Simulation Investigation Of The Effect Of Flexibility Of DNA Chains On Its Knotting Probability Under Spherical Constraints

Knots appear in a wide variety of biophysical systems,from biopolymers,such as DNA and proteins,to macroscopic objects,such as umbilical cords and catheters.In a recent study,an interesting result regarding the knotting probability _kP is that in free space _kP exhibits non-monotonic dependence on the bending stiffness and appears as a significant maximum.We use Monte Carlo simulations to sample the conformation of DNA in a confined space and investigate the origin of the non-monotonicity of the knotting probability concerning the bending stiffness.The worm chain model and the flexible defect model were used in the simulations to describe the ring DNA confined in a spherical space,and the results of the simulations showed that the knotting probability of the ring DNA as a function of the bending stiffness depends strongly on its contour length and confinement radius.In the worm chain model,when the contour length is fixed,the knotting probability of the chain increases gradually with increasing confinement strength,but remains single-peaked non-monotonic at the short chain,while the properties of the long chain are quite complex.After that,we further investigated the knotting probabilities of ring DNA chains with different contour lengths at a fixed confinement radius.In the case of strong confinement,the short chain still has single-peaked non-monotonicity,while the change of knotting probability becomes more complicated as the chain contour length gradually increases.In the case of weak constraint,the knotting probabilities all have single-peaked non-monotonicity,and the distribution of the knotting probabilities shifts significantly to the right as the contour length of the ring DNA chain increases.Then,the interaction between the geometry and topology of the ring DNA chain is explored,with the ratio of the radius of gyration of the knotted to the unknotted ring having an anti-correlation with the average knotted length of the chain in the case of short chains,while it will no longer have this property in the case of long chains.Finally,the general rule of the average knot length as a function of the bending stiffness is investigated,i.e.,the average knot length decreases to a local minimum at a bending stiffness of 5 and then gradually increases to a constant value for any length of ring DNA,the existence of the local minimum being determined by the cut-off distance of the repulsive L-J potential.The bending stiffness corresponding to a constant start of the average knot length is consistent with the bending stiffness corresponding to the maximum of the knotting probability distribution,where the knot size effect balances with the fragment free energy effect and the knotting length breathe around its average knot length at larger bending stiffnesses.In comparison,in DNA rings described by the flexible defect model,the difference in the kink number of DNA chains with different profile lengths in free space and restricted space was first investigated,and the restriction led to a significant increase in the kink number of DNA chains compared to free space.Next,the effect of kink structure on the knotting probability of ring DNA at a fixed confinement radius was continued to be investigated.Under weak confinement,the knotting probability shows a non-monotonic distribution as a function of excitation energy,with a maximum around the excitation energy of 6k _BT,while the knotting probability gradually loses its monoclinic nature as the confinement strength increases.Finally,we also investigate the interaction between the geometry and topology of the ring DNA chain in the presence of the kink structure.Under weak confinement,the ratio of the radius of gyration of the knotted to the unknotted ring for all lengths shows an anti-correlation with the knotted length,and this anti-correlation gradually disappears as the confinement strength increases.In the end,we will construct two generic phase diagrams in the phase space of total knotting probability _kP,scalar bending stiffness k _b/R and scalar contour length L/R under the worm chain model and in the phase space of total knotting probability _kP,excitation energyμand scalar contour length L/R under the flexible defect model,where the knotting behavior in each phase diagram is different.