Planar diagrams from optimization

TL;DR

This paper introduces a new model for heteropolymer chains that can form planar RNA-like structures, using optimization techniques to analyze the ground state energy and topological transitions between different configurations.

Contribution

It develops a novel analytical approach based on optimal transport to study the ground state of heteropolymer chains with planar structures, revealing topological crossovers.

Findings

Existence of a topological crossover from sequential to nested configurations.

Application of optimal transport methods to heteropolymer modeling.

Analysis of different interval distributions like Gaussian and scale-free.

Abstract

We propose a new 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. 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 embedded (nested) configurations of paired links.