VASSILIEV KNOT INVARIANTS AND THE STRUCTURE OF RNA FOLDING

Abstract

It is the purpose of this paper to introduce certain combinatorial structures into the study of RNA folding. These structures are useful for the classification of foldings and for the topological classification of the embeddings of these foldings into three-dimensional space. Both the abstract classification and the topological classification are highly relevant to problems in molecular biology where these folded structures are instantiated as molecules in a three dimensional ambient physical space. The paper is organized as follows. In section 2 we discuss the basic idea of a folding (folded molecule) and graphical models for such foldings. We introduce the use of the Brauer monoid for the classification of nonembedded foldings. This introduces a multiplicative structure into the set of foldings and we discuss the structure of the resulting algebra. Section 3 discusses the relationship of foldings and topological invariants of embedded rigid vertex graphs. Vertices arise in foldings as loci of a linear sequence of base pairs. We translate these folding vertices into standard 4-valent vertices and thereby obtain a translation of rigid vertex invariants to the category of folded molecular structures. This section discusses both Vassiliev invariants and a more general scheme that produces invariants of embedded foldings from any topological invariant of knots and links. Section 4 gives specific information about the Vassiliev invariants. In particular, we show how to construct a Vassiliev invariant of type 3, and we illustrate how Lie algebras give rise to Vassiliev invariants. The appendix discusses this last point in more detail. This key relationship between Lie algebras and Vassiliev invariants provides an interconnection among topological invariants, Lie algebras, Feynman diagrams and significant indices for protein