We continue our investigation on harnessing DNA to perform computation and organized self-assembly.

1. Gene expression arrays. A basic tool in determining switches.
2. Review: polymerase chain reaction. An amplification technique.
3. Review: gel electrophoresis.
4. Adleman's hamiltonian path experiment: use DNA to find the a path through a graph, from v1 to vn, that visits each of the vertices exactly once. (The problem is NP-complete; think about this as a maze that visits each of several points.) Here's Adleman's little graph:

   The basic algorithm:

   1. Generate random paths.
   2. Pick those that start and top at the right vertices.
   3. Of those, keep those that visit n vertices.
   4. Of those, keep those that enter exactly the n vertices in the graph.

   1. First, he encodes each vertex as a "random" 20-base nucleotide. The edges are encoded as the 10-base section on the 3' end of the source vertex, followed by the 10-base section on the 5' end of the destination vertex.

      v2 =	TATCGGATCG GTATATCCGA
      v3 =	GCTATTCGAG CTTAAAGCTA
      v4 =	GGCTAGGTAC CAGCATGCTT
      e2->3 =	GTATATCCGA GCTATTCGAG
      e3->4 =	CTTAAAGCTA GGCTAGGTAC
      comp(v3) =	CGATAAGCTC GAATTTCGAT

   2. (Step 1) Each edge of the graph, and the complement of each vertex is poured into a solution. The result is a ligation reaction, where random paths throug the graph are formed.
   3. (Step 2) Now, we perform PCR using v1 and comp(vn). This amplifies all paths that go from v1 through vn. This includes short paths and cycles.
   4. (Step 3) We now run the result through a gel, looking for strands that are 20*n bases long. These correspond to paths through n vertices (including some cycles). Figures in the paper are worth looking at. Those strands that were 20*n bases long were cut from the gel and re-placed into solution.
   5. (Step 4) The solution now contains only strands that are the correct length. We now draw out those single strands that have an affinity for comp(v2), comp(v3), ..., comp(vn-1). This is accomplished with a specially constructed complement strand with an attached metal bead.
   6. (Final printing step.) We run n-1 PCR reactions with V1 and comp(vi) where 1 ≤ i ≤ n. The product of each reaction is run in separate gel lane. The length of the product in the lane i indicates the position of vertex i in the path.
5. The ligation computation was 1000 times faster than current computation.
6. The number of operations per joule was 2e19, whereas for common computers the value is at most 1e9, 10 billion times more efficient.
7. Information density in DNA is 1 bit per cubic nm, whereas the density for standard technologies was 100 billion times less dense.
8. Work of Hao Wang: understanding some tilings is equivalent to general computation. He constructed a universal Turing Machine from square tiles with edge colors. The "solitare game with dominoes" (ie. a consistent assembly problem) is difficult.
9. Work of Burger through Penrose and Conway: it is possible that each neighborhood in a tiling is forced to be unique; ie. that the entire tiling is aperiodic . Some of these tiling systems model real quasicrystalline structures. The simplest solid aperiodic tiling sets have two tiles.

   

10. Work by Feng and Bailey suggest that even one tile can force aperiodicity, though the tile is not solid.

    

11. Work by Erik Winfree suggests that tiles may be constructed from 4-stranded DNA; these systems are capable of counting.
12. Work by Bailey and Chen: Simple dynamic programming may be possible. This would allow us to harness DNA to make statements about homology.

Resources needed for this class:

• The first paper on DNA computation, by Len Adleman.
• Wang's Scientific American article on the undecidability of tiling problems.
• Erik Winfree's paper on self assembly.

Resources needed for the next class:

• The clone program. What features of this program make it autocatalytic? With what does it react?

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