Planar diagrams from optimization for concave potentials

We propose a toy model of a heteropolymer chain capable of forming planar secondary structures typical for RNA molecules. In this model, the sequential intervals between neighboring monomers along a chain are considered as quenched random variables, and energies of nonlocal bonds are assumed to be concave functions of those intervals. A few factors are neglected: the contribution of loop factors to the partition function, the variation in energies of different types of complementary nucleotides, the stacking interactions, and constraints on the minimal size of loops. However, the model captures well the formation of folded structures without pseudoknots in an arbitrary sequence of nucleotides. Using the optimization procedure for a special class of concave-type potentials, borrowed from optimal transport analysis, we derive the local difference equation for the ground state free energy of the chain with the planar (RNA-like) architecture of paired links. We consider various distribution functions of intervals between neighboring monomers (truncated Gaussian and scale free) and demonstrate the existence of a topological crossover from sequential to essentially nested configurations of paired links.