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C A A C C A A A A A A- T T A C C T C A A A C C- T T C A A T C C A C A A- T C A C T C T C T C A A - C A T C T C A C C A T C

Outline

- DNA Hybridization/Cross Hybridization
- DNA Codes
- Nearest Neighbor Thermodynamics
- complete computations
- bounds
- Overview of Applications and Purposes
- DNA Bitstring Library
- Biomolecular Computing

- DNA strands are modeled by directed 3’--> 5’ sequences of letters from the alphabet {A, C, G, T}
- (A, T) and (C, G) are complementarypairs.
- Two oppositely directed DNA sequences are capable of coalescing into a duplex.
- Because an A (C) in one strand can (usually) only bind to a T (G) in the oppositely directed strand, the greatest energy of duplex formation is obtained when the two sequences are reverse-complements (complements)

Orientation of single DNA strands is important for hybridization.

Coding Strands

for Ligation

Probing Complement Strands

for Reading

TACGCGACTTTC GAAAGTCGCGTA

ATCAAACGATGC GCATCGTTTGAT

TGTGTGCTCGTC GACGAGCACACA

ATTTTTGCGTTA, TAACGCAAAAAT

CACTAAATACAA TTGTATTTAGTG

GAAAAAGAAGAA, TTCTTCTTTTTC

5’ 3’ 5’ 3’

Watson Crick

(WC) Duplexes

5’TACGCGACTTTC3’

ATCAAACGATGC

Must Have

5’GAAAGTCGCGTA3’

GCATCGTTTGAT

TACGCGACTTTC

Cross Hybridized

(CH) Duplexes

ATTTTTGCGTTA

Must Avoid

GCATCGTTTGAT

GAAAAAGAAGAA

5’ggCaCaTcatAct3’

5’ AggTTaaCcatct3’

y=5’agatgGttAAccT3’

5’ggCaCaTcatAct3’

3’ TccAAttGgtaga5’

5’agatgGttAAccT3’ =y

DNA codes serve as universal components for biomolecular computing. DNA codes are closed under reverse-complementation. The strands in a DNA code have such binding specificity that a code strand will only hybridize with its reverse-complement and will not cross hybridize with any other code strand in the DNA code

Such collections of strands are crucial to the success biomolecular computing and biomolecular nanotechnology.

Basic idea is to have correct, parallel and autonomous addressing

Characterization of synthetic DNA bar codes in Saccharomyces cervisiae gene-deletion strains”

(Eason et al., PNAS).

DNA codes for self-assembly of any components that can be attached

to DNA. Their size presents the potential for increased complexity and location control in nanostructures produced by assembly that is driven by DNA duplex formation. Fundamentalphysical limits and increasing costs of fabrication facilities will force alternatives to conventional microelectronics manufacturing to be developed.

In self-assembly, weak, local interactions among molecular components spontaneously organize those components into aggregates with properties that range from simple to complex

DNA memory:The capacity and storage density of such memories is potentially very large. Information could

be mined through massively parallel template-matching reactions. In addition, information could be processed based upon context, and information matched associatively based upon content.

Interest into DNA computing was sparked in 1994 by Len Adleman.

Adleman showed how we can use DNA molecules to solve a mathematical problem. (Hamiltonian path problem).

DNA computing relies on the fact that DNA strands can be represented as sequences of bases (4-ary sequences) and the property of hybridization.

In Hybridization, errors can occur. Thus, error-correcting codes are required for efficient synthesis of DNA strands to be used in computing.

DNA Computing Strand Engineering

No codeword-codewode CH (cc-CH)

No codeword-probe CH (cp-CH)

No probe-probe CH (pp-CH)

A A A A A A A A C C=T1

G G T T T T T T T T =BEAD PROBE (T1)

T T T C C A A A A A =F1

T T T T T G G A A A = BEAD PROBE (F1)

T T T C T T A A C C=T2

G G T T A A G A A A= BEAD PROBE (T2)

A C T A A C A A A A=F2

T T T T G T T A G T= BEAD PROBE (F2)

C A T A A A A C A C=T3

G T G T T T T A T G= BEAD PROBE (T3)

A T C T T T T C A A=F3

T T G A A A A G A T= BEAD PROBE (F3)

C A A T C C A T T A=T4

T A A T G G A T T G= BEAD PROBE (T4)

C C T T C T A A A T=F4

A T T T A G A A G G= BEAD PROBE (F4)

A C T C C T A A T A=T5

T A T T A G G A G T= BEAD PROBE (T5)

T C T C T C T A C T=F1

A G T A G A G A G A= BEAD PROBE (F5)

Only Allowed Hybridizations

T T T T T T G G T T G G=Probe(T1)

G G T G G T T T T T T T=Probe(F1)

T G G A A G G A A A A A=Probe(T2)

G G T T T G A G G T A A =Probe(F2)

G G A G T T G T G A A A=Probe(T3)

C C A A C C A A A A A A = T1

A A A A A A A C C A C C=F1

T T T T T C C T T C C A =T2

T T A C C T C A A A C C =F2

T T T C A C A A C T C C=T3

No cp-CH

T T G T G G A T T G A A=Probe(F3)

T T G A G A G A G T G A=Probe(T4)

A G A G G A G A A A G A=Probe(F4)

G A T G G T G A G A T G=Probe(T5)

G T G T G T A G T G T T=Probe(F5)

T T C A A T C C A C A A =F3

T C A C T C T C T C A A =T4

T C T T T C T C C T C T=F4

C A T C T C A C C A T C =T5

A A C A C T A C A C A C =F5

No cc-CH

No pp-CH

DNA Computing Strand Engineering

No codeword cp-CH

T T C A A T C C A C A A =F3

T T G T G G A T T G A A=Probe(F3)

T C A C T C T C T C A A =T4

T T G A G A G A G T G A=Probe(T4)

T C T T T C T C C T C T=F4

A G A G G A G A A A G A=Probe(F4)

C A T C T C A C C A T C =T5

G A T G G T G A G A T G=Probe(T5)

A A C A C T A C A C A C =F5

G T G T G T A G T G T T=Probe(F5)

C C A A C C A A A A A A = T1

T T T T T T G G T T G G=Probe(T1)

A A A A A A A C C A C C=F1

G G T G G T T T T T T T=Probe(F1)

T T T T T C C T T C C A =T2

T G G A A G G A A A A A=Probe(T2)

T T A C C T C A A A C C =F2

G G T T T G A G G T A A =Probe(F2)

T T T C A C A A C T C C=T3

G G A G T T G T G A A A=Probe(T3)

PROBE(F2)

G G T T T G A G G T A A

C A A C C A A A A A A- T T A C C T C A A A C C- T T C A A T C C A C A A- T C A C T C T C T C A A - C A T C T C A C C A T C

Yes WC bonding

Yes, bitstring is F2

Good read

T1-F2-F3-T4-T5

1 0 0 1 1

PROBE(T2)

G G A G T T G T G A A

C A A C C A A A A A A- T T A C C T C A A A C C- T T C A A T C C A C A A- T C A C T C T C T C A A - C A T C T C A C C A T C

Darn! CH bonding

No, bitstring is not T2 Bad read

T1-F2-F3-T4-T5

DNA Computing Strand Engineering

No codeword pp-CH, cc-CH

PROBE(F2)

pp-CH

interferes with reading

G G T T T G A G G T A A

T T G A G A G A GT G

PROBE(T4)

PROBE(F2)

G G T T T G A G G T A A

bonding site

competition

T T G A G A G A GT G

C A A C C A A A A A A- T T A C C T C A A A C C- T T C A A T C C A C A A- T C A C T C T C T C A A - C A T C T C A C C A T C

cc-CH

interferes with separation and leads to unwanted library strand interaction

T1-F2-F3-T4-T5

F1-F2-T3-T4-f5

T T T C C A A A A-AT T A C C T C A A A C C- T T T C A C A A C T C C-T C A C T C T C T C A A - A A C A C T A C A C A C

Watson-Crick Nearest Neighbor Computation

1.44

2.24

WC

Duplex

5’g g c a c a3’

3’c c g t g t 5’

5’g g c a c a3’

5’g g c a c a3’

NNFE=8.42

5’g g c a c a3’

5’g g c a c a3’

1.84

1.45

1.45

Cross Hybridized Nearest Neighbor Upper Bound Computation

1.45

1.28

5’ggCaCaTcatAct3’

3’ TccAAttGgtaga5’

5’g gC aCaTcatAct3’

3’ Tc cA AttGgtaga5’

5’g gC aC a T c a t A ct3’

5’ A g g T T a a C c a t ct3’

.27

1.84

0.88

NNFE~<5.45

5’ggCaCaTcatAct3’

5’ AggTTaaCcatct3’

NNFE~<5.72

NNFE~<5.66+ .59 + .32=6.57

3’ T cc AA t t G g t aga5’

5’ggC a C a T c a t A ct3’

5’ A ggTT a aC c a t ct3’

5’ggCaCaTcatAct3’

3’ TccAAttGgtaga5’

Virtual Stacked Pairs

Virtual Duplex

5’ggCaCaTcatAct3’

5’ AggTTaaCcatc3’

5’AGTATGATGTGCC 3’

5’AGGTTAACCATCT3’

5’AGATGGTTAACCT3’

5’GGCACATCATACT3’

Neareast Neighbor Appr. Free Energy

of duplex formation (WC)

5’AGTATGATGTGCC 3’

…= 18.8

2.24

1.84

1.45

5’GGCACATCATACT3’

5’AGGTTAACCATCT3’

5’ggCaCaTcatAct3’

3’ TccAAttGgtaga5’

5’ggC aC aT c a tA ct3’

3’ T cc AA t t G g t aga5’

1.28

1.45

5’ggCaCaTcatAct3’

5’ AggTTaaCcatct3’

5’ggC a C a T c a t A ct3’

5’ A ggTT a aC c a t ct3’

NNFE

CH =6.45

0.88

1.84

Let denote a set consisting of all vectors (codewords) of length n built over

i.e.

Let such that:

1)

2)

3)

Let be such that:

is referred to as a Code of length n, size M, and minimum distance d.

A sphere in centered at x having radius d:

Volume of the sphere around x, of radius d:

Spaces

A space is HOMOGENEOUS when the volume of a sphere does not depend on where it is centered i.e.

A space is NON - HOMOGENEOUS when the volume of a sphere does depend on where it is centered.

Sequence

is a subsequence of

if and only if there exists a strictly increasing sequence of indices:

Such that:

is defined to be the set of longest common subsequences of

and

is defined to be the length of the longest common subsequenceof

and

Just what it says:

x = <A C G T C G A G C>

y = <G A C G C T G A G>

LCS(x, y) = {<A C G C G A G>}

|LCS(x, y)| = 7

Original Insertion-Deletion metric (Levenshtein 1966):

This metric results from the number of deletions and insertions that need to be made to obtain ‘ y ’from ‘ x ’.

For vectors that have the same length:

the number of deletions that will be made is:

likewise, the number of insertions that will be made is:

Better Metric ?

- LCS is simple and easy to compute.
- LCS essentially is a count of the number of base pairings between two sequences, and thus does approximate bonding energy.
- Clue: if two base pairs bond, but neither their neighbors to the right or left bond, it really doesn’t contribute much.
- We might call such inconsequential bonds “lone” bonds.

“Lone Bonds”

B B B B B B B B B B B B B B B B

B B B B B B B B B B B B B B B B

The red bonds are “lone bonds” that don’t contribute to the binding energy.

Block LCS

The longest common subsequence SUCH THAT:

If xi is matched to yj, THEN EITHER

xi-1 is matched to yj-1, OR

xi+1 is matched to yj+1

Longest Common Stacked Pair Subsequence

A common subsequence is called a common stacked pair subsequence of length between x and y if two elements , are consecutive inx and consecutive inyor if they are non -consecutive in xand ornon-consecutiveiny, then and are consecutive in xandy.

Let , denote the length of the longest sequence occurring as a common stacked pair subsequence subsequence zbetween sequences x and y. The number , is called a similarityof blocks between xand y.The metric is defined to be

We will be working in a NON-HOMOGENEOUS space making the obtainment of exact formulas for sphere volumes and code sizes VERY HARD.

[6] L. M. G. M. Tolhuizen (1997): The Generalized Varshamov-Gilbert Bound is Implied

by Turan’s Theorem, IEEE Transactions on Information Theory, 43:05.

Varshamov-Gilbert Lower Bound on Code Size in with any metric:

The edge set of G is constructed as follows; an edge (x, y) exists in G if and only if

d(x, y) > d. The first question is; how many edges does G have? This can be found by taking spheres of radius d − 1 around each

vector and counting how many vectors are outside the particular sphere. Since

edges will be double counted, we must divide by 2:

The upper bound for the average sphere volume in this metric will be:

The Varshamov-Gilbert bound becomes:

Insertion-deletion stacked pair

thermodynamic metric

Thermodynamic weight of virtual stacked pairs.

- Can use statistical estimation of sphere volume.

A C G C G T T A

C T G A T A C A

Get LCS of this and add 1 for the

A’s that have to match

Case 1:sequences end with the same symbolA C G C G T T A

C T G A T A C C

Take the best LCS of these two

Case 2:sequences end with different symbolsSolve Problem Recursively

- If x(i) and y(j) end with the same symbol, say A, then: LCS(x(i), y(j)) = LCS(x(i – 1), y(j – 1) + A
- If xi and yj do NOT end with the same symbol, then: LCS(x(i), y(j)) = max[LCS(x(i – 1), y(j)), LCS(x(i), y(j – 1))]

Inefficient: we keep evaluating the same LCS(i, j) over and over.

Instead, use dynamic programming.

Fill in a table of LCS(i, j) values by i and j.

You only have to figure each LCS(i, j) once.

O(n2).

Dynamic Programming

Stacked pair metric

Algorithm for Stacked Pair Metric

The longest common subsequence SUCH THAT there are no lone bonds.

If xi is matched to yj, THEN EITHER

xi-1 is matched to yj-1, OR

xi+1 is matched to yj+1

Cannot “break” a block LCS

Big regular LCS:

A C T G C T

G A C G C T

Break to get two smaller regular LCS’s:

A C T G C T

G A C G C T

Cannot “break” a block LCS

Big block LCS:

G G T A G G

C C T A C C

CANNOT break to get two smaller block LCS’s:

G G T A G G

C C T A C C

Adding a single symbol to a string can have effects arbitrarily far back

A C T C C C C T

G G G G G A C T G

A C T C C C C T G

G G G G G A C T G

These three bonds make the LCSP.

Add just one symbol, G, and the red bond must be moved to make the new LCSP.

Tail equality of two sequences

Tail equality 3:

A G C T C

A T C T C

Tail equality 0:

A G C T G

A T C T A

End count of a matching

End count 2:

A G C T C

A T C T C

End count 0:

A G C T G

A T C T A

The end count of a matching between x and y cannot exceed the tail equality of x and y.

- Let LCSP(k)(i, j) be the length of the longest LCSP(i, j) achievable with a matching of end count k.
- where e is the tail equality of x and y.

Substituting:

- O(n) worst case for one cell.
- O(n3) for algorithm.

- In practice, only 56% more time.
- “Efficient” algorithm takes O(n) memory.
- “Simple” algorithm takes O(n2) memory.

- Start with empty code.
- Repeatedly generate random codewords and add them if they meet the distance requirement.

When to stop?

- After “n” trials?
- When “n” trials in a row have failed?
- When fewer than “i” of the last “n” trials have succeeded?
- When the size of the code is near a maximum predicted by theory?

The identification of maximal frequent sets in data fields are the computational bottleneck in association rule discovery. This is an important problem and the independent sets and maximal cliques problems fit this paradigm.

DNA Code and DNA Bitstring Library

A A A A A A A A C C=T1

G G T T T T T T T T =BEAD PROBE (T1)

T T T C C A A A A A =F1

T T T T T G G A A A = BEAD PROBE (F1)

T T T C T T A A C C=T2

G G T T A A G A A A= BEAD PROBE (T2)

A C T A A C A A A A=F2

T T T T G T T A G T= BEAD PROBE (F2)

C A T A A A A C A C=T3

G T G T T T T A T G= BEAD PROBE (T3)

A T C T T T T C A A=F3

T T G A A A A G A T= BEAD PROBE (F3)

C A A T C C A T T A=T4

T A A T G G A T T G= BEAD PROBE (T4)

C C T T C T A A A T=F4

A T T T A G A A G G= BEAD PROBE (F4)

A C T C C T A A T A=T5

T A T T A G G A G T= BEAD PROBE (T5)

T C T C T C T A C T=F5

A G T A G A G A G A= BEAD PROBE (F5)

1. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-T4-T5

2. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-T4-F5

3. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-F4-T5

4. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-F4-F5

5. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-T4-T5

6. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-T4-F5

7. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-F4-T5

8. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-F4-F5

9. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-T4-T5

10. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-T4-F5

11. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-F4-T5

12. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-F4-F5

13. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-T4-T5

14. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-T4-F5

15. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-F4-T5

16. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-F4-F5

17. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-T4-T5

18. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-T4-F5

19. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-F4-T5

20. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-F4-F5

21. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-T4-T5

22. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-T4-F5

23. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-F4-T5

24. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-F4-F5

25. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-T4-T5

26. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-T4-F5

27. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-F4-T5

28. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-F4-F5

29. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-T4-T5

30. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-T4-F5

31. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-F4-T5

32. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-F4-F5

DNA LIBRARY=

DNA BITSTRINGS

DNA CODE

Example: Independent Sets and Cliques

3

Edges in G are

{1,2}, {2,3}, {3,4},

{4,5},{1,4},{2,5}

Edges in G’ are

{1,3}, {1,5}, {2,4},

{3,5},

4

2

3

3

2

2

4

G =

G’=

4

5

1

1

5

1

5

An independent set is a collection of vertices that contains no edge.

A clique is a subgraph were every pair of vertices has an edge between them.

For a graph G, its complement G’ is the set of edges not in G

A maximal independent set in G is a maximal clique in G’, e.g., {1,3,5}.

3

1

5

1. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-T4-T5

2. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-T4-F5

3. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-F4-T5

4. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-F4-F5

5. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-T4-T5

6. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-T4-F5

7. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-F4-T5

8. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-F4-F5

9. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-T4-T5

10. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-T4-F5

11. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-F4-T5

12. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-F4-F5

13. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-T4-T5

14. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-T4-F5

15. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-F4-T5

16. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-F4-F5

17. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-T4-T5

18. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-T4-F5

19. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-F4-T5

20. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-F4-F5

21. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-T4-T5

22. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-T4-F5

23. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-F4-T5

24. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-F4-F5

25. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-T4-T5

26. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-T4-F5

27. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-F4-T5

28. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-F4-F5

29. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-T4-T5

30. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-T4-F5

31. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-F4-T5

32. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-F4-F5

DNA Library

2^( # Coding Strands / 2)

# Coding Strands / 2 Bits

T T T T T G G A A A

24. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-F4-F5

T T T T G T T A G T

10.A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-T4-F5

X1=F or X2=F

T T T T G T T A G T

T T T T T G G A A A

29. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-T4-T5

All subsets

not containing

{1,2}

T T T T T G G A A A=Probe(F1)

T T T T G T T A G T=Probe(F2)

Edge {1,2} STM

9. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-T4-T5

10. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-T4-F5

11. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-F4-T5

12. A A A A A A A A C C-A C T A A C A A A A-C A T A A A A C A C-F4-F5

13. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-T4-T5

14. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-T4-F5

15. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-F4-T5

16. A A A A A A A A C C-A C T A A C A A A A-A T C T T T T C A A-F4-F5

17. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-T4-T5

18. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-T4-F5

19. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-F4-T5

20. T T T C C A A A A A -T T T C T T A A C C-C A T A A A A C A C-F4-F5

21. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-T4-T5

22. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-T4-F5

23. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-F4-T5

24. T T T C C A A A A A -T T T C T T A A C C -A T C T T T T C A A-F4-F5

25. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-T4-T5

26. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-T4-F5

27. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-F4-T5

28. T T T C C A A A A A -A C T A A C A A A A-C A T A A A A C A C-F4-F5

29. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-T4-T5

30. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-T4-F5

31. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-F4-T5

32. T T T C C A A A A A -A C T A A C A A A A-A T C T T T T C A A-F4-F5

X1=T and X2=T

1. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-T4-T5

2. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-T4-F5

3. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-F4-T5

4. A A A A A A A A C C -T T T C T T A A C C-C A T A A A A C A C-F4-F5

5. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-T4-T5

6. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-T4-F5

7. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-F4-T5

8. A A A A A A A A C C -T T T C T T A A C C -A T C T T T T C A A-F4-F5

{1,2}

{2,4}

{1,3}

{2,5}

{1,4}

{3,4}

{1,5}

{3,5}

{2,3}

{4,5}

Black ON, Red OFF =Independent Sets in G

Black OFF, Red ON =Cliques in G

Universal DNA Computer for any

Graph on n Vertices

DNA Library

Every Graph G on n vertices has

G union G’= all possible pairs on

n vertices. This enables the construction

of a universal device.

{1,2}

{1,3}

Each possible edge is an STM. Then depending

on the problem, the flow is directed by the edges

present (or absent) in the given graph

.

{n-2,n}

{n-1,n}

Edges in G ON, Edges in G’ OFF =Independent Sets in G

when flow completed

Edges in G OFF, Edges in G’ ON =Cliques in G

when flow completed

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