Revisiting the Non-monotonic Dependence of Polymer Knotting Probability on the Bending Stiffness

Activity-driven polymer knotting for macromolecular topology engineering

Macromolecules can gain special properties by adopting knotted conformations, but engineering knotted macromolecules is a challenging task. Here, we unexpectedly find that knots can be efficiently generated in active polymer systems. When one end of an actively reptative polymer is anchored, it undergoes continual self-knotting as a result of intermittent giant conformation fluctuations and the outward reptative motion. Once a knot is formed, it migrates to the anchoring point due to a nonequilibrium ratchet effect. Moreover, when the active polymer is grafted on a passive polymer, it can function as a self-propelling soft needle to either transfer its own knots or directly braid knots on the passive polymer. We further show that these active needles can create intermolecular bridging knots between two passive polymers. Our finding highlights the nonequilibrium effects in modifying the dynamic pathways of polymer systems, which have potential applications in macromolecular topology engineering.

Compression of a confined semiflexible polymer under direct and oscillating fields

The folding transition of biopolymers from the coil to compact structures has attracted wide research interest in the past and is well studied in polymer physics. Recent seminal works on DNA in confined devices have shown that these long biopolymers tend to collapse under an external field, which is contrary to the previously reported stretching of the chain. In this work, we capture the compression of a confined semiflexible polymer under direct and oscillating fields using a coarse-grained computer simulation model in the presence of long-range hydrodynamics. In the case of a semiflexible polymer chain, the inhomogeneous hydrodynamic drag from the center to the periphery of the coil couples with the chain bending to cause a swirling movement of the chain segments, leading to structural intertwining and compaction. Contrarily, a flexible chain of the same length lacks such structural deformation and forms a well-established tadpole structure. While bending rigidity profoundly influences the chain's folding favorability, we also found that subject to the direct field, chains in stronger confinements exhibit substantial compaction, contrary to the one in moderate confinements or bulk where such compaction is absent. However, an alternating field within an optimum frequency can effectuate this compression even in moderate or no confinement. This field-induced collapse is a quintessential hydrodynamic phenomenon, resulting in intertwined knotted structures even for shorter chains, unlike other spontaneous knotting experiments where it happens exclusively for longer chains.

Phase transition of DNA knotting in spherical space

Knots have been discovered in various biological systems, such as DNA. The knotting probability of DNA in free space depends non-monotonically on its bending rigidity and has a prominent peak. The current work aims to understand the underlying mechanism of the non-monotonic dependence of DNA knotting probability on bending rigidity. Monte Carlo simulations are performed on a closed DNA molecule confined in spherical space described by a worm-like chain model and a flexible kink model, respectively. The closed DNA's contour length and the spherical space radius both increase knotting probability, but also alter the unimodal dependence of knotting probability on bending rigidity. This is generalized using universal phase diagrams based on the two models. Under the flexible kink model, the total knotting probability of closed DNA is obviously increased at a relatively high excited energy. This supports the expectation that the entropy effect of knot size favours knot formation at a relatively low bending rigidity. In a given spherical space, the increasing contour length of closed DNA described by the worm-like chain model results in a visible shift in the knotting probability distribution. At the same time, the gyration radius of non-trivial closed DNA becomes comparable to that of trivial closed DNA, so that their ratio is not anti-correlated with average knot length. For closed DNA of various contour lengths, the relationship between average knot length and bending rigidity has a universal behaviour: the average knot length decreases to a local minimum at a bending rigidity of ∼5 and then gradually increases to a constant value. The existence of the local minimum is determined by the cut-off distance in repulsive Lennard-Jones potential. The bending rigidity corresponding to the beginning of the constant average knot length is consistent with that at the peak in the knotting distribution. At this point, the knot-size effect balances with the fragment free-energy effect and, at an even greater bending rigidity, knot length breathes around the average knot length value.

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.

Crowding-induced polymer trapping in a channel

In this work, we report an intriguing phenomenon: crowding-induced polymer trapping in a channel. Using Langevin dynamics simulations and analytical calculations, we find that for a polymer confined in a channel, crowding particles can push a polymer into the channel corner through inducing an effective polymer-corner attraction due to the depletion effect. This phenomenon is referred to as polymer trapping. The occurrence of polymer trapping requires a minimum volume fraction of crowders, ϕ^{*}, which scales as ϕ^{*}∼(a_{c}/L_{p})^{1/3} for a_{c}≫a_{m} and ϕ^{*}∼(a_{c}/L_{p})^{1/3}(a_{c}/a_{m})^{1/2} for a_{c}≪a_{m}, where a_{c} is the crowder diameter, a_{m} is the monomer diameter, and L_{p} is the polymer persistence length. For DNA, ϕ^{*} is estimated to be around 0.25 for crowders with a_{c}=2nm. We find that ϕ^{*} also strongly depends on the shape of the channel cross section, and ϕ^{*} is much smaller for a triangle channel than a square channel. The polymer trapping leads to a nearly fully stretched polymer conformation along a channel corner, which may have practical applications, such as full stretching of DNA for the nanochannel-based genome mapping technology.