Configurational entropy of randomly double-folding ring polymers

Topologically constrained genome-like polymers often double-fold into tree-like configurations. Here, we calculate the exact number of tightly double-folded configurations available to a ring polymer in ideal conditions. For this purpose, we introduce a scheme that allows us to define a "code" specifying how a ring wraps a randomly branching tree and calculate the number of admissible wrapping codes via a variant of Bertrand's ballot theorem. As a validation, we demonstrate that data from Monte Carlo simulations of an elastic lattice model of non-interacting tightly double-folded rings with controlled branching activity are in excellent agreement with exact expressions for branch-node and tree size statistics that can be derived from our expression for the ring entropy.