1. Impact on current processes Acknowledgments Literature Cited Abstract Minimum Pots for the Usual Suspects Recent exciting advances in DNA self-assembly of mathematical constructs have resulted in nanoscale cubes, octahedrons, truncated octahedra, and even buckyballs, as well as ultra-fine meshes. These constructs serve emergent applications in biomolecular computing, nanoelectronics, biosensors, drug delivery systems, and organic synthesis. One construction method uses k-armed branched junction molecules, called tiles, whose arms are double strands of DNA with one strand extending beyond the other, forming a ‘sticky end’ at the end of the arm that can bond to any other sticky end with complementary Watson-Crick bases. A vertex of degree k is formed from a k-armed branched molecule, and joined sticky ends form the edges of the target graph. We use graph theory to model this application and to determine optimal design strategies for biologists producing these nanostructures. We find the minimum number of tiles and edge types necessary to create a given graph under three different laboratory scenarios for common graph classes (complete, bipartite, trees, regular, etc.). For these classes of graphs, we provide either explicit descriptions of the set of tiles achieving the minimum number of tile and bond edge types, or efficient algorithms for generating the desired set. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Algorithms, Addison Wesley, (1974), pp. 84-86. J. Chen, N.C. Seeman, “Synthesis from DNA of a molecule with the connectivity of a cube,” Nature, 350 (1991), 631-633. N. Jonoska, G. McColm, A. Staninska, “Spectrum of a pot for DNA complexes,” in DNA Computing 12 (editors: C. Mao, T. Yokomori), Springer LNCS, 4287 (2006), 83- 94. N. Jonoska, G. McColm, A. Staninska, “Expectation and Variance of Self-Assembled Graph Structures,” in DNA Computing (DNA11), (editors: A. Carbone N.A. Pierce), Springer LNCS, 3892 (2006), 144--157. T.H. LaBean, H. Li, “Constructing novel materials with DNA,” nanotoday, 2 no. 2 (2007), 26-35. A. Staninska, “The Graph of a Pot with DNA molecules,” Proceedings of the 3rd annual conference on Foundations of Nanoscience (FNANO'06), April 2006, 222-226. W.M. Shih, J.D. Quispe, G.F. Joyce, “A 1.7 kilobase single-stranded DNA that folds into a nanoscale octahedron,” Nature, 427 (2004), 618-621. B. Steele, “Buckyballs demonstrate DNA as building material,” Cornell Chronical, September 1 (2005), 9. H. Yan, S.H. Park, G. Finkelstein, J. Reif, T. LaBean, “DNA-templated self-assembly of protein arrays and highly conductive nanowires,” Science, 301 (2003), 1882-1884. Y. Zhang, N.C. Seeman, “Construction of a DNA-truncated octahedron,” J. Am. Chem. Soc., 116 (1994), 1661-1669. The project described was supported in by the Vermont Genetics Network through NIH Grant Number 1 P20 RR16462 from the INBRE program of the National Center for Research Resources, and by a National Security Agency Standard Grant. Table A: Minimum Tile Types Scenario 1T 1 (G) = Minimum number of tile types required if complexes of smaller size than the target graph are allowed General graph G The number of different vertex degrees ≤ T 1 (G) ≤ The number of different even vertex degrees + 2*(The number of different odd vertex degrees). TreesThe number of different vertex degrees ≤ T 1 (T ) ≤ The number of different vertex degrees + 1 CnCn T 1 (C n ) = 1 KnKn T 1 (K n ) = 1 if n is even, and T 1 (K n ) = 2 if n is odd Kn,mKn,m T 1 (K n,m ) = 1 if n=m and even, and T 1 (K n,m ) = 2 otherwise K-regular graphsT 1 (G) = 1 if n is even, and T 1 (G) = 2 if n is odd Scenario 2T 2 (G) = Minimum number of tile types required if allow complexes of the same size as the target graph TreesT 2 (T) = The number of different lesser size subtree sequences CnCn T 2 (C n ) = ceiling(n/2)+1 KnKn T 2 (K n ) = 2 if n is even, and T 2 (K n ) = 3 if n is odd Kn,mKn,m T 2 (K n,m ) = 2 if gcd(m,n)=1, and T 2 (K n,m ) = 3 if gcd(m,n)>1 Scenario 3T 3 (G) = Minimum number of tile types required if do not allow complexes of the same size as (or smaller than) the target graph TreesT 3 (T) = the number of induced subtree isomorphisms CnCn T 3 (C n ) = ceiling(n/2)+1 KnKn T 3 (K n ) = n Kn,mKn,m T 3 (K n,m ) = min(n,m)+1 Table B: Minimum Bond-Edge Types Scenario 1B 1 (G) = Minimum number of bond-edge types required if allow complexes of smaller size than the target graph General graph G B 1 (G) = 1 for all graphs Scenario 2B 2 (G) = Minimum number of bond-edge types required if allow complexes of the same size as the target graph TreesB 2 (T) = The number of different sizes of lesser size subtrees CnCn B 2 (C n ) = ceiling(n/2) KnKn B 2 (K n ) = 1 if n is even, and B 2 (K n ) = 2 if n is odd Kn,mKn,m B 2 (K n,m ) = 1 if gcd(m,n)=1, and B 2 (K n,m ) = 2 if gcd(m,n)>1 Scenario 3B 3 (G) = Minimum number of bond-edge types required if do not allow complexes of the same size as (or smaller than) the target graph TreesB 3 (T) = The number of induced subtree isomorphisms -1 CnCn B 3 (C n ) = ceiling(n/2) KnKn B 3 (K n ) = n – 1 Kn,mKn,m B 3 (K n,m ) = min(m,n) Why self- assembling nanostructures? Biomolecular computing (Hamilton Cycle/3-Sat) Nanoelectronics Fine screen filters (lattices) at the nano-size scale Biosensors and drug delivery mechanisms http://www.nanopicoftheday.org/2004Pics/April2004/DNAmesh.htm http://www.news.cornell.edu/stories/ Aug05/DNABuckyballs.ws.html self-assembled DNA cube and Octahedron http://seemanlab4.chem.nyu.edu/nanotech.html 22 nanometers molecular building blocks K-armed branched junction molecules Y-shaped DNA, Schematic diagrams of the structure (left) and sequence (middle) of Y-DNA, and dendrimer-like DNA (right). D. Luo, “The road from biology to materials,” Materials Today, 6 (2003), 38-43 combinatorial abstraction A pot P representing branched junction molecules a c s â ĉ ŝ â ĉ s t1t1 t4t4 a c ŝ t3t3 t2t2 ATTCG TAAGCCCATTG GGTAACATTCG TAAGC a â Two complete complexes built from this pot t1t1 t2t2 t3t3 ŝ t2t2 ac t1t1 t4t4 Three separate laboratory constraints 1.The incidental construction of a graph smaller than G is acceptable 2.The incidental construction of a graph smaller than G is not acceptable but a graph with the same size as G (same number of edges and vertices) is acceptable 3.Any graph incidentally constructed must be larger than G. In all cases, we assume flexible armed molecules (abstract, not embedded, graphs). Results The original grid design paradigm for a square lattice involved over 12 tile types (at over $1000 per tile), and hand intervention at the corners. New hierarchical design uses only two tiles and no interventions. Courtesy Seeman Laboratory Laura Beaudin, Jo Ellis-Monaghan*, Natasha Jonoska, David Miller, and Greta Pangborn http://www.nature.com/nature/journal/v427/n6975/full/nat ure02307.html Optimal design parameters for the ‘usual suspects’, important basic classes of graphs in each scenario Individual letters represent unmatched sequences of bases