Solving Edge-Matching Puzzles Using DNA Computing

Solving Edge-Matching Puzzles Using DNA Computing

Mohammed AlShamrani Department of Computer Science

Concordia University March 23, 2011

Solving Edge-Matching Puzzles Using DNA Computing

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Solving Edge-Matching Puzzles Using DNA Computing

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Solving Edge-Matching Puzzles Using DNA Computing

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Solving Edge-Matching Puzzles Using DNA Computing

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Solving Edge-Matching Puzzles Using DNA Computing

Solving Edge-Matching Puzzles Using DNA Computing

Solving Edge-Matching Puzzles Using DNA Computing

Solving Edge-Matching Puzzles Using DNA Computing

Phosphate

Sugar (deoxyribose )

Hydrogen bonds

Solving Edge-Matching Puzzles Using DNA Computing

5’-GACACTCACTGTCA-3’

3’-CTGTGAGTGACAGT-5’

=

5’-CCAAGTTGATTGAGAA

Solving Edge-Matching Puzzles Using DNA Computing

5’-TAACTCT T T T CTCAAT-3’

1. Synthesis

What we can do with DNA …

2. Hybridization

pH AAGAGTTATATGGGCT-3’

•

3. Ligation

4. Replication (PCR)

………..

Exponential Growth

Solving Edge-Matching Puzzles Using DNA Computing

1 copy

2 copies

4 copies 8 copies

Sd

Sd

Sd

Jj

Solving Edge-Matching Puzzles Using DNA Computing

DNA Computing is new perspective

 If DNA strands are made to represent objects/relations, then new knowledge

can result from the application of these operations (synthesis, hyb., ligation, PCR, etc).

This is DNA Computing.

Solving Edge-Matching Puzzles Using DNA Computing

Example: Six degrees of Separation

• The claim is that any two people in the world are connected, on average, by 6 people who are connected by the “is-a-friend-of” relation.

• So

you

Nelson Mandela

Solving Edge-Matching Puzzles Using DNA Computing

Example: Six degrees of Separation

• So

you

Nelson Mandela

5’-CCAAGTTGATTGAGAA

1. Synthesis

2. Hybridization 3. Ligation

Unit Computation

Solving Edge-Matching Puzzles Using DNA Computing

Example: Six degrees of Separation

• So

you

Nelson Mandela

1. Synthesis

2. Hybridization 3. Ligation

Post-Ligation Product

4. Replication: Replicate sequences that begin with

“You” and end with “Nelson Mandela”

Solving Edge-Matching Puzzles Using DNA Computing

Example: Six degrees of Separation

• So

you

Nelson Mandela

1. Synthesis

2. Hybridization 3. Ligation

4. Replication

Sequence encoding 6 people between you and Mandela

Post-Replication product:

sequences of different lengths but all begin with you and end with Mandela

Solving Edge-Matching Puzzles Using DNA Computing

Example: Six degrees of Separation

• So

you

Nelson Mandela

1. Synthesis

2. Hybridization 3. Ligation

4. Replication

Sequence encoding 6 people between you and Mandela

Post-Replication product:

sequences of different lengths but all begin with you and end with Mandela

Solving Edge-Matching Puzzles Using DNA Computing

Example: Six degrees of Separation

• So

you

Nelson Mandela

1. Synthesis

2. Hybridization 3. Ligation

4. Replication

5. Gel Electrophoresis

Your best chance of meeting Mandela

Reference ladder

Solving Edge-Matching Puzzles Using DNA Computing

Challenge:

1) All abutting edges match in color 2) All boundary edges are grey

10

13 8

16 4

9 2

5 7

15

6

3

12 1

11 14

8 13 16

10

4 9 2 7

14 1 11 5

12 3 6 15

Solving Edge-Matching Puzzles Using DNA Computing

Relevance:

 Complexity: EMPs are NP-Complete

 Turing-universality: 2-Dimensional growth of tiles can simulate the execution of any Turing machine.

 Nanotechnology: Tiles can serve as definitional motifs for nano-technological constructions: given a desired 2D shape, what set of tiles (preferably minimal) can grow to that

shape?

5 + 9 = 14

5 9 14

10

13 8

16 4

9 2

7 5

15 6

3

12 1

11 14

Solving Edge-Matching Puzzles Using DNA Computing

To solve an EMP with DNA, we need to:

1. Define the problem 2. Formulate an algorithm 3. Implement a DNA lab protocol

To solve an EMP with DNA, we need to:

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

To solve an EMP with DNA, we need to:

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

h1 h2 h3

h4 h5

S0

h6 h7

h8 h9 h10 h11

h12 h13 h14 S15

12

13

S1 h2 h3

S4 h5

S0

h6 h7

h8 h9 h10 S11

h12 h13 S14 S15

12

13

3 14

7 8

S1 S2 h3

S4 S5

S0

h6 S7

S8 h9 S10 S11

h12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

S1 S2 S3

S4 S5

S0

S6 S7

S8 S9 S10 S11

S12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

15 11

9 10

Diagonal-wise tile stacking:

The algorithm succeeds if it makes a series of correct choices:

at each step, find diagonal sets of tiles that can fit legally.

To solve an EMP with DNA, we need to:

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

h1 h2 h3

h4 h5

S0

h6 h7

h8 h9 h10 h11

h12 h13 h14 S15

12

13

S1 h2 h3

S4 h5

S0

h6 h7

h8 h9 h10 S11

h12 h13 S14 S15

12

13

3 14

7 8

S1 S2 h3

S4 S5

S0

h6 S7

S8 h9 S10 S11

h12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

S1 S2 S3

S4 S5

S0

S6 S7

S8 S9 S10 S11

S12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

15 11

9 10

Diagonal-wise tile stacking:

But couldn’t there be more than

“correct” choice at each step?

Yes: non-determinism.

S1 S2

S4 h5

S0

h⁶ h7

h8 h9 h10 S11

h13 S14 S15

12 14

8 16

13 3

7 h12

h3

S1 S2 h³

S4 S5

S0

h6 S7

S8 h⁹ S10 S11

h12 S13 S14 S15

12 14 4

2

8 16

13

7 3

5 1

6

S1 S2 h3

S4 S5

S0

h6 S7

S8 h9 S10 S11

h12 S13 S14 S15

12 14 4

2

8 16

13

7 3

5 1

6

NP-Complete

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise tile stacking:

Observation: at any given step, only two edges of each tile are involved in constraint validation.

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise tile stacking:

Observation: at any given step, only two edges of each tile are involved in constraint validation

Conceptually: a tile is the union of two half tiles.

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise half-tile stacking:

Conceptually: a tile is the union of two half tiles.

h1 h2 h3

h4 h5

S0

h6 h7

h8 h9 h10 h11

h12 h13 h14 S15

12

13

S1 h2 h3

S4 h5

S0

h6 h7

h8 h9 h10 S11

h12 h13 S14 S15

12

13

3 14

7 8

S1 S2 h3

S4 S5

S0

h6 S7

S8 h9 S10 S11

h12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

S1 S2 S3

S4 S5

S0

S6 S7

S8 S9 S10 S11

S12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

15 11

9 10

The algorithm succeeds if it makes a series of correct choices:

at each step, find diagonal sets of half-tiles that can fit perfectly.

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise half-tile stacking:

2

Tile dissection along the diagonals produces two pairs of half tiles

Pairs of half-tiles

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise half-tile stacking: The union of a pair on the tiling grid reproduces the tile in one orientation

UU U U

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise half-tile stacking:

Formalizations What is a half-tile?

What is the relation between half-tiles? Is the solution with the set of half-tiles equivalent to that of tiles?

Can we proof it?

What is the union of two half-tiles?

What is the relation (“bridging”) between diagonal sets of half-tiles (“lanes”)?

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7

10 11

S

What is a valid lane ?

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise half-tile stacking:

Formalizations

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise half-tile stacking:

Why ?

In the correct solution, diagonal sets of half-tiles have two useful properties:

1. All diagonal sets of half-tiles are of odd length

h1 h2 h3

h4 h5

S0

h6 h7

h8 h9 h10 h11

h12 h13 h14 S15

12

13

1 half-tile

1 half-tile

S1 h2 h3

S4 h5

S0

h6 h7

h8 h9 h10 S11

h12 h13 S14 S15

12

13

3 14

7 8

3 half-tiles

3 half-tiles

S1 S2 h3

S4 S5

S0

h6 S7

S8 h9 S10 S11

h12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

5 half-tiles 5 half-tiles

S1 S2 S3

S4 S5

S0

S6 S7

S8 S9 S10 S11

S12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

15 11

9 10

7 half-tiles

7 half-tiles

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise half-tile stacking:

Why ?

In the correct solution, diagonal sets of half-tiles have two useful properties:

1. All diagonal sets of half-tiles are of odd length

2. All diagonal sets of half-tiles are begin and end with “grey”

S1 S2 S3

S4 S5

S0

S6 S7

S8 S9 S10 S11

S12 S13 S14 S15

12

13

3 14

7 8

16

6 1

4 2

5

15 11

9 10

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Diagonal-wise half-tile stacking:

Why ?

In the correct solution, diagonal sets of half-tiles have two useful properties:

1. All diagonal sets of half-tiles are of odd length

2. All diagonal sets of half-tiles are begin and end with “grey”

Gel Electrophoresis as a computational heuristic

Polymerase Chain Reaction (PCR) as a computational heuristic and a processing power

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Ultimately we seek to find the set of DNA lanes that encode the full solution to the puzzle:

1. Enumerate DNA lanes (“stapling”)

2. Build DNA grid by stacking lanes (“bridging”) DNA Grid

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Step 1: Associate half-tiles to random ssDNA sequences (synthesis)

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5’-ATGGGTGAAGAAGATG GTAGAAGAGAAATAAG -3’ GAATAAAGCTAGCGGC-3’

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

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5’-ATGGGTGAAGAAGATG GTAGAAGAGAAATAAG-3’ GAATAAAGCTAGCGGC-3’

3’-TCTTCTACCATCTTCT-5’3’-CTTTATTCCTTATTTC-5’

“Red” staple

“Blue” staple

Step 2: Mix half-tiles strands with stapler strands to generate random lanes

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Step 3: Keep only lanes of beginning and ending with grey (PCR)

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Step 4: Keep only lanes of length 1, 3, 5, and 7 half-tiles (Gel Electrophoresis).

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Step 5: Bridge lanes.

5’………..XXXXXTGTATATGTGTGGGAACAGGTTTAATXXXXX………3’

3’-...XXXXXXAAGAGTTATATGA CTCCTGAAATGGAXXXXX…..5’

CC AC AC AT AT AC A-

3’ C TT TG CC AA TA AT ‘5-

5’-TTCTCAATATACT GAGGACTTTACCT-3’

Bridging strands

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Step 5: Bridge lanes.

 Bridging is a sensitive and labor-intensive process:

 3-D shape of the double-helix must be taken into account: design constraints

 Migration of bridged assemblies on gel is tricky

2. Formulate an algorithm 3. Implement a DNA lab protocol 1. Define the problem

Solving Edge-Matching Puzzles Using DNA Computing

Puzzle Solution

=

8 16

10 4

9

2

7 14

1

11

5 12

3

6

15

Solving Edge-Matching Puzzles Using DNA Computing

Concluding remarks:

 NP-Completeness of EMPs: we can measure processing power of DNA Computing

 Half-tile Assembly Model:

 Turing-complete

 PCR-powered model for DNA nanotechnological fabrication

Solving Edge-Matching Puzzles Using DNA Computing

http://users.encs.concordia.ca/~mo_alsha/thesis/