experience spatial confinement in biological systems and experiments. While confinement dramatically affects the knot behavior, the physical mechanisms underlying the confinement effects are not fully understood. In this work, we provide a simple physical picture of the polymer knots in slit confinement using the tube model. In the tube model, the polymer segments in the knot core are assumed to be confined in a virtual tube due to the topological restriction. We first perform Monte Carlo simulation of a flexible knotted chain confined in a slit. We find that with the decrease of the slit height from H=+∞ (the 3D case) to H=2a (the 2D case), the most probable knot size L_{knot}^{*} dramatically shrinks from (L_{knot}^{*})_{3D}≈140a to (L_{knot}^{*})_{2D}≈26a, where a is the monomer diameter of the flexible chain. Then we quantitatively explain the confinement-induced knot shrinking and knot deformation using the tube model. Our results for H=2a can be applied to a polymer knot on a surface, which resembles DNA knots measured by atomic force microscopy under the conditions that DNA molecules are weakly absorbed on the surface and reach equilibrium 2D conformations. This work demonstrates the effectiveness of the tube model in understanding polymer knots.

Tunable Knot Segregation in Copolyelectrolyte Rings Carrying a Neutral Segment

We use Langevin dynamics simulations to study the knotting properties of copolyelectrolyte rings carrying neutral segments. We show that by solely tuning the relative length of the neutral and charged blocks, one can achieve different combinations of knot contour position and size. Strikingly, the latter is shown to vary nonmonotonically with the length of the neutral segment; at the same time, the knot switches from being pinned at the block's edge to becoming trapped inside it. Model calculations relate both effects to the competition between two adversarial mechanisms: the energy gain of localizing one or more of the knot's essential crossings on the neutral segment and the entropic cost of such localization. Tuning the length of the neutral segment sets the balance between the two mechanisms and hence the number of localized essential crossings, which in turn modulates the knot's size. This general principle ought to be useful in more complex systems, such as multiblock copolyelectrolytes, to achieve a more granular control of topological constraints.

Can Soft Models Describe Polymer Knots?

Similar to macroscopic ropes and cables, long polymers create knots. We address the fundamental question whether and under which conditions it is possible to describe these intriguing objects with crude models that capture only mesoscale polymer properties. We focus on melts of long polymers which we describe by a model typical for mesoscopic simulations. A worm-like chain model defines the polymer architecture. To describe nonbonded interactions, we deliberately choose a generic "soft" repulsive potential that leads to strongly overlapping monomers and coarse local liquid structure. The soft model is parametrized to accurately reproduce mesoscopic structure and conformations of reference polymer melts described by a microscopic model. The microscopically resolved samples retain all generic features affecting polymer topology and provide, therefore, reliable reference data on knots. We compare characteristic knotting properties in mesoscopic and microscopically resolved melts for different cases of chain stiffness. We conclude that mesoscopic models can reliably describe knots in those melts, where the length scale characterizing polymer stiffness is substantially larger than the size of monomer-monomer excluded volume. In this case, simplified local liquid structure influences knotting properties only marginally. In contrast, mesoscopic models perform poorly in melts with flexible chains. We qualitatively explain our findings through a free energy model of simple knots available in the literature.

Interface Counterion Localization Induces a Switch between Tight and Loose Configurations of Knotted Weak Polyacid Rings despite Intermonomer Coulomb Repulsions

Stochastic simulations have been used to investigate the conformational behavior of knotted weak polyacid rings as a function of pH. Different from the commonly expected ionization-repulsion-expansion scheme upon increasing pH, theoretical results suggest a nonmonotonic behavior of the gyration radius R_g². Polyelectrolyte recontraction at high ionization is induced by the weakening of Coulomb repulsion due to counterions (CIs) localizing at the interphase between the polymer and solvent, and the more marked it appears, the more complex is the knot topology. Compared with strong polyelectrolytic species of identical ionization, weak polyacids present tighter knots due to their ability to localize neutral monomers inside the tangled part. Increasing the solvent Bjerrum length enhances CIs localization, lowering the pH at which polyacids start decreasing their average size. A similar effect is also obtained by increasing the amount of "localizable" cations by adding salts.

Nanomechanical properties of steric zipper globular structures

The term amyloid defines a group of proteins that aggregate into plaques or fibers. Amyloid fibers gained their fame mostly due to their relation with neurodegenerative diseases in humans. However, secreted by lower organisms, such as bacteria and fungi, amyloid fibers play a functional role: for example, when they serve as cement in the extracellular matrix of biofilms. Originating either in humans or in microorganisms, the sequence of amyloid proteins is decorated with hexapeptides with high propensity to form fibers, known as steric zippers. We have found that steric zippers form globular structures on route to making fibers and exhibit a characteristic force-distance (F-D) fingerprint when pulled with an atomic force microscope (AFM) tip. Particularly, the F-D pulling curves showed force plateau steps, suggesting that the globular structures were composed of chains that were unwound like a yarn ball. Force plateau analysis showed that the F-D characteristic parameters were sequence sensitive, representing differences in the packing of the hexapeptides within the globules. These unprecedented findings show that steric zippers exhibit a characteristic nanomechanical signature in solution in addition to previously observed characteristic crystallographic structure. Getting to the fundamental interactions that govern the unzipping of full-length amyloid fibers may initiate the development of antiamyloid methods that target the physical in addition to the structural properties of steric zippers.

The folding pathways and thermodynamics of semiflexible polymers

Inspired by the protein folding and DNA packing, we have systematically studied the thermodynamic and kinetic behaviors of single semiflexible homopolymers by Langevin dynamics simulations. In line with experiments, a rich variety of folding products, such as rod-like bundles, hairpins, toroids, and a mixture of them, are observed in the complete diagram of states. Moreover, knotted structures with a significant population are found in a certain range of bending stiffness in thermal equilibrium. As the solvent quality becomes poorer, the population of the intermediate occurring in the folding process increases, which leads to a severe chevron rollover for the folding arm. However, the population of the intermediates in the unfolding process is very low, insufficient to induce unfolding arm rollover. The total types of folding pathways from the coil state to the toroidal state for a semiflexible polymer chain remain unchanged by varying the solvent quality or temperature, whereas the kinetic partitioning into different folding events can be tuned significantly. In the process of knotting, three types of mechanisms, namely, plugging, slipknotting, and sliding, are discovered. Along the folding evolution, a semiflexible homopolymer chain can knot at any stage of folding upon leaving the extended coil state, and the probability to find a knot increases with chain compactness. In addition, we find rich types of knotted topologies during the folding of a semiflexible homopolymer chain. This study should be helpful in gaining insight into the general principles of biopolymer folding.

Trefoil Knot Hydrodynamic Delocalization on Sheared Ring Polymers

The behavior of unknotted and trefoil-knotted ring polymers under shear flow is here examined by means of mesoscopic simulations. In contrast to most polymers, ring polymers in a hydrodynamic solvent at high shear rates do not get shortened in the vorticity direction. This is a consequence of the backflow produced by the interaction of the sheared solvent with the end-free polymer topology. The extended structures of the ring in the vorticity-flow plane, when they are aligned in a constant velocity plane, favor ring contour fluctuations. This variety of conformations largely suppresses the tank-treading type of rotation with extended conformations in favor of the tumbling type of rotations, where stretched and collapsed conformations alternate. The extension of trefoil knots is also enhanced, so that the knots become delocalized. We anticipate that these effects, which disappear in the absence of hydrodynamic interactions, will have a crucial impact on the rheological properties of concentrated ring solutions, and will also influence the behavior of more complicated systems such as mixtures of polymers with different topologies.

Trapping a Knot into Tight Conformations by Intra-Chain Repulsions

Knots can occur in biopolymers such as DNA and peptides. In our previous study, we systematically investigated the effects of intra-chain interactions on knots and found that long-range repulsions can surprisingly tighten knots. Here, we use this knowledge to trap a knot into tight conformations in Langevin dynamics simulations. By trapping, we mean that the free energy landscape with respect to the knot size exhibits a potential well around a small knot size in the presence of long-range repulsions, and this potential can well lead to long-lived tight knots when its depth is comparable to or larger than thermal energy. We tune the strength of intra-chain repulsion such that a knot is weakly trapped. Driven by thermal fluctuations, the knot can escape from the trap and is then re-trapped. We find that the knot switches between tight and loose conformations-referred to as "knot breathing". We use a Yukawa potential to model screened electrostatic interactions to explore the relevance of knot trapping and breathing in charged biopolymers. We determine the minimal screened length and the minimal strength of repulsion for knot trapping. We find that Coulomb-induced knot trapping is possible to occur in single-stranded DNA and peptides for normal ionic strengths.