Composite spring model in stretched polymer knots

The deformation of a knotted polymer under a stretching force is studied by modeling the deformed knot as a composite spring system. Our results predict that the elastic modulus of a knotted polymer is larger when compared to an equal-length linear chain. More complex knots are in general stiffer. The increase in stiffness of knots relative to the linear chain is also derived. Monte Carlo simulations are also performed to investigate the streching of polymer knots. Chain lengths up to N = 82 and prime knots 0₁, 3₁, 4₁, 5₁, 6₁ and 8₁ are considered. Segegation of the crossings into a small tight region of the knot structure at strong forces is observed. The increase in stiffness predicted by the composite spring model agree well with the simulation data. Our simulation data show that the scaling laws proposed by de Gennes and Pincus for a single linear chain under traction force still hold for the knotted type polymers.