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Knotting is common in long polymers and significantly affects the polymer behavior. It is generally believed that knotting should increase polymer bending energy. However, many recent studies found...

As an intriguing phenomenon, knotting in DNA, proteins, and other molecules has been extensively investigated in recent years, not only because of fundamental interest in knotting but also because of significant effects of knotting on mechanical, rheological, chemical, and biological properties. Some polymers contain side chains, and the effects of side chains on polymer knots are not clear. In this work, we investigate the effects of side chains on knots through Langevin dynamics simulations of knot diffusion along a stretched polymer with side chains. When the gap between side chains is larger than the knot size; each side chain acts as a barrier to slow down the knot diffusion. We find that how a knot crosses the barrier by a side chain resembles how a polymer translocates a pore. Inspired by the latter, we find a simple empirical expression for the free energy barrier caused by a side chain: Fbarrier/kBT ≈ 75Rg/Lknot, where kB is the Boltzmann constant, T is the temperature, Rg is the radius of gyration of a side chain, and Lknot is the contour length of the knot core. The equations and numerical results obtained in this work can guide the rational control of knot diffusivity along a polymer by side chains. In addition, our results provide insights into the understanding of knots in peptides and single-stranded DNA where side chains are common. Furthermore, this work makes a connection between two interesting phenomena for polymers - knotting and pore translocation.

Knotting is a prevalent phenomenon which occurs at the macroscale (e.g., in headphone cords) and at the microscale (e.g., in DNA and proteins). For a confined polymer, the knotting probability can rapidly approach 100% as the degree of confinement increases, while the mechanism of knot formation in a confined space is unclear. In this work, we use computer simulation to generate equilibrium conformations of a polymer confined in a sphere or a tube and then calculate the knotting probability, pknot, and the knot complexity that is quantified by the minimal crossing number, Ncross. Surprisingly, we find a universal correlation between pknot and ⟨Ncross⟩. Further analysis reveals that the universal correlation is caused by the fact that the distribution of knot types, i.e., the knot spectrum, of a confined polymer follows a universal behavior, only depending on the total knotting probability, regardless of the polymer length, bending stiffness, and degree of confinement. Such universal behavior reveals a pos...

Knots occur frequently in our daily life and also in long polymers. Studies of spontaneously occurring knots in a linear polymer have focused on their size and probabilities. In this work, we report counterintuitive phenomena about knot sizes and explain them by an analytic theory. Our simulations show that short-range intrachain attractions (repulsions) lead to shrinking (swelling) of knot sizes, trends which follow intuition. However, long-range attractions (repulsions) surprisingly lead to swelling (shrinking) of knot sizes. Such counterintuitive trends are because larger knots contain more interacting monomer pairs if the interaction range is larger than a critical value. The critical interaction range varies from a fraction of to multiple persistence lengths, and so the long-range interaction regime can be reached for DNA, peptide chains, or synthetic polymers under depletion attractions induced by colloids (e.g., proteins) or via Coulomb repulsions. Our results suggest that probabilities and sizes o...

Singapore. National Research Foundation (Singapore-MIT Alliance for Research and Technology)

Knotted structures can spontaneously occur in polymers such as DNA and proteins, and the formation of knots affects biological functions, mechanical strength and rheological properties. In this work, we calculate the equilibrium size distribution of trefoil knots in linear DNA using off-lattice simulations. We observe metastable knots on DNA, as predicted by Grosberg and Rabin. Furthermore, we extend their theory to incorporate the finite width of chains and show an agreement between our simulations and the modified theory for real chains. Our results suggest localized knots spontaneously occur in long DNA and the contour length in the knot ranges from 600 to 1800 nm.

Scaling regimes for polymers confined to tubular channels are well established when the channel cross-sectional dimension is either very small (Odjik regime) or large (classic de Gennes regime) relative to the polymer Kuhn length. However, experiments of confined polymers using DNA as a model system are usually located in the intermediate region between these two regimes. In the literature, controversy exists regarding the existence of the extended de Gennes regime in this intermediate region. Here we use simulations and theory to reconcile conflicting theories and confirm the existence of extended de Gennes regime. We show that prior work did not support the notion of this regime because of the use of a wrong confinement free energy. In a broad sense, the extended de Gennes regime corresponds to the situation when excluded volume interaction is weaker than thermal energy. Such a situation also occurs in many other cases, such as semidilute polymer solutions and polymers under tension. This work should be...

The similarities and differences between geometric and tensile constraints on polymers have not been fully investigated. Here we use theory (blob models) and simulations to present a comprehensive comparison between polymers in these two situations. For a polymer in good solvent, the effect of tensile force f on extension in the Pincus regime is similar to the effect of cylindrical confinement in the de Gennes regime after mapping the characteristic length kBT/f to the cylindrical diameter D, where kBT is the thermal energy. However, the comparison of the effects of tension and confinement on extension is lacking when kBT/f and D are less than the thermal blob size b, referred to as extended Pincus regime and extended de Gennes regime, respectively. In the extended Pincus regime, force can still segregate the ideal-coils with the size of ∼kBT/f, resulting in the scaling of extension L∥ ∼ (kBT/f)−1. In the extended de Gennes regime, excluded volume interaction is not sufficient to segregate the ideal-coils...