Solution of Satisfiability Problem on a GelBased DNA computer

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Solution of Satisfiability Problem on a
Gel-Based DNA computer

• Ji Yoon Park
• Dept. of Biochem
• Hanyang University

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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

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I. Introduction

• ? (x1?? x2?? x3)?(x2?? x3?? x4)?(x3??
  x4?x5) ?
• (x4?? x5?? x6)?(x5??x6??x1)?(x6??x1??x2)
  ?
• (x1?x2?x3)?(x1?x2??x3)? (?x1?x2 ?x3)?
• (?x1?x2??x3) ?(x1??x2?x3)
• ? has a unique solution x1 x2
  x6 true

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? 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

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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

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Sequences were computer-generated to satisfy the
following constraints

• 1. Library seqs contain only As, Ts, Cs.
• 2. All libray and probe seqs have no occurrence
  of 5 or more consecutive identical nucleotides
  i.e. no runs of more than 4 As, 4 Ts, 4 Cs or
  4 Gs 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

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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, k1. . .
  6 and modified at the 5-end with AcryditeTM

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2.3 Library capture analysis

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

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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

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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

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• 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.

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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
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2.7 Computation
Fig 2. Apparatus assembled for computation
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2.8 Determination of answer strand

• - PCR
• - Sequencing

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Fig 3. Capture of the library by gel-embedded
probes
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X1F, X2T, . . . ,X6T probe
Fig 4. PCR analysis of the original library
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Fig 5. Readout of the answer by PCR
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Fig 6. Sequencing of the diluted answer strands
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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

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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