Spectral decomposition of the Laplacian matrix applied to RNA folding prediction

Abstract

RNA secondary structure consists of elements such as stems, bulges, loops. The most obvious and important scalar number that can be attached to an RNA structure is its free energy, with a landscape that governs the folding pathway. However, because of the unique geometry of RNA secondary structure, another interesting single-signed scalar number based on geometrical scales exists that can assist in RNA structure computations. This scalar number is the second eigenvalue of the Laplacian matrix corresponding to a tree-graph representation of the RNA secondary structure. Because of the mathematical properties of the Laplacian matrix, the first eigenvalue is always zero, and the second eigenvalue (often denoted as the Fiedler eigenvalue) is a measure of the compactness of the associated tree-graph. The concept of using the Fiedler eigenvalue/eigenvector is borrowed from domain decomposition in parallel computing. Thus, along with the free energy, the Fiedler eigenvalue can be used as a signature in a clever search among a collection of structures by providing a similarity measure between RNA secondary structures. This can also be used for mutation predictions, classification of RNA secondary folds, filtering and clustering. Furthermore, the Fiedler eigenvector may be used to chop large RNAs into smaller fragments by using spectral graph partitioning, based on the geometry of the secondary structure. Each fragment may then be treated differently for the folding prediction of the entire domain.