  Ring polymers are prevalent in natural and engineered systems, including
circular bacterial DNA, crown ethers for cation chelation, and mechanical
nanoswitches. The morphology and dynamics of ring polymers are governed by the
chemistry and degree of polymerization of the ring, and intramolecular and
supramolecular topological constraints such as knots or
mechanically-interlocked rings. In this study, we perform molecular dynamics
simulations of polyethylene ring polymers at two different degrees of
polymerization and in different topological states, including a trefoil knot,
catenane state (two interlocked rings), and Borromean state (three interlocked
rings). We employ nonlinear manifold learning to extract the low-dimensional
free energy surface to which the structure and dynamics of the polymer chain
are effectively restrained. The free energy surfaces reveal how degree of
polymerization and topological constraints affect the thermally accessible
conformations, chiral symmetry breaking, and folding and collapse pathways of
the rings, and present a means to rationally engineer ring size and topology to
stabilize particular conformational states and folding pathways. We compute the
rotational diffusion of the ring in these various states as a crucial property
required for the design of engineered devices containing ring polymer
components.
