Asymptotic distribution of motifs in a stochastic context-free grammar model of RNA folding

RNA Secondary Structures with Given Motif Specification: Combinatorics and Algorithms

The study of native motifs of RNA secondary structures helps us better understand the formation and eventually the functions of these molecules. Commonly known structural motifs include helices, hairpin loops, bulges, interior loops, exterior loops and multiloops. However, enumerative results and generating algorithms taking into account the joint distribution of these motifs are sparse. In this paper, we present progress on deriving such distributions employing a tree-bijection of RNA secondary structures obtained by Schmitt and Waterman and a novel rake decomposition of plane trees. The key feature of the latter is that the derived components encode motifs of the RNA secondary structures without pseudoknots associated with the plane trees very well. As an application, we present an algorithm (RakeSamp) generating uniformly random secondary structures without pseudoknots that satisfy fine motif specifications on the length and degree of various types of loops as well as helices.

Distribution of genotype network sizes in sequence-to-structure genotype-phenotype maps

An essential quantity to ensure evolvability of populations is the navigability of the genotype space. Navigability, understood as the ease with which alternative phenotypes are reached, relies on the existence of sufficiently large and mutually attainable genotype networks. The size of genotype networks (e.g. the number of RNA sequences folding into a particular secondary structure or the number of DNA sequences coding for the same protein structure) is astronomically large in all functional molecules investigated: an exhaustive experimental or computational study of all RNA folds or all protein structures becomes impossible even for moderately long sequences. Here, we analytically derive the distribution of genotype network sizes for a hierarchy of models which successively incorporate features of increasingly realistic sequence-to-structure genotype-phenotype maps. The main feature of these models relies on the characterization of each phenotype through a prototypical sequence whose sites admit a variable fraction of letters of the alphabet. Our models interpolate between two limit distributions: a power-law distribution, when the ordering of sites in the prototypical sequence is strongly constrained, and a lognormal distribution, as suggested for RNA, when different orderings of the same set of sites yield different phenotypes. Our main result is the qualitative and quantitative identification of those features of sequence-to-structure maps that lead to different distributions of genotype network sizes.