1. Solution of Satisfiability Problem on a Gel-Based DNA computer
Ji Yoon Park
Dept. of Biochem
Hanyang University

2. Abstract 1. Succeeded in solving an instance of a 6-variable 11-
clause 3-SAT problem on a gel-based DNA computer
2. Separation were performed using probes covalently
bound to polyacrylamide gel
3. During the entire computation, DNA was retained
within a single gel and moved via electrophoresis
4. To be readily automatable and should be suitable for
problems of a significantly larger size

3. I. Introduction  = (x1∨￢ x2∨￢ x3)∧(x2∨￢ x3∨￢ x4)∧(x3∨￢ x4∨x5) ∧
 has a unique solution: x1 = x2 = … x6 = true

4. - two distinct 15 base value sequences were designed
◈ To represent all possible variable assignments for the chosen 6-variable SAT problem, a Lipton encoding was used
- For each of the 6 variables x1, x2, · · · , x6
- two distinct 15 base value sequences were designed
: true (T) XkT , false(F) XkF
- Each of the 26 truth assignments was represented by a library sequence of 90 bases consisting of the concatenation of one value sequence for each variable.
- DNA molecules with library sequences are termed library strand
- Combinatorial pool containing library strands is termed a library
- The probes used for separating the library strands have sequences complementary to the value sequences
- Errors in the separation of the library strands are errors in the computation
- Sequences must be designed to ensure that library strands have little secondary structure which might inhibit intended probe-library hybridization

5. 2.1 Design of the library
The value sequences generated to represent x1= F, x2= F, · · · , x6= F were:
X1F= 5’ - TATTCTCACCCATAA - 3’ X2F= 5’ - ACACTATCAACATCA - 3’
X3F= 5’ - CCTTTACCTCAATAA - 3’ X4F= 5’ - CTCCCAAATAACATT - 3’
X5F= 5’ - AACTTCACCCCTATA - 3’ X6F= 5’ - TCATATCAACTCCAC - 3’
The value sequences generated to represent x1= T, x2= T, · · · , x6= T were:
X1T= 5’ - CTATTTATATCCACC - 3’ X2T= 5’ – ACACCTAACTAAACT - 3’
X3T= 5’ - CTACCCTATTCTACT - 3’ X4T= 5’ – ATCTTTAAATACCCC - 3’
X5T= 5’ - TCCATTTCTCCATAT - 3’ X6T= 5’ – TTTCTTCCATCACAT - 3’

6. Sequences were computer-generated to satisfy the following constraints:
1. Library seqs contain only A’s, T’s, C’s.
2. All libray and probe seqs have no occurrence of 5 or more consecutive identical nucleotides; i.e. no runs of more than 4 A’s, 4 T’s, 4 C’s or 4 G’s occur in any library or probe seqs.
3. Every probe seq has at least 4 mismatches with all 15 base alignment of any library seq(except for with its matching value seq)
4. Every 15 base subseq of a library seq has at least 4 mismatches with all 15 base alignment of itself or any other library seq
5. No probe seq has a run of more than 7 matches with any 8 base alignment of any library seq(except for with its matching value seq)
6. No library seq has a run of more than 7 matches with any 8 base alignment of itself or any other library seq
7. Every probe seq has 4, 5, or 6 Gs in its seq

7. 2.2 Synthesis of the library and probes
- Mix-and-split combinatorial synthesis technique
- A dual column ABI 392 DNA/RNA synthesizer at a 1µmole scale on CPG solid support.
- The library strands: (5’-X1-X2-X3-X4-X5-X6-3’)
- Synthesis began by assembling the two 15 bases oligonucleotides with sequences X6T and X6F in separate columns
- The columns were then removed from the synthesizer and opened.
; the CPG beads in the columns were removed and mixed together.
One half of the beads were retruned to the first column and the other half
to the second
- Synthesis continued with sequences X5T and X5F. This process was repeated until all 6 variables had been treated.
- 12 probes, having sequences XkF, XkT, k= and modified at the 5’-end with AcryditeTM

8. 2.3 Library capture analysis
To determine the efficiency of library capture and release by gel-embedded probes
- preparation of gels
- Running the gels

9. 2.4 Confirming integrity of the library via PCR
To verify the degeneracy and integrity of the library, the library was amplified via PCR
20 PCR reactions were performed on the library using 5’- end primers with sequences X1T or X1F and 3’- end primers with sequences X2T, , X6T or X2F,…, X6F

10. 2.5 The algorithm
- Coupling of the AcryditeTM phosphoramidite to DNA probes allows the probes to be immobilized in a polyacrylamide gel matrix
- During electrophoresis at low temp, such probes hybridize with and capture passing DNA molecules bearing complementary subsequences.
- DNA molecules without complementary subsequences pass through the gel relatively unhindered.
- Captured DNA strands can be released by running electrophoresis
- Released molecules can be used in subsequent steps as required

11. 1. For each of the 11 clauses of  prepare a polyacrylamide gel capture layer
containing three AcryditeTM modified probes, on for each literal in the
clause( If xk appears in the clause, add a probe with sequence Xk; if ￢ xk
appears add a probe with sequence for XkF )
2. Layer while heating the areas of the gel preceding and following it. Begin electrophoresis to move the library through the first capture layer. Molecules encoding truth assignments satisfying the first clause will be captured in the first capture layer, while molecules encoding non-satisfying assignments will run through the first capture layer and continue beyond the second capture layer.
3. Cool the area of the gel containing the second capture layer while heating the areas of the gel preceding and following it. Molecules captured in the first capture layer will be released to move through the second capture layer. Released molecules encoding truth assignments satisfying the second clause will be captured in the second capture layer, while molecules encoding non-satisfying assignments will run through the second capture layer and continue beyond the third capture layer.

12. 2.6 Construction and running of the computer
- Preparation of the modules
- Loading the modules
- Heating and cooling the capture layers
Fig 1. Preparation of a clause module

13. 2.7 Computation
Fig 2. Apparatus assembled for computation

14. 2.8 Determination of answer strand
- PCR
- Sequencing

15. Fig 3. Capture of the library by gel-embedded probes

16. Fig 4. PCR analysis of the original library
X1T, X2T, X6T
X1F, X2T, … X6T
X1F, X2T, ,X6T probe
X1T, X2F, …, X6F
X2F and X2F, … X6F
Fig 4. PCR analysis of the original library

17. Fig 5. Readout of the answer by PCR

18. Fig 6. Sequencing of the diluted answer strands

19. Prospects for scaling up
Whether SAT problems of greater size may be solved depends on the difficulty of scaling up each of three procedures
1) design of the library strands
- X1T, …, X6T and X1F, …, X6F
- Longer library strands composed of these sequences performs,
sequence design does not seem to be a limiting factor
2) synthesis of the library strands
- variable library strands synthesized by a mix-and-spilt synthesis
- Each library is tested separately by running a capture analysis and
simple computation
3) execution of the computation
- Enough to complete a successful 20-variable computation

20. Discussion - Successful DNA computation on a 6-variable SAT problem
- The correct solution was culled from 64 alternatives
- Optimistic about the prospects of building an automated device for carrying out such computations