Nicholas Bruno, Akie Hashimoto, and Paul Jarvis Saint Michael’s College

1. a b b c Nicholas Bruno, Akie Hashimoto, and Paul Jarvis Saint Michael’s College Mathematical Models for Building Graphical Complexes from DNA Results in the Three Separate Scenarios Abstract We examine the use of flexible branched junction molecules with free ‘sticky ends’ created from DNA as the building blocks for graphical complexes. These molecules self-assemble via Watson-Crick complementary pairing. We will determine the minimum number of branched junction molecules, called tiles, and ‘sticky end’ types, called bond-edge types, needed in order to create different polyhedrons under three scenarios: Scenario One: Where smaller complexes can be formed from the set of tiles used to create the target graph; Scenario Two: Where the formation of complexes of the same size, but not smaller, from the set of tiles is allowed; Scenario Three: Where no complexes of equal or lesser size are allowed. In each of the cases we give bounds for the minimum number of bond-edge and tile types needed in order to create the target graphs, and explicit sets of tiles that create the target graphs. Scenario One Scenario Two Scenario Three Pot for a Bipyramid with n even Observation 3.1. Complementary sticky end types can not be on the same tile because this allows for the formation of a loop. Spectrum of a Pot with : If the number of bond-edge types equals the number of tile types you get this: (where av(G) is the number of unique vertex degrees and ev(G) and ov(G) are the numbers of unique even and odd vertex degrees, respectively.) Pot for a Bipyramid with n odd contradiction Pot for a Pyramid Theorem 3.2. Adjacent vertices can not be of the same tile type, so the chromatic number gives a lower bound. Pot for a Trapezohedron Background Sticky end cohesion Pot for a Pyramid with n odd Pyramid Bipyramid The combinatorial representation of a branched junction molecule is a tile and the representation of a sticky end is a bond-edge [B+]. Tiles are then vertices with labeled half edges. Different bond-edge types are labeled with different letters, and complementary bond-edge types are labeled as “hatted” letters. Tile types are distinguished by the number and type of bond-edges they have. A collection of tile types, such that for each tile containing a sticky end of type h there exists a tile with a sticky end of type is called a pot [JMS06]. Pot for a Pyramid with n even Graphical Example of Making a Complex Using Branched Junction Molecules With Sticky Ends Trapezohedron a Conclusion b d = c Using mathematical tools, we determined the minimum tile and bond-edge types needed in order to create several different polyhedrons. This process was based on self-assembling DNA branched junction molecules with flexible arms. Recent research has shown that these molecules and other self-assembling DNA molecules have several practical real-life applications, especially in the areas of nanotechnology and computation. Ignoring the laboratory complications associated with DNA self-assembly, we sought to apply mathematical tools in order to develop the theory determining several conditions necessary for the formation of three-dimensional polyhedron complexes. This in turn is to make the formation of these complexes as efficient as possible. + a b d Spectrum of a Pot [JMS06] proportion of tile type needed to create a complete complex Literature Cited number of sticky ends of type h on tile [B+] Beaudin, Lauren; Ellis-Monaghan, Johanna; Jonoska, Natasha; Miller, David; Pangborn, Greta. Minimal Tile and Bond-EdgeTypes for Self-AssemblingDNA Graphs. In progress. [JMS06] Jonoska, Natasha; McColm, Gregory L.; Staninska, Ana. Spectrum of a Pot for DNA Complexes, in DNA Computing 12 (editors: C Mao, T.Yokomori), Springer LNCS, 4287 (2006), 83-94. Spectrum Example Acknowledgments The project was supported by the Vermont Genetics Network through NIH Grant Number 1 P20 RR16462 from the INBRE program of the National Center for Research Resources, by a National Security Agency Standard Grant, and by a Saint Michael’s College Provost Grant. is a solution to the system.