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

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

I. Introduction  = (x 1 ∨ ￢ x 2 ∨ ￢ x 3 ) ∧ (x 2 ∨ ￢ x 3 ∨ ￢ x 4 ) ∧ (x 3 ∨ ￢ x 4 ∨ x 5 ) ∧  = (x 1 ∨ ￢ x 2 ∨ ￢ x 3 ) ∧ (x 2 ∨ ￢ x 3 ∨ ￢ x 4 ) ∧ (x 3 ∨ ￢ x 4 ∨ x 5 ) ∧ (x 4 ∨ ￢ x 5 ∨ ￢ x 6 ) ∧ (x 5 ∨ ￢ x 6 ∨ ￢ x 1 ) ∧ (x 6 ∨ ￢ x 1 ∨ ￢ x 2 ) ∧ (x 4 ∨ ￢ x 5 ∨ ￢ x 6 ) ∧ (x 5 ∨ ￢ x 6 ∨ ￢ x 1 ) ∧ (x 6 ∨ ￢ x 1 ∨ ￢ x 2 ) ∧ (x 1 ∨ x 2 ∨ x 3 ) ∧ (x 1 ∨ x 2 ∨ ￢ x 3 ) ∧ ( ￢ x 1 ∨ x 2 ∨ x 3 ) ∧ (x 1 ∨ x 2 ∨ x 3 ) ∧ (x 1 ∨ x 2 ∨ ￢ x 3 ) ∧ ( ￢ x 1 ∨ x 2 ∨ x 3 ) ∧ ( ￢ x 1 ∨ x 2 ∨ ￢ x 3 ) ∧ (x 1 ∨ ￢ x 2 ∨ x 3 ) ( ￢ x 1 ∨ x 2 ∨ ￢ x 3 ) ∧ (x 1 ∨ ￢ x 2 ∨ x 3 )  has a unique solution: x 1 = x 2 = … x 6 = true  has a unique solution: x 1 = x 2 = … x 6 = true

◈ To represent all possible variable assignments for the chosen 6-variable SAT problem, a Lipton encoding was used - For each of the 6 variables x 1, x 2, · · ·, x 6 - For each of the 6 variables x 1, x 2, · · ·, x 6 - two distinct 15 base value sequences were designed - two distinct 15 base value sequences were designed : true (T) X k T, false(F) X k F : true (T) X k T, false(F) X k F - Each of the 2 6 truth assignments was represented by a library sequence of 90 bases consisting of the concatenation of one value sequence for each variable. - Each of the 2 6 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 - DNA molecules with library sequences are termed library strand - Combinatorial pool containing library strands is termed a library - Combinatorial pool containing library strands is termed a library - The probes used for separating the library strands have sequences complementary to the value sequences - 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 - 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 - Sequences must be designed to ensure that library strands have little secondary structure which might inhibit intended probe-library hybridization

2.1 Design of the library The value sequences generated to represent x 1 = F, x 2 = F, · · ·, x 6 = F were: The value sequences generated to represent x 1 = F, x 2 = F, · · ·, x 6 = F were: X 1 F = 5’ - TATTCTCACCCATAA - 3’ X 2 F = 5’ - ACACTATCAACATCA - 3’ X 1 F = 5’ - TATTCTCACCCATAA - 3’ X 2 F = 5’ - ACACTATCAACATCA - 3’ X 3 F = 5’ - CCTTTACCTCAATAA - 3’ X 4 F = 5’ - CTCCCAAATAACATT - 3’ X 3 F = 5’ - CCTTTACCTCAATAA - 3’ X 4 F = 5’ - CTCCCAAATAACATT - 3’ X 5 F = 5’ - AACTTCACCCCTATA - 3’ X 6 F = 5’ - TCATATCAACTCCAC - 3’ X 5 F = 5’ - AACTTCACCCCTATA - 3’ X 6 F = 5’ - TCATATCAACTCCAC - 3’ The value sequences generated to represent x 1 = T, x 2 = T, · · ·, x 6 = T were: The value sequences generated to represent x 1 = T, x 2 = T, · · ·, x 6 = T were: X 1 T = 5’ - CTATTTATATCCACC - 3’ X 2 T = 5’ – ACACCTAACTAAACT - 3’ X 1 T = 5’ - CTATTTATATCCACC - 3’ X 2 T = 5’ – ACACCTAACTAAACT - 3’ X 3 T = 5’ - CTACCCTATTCTACT - 3’ X 4 T = 5’ – ATCTTTAAATACCCC - 3’ X 3 T = 5’ - CTACCCTATTCTACT - 3’ X 4 T = 5’ – ATCTTTAAATACCCC - 3’ X 5 T = 5’ - TCCATTTCTCCATAT - 3’ X 6 T = 5’ – TTTCTTCCATCACAT - 3’ X 5 T = 5’ - TCCATTTCTCCATAT - 3’ X 6 T = 5’ – TTTCTTCCATCACAT - 3’

1. Library seqs contain only A’s, T’s, C’s. 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. 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) 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 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) 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 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. Every probe seq has 4, 5, or 6 Gs in its seq Sequences were computer-generated to satisfy the following constraints:

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’-X 1 -X 2 -X 3 -X 4 -X 5 -X 6 -3’) - Synthesis began by assembling the two 15 bases oligonucleotides with sequences X 6 T and X 6 F in separate columns - The columns were then removed from the synthesizer and opened. ; the CPG beads in the columns were removed and mixed together. ; 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 One half of the beads were retruned to the first column and the other half to the second to the second - Synthesis continued with sequences X 5 T and X 5 F. This process was repeated until all 6 variables had been treated probes, having sequences X k F, X k T, k= and modified at the 5’-end with Acrydite TM

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

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 X 1 T or X 1 F and 3’- end primers with sequences X 2 T,..., X 6 T or X 2 F,…, X 6 F 20 PCR reactions were performed on the library using 5’- end primers with sequences X 1 T or X 1 F and 3’- end primers with sequences X 2 T,..., X 6 T or X 2 F,…, X 6 F

2.5 The algorithm Coupling of the Acrydite TM phosphoramidite to DNA probes allows the probes to be immobilized in a polyacrylamide gel matrix - Coupling of the Acrydite TM 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. - 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. - DNA molecules without complementary subsequences pass through the gel relatively unhindered. - Captured DNA strands can be released by running electrophoresis - Captured DNA strands can be released by running electrophoresis - Released molecules can be used in subsequent steps as required - Released molecules can be used in subsequent steps as required

1. For each of the 11 clauses of  prepare a polyacrylamide gel capture layer 1. For each of the 11 clauses of  prepare a polyacrylamide gel capture layer containing three Acrydite TM modified probes, on for each literal in the containing three Acrydite TM modified probes, on for each literal in the clause( If x k appears in the clause, add a probe with sequence X k ; if ￢ x k clause( If x k appears in the clause, add a probe with sequence X k ; if ￢ x k appears add a probe with sequence for X k F ) appears add a probe with sequence for X k F ) 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. 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. 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.

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

2.7 Computation Fig 2. Apparatus assembled for computation

2.8 Determination of answer strand - PCR - PCR - Sequencing - Sequencing

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

Fig 4. PCR analysis of the original library X 1 F, X 2 T,...,X 6 T probe X 1 T, X 2 T,... X 6 T X 1 F, X 2 T, … X 6 T X 1 T, X 2 F, …, X 6 F X 2 F and X 2 F, … X 6 F

Fig 5. Readout of the answer by PCR

Fig 6. Sequencing of the diluted answer strands

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 1) design of the library strands - X 1 T, …, X 6 T and X 1 F, …, X 6 F - X 1 T, …, X 6 T and X 1 F, …, X 6 F - Longer library strands composed of these sequences performs, - Longer library strands composed of these sequences performs, sequence design does not seem to be a limiting factor sequence design does not seem to be a limiting factor 2) synthesis of the library strands 2) synthesis of the library strands - variable library strands synthesized by a mix-and-spilt synthesis - variable library strands synthesized by a mix-and-spilt synthesis - Each library is tested separately by running a capture analysis and - Each library is tested separately by running a capture analysis and simple computation simple computation 3) execution of the computation 3) execution of the computation - Enough to complete a successful 20-variable computation - Enough to complete a successful 20-variable computation

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