A MULTIFRACTAL WALK DOWN WALL STREET FREE
His method allows comparison of two different RNA 2D structures by approximating the free energy based on the adjacency matrix representations. These planar graphs were analyzed for base pairing using an adjacency matrix. Specifically, he offered the first graph-theoretic definition of secondary structure and classified graphs of RNA secondary structures with the goal of finding stable 2D structures. Waterman developed graphical representations of RNA in 1978 with the aim of analyzing the secondary structure of tRNA ( Figure 2b). Pioneering modeling of graphs for RNA began in the late 1970s. Long-range interactions between secondary structural motifs form complex tertiary network-like structures.ģ.1 Early Graph Approaches by Waterman, Nussinov and Shapiro When two single-stranded regions flanked by a stem are base-paired, an interwined RNA structure called pseudoknot forms (see Figure 1 top middle, and top right, for the interwining green and red stems). This single-stranded polymer folds upon itself, to form GC, AU, or GU (“wobble”) base pairs which define double-helical regions (“stems”), imperfect with single-stranded regions named “hairpin loops”, “internal loops”, and “junctions”, which have one, two, or more adjacent helical arms, respectively (see 2D structure elements in Figure 2a and Figure 3). RNA is a single-stranded polymer whose sugar-phosphate backbone contains four standard bases, Adenine (A) and Guanine (G), Uracil (U) and Cytosine (C), and their modified bases in various order. These structures and their analyses have established RNA’s hierarchical structure, in which building blocks (motifs) combine stepwise to form complex active shapes of RNA : The primary (1D) structure or sequence leads to the seconday (2D) structure – pattern of hydrogen bonding arrangements – which in turn triggers tertiary (3D) structure formation, created by all interactions including long-range contacts between 2D substructures (see, Figure 1). Information on RNA structure comes from experimental information (X-ray crystallography, NMR, chemical probing) as well as modeling (e.g., as reviewed in ). These new discoveries have also opened new avenues for thinking about therapeutic biotechnology applications of RNA, because RNA’s editing, silencing, and other regulatory capabilities can be manipulated to shut off and turn on genes, deliver drugs, diagnose gene activity, and design new materials. Beyond protein synthesis, RNA can regulate gene expression and modify the genetic message by gene silencing, chemical modifications of ribosomal RNAs, control of protein stability, and changing conformations of ligand-binding sites of messenger RNA. RNA has come to the forefront of science with recent discoveries of its regulatory roles. In this work, we describe how graphs can be used to model and study RNA structure. If E is composed instead of ordered pairs of vertices (each representing direction of an edge), G is called a directed graph (digraph). In mathematical terms, an undirected graph G = ( V, E) is a discrete object described by a finite set of vertices V and a set E of unordered pair of vertices called edges, where each edge represents a connection between two vertices. Specific examples include chemical structures (e.g., hydrocarbons, drug compounds), genetic and cellular relationship, transportation arrays, Internet linkages, and social media communications. Essentially, foundations of graph the ory can be used to enumerate and analyze combinatorial properties of various systems, such as communication, chemical, and biological networks. Network or graph theory is a well-established field of mathematics that has been used extensively in a variety of economic, social, biological, and medical contexts. Examples include quantum computing using DNA, analyzing DNA recombination products using knot theory, storing digital information within DNA, modeling economic scenarios using game theory, and applications of Mandelbrot’s fractal geometry to architecture, financial markets and turbulence. Many successes in science can be credited to borrowing tools from seemingly disparate fields and applying them in new ways. Modeling biological systems involves not only choices of what to approximate and how, as well as what can be neglected, but also selection of appropriate tools, both existing and new, to design and apply to complex problems.