Solution of Satisfiability Problem Solution of Satisfiability Problem on a Gel-Based DNA computeron a Gel-Based DNA computer

Ji Yoon ParkJi Yoon ParkDept. of BiochemDept. of Biochem

Hanyang UniversityHanyang University

AbstractAbstract

1. 1. Succeeded inSucceeded in solving an instance of a 6-variable 11- solving an instance of a 6-variable 11- clause 3-SAT problemclause 3-SAT problem on a on a gel-based DNA computergel-based DNA computer

2. 2. SeparationSeparation were performed were performed using probes covalentlyusing probes covalently bound to polyacrylamide gelbound to polyacrylamide gel

3. During the entire computation, 3. During the entire computation, DNA was retained DNA was retained within a single gelwithin a single gel and moved via electrophoresis and moved via electrophoresis

4. To be readily4. To be readily automatable automatable and should be and should be suitable forsuitable for problems of a significantly larger size problems of a significantly larger size

I. IntroductionI. Introduction

= (= (xx11∨∨ ￢￢ xx22∨∨ ￢￢ xx33) (∧) (∧ xx22∨∨ ￢￢ xx33∨∨ ￢￢ xx44) (∧) (∧ xx33∨∨ ￢￢ xx44∨∨xx55) ∧) ∧

((xx44∨∨ ￢￢ xx55∨∨ ￢￢ xx66) (∧) (∧ xx55∨∨ ￢￢ xx66∨∨ ￢￢ xx11) (∧) (∧ xx66∨∨ ￢￢ xx11∨∨ ￢￢ xx22) ∧) ∧

((xx11∨∨xx22∨∨xx33) (∧) (∧ xx11∨∨xx22∨∨ ￢￢ xx33) (∧) (∧ ￢￢ xx11∨∨xx2 2 ∨∨xx33)∧)∧

(( ￢￢ xx11∨∨xx22∨∨ ￢￢ xx33)) ∧∧((xx11∨∨ ￢￢ xx22∨∨xx33))

has a unique solution:has a unique solution: xx1 1 = = xx2 2 = …= … xx6 6 = true= true

◈ ◈ To represent all possible variable assignments To represent all possible variable assignments for the chosen 6-variable SAT problem, for the chosen 6-variable SAT problem,

a Lipton encoding was useda Lipton encoding was used

- For each of the - For each of the 6 variables 6 variables xx11, , xx22, , · · · ,· · · , x x66

- - two distinct 15 base two distinct 15 base value sequencesvalue sequences were designed were designed : : true (T) Xtrue (T) Xkk

T T ,, false(F) Xfalse(F) XkkFF

- Each of the - Each of the 226 6 truth assignmentstruth assignments was represented by a library sequence of was represented by a library sequence of 90 90 basesbases consisting of the concatenation of one value sequence for each variable. consisting of the concatenation of one value sequence for each variable.

- - DNA molecules with library sequencesDNA molecules with library sequences are termed are termed library strandlibrary strand - - Combinatorial pool containing library strandsCombinatorial pool containing library strands is termed is termed a librarya library - - The probesThe probes used for used for separating the library strands have sequences complemeseparating the library strands have sequences compleme

ntary to the value sequencesntary 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 designedSequences must be designed to ensure that library strands to ensure that library strands have little secondahave little seconda

ry structurery structure which might which might inhibit intended probe-library hybridizationinhibit intended probe-library hybridization

2.1 Design of the library2.1 Design of the library

The value sequences generated to represent The value sequences generated to represent xx11= F, = F, xx22= F, = F, · · · , · · · , xx66= F were:= F were:

XX11FF= 5’ - TATTCTCACCCATAA - 3’ X= 5’ - TATTCTCACCCATAA - 3’ X22

FF= 5’ - ACACTATCAACATCA - 3’= 5’ - ACACTATCAACATCA - 3’

XX33FF= 5’ - CCTTTACCTCAATAA - 3’ X= 5’ - CCTTTACCTCAATAA - 3’ X44

FF= 5’ - CTCCCAAATAACATT - 3’= 5’ - CTCCCAAATAACATT - 3’

XX55FF= 5’ - AACTTCACCCCTATA - 3’ X= 5’ - AACTTCACCCCTATA - 3’ X66

FF= 5’ - TCATATCAACTCCAC - 3’= 5’ - TCATATCAACTCCAC - 3’

The value sequences generated to representThe value sequences generated to represent x x11= T, = T, xx22= T, = T, · · · , · · · , xx66= T were:= T were:

XX11TT= 5’ - CTATTTATATCCACC - 3’ X= 5’ - CTATTTATATCCACC - 3’ X22

TT= 5’ – ACACCTAACTAAACT - 3’= 5’ – ACACCTAACTAAACT - 3’

XX33TT= 5’ - CTACCCTATTCTACT - 3’ X= 5’ - CTACCCTATTCTACT - 3’ X44

TT= 5’ – ATCTTTAAATACCCC - 3’= 5’ – ATCTTTAAATACCCC - 3’

XX55TT= 5’ - TCCATTTCTCCATAT - 3’ X= 5’ - TCCATTTCTCCATAT - 3’ X66

TT= 5’ – TTTCTTCCATCACAT - 3’= 5’ – TTTCTTCCATCACAT - 3’

1. 1. Library seqsLibrary seqs contain contain only A’s, T’s, C’s.only A’s, T’s, C’s. 2. 2. All libray and probe seqsAll libray and probe seqs have have no occurrence of 5 or more consecutive ideno occurrence of 5 or more consecutive ide

ntical nucleotidesntical nucleotides; i.e. no runs of more than 4 A’s, 4 T’s, 4 C’s or 4 G’s ; 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.occur in any library or probe seqs.

3. 3. Every probe seqEvery probe seq has at has at least 4 mismatches with all 15 baseleast 4 mismatches with all 15 base alignment of a alignment of any library seq(except for with its matching value seq)ny library seq(except for with its matching value seq)

4. 4. Every 15 base subseqEvery 15 base subseq of a library seq has at of a library seq has at least 4 mismatches with all 15 least 4 mismatches with all 15 base alignmentbase alignment of itself or any other library seq of itself or any other library seq

5. 5. No probe seq has a run of more than 7 matches with any 8 baseNo probe seq has a run of more than 7 matches with any 8 base alignment alignment of any library seq(except for with its matching value seq)of any library seq(except for with its matching value seq)

6. 6. No library seq has a run of more than 7 matches with any 8 baseNo library seq has a run of more than 7 matches with any 8 base alignmen alignment of itself or any other library seqt of itself or any other library seq

7. 7. Every probeEvery probe seq seq has 4, 5, or 6 Gs in its seqhas 4, 5, or 6 Gs in its seq

Sequences were computer-generated to Sequences were computer-generated to satisfy the following constraints: satisfy the following constraints:

2.2 Synthesis of the library and probes2.2 Synthesis of the library and probes

- - Mix-and-split combinatorial synthesis techniqueMix-and-split combinatorial synthesis technique- - A dual column ABI 392 DNA/RNA synthesizerA dual column ABI 392 DNA/RNA synthesizer at a 1 at a 1µmole scale on CPG soliµmole scale on CPG soli

d support.d support.- The library strands: (- The library strands: (5’-X5’-X11-X-X22-X-X33-X-X44-X-X55-X-X66-3’-3’))- - SynthesisSynthesis began began by assembling the two 15 bases oligonucleotides withby assembling the two 15 bases oligonucleotides with sequencsequenc

es Xes X66TT and X and X66

FF in separate columns in separate columns- The columns were then removed from the synthesizer and opened.- 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 secondto the second- Synthesis continued with sequences X- Synthesis continued with sequences X55

TT and X and X55FF. This process was repeated un. This process was repeated un

til all 6 variables had been treated. til all 6 variables had been treated. - 12 probes, having sequences X- 12 probes, having sequences Xkk

FF, X, XkkTT,, k k=1. . . 6 and modified at the 5’-end wit=1. . . 6 and modified at the 5’-end wit

h Acryditeh AcryditeTMTM

2.3 Library capture analysis2.3 Library capture analysis

To determine the efficiency of library capture To determine the efficiency of library capture andand release by gel-embedrelease by gel-embedded probesded probes

- preparation of gels - preparation of gels - Running the gels- Running the gels

2.4 Confirming integrity of2.4 Confirming integrity of the library via PCR the library via PCR

To verify the degeneracy and integrity of the libraryTo verify the degeneracy and integrity of the library, , the library was the library was amplified via PCRamplified via PCR

20 PCR reactions20 PCR reactions were performed on the library using were performed on the library using 5’- end5’- end prime primers with sequences rs with sequences XX11

TT or X or X11FF and and 3’- end3’- end primers with sequences primers with sequences XX

22TT, . . . , X, . . . , X66

TT or X or X22FF,…, X,…, X66

FF

2.5 The algorithm2.5 The algorithm

- Coupling of the AcryditeCoupling of the AcryditeTMTM phosphoramidite to DNA probes phosphoramidite to DNA probes allows the pr allows the probes to be obes to be immobilized in a polyacrylamide gel matriximmobilized in a polyacrylamide gel matrix

- During electrophoresis at low tempDuring electrophoresis at low temp, such , such probes hybridize with and capturprobes hybridize with and capture passing DNA molecules bearing complementary subsequences.e passing DNA molecules bearing complementary subsequences.

- DNA molecules without complementary subsequences pass through the gel - DNA molecules without complementary subsequences pass through the gel relatively unhindered.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 1. For each of the 11 clauses of each of the 11 clauses of prepare a polyacrylamide gel capture layer prepare a polyacrylamide gel capture layer containing containing three Acryditethree AcryditeTMTM modified probes modified probes, on for each literal in the, on for each literal in the clause( clause( If If xxkk appears in the clause, add a probe with sequence X appears in the clause, add a probe with sequence Xkk; ; ifif ￢￢ xxkk

appears add a probe with sequence for Xappears add a probe with sequence for XkkFF ))

2. Layer while heating the areas of the gel preceding and following it. Begin ele2. Layer while heating the areas of the gel preceding and following it. Begin electrophoresis to move the library through the first capture layer. Moleculctrophoresis to move the library through the first capture layer. Molecules encoding truth assignments satisfying the first clause will be captured es encoding truth assignments satisfying the first clause will be captured in the first capture layer, while molecules encoding non-satisfying assignin the first capture layer, while molecules encoding non-satisfying assignments will run through the first capture layer and continue beyond the sements will run through the first capture layer and continue beyond the second capture layer. cond capture layer.

3. Cool the area of the gel containing the second capture layer while heating th3. 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 fie areas of the gel preceding and following it. Molecules captured in the first capture layer will be released to move through the second capture layrst capture layer will be released to move through the second capture layer. Released molecules encoding truth assignments satisfying the second er. Released molecules encoding truth assignments satisfying the second clause will be captured in the second capture layer, while molecules encoclause will be captured in the second capture layer, while molecules encoding non-satisfying assignments will run through the second capture layeding non-satisfying assignments will run through the second capture layer and continue beyond the third capture layer.r and continue beyond the third capture layer.

2.6 Construction and running of 2.6 Construction and running of the computerthe 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 moduleFig 1. Preparation of a clause module

2.7 Computation2.7 Computation

Fig 2. Apparatus assembled for Fig 2. Apparatus assembled for computationcomputation

2.8 Determination of answer strand2.8 Determination of answer strand

- PCR- PCR

- Sequencing- Sequencing

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

Fig 4. PCR analysis of the original Fig 4. PCR analysis of the original librarylibrary

XX11FF, X, X22

TT, . . . ,X, . . . ,X66TT probe probe

XX11TT, X, X22

TT, ... X, ... X66TT XX11

FF, X, X22TT, … X, … X66

TT

XX11TT, X, X22

FF, …, X, …, X66FF

XX22FF and X and X22

FF, … X, … X66FF

Fig 5. Readout of the answer by PCRFig 5. Readout of the answer by PCR

Fig 6. Sequencing of the diluted answer strandsFig 6. Sequencing of the diluted answer strands

Prospects for scaling upProspects for scaling up

Whether SAT problems of greater size may be solved depends on the difficultWhether SAT problems of greater size may be solved depends on the difficulty of scaling up each of three proceduresy of scaling up each of three procedures

1) design of the library strands1) design of the library strands - - XX11

TT, …, X, …, X66TT and X and X11

FF, …, X, …, X66FF

- 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 strands2) synthesis of the library strands - variable library strands synthesized by a - variable library strands synthesized by a mix-and-spilt synthesismix-and-spilt synthesis - Each library is tested - Each library is tested separately by running a capture analysisseparately by running a capture analysis and and simple computation simple computation 3) execution of the computation3) execution of the computation - - Enough to complete a successful 20-variable computation Enough to complete a successful 20-variable computation

DiscussionDiscussion

- - Successful DNA computation on a 6-variable SAT problemSuccessful DNA computation on a 6-variable SAT problem- - The correct solution was culled from 64 alternativesThe correct solution was culled from 64 alternatives - - Optimistic about the prospects of building an automated deviceOptimistic about the prospects of building an automated device for for

carrying out such computationscarrying out such computations