The RNA Newton Polytope and Learnability of Energy Parameters

Despite nearly two scores 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. Researchers have proposed increasingly complex energy models and improved parameter estimation methods 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. In this paper, we introduce the notion of learnability of the parameters of an energy model as a measure of its 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. We demonstrated the application of our theory to a simple energy model consisting of a weighted count of A-U and C-G base pairs. Our results show that this simple energy model satisfies the necessary condition for less than one third of the input unpseudoknotted sequence-structure pairs chosen from the RNA STRAND v2.0 database. For another one third, the necessary condition is barely violated, which suggests that augmenting this simple energy model with more features such as the Turner loops may solve the problem. The necessary condition is severely violated for 8%, which provides a small set of hard cases that require further investigation

A Seeded Genetic Algorithm for RNA Secondary Structural Prediction with Pseudoknots

This work explores a new approach in using genetic algorithm to predict RNA secondary structures with pseudoknots. Since only a small portion of most RNA structures is comprised of pseudoknots, the majority of structural elements from an optimal pseudoknot-free structure are likely to be part of the true structure. Thus seeding the genetic algorithm with optimal pseudoknot-free structures will more likely lead it to the true structure than a randomly generated population. The genetic algorithm uses the known energy models with an additional augmentation to allow complex pseudoknots. The nearest-neighbor energy model is used in conjunction with Turner’s thermodynamic parameters for pseudoknot-free structures, and the H-type pseudoknot energy estimation for simple pseudoknots. Testing with known pseudoknot sequences from PseudoBase shows that it out performs some of the current popular algorithms

Safe and Complete Prediction of RNA Secondary Structure

Ribonucleic acid, RNA, is an essential type of molecule for all known forms of life. It is a nucleic acid, like DNA. However, where DNA appears as two complementary strands that join and twist into a double helix structure, RNA has only a single strand. This strand can fold upon itself, pairing complementary bases. The resulting set of base pairs is the RNA secondary structure, also known as folding. It is typical that a prediction algorithm gives a large number of optimal or near-optimal foldings for an RNA sequence. Only in the simplest cases it is possible to manually go through all of these foldings, and in hard cases it is infeasible to even generate the full set of optimal foldings. In fact, we observe that the number of optimal foldings may be exponential in the sequence length, and that some naturally occurring RNA sequences of 2000–3000 bases in length have well over 10^100 optimal foldings, under the model of maximizing the number of base pairs. To help analyze the full set of optimal foldings, we apply the concept of safe and complete algorithms. In the presence of multiple optimal solutions, any partial solution that appears in all optimal solutions is called a safe part, and a safe and complete algorithm finds all of the safe parts. We show a trivial safe and complete algorithm that computes safety by going through the full set of optimal foldings. However, this algorithm is only practical for short RNA sequences that do not have too many optimal foldings. In order to analyze the harder RNA sequences, we develop and implement a novel polynomial-time safe and complete algorithm for RNA secondary structure prediction, using the model of maximizing base pairs. Using the dynamic programming approach, this new algorithm can compute how often each base pair and unpaired base appears in the full set of optimal foldings without having to produce the actual foldings. Our experimental evaluation shows that the safe parts of a folding are more likely to be biologically correct than the non-safe parts. We observe this both by using our implementation of the efficient safe and complete algorithm and by combining an existing predictor program with the trivial algorithm. As this existing predictor uses a modern minimum free energy model for predicting the RNA foldings, tests using this combination show that safety is a useful property, even beyond the simple maximum pairs model in our implementation

Thermodynamic Analysis of Interacting Nucleic Acid Strands

Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single-stranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium base-pairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality

Geometric combinatorics and computational molecular biology: branching polytopes for RNA sequences

Questions in computational molecular biology generate various discrete optimization problems, such as DNA sequence alignment and RNA secondary structure prediction. However, the optimal solutions are fundamentally dependent on the parameters used in the objective functions. The goal of a parametric analysis is to elucidate such dependencies, especially as they pertain to the accuracy and robustness of the optimal solutions. Techniques from geometric combinatorics, including polytopes and their normal fans, have been used previously to give parametric analyses of simple models for DNA sequence alignment and RNA branching configurations. Here, we present a new computational framework, and proof-of-principle results, which give the first complete parametric analysis of the branching portion of the nearest neighbor thermodynamic model for secondary structure prediction for real RNA sequences.Comment: 17 pages, 8 figure

Target prediction and a statistical sampling algorithm for RNA-RNA interaction

It has been proven that the accessibility of the target sites has a critical influence for miRNA and siRNA. In this paper, we present a program, rip2.0, not only the energetically most favorable targets site based on the hybrid-probability, but also a statistical sampling structure to illustrate the statistical characterization and representation of the Boltzmann ensemble of RNA-RNA interaction structures. The outputs are retrieved via backtracing an improved dynamic programming solution for the partition function based on the approach of Huang et al. (Bioinformatics). The $O(N^6)$ time and $O(N^4)$ space algorithm is implemented in C (available from \url{http://www.combinatorics.cn/cbpc/rip2.html})Comment: 7 pages, 10 figure

RNA secondary structure prediction from multi-aligned sequences

It has been well accepted that the RNA secondary structures of most functional non-coding RNAs (ncRNAs) are closely related to their functions and are conserved during evolution. Hence, prediction of conserved secondary structures from evolutionarily related sequences is one important task in RNA bioinformatics; the methods are useful not only to further functional analyses of ncRNAs but also to improve the accuracy of secondary structure predictions and to find novel functional RNAs from the genome. In this review, I focus on common secondary structure prediction from a given aligned RNA sequence, in which one secondary structure whose length is equal to that of the input alignment is predicted. I systematically review and classify existing tools and algorithms for the problem, by utilizing the information employed in the tools and by adopting a unified viewpoint based on maximum expected gain (MEG) estimators. I believe that this classification will allow a deeper understanding of each tool and provide users with useful information for selecting tools for common secondary structure predictions.Comment: A preprint of an invited review manuscript that will be published in a chapter of the book `Methods in Molecular Biology'. Note that this version of the manuscript may differ from the published versio