1. DNA Self-Assembly Robert Schweller Northwestern University
Speaking of Science talk
Buena Vista University
February 28, 2005

2. Outline Importance of DNA Self-Assembly Tile Self-Assembly
Synthesis of Nanostructures
DNA Computing
Tile Self-Assembly
DNA Word Design

3. Smart Bricks

4. Wang Tiles
TILE

5. TILE

6. TILE
G C A T C G
C G T A G C

7. TILE
G C A T C G
C G T A G C

8. TILE

10. TILE

13. Super Small Circuits, Built Autonomously

14. Molecular-scale pattern for a RAM memory with demultiplexed addressing
(Winfree, 2003)

15. DNA Computers +
Output!
Computer Program
Input

16. DNA Computers +
Output!
Computer Program
Input
Program

17. DNA Computers +
Output!
Computer Program
Input +
Input
Program

18. DNA Computers +
Output!
Computer Program
Input +
Output!
Input
Program

19. Outline Importance of DNA Self-Assembly
Tile Self-Assembly (Generalized Models)
Tile Complexity
Shape Verification
Error Resistance
DNA Word Design

20. Tile Model of Self-Assembly
(Rothemund, Winfree STOC 2000)
Tile System:
t : temperature, positive integer
G: glue function
T: tileset
s: seed tile

21. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

22. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

23. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

24. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

25. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

26. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

27. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

28. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

29. How a tile system self assembles
G(y,y) = 2
G(g,g) = 2
G(r, r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

30. New Models Multiple Temperature Model Flexible Glue Model
temperature may go up and down
Flexible Glue Model
Remove the restriction that G(x, y) = 0 for x!=y
Multiple Tile Model
tiles may cluster together before being added
Unique Shape Model
unique shape vs. unique supertile

31. New Models Multiple Temperature Model Flexible Glue Model
temperature may go up and down
Flexible Glue Model
Remove the restriction that G(x, y) = 0 for x!=y
Multiple Tile Model
tiles may cluster together before being added
Unique Shape Model
unique shape vs. unique supertile

32. New Models Multiple Temperature Model Flexible Glue Model
temperature may go up and down
Flexible Glue Model
Remove the restriction that G(x, y) = 0 for x!=y
Multiple Tile Model
tiles may cluster together before being added
Unique Shape Model
unique shape vs. unique supertile

33. New Models Multiple Temperature Model Flexible Glue Model
temperature may go up and down
Flexible Glue Model
Remove the restriction that G(x, y) = 0 for x!=y
Multiple Tile Model
tiles may cluster together before being added
Unique Shape Model
unique shape vs. unique supertile

34. Reduce Tile Complexity
Focus
Multiple Temperature Model
Adjust temperature during assembly
Flexible Glue Model
Remove the restriction that G(x, y) = 0 for x!=y
Goal:
Reduce Tile Complexity

35. Our Tile Complexity Results
Multiple temperature model:
k x N rectangles:
(our paper)
beats standard model:
(our paper)
Flexible Glue:
N x N squares:
(our paper)
(Adleman, Cheng,
Goel, Huang STOC 2001)
beats standard model:

36. Building k x N Rectangles
k-digit, base N(1/k) counter: k N

37. Building k x N Rectangles
k-digit, base N(1/k) counter: k
If N is the kth power of some integer, then you choose a base that is big enough and then seed the counter to an appropriate value.
Note that for k<<N, N^1/k dominates. N
Tile Complexity:

38. Build a 4 x 256 rectangle:
t = 2 S3 S2 S1 S g g g p C0 C1 C2 C3 S

39. t = 2 Build a 4 x 256 rectangle: S3 g S2 1 2 3 g S1 S g g g p C0 C1 C2g S2 1 2 3 g S1 S g g g p C0 C1 C2 C3 S3 S2 S1 g g p S C1 C2 C3

40. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 g S1 S g g1 1 S3 p r g S2 1 2 3 g S1 S g g g p C0 C1 C2 C3 S3 S2 S1 p S C1 C2 C3

41. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 g S1 S g g1 1 S3 p r g S2 1 2 3 g S1 S g g g p C0 C1 C2 C3 S3 S2 g g S1 1 S C1 C2 C3

42. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 g S1 S g g1 1 S3 p r g S2 1 2 3 g S1 S g g g p C0 C1 C2 C3 S3 S2 S1 1 p S C1 C2 C3 C0 C1 C2 C3

43. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 1 2 g S1 S1 1 S3 p r g S2 1 2 3 1 2 g S1 S g g g p 2 3 C0 C1 C2 C3 S3 S2 S1 1 1 1 p S C1 C2 C3 C0 C1 C2 C3

44. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 1 2 g S1 p1 1 S3 p r g S2 1 2 3 1 2 g S1 p r S g g g p 3 P R 2 3 p r C0 C1 C2 C3 S3 S2 S1 1 1 1 1 2 2 2 2 3 3 3 p S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3

45. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 1 2 g S1 p1 1 S3 p r g S2 1 2 3 1 2 g S1 p r S g g g p 3 P R 2 3 p r C0 C1 C2 C3 S3 S2 S1 1 1 1 1 2 2 2 2 3 3 3 P S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3

46. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 1 2 g S1 p1 1 S3 p r g S2 1 2 3 1 2 g S1 p r S g g g p 3 P R 2 3 p r C0 C1 C2 C3 S3 S2 1 S1 1 1 1 1 2 2 2 2 3 3 3 P S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3

47. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 1 2 g S1 p1 1 S3 p r g S2 1 2 3 1 2 g S1 p r S g g g p 3 P R 2 3 p r C0 C1 C2 C3 S3 S2 1 S1 1 1 1 1 2 2 2 2 3 3 3 P R S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3

48. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 1 2 g S1 p1 1 S3 p r g S2 1 2 3 1 2 g S1 p r S g g g p 3 P R 2 3 p r C0 C1 C2 C3 S3 S2 1 S1 1 1 1 1 2 2 2 2 3 3 3 P R … S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2

49. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 1 2 g S1 p1 1 S3 p r g S2 1 2 3 1 2 g S1 p r S g g g p 3 P R 2 3 p r C0 C1 C2 C3 S3 S2 1 1 1 … S1 1 1 1 1 2 2 2 2 3 3 3 P R S C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2

50. t = 2 Build a 4 x 256 rectangle: g g 1 1 S3 p r g S2 1 2 3 1 2 g S1 p1 1 S3 p r g S2 1 2 3 1 2 g S1 p r S g g g p 3 P R 2 3 p r C0 C1 C2 C3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 P 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 P 3 3 P R 1 1 1 1 2 2 2 2 3 3 3 P C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3

51. Building k x N Rectangles
k-digit, base N(1/k) counter: k
If N is the kth power of some integer, then you choose a base that is big enough and then seed the counter to an appropriate value.
Note that for k<<N, N^1/k dominates. N
Tile Complexity:

52. 2-temperature model
t = 4 3 1 3 3

53. 2-temperature model
t =

54. 2-temperature model Kolmogorov Complexity Beats Standard Model
(our paper)
Kolmogorov Complexity
(Rothemund,
Winfree STOC 2000)
Beats Standard Model
(our paper)

55. Assembly of N x N Squares

56. Assembly of N x N Squares
N - k k
N - k k

57. Assembly of N x N Squares
Complexity:
N - k X
(Adleman, Cheng,
Goel, Huang
STOC 2001) k
N - k Y k

58. N x N Squares --- Flexible Glue Model
Kolmogorov lower bounds:
Standard
(Rothemund, Winfree
STOC 2000)
Flexible
Standard Glue Function
Flexible Glue Function
a b c d e f a b c d e f
a b c d e f a b c d e f

59. N x N Square --- Flexible Glue Model
N – log N
All the complexity is coming from that damn seed row!
seed row
log N

60. N x N Square --- Flexible Glue Model
N – log N
Complexity:
All the complexity is coming from that damn seed row!
seed row
log N

61. N x N Square --- Flexible Glue Model
goal:
- seed binary counter to a given value - 1 1 1 1 1 1 1 1 1 1 1
All the complexity is coming from that damn seed row! 2
log N

62. N x N Square --- Flexible Glue Model5 3 3 3 4 4 4 4 4 4 5 5 5 5
. . . 3 4 5 1 2 3 4 5 1 2 3 4 5
All the complexity is coming from that damn seed row!

63. N x N Square --- Flexible Glue Model
key idea: 5
| | | | | | | | | | | | | 5 3 3 3 4 4 4 4 4 4 5 5 5 5
. . . 3 4 5 1 2 3 4 5 1 2 3 4 5
All the complexity is coming from that damn seed row!

64. N x N Square --- Flexible Glue Model
G(b4, p5) = 1
G(b4, w5) = 0 5 p5 5 5 5 5 w5 b4 1 2 3 4 5

65. N x N Square --- Flexible Glue Model5
given B = …
encode B into glue function p5 b4 4
p0 p1 p2 p3 p4 p5 b b b b b b
B = …

66. N x N Square --- Flexible Glue Model
build block
Complexity:

68. N – log N
2 x log N block
log N

69. N – log N
N – log N
log N
log N

70. X
N – log N
Complexity:
N – log N
log N Y
log N

71. Our Tile Complexity Results
Multiple temperature model:
k x N rectangles:
(our paper)
beats standard model:
(our paper)
Flexible Glue:
N x N squares:
(our paper)
(Adleman, Cheng,
Goel, Huang STOC 2001)
beats standard model:

72. Molecular-scale pattern for a RAM memory with demultiplexed addressing
(Winfree, 2003)

73. Outline Importance of DNA Self-Assembly
Tile Self-Assembly (Generalized Models)
Tile Complexity
Shape Verification
Error Resistance
DNA Word Design

74. Shape Verification Unique Shape Problem Input: T, a tile system
S, a shape
Question: Does T uniquely assemble S.
Standard: P (Adleman, Cheng, Goel, Huang, Kempe,
Flexible Glue: P Espanes, Rothemund, STOC 2002)
Unique Shape: Co-NPC (our paper)
Multiple Temperature: NP-hard (our paper)
Multiple Tile: NP-hard (our paper)

75. 3-SAT Problem
Clause 1:
Clause 2:
Clause 3:

76. Unique-Shape Model *

77. Unique-Shape Model * x3 x2 x1 *

78. Unique-Shape Model * x3 x2 x1 * * c1 c2 c3 *

79. Unique-Shape Model * 1 x x3 x x x2 x x1 x * * c1 c2 c3 *

80. Unique-Shape Model * x3 1 x2 1 x1 * * c1 c2 c3 *

81. Unique-Shape Model * x3 1 x2 1 x1 c1 * * c1 c2 c3 *

82. Unique-Shape Model * x3 1 x2 1 ok x1 c1 * * c1 c2 c3 *

83. Unique-Shape Model * x3 1 ok x2 1 ok x1 c1 * * c1 c2 c3 *

84. Unique-Shape Model * x3 1 ok x2 1 ok x1 c1 c2 * * c1 c2 c3 *

85. Unique-Shape Model * x3 1 ok x2 1 ok c2 x1 c1 c2 * * c1 c2 c3 *

86. Unique-Shape Model * x3 1 ok ok x2 1 ok c2 x1 c1 c2 * * c1 c2 c3 *

87. Unique-Shape Model * x3 1 ok ok x2 1 ok c2 x1 c1 c2 ok * * c1 c2 c3 *

88. Unique-Shape Model * x3 1 ok ok ok x2 1 ok c2 ok x1 c1 c2 ok * * c1 c2c1 c2 ok * * c1 c2 c3 *

89. Unique-Shape Model * * x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * *c1 c2 ok * * * c1 c2 c3 *

90. Unique-Shape Model * * T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok *c1 c2 ok * * * c1 c2 c3 *

91. Unique-Shape Model * * T T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 okc1 c2 ok * * * c1 c2 c3 *

92. Unique-Shape Model * * T T T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2c1 c2 ok * * * c1 c2 c3 *

93. Satisfied Unique-Shape Model * * T T T SAT x3 1 ok ok ok * x2 1 ok c2c1 c2 ok * * * c1 c2 c3 *
Satisfied
(LaBean and Lagoudakis, 1999)

94. Satisfied Unique-Shape Model * * T T T SAT * * x3 1 ok ok ok * x3 okok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
(LaBean and Lagoudakis, 1999)

95. Satisfied Unique-Shape Model * * T T T SAT * * T x3 1 ok ok ok * x3 okok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
(LaBean and Lagoudakis, 1999)

96. Satisfied Unique-Shape Model * * T T T SAT * * T F x3 1 ok ok ok * x3ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
(LaBean and Lagoudakis, 1999)

97. Not Satisfied Satisfied Unique-Shape Model * * T T T SAT * * T F F x31 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
Not Satisfied
(LaBean and Lagoudakis, 1999)

98. Multiple Temperature Model* * * * * * * * * * x3 x3 x2 x2 x1 x1 * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
Not Satisfied

99. Multiple Temperature Model* * * * * * * * * T T T T
SAT * T T F F NO x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
Not Satisfied

100. Multiple Temperature Model* * * * * * * * * T T T T
SAT * T T F F NO x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
Not Satisfied

101. Multiple Temperature Model* * * * * * * * * * x3 x3 x2 x2 x1 x1 * *
Satisfied
Not Satisfied

102. Unique Shape Problem Results
Standard P
Flexible Glue P
Multiple Temperature NP-hard
Unique Shape Co-NPC
Multiple Tile NP-hard
(Adleman, Cheng,
Goel, Huang, Kempe,
Espanes, Rothemund,
STOC 2002)
(our paper)
(our paper)
(our paper)

103. Outline Importance of DNA Self-Assembly
Tile Self-Assembly (Generalized Models)
Tile Complexity
Shape Verification
Error Resistance
DNA Word Design

104. Further Research t = 2 Error Resistance: Insufficient Bindings
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

105. Further Research t = 2 Error Resistance: Insufficient Bindings
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

106. Further Research t = 2 Error Resistance: Insufficient Bindings
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

107. Further Research t = 2 Error Resistance: Insufficient Bindings
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

108. Further Research t = 2 Error Resistance: Insufficient Bindings
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

109. Further Research t = 2 Error Resistance: Insufficient Bindings
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

110. Further Research t = 2 Error Resistance: Insufficient Bindings
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

111. Further Research Error Resistance: Insufficient Bindings Standard
Fluctuating b
temperature
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity a

112. Further Research 2 1 Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

113. Further Research 2 1 Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

114. Further Research 2 1 Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

115. Further Research 2 1 Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

116. Outline Importance of DNA Self-Assembly
Tile Self-Assembly (Generalized Models)
DNA Word Design

117. DNA Word Design 1 2 3 4 5 6 7 8 9 3 4
ACCT
TGGA
GCTA
CGAT 5

118. DNA Word Design 1 2 3 4 5 6 7 8 9 green: red: yellow: blue: purple:
white:
black:
teal:
ACCT
GAAA
GCTA
CGTA
CTCG
CATG
ACGA
TTTA
Must be sufficiently
different
-Must have similar
thermodynamic properties
-Must be short

119. Hamming Constraint (k)
ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTAGCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGCCCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTAGGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCTAGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATCAGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC
X= GCTTCGTAGCATAG
| | |
Y= TTAGCCGCGTAGCT
n strings
HAMM(X,Y) = 11 > k
length L = 14

120. Free Energy Constraint
A C G T A C G T
ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTAGCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGCCCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTAGGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCTAGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATCAGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC
Pairwise
free energies =
n strings
length L = 14

121. Free Energy Constraint
A C G T A C G T
ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTAGCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGCCCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTAGGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCTAGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATCAGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC
Pairwise
free energies =
n strings
X= AGCATTATAGATAC
FE(X) =
length L = 14

122. Free Energy Constraint
A C G T A C G T
ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTAGCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGCCCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTAGGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCTAGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATCAGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC
Pairwise
free energies =
n strings
X= AGCATTATAGATAC
FE(X) =
For all strings X and Y:
|FE(X) – FE(Y)| < C
length L = 14

123. DNA Word Design Word Design Problem Input: integers n and k
Output: n strings of length L such that
for all strings X and Y:
1) HAMM(X,Y) > k
2) |FE(X) – FE(Y)| < C
Minimize L

124. DNA Word Design
Simple Lower Bound:
L > log n
L > k
L > ½(k + log n)

125. DNA Word Design
Word Length:
Run-Time:

126. DNA Word Design Hamming Constraint k: -Set L = 5*(k + log n)
-Generate all random strings
Pr[FAILURE] =
All Random
length L = 5*(k+log n)

127. Free Energy Constraint:
length L = O(k+log n)

128. Free Energy Constraint:
All length
L strings n
length L = O(k+log n)

129. Free Energy Constraint:
Low FE
All length
L strings n
length L = O(k+log n)

130. Free Energy Constraint:
Low FE
All length
L strings n
High FE
length L = O(k+log n)

131. Free Energy Constraint:
Low FE
All length
L strings n
High FE
length L = O(k+log n)

132. Free Energy Constraint:
All length
L strings n
length L = O(k+log n)
Fact: Strings can be chosen
to satisfy the Free Energy Constraint

133. Free Energy Constraint:
For each string X:
a < FE(X) < b n
How do you get
these strings?
length L = O(k+log n)

134. Free Energy Constraint:
Given:

135. Free Energy Constraint:
Given:
Find:

136. Free Energy Constraint: Problem: 4^L length L strings
Given:
Find:
a < FE < b
Problem: 4^L length L strings

137. Free Energy Constraint:
Fixed Energy String Problem
Input: Length L, Energy E
Output: a string with:
1) length L
2) free energy E

138. Free Energy Constraint:
Consider bases a,b in {A,C,G,T}
ci = # of length L strings such that:
1) FE = i
2) First character is a
3) Last Character is b a b L

139. fLa,b, fL/2a,b, fL/4a,b, …, f1a,b for all
What if we knew…
fLa,b, fL/2a,b, fL/4a,b, …, f1a,b for all
a,b in {A,C,G,T}

140. fLa,b, fL/2a,b, fL/4a,b, …, f1a,b for all
What if we knew…
fLa,b, fL/2a,b, fL/4a,b, …, f1a,b for all
a,b in {A,C,G,T} a b L

141. fLa,b, fL/2a,b, fL/4a,b, …, f1a,b for all
What if we knew…
fLa,b, fL/2a,b, fL/4a,b, …, f1a,b for all
a,b in {A,C,G,T} a c d b
FEc,d
L/2 L

142. fLa,b, fL/2a,b, fL/4a,b, …, f1a,b for all
What if we knew…
fLa,b, fL/2a,b, fL/4a,b, …, f1a,b for all
a,b in {A,C,G,T}
SOLUTION: in O(L log L)
time complexity a c d b
FEc,d
L/2 L

143. Recursive Property: a c d b
FEc,d
L/2 L

144. Recursive Property:
T(L) = a c d b
FEc,d
L/2 L

145. Recursive Property:
T(L) = T(L/2) + a c d b
FEc,d
L/2 L

146. Recursive Property:
T(L) = T(L/2) + L log L a c d b
FEc,d
L/2 L

147. T(L) = T(L/2) + L log L = O(L log L) FEc,d Recursive Property: a c d b

148. Summary for Word Design
Hamming Constraint (k):
-Randomly generate words
of length L = O(k + log n) n
length L = O(k+log n)

149. Summary for Word Design
Hamming Constraint (k):
-Randomly generate words
of length L = O(k + log n)
Free Energy Constraint:
-Append new strings n
length L = O(k+log n)

150. Summary for Word Design
Hamming Constraint (k):
-Randomly generate words
of length L = O(k + log n)
Free Energy Constraint:
-Append new strings
Run-Time: n
Word Length:
length L = O(k+log n)

151. Questions? DNA Self-Assembly Importance of DNA Self-Assembly
Tile Self-Assembly
DNA Word Design
Questions?