1. DNA Solution of the Maximal Clique Problem
Cell and Microbial Engineering Laboratory
Lee Ji Youn

2. Introduction Clique Maximal clique problem
: defined as a set of vertices in which every vertex is connected to every other vertex by an edge
Maximal clique problem
: Given a network containing N vertices and M edges, how many vertices are in the largest clique?
Finding the size of the largest
clique has been proven to be
an NP-complete problem

3. Algorithm Step 1. Make the complete data pool Step 2.
For a graph with N vertices, each possible clique is represented by an N-digit binary number
1: a vertex in the clique : a vertex out of the clique
example) clique (4,1,0)  binary number
Step 2.
Find pairs of vertices in the graph that are not connected by an edge
(0,2) (0,5) (1,5) (1,3)
The complementary graph

4. Step 3.
Eliminate from the complete data pool all numbers containing connections in the complementary graph
 xxx1x1 or 1xxxx1 or 1xxx1x or xx1x1x
Step 4.
Sort the remaining data pool to find the data containing the largest number of 1’s
 the clique with the largest number of 1’s tells us the size of the maximal clique

5. Experiment - Construction of DNA molecules
Complete data pool
two DNA sections
bit’s value (Vi) V0~V bp when Vi =1
10 bp when Vi =0
position value (Pi) P0~P6 20 bp
Longest = 610 + 720 = 200bp (000000)
Shortest = 60 + 720 = 140bp (111111)
dsDNA

6. sequence construction - randomly generated
to avoid mispairing, avoid accidental homologies longer than 4bp
embedded restriction sequences within each Vi =1
POA (parallel overlap assembly)

7. Experiment - POA POA (parallel overlap assembly) 12 oligonucleotides
PiViPi+1 for even i
<Pi+1 Vi Pi> for odd i
P0V0P1 P2V2P3
P4V4P5
<P2V1P1> <P4V3P3>
<P6V5P5>
PCR with P0 and <P6>
as primers (lane2 in fig3)

10. Experiment - Digestion with restriction enzyme
Break DNA : internal sequence Vi =1
PCR with P0 and <P6> as primers
broken sting were not amplified
Division of the data pool into two test tube
t0 : Alf II cut Vo=1
t1 : Spe I cut V2=1
 combine t0 and t1 into
test tube t, which did not
contain xxx1x1

11. Experiment - Digestion and PCR amplification
Elimination all strings connected by edges
xxx1x1, 1xxxx1, 1xxx1x, xx1x1x
PCR amplification of remaining data DNA
Fig 3,
Lane 5: digestion result
Lane 6: PCR result

12. Experiment - Readout the data
Reading the size of the largest clique(s)
shortest length : 160bp
 four vertices
What is the maximal clique?
6C4 = 15, 15 different strings
read the answer by molecular cloning
1) insertion the DNA into M13 bacteriophage through site-directed mutagenesis
2) transfection of the mutagenized M13 phase DNA into E.coli
3) cloning
4) DNA extraction and sequencing

13. Result
correct answer

14. Discussion - Major error
Production of ssDNA during PCR
cannot be cut by restriction enzymes
solution : digestion of the ssDNA with S1 nuclease before restriction digestion
Incomplete cutting by restriction enzymes
repetition of digestion-PCR process
increase the signal-to-noise

15. Discussion - Strengths and Weaknesses
high parallelism
Weaknesses
limitation on the number of vertices that this algorithm can handle
maximum number of vertices with picomole operations = 27 (36 vertices with nanomole)
exponential increase in the size of the pool with the size of the problem
 Further scale-up becomes impractical
 New algorithms are needed

16. Discussion - Future direction
Rapid and accurate data access is needed
biotin-avidin purification
electrophoresis
DNA cloning
 too slow/ too noising
 biochip is needed to accelerate readout