Coherent modeling of double-folded ring polymers and their underlying random tree structure

Topologically constrained genome-like polymers often double-fold into tree-like configurations, which can be modelled on the level of folded (ring) polymers or on the level of the underlying random trees. For both descriptions, we have recently obtained expressions for the configurational entropy in ensembles with controlled branching activity. Here we demonstrate that they are equivalent up to a contribution originating from the number of distinct wrappings of a single tree. This allows us to develop a coherent framework for freely switching between the two representations. Importantly, the equivalence extends to interacting systems provided the interactions are treated consistently on the tree and on the ring level. To demonstrate the utility of the scheme, we introduce a generalization of the Amoeba Monte Carlo algorithm capable of generating the required ensembles of trees with fluctuating sizes. While the tree algorithm reproduces results obtained by dynamic simulations of the corresponding ring model, it is $O(N)$ faster for the purpose of sampling static properties and leverages the utility of the ring model for the study of dynamical properties, when used for the preparation of equilibrated starting states.