A steepest descent calculation of RNA pseudoknots

We enumerate possible topologies of pseudoknots in single-stranded RNA molecules. We use a steepest-descent approximation in the large N matrix field theory, and a Feynman diagram formalism to describe the resulting pseudoknot structure. An RNA molecule is a heteropolymer strand made up of four types of nucleotides, uracil (U), adenine (A), guanine (G), and cytosine (C). The sequence of these nucleotides, or bases, makes up the molecule's primary structure. Bases form hydrogen bonds with each other to give the molecule a stable shape in three dimensions, with U bonding to A, and C to G. Calculating the shape a given primary structure will fold into is important in molecular biology. We can associate −U ij with the energy of forming a hydrogen bond between the ith and jth bases, and let V ij = exp(U ij /T) where T is the temperature. This is a minimalist model: we make no attempt to account for loop penalties or stacking interactions. There is some rigidity in the chain of nucleotides, as well as steric constraints , which prevent hydrogen bonding between nucleotides that are within four bases of each other, so we let V i,i+k = 0 if k < 4. The partition function associated with this bonding is given by