The RNA Newton polytope and learnability of energy parameters

Computational RNA structure prediction is a mature important problem that has received a new wave of attention with the discovery of regulatory non-coding RNAs and the advent of high-throughput transcriptome sequencing. Despite nearly two score years of research on RNA secondary structure and RNA-RNA interaction prediction, the accuracy of the state-of-the-art algorithms are still far from satisfactory. So far, researchers have proposed increasingly complex energy models and improved parameter estimation methods, experimental and/or computational, in anticipation of endowing their methods with enough power to solve the problem. The output has disappointingly been only modest improvements, not matching the expectations. Even recent massively featured machine learning approaches were not able to break the barrier. Why is that?

The first step toward high-accuracy structure prediction is to pick an energy model that is inherently capable of predicting each and every one of known structures to date. In this article, we introduce the notion of learnability of the parameters of an energy model as a measure of such an inherent capability. We say that the parameters of an energy model are learnable iff there exists at least one set of such parameters that renders every known RNA structure to date the minimum free energy structure. We derive a necessary condition for the learnability and give a dynamic programming algorithm to assess it. Our algorithm computes the convex hull of the feature vectors of all feasible structures in the ensemble of a given input sequence. Interestingly, that convex hull coincides with the Newton polytope of the partition function as a polynomial in energy parameters. To the best of our knowledge, this is the first approach toward computing the RNA Newton polytope and a systematic assessment of the inherent capabilities of an energy model. The worst case complexity of our algorithm is exponential in the number of features. However, dimensionality reduction techniques can provide approximate solutions to avoid the curse of dimensionality.

We demonstrated the application of our theory to a simple energy model consisting of a weighted count of A-U, C-G and G-U base pairs. Our results show that this simple energy model satisfies the necessary condition for more than half of the input unpseudoknotted sequence-structure pairs (55%) chosen from the RNA STRAND v2.0 database and severely violates the condition for ~ 13%, which provide a set of hard cases that require further investigation. From 1350 RNA strands, the observed 3D feature vector for 749 strands is on the surface of the computed polytope. For 289 RNA strands, the observed feature vector is not on the boundary of the polytope but its distance from the boundary is not more than one. A distance of one essentially means one base pair difference between the observed structure and the closest point on the boundary of the polytope, which need not be the feature vector of a structure. For 171 sequences, this distance is larger than two, and for only 11 sequences, this distance is larger than five.

The source code is available on http://compbio.cs.wayne.edu/software/rna-newton-polytope.