A growth model for RNA secondary structures

A hierarchical model for the growth of planar arch structures for RNA secondary structures is presented, and shown to be equivalent to a tree-growth model. Both models can be solved analytically, giving access to scaling functions for large molecules, and corrections to scaling, checked by numerical simulations of up to 6500 bases. The equivalence of both models should be helpful in understanding more general tree-growth processes.

💡 Research Summary

The paper introduces a hierarchical growth model for RNA secondary structures that represents the molecule as a set of non‑crossing planar arches. Each base pair is mapped to an arc connecting two points on a line, and the collection of arcs forms a planar graph that respects the biological “no‑pseudoknot” constraint. Growth proceeds by sequentially inserting new arcs into the gaps between existing ones; the insertion site is chosen uniformly among all admissible positions, ensuring that the planar‑arch property is preserved at every step.

A central contribution is the rigorous proof that this planar‑arch model is mathematically equivalent to a binary‑tree growth process. In the mapping, each arc corresponds to an internal node of a binary tree, while the left and right endpoints of the arc become the left and right children of that node. Consequently, the entire arch configuration is isomorphic to a rooted binary tree, and any structural observable of the RNA (e.g., maximum stack depth, number of loops, leaf count) can be expressed as a tree statistic. This equivalence allows the authors to import the extensive toolbox developed for random tree growth into the RNA context.

The analytical treatment relies on generating‑function techniques. Let (P_n(k)) denote the probability that a structure of size (n) (i.e., (n) base pairs) possesses a given property (k) (such as tree height). By writing a master equation for the insertion process and applying Laplace transforms, the authors obtain an exact closed‑form expression for the generating function. Asymptotic analysis of this function yields a universal scaling law

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