Chapter 13 RNA Secondary Structures: Combinatorial Models and Folding Algorithms Qijun He 1 , Matthew Macauley 1 and Robin Davies 2 1 Department of Mathematical Sciences, Clemson University, Clemson, SC, USA, 2 Department of Biology, Sweet Briar College, Sweet Briar, VA, USA 13.1 INTRODUCTION Deoxyribonucleic acid (DNA) and ribonucleic acid (RNA) are examples of nucleic acids—large organic molecules that are polymers of smaller molecules called nucleotides. Nucleotides consist of a fve-carbon sugar to which are linked a nitrogenous base and a phosphate. The nucleotides are linked together through phosphodiester bonds that link the sugars to phosphates on adjacent nucleotides, resulting in a sugar-phosphate backbone. DNA is better known than RNA to the general public, as it encodes the essential genetic information for all cells, is standard material in most high school biology classes, and has even made its way into common parlance, such as “it is in my DNA.” In cells, as well as in the double-stranded DNA viruses, DNA consists of two chains of nucleotides that pair to each other via hydrogen bonds, forming a double-helix structure. In contrast, RNA consists of a single strand of nucleotides that can fold and bond to itself. The specifc shape into which RNAs fold plays a major role in their function, which makes RNA folding of prime interest to scientists. Initially, RNA was regarded as a simple messenger—the conveyor of genetic information from its repository in DNA to the ribosomes. The information encoded in the RNA was then used to direct the construction of proteins, which were then thought to be the only actively functional molecules of the cell. Over the last several decades, however, researchers have discovered an increasing number of important roles for RNA. RNAs have been found to have catalytic activities, to participate in processing of messenger RNAs, to help maintain the telomers (ends) of eukaryotic chromosomes, and to infuence gene expression in multiple ways. Clearly, RNA is more than a simple messenger. For more information on the growing recognition of the role of RNA, see the account by Darnell [1]. Not surprisingly, as is true of proteins, the three-dimensional structure of RNA is critically important to its function. Structure determines function, so an understanding of RNA’s three-dimensional structure will allow a greater understanding of RNA function. This should lead to the discovery of additional RNA-encoding genes and to the development of RNA-based therapeutic agents. One can casually think of a nucleic acid chain as a length of ribbon, to which squares of a hook and loop fastener, such as Velcro  , are attached. Furthermore, suppose this ribbon has some thickness and rigidity— like that of a stiff belt one would wear, rather than a decorative ribbon used to wrap a gift. In a DNA model, two separate strands of ribbon are attached together, through the linking of hooks on one ribbon and loops in the corresponding position on the other ribbon. In an RNA model, the position of hooks and loops would allow the ribbon to fold and attach to itself. In this metaphor, the ribbon represents the sugar-phosphate backbone, the squares of Velcro  represent nucleotides, and the attachment of the hooks and loops represents the formation Algebraic and Discrete Mathematical Methods for Modern Biology. http://dx.doi.org/10.1016/B978-0-12-801213-0.00013-7 Copyright © 2015 Elsevier Inc. All rights reserved. 321