Do Empirical and Abstract Shannon Entropies Converge in Value? A Case in RNA Molecular Structure

The RNA molecule is capable of folding into different shapes, with some being more stable than others. The structural space of the RNA can be described by Stochastic Context-free Grammars (SCFG), offering a probabilisitic distribution of structural scenarios. In a more accurate folding model, the probability associated with a structural scenario is more informative of its stability. Here, we offer two different ways of calculating the Shannon Entropy of the SCFG-modeled RNA structural space; Grammar Space (GS) Entropy and SCFG Entropy. The former is the Shannon Entropy of the infinite number of structures produced by the model and the latter is the Shannon Entropy of a limited subset of structures all of which belong to the same RNA sequence. After a brief introduction on the two measures, we explore the relationship between these measures on a given set of RNA folding models and biologically functional RNA sequences. We show that these two measures of entropy are indeed correlated. While more theoretical work is needed in understanding the convergence behavior between the two, this observation suggests that GS Entropy is a promising characteristic in future model training approaches.