1

Minimum Pots for the Usual Suspects
Laura Beaudin, Jo Ellis-Monaghan, Natasha
Jonoska, David Miller, and Greta Pangborn
Impact on current processes
Abstract
self-assembled DNA cube and Octahedron
Three separate laboratory constraints

• 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.

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.

• The incidental construction of a graph smaller
  than G is acceptable
• 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
• Any graph incidentally constructed must be larger
  than G.
• In all cases, we assume flexible armed molecules
• (abstract, not embedded, graphs).

http//seemanlab4.chem.nyu.edu/nanotech.html
http//www.nature.com/nature/journal/v427/n6975/fu
ll/nature02307.html
molecular building blocks
Results
Optimal design parameters for the usual
suspects, important basic classes of graphs in
each scenario
Table A Minimum Tile Types Table A Minimum Tile Types
Scenario 1 T1(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 T1(G) The number of different even vertex degrees 2(The number of different odd vertex degrees).
Trees The number of different vertex degrees T1(T ) The number of different vertex degrees 1
Cn T1(Cn) 1
Kn T1(Kn) 1 if n is even, and T1(Kn) 2 if n is odd
Kn,m T1(Kn,m) 1 if nm and even, and T1(Kn,m) 2 otherwise
K-regular graphs T1(G) 1 if n is even, and T1(G) 2 if n is odd
Scenario 2 T2(G) Minimum number of tile types required if allow complexes of the same size as the target graph
Trees T2(T) The number of different lesser size subtree sequences
Cn T2(Cn) ceiling(n/2)1
Kn T2(Kn) 2 if n is even, and T2(Kn) 3 if n is odd
Kn,m T2(Kn,m) 2 if gcd(m,n)1, and T2(Kn,m) 3 if gcd(m,n)gt1
Scenario 3 T3(G) Minimum number of tile types required if do not allow complexes of the same size as (or smaller than) the target graph
Trees T3(T) the number of induced subtree isomorphisms
Cn T3(Cn) ceiling(n/2)1
Kn T3(Kn) n
Kn,m T3(Kn,m) min(n,m)1
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
K-armed branched junction molecules
Courtesy Seeman Laboratory
Why self-assembling nanostructures?
Literature Cited
 

• 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.
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  conference on Foundations of Nanoscience
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  kilobase single-stranded DNA that folds into a
  nanoscale octahedron, Nature, 427 (2004),
  618-621.
• B. Steele, Buckyballs demonstrate DNA as
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combinatorial abstraction
Individual letters represent unmatched sequences
of bases

• Biomolecular computing (Hamilton Cycle/3-Sat)
• Nanoelectronics
• Fine screen filters (lattices) at the nano-size
  scale
• Biosensors and drug delivery mechanisms

Table B Minimum Bond-Edge Types Table B Minimum Bond-Edge Types
Scenario 1 B1(G) Minimum number of bond-edge types required if allow complexes of smaller size than the target graph
General graph G B1(G) 1 for all graphs
Scenario 2 B2(G) Minimum number of bond-edge types required if allow complexes of the same size as the target graph
Trees B2(T) The number of different sizes of lesser size subtrees
Cn B2(Cn) ceiling(n/2)
Kn B2(Kn) 1 if n is even, and B2(Kn) 2 if n is odd
Kn,m B2(Kn,m) 1 if gcd(m,n)1 , and B2(Kn,m) 2 if gcd(m,n)gt1
Scenario 3 B3(G) Minimum number of bond-edge types required if do not allow complexes of the same size as (or smaller than) the target graph
Trees B3(T) The number of induced subtree isomorphisms -1
Cn B3(Cn) ceiling(n/2)
Kn B3(Kn) n 1
Kn,m B3(Kn,m) min(m,n)
Acknowledgments
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.
A pot P representing branched junction molecules
http//www.news.cornell.edu/stories/Aug05/DNABucky
balls.ws.html
http//www.nanopicoftheday.org/2004Pics/April2004/
DNAmesh.htm
Two complete complexes built from this pot