DNA Solution of the Maximal Clique Problem

1
DNA Solution of the Maximal Clique Problem

• Q. Ouyang, P. D. Kaplan, S. Liu, A. Libchaber
• ??? ??

2
The Clique Problem

• Every vertex is connected to every other vertex.
• Represented by an N-digit binary number.
• 1 a vertex in the clique, 0 a vertex not in
  the clique
• Complementary graph containing all edges
  missing in the original graph

3
Algorithm

1. Transform the complete set of possible cliques
   into an ensemble of all N-digit binary numbers
   (complete data pool.)
2. Find pairs of vertices that are not connected by
   an edge.
3. Eliminate numbers containing connections in the
   complementary graph.
4. Find the data containing the largest numbers of
   1s.

4
DNA encoding

• Form double-stranded DNA (dsDNA).
• Each bit in a binary number is represented by two
  DNA sections
• Pi position, P6 is needed for PCR
  amplification.
• Vi bits value
• Length
• Pi 20bp (base pairs)
• Vi 0 bp if the value is 1, 10 bp if 0.
• Longest DNA(000000) has 200bp, shortest(111111)
  has 140bp.

5
Constructing the DNA data poolParallel overlap
assembly (POA)

• Each oligonucleotide consists of two position and
  one value
• for even i , for odd I
• Starts with the 12 oligonucleotides listed in
  next table.
• During each thermal cycle, the position strings
  annealed to the complementary string of the next
  oligonucleotide.
• After a few thermal cycles, a data pool with all
  combinations of V0V1V2V3V4V5 was built.

6
Oligonucleotides used to construct the DNA data
pool

• Value sequences are written with lowercase
  letters.
• Restriction enzyme sites are indicated by
  underlining.
• Each restriction enzymes were applied to cut each
  values.
• Ex) Afl II V0, Hind III V1, Spe I V2, etc.

7
Parallel overlap assembly (POA)
8
Eliminating vertices not in clique

• Ex) Eliminate
• 0-2 connection
• Divide the data pool into t0 and t1.
• In t0, cut strings containing v01
• In t1, cut strings containing v21
• Combine t0, t1.

9
Reading the answer

• The lowest band is the answer
• The clique of largest size is represented by the
  shortest length of DNA.
• Use molecular cloning
• Insert the DNA of the answer into M13
  bacteriophage.
• The mutagenized M13 phage DNA was transfected
  into E. coli bacteria, cloned, and its DNA
  extracted and sequenced.
• The result is the correct answer.

10
Error Handling

• Production of ssDNA during PCR
• This cannot be cut by restriction enzymes
• Avoid this error by digesting the ssDNA with S1
  nuclease before restriction digestions.
• Incomplete cutting by restriction enzymes
• Repeating the digestion-PCR process increase the
  s/n ratio arising from incomplete digestion.
• The selected restriction enzymes worked well.

11
Conclusion Future work

• Advantage of DNA computing.
• High parallelism.
• Size is limited.
• 27 with picomole, 36 with nanomole chemistry.
• New algorithms are needed.
• Resembling in vitro evolution.
• Initial data pool need not contain every possible
  final answer.
• Rapid and accurate data access are needed.
• Current technology are either too slow or too
  noisy.