DNA Self-Assembly

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
• Synthesis of Nanostructures
• DNA Computing
• Tile Self-Assembly
• DNA Word Design

3
Smart Bricks
4
Wang Tiles
5
(No Transcript)
6
G C A T C G
C G T A G C
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11


12
(No Transcript)
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
• 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
• 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
• 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
• 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
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
N
Tile Complexity
38
Build a 4 x 256 rectangle
t 2
S3
0
S2
0
S1
0
S
g
g
p
g
C1
C2
C3
C0
S
39
Build a 4 x 256 rectangle
t 2
S3
0
g
S2
0
0
1
2
3
0
0
g
S1
0
S
g
g
p
g
C1
C2
C3
C0
0
S3
0
S2
0
0
S1
g
g
p
S
C1
C2
C3
40
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
g
S1
0
S
g
g
p
g
C1
C2
C3
C0
S3
0
0
S2
0
0
S1
0
0
p
S
C1
C2
C3
41
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
g
S1
0
S
g
g
p
g
C1
C2
C3
C0
S3
0
0
S2
0
0
g
g
S1
0
0
0
1
S
C1
C2
C3
42
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
g
S1
0
S
g
g
p
g
C1
C2
C3
C0
S3
0
0
0
0
S2
0
0
0
0
S1
0
0
0
1
p
S
C1
C2
C3
C0
C1
C2
C3
43
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
1
2
g
S1
0
S
g
g
p
g
2
3
C1
C2
C3
C0
S3
0
0
0
0
0
0
S2
0
0
0
0
0
0
S1
0
0
0
1
1
1
p
S
C1
C2
C3
C0
C1
C2
C3
44
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
1
2
g
S1
0
p
r
S
P
R
g
g
p
g
3
0
2
3
p
r
C1
C2
C3
C0
S3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S1
0
0
0
1
1
1
2
2
3
3
1
2
2
3
p
S
C0
C1
C2
C3
C1
C2
C3
C0
C1
C2
C3
C0
C1
C2
C3
45
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
1
2
g
S1
0
p
r
S
P
R
g
g
p
g
3
0
2
3
p
r
C1
C2
C3
C0
S3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S1
0
0
0
1
1
1
2
2
3
3
1
2
2
3
P
S
C0
C1
C2
C3
C1
C2
C3
C0
C1
C2
C3
C0
C1
C2
C3
46
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
1
2
g
S1
0
p
r
S
P
R
g
g
p
g
3
0
2
3
p
r
C1
C2
C3
C0
S3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
S1
0
0
0
1
1
1
2
2
3
3
1
2
2
3
P
S
C0
C1
C2
C3
C1
C2
C3
C0
C1
C2
C3
C0
C1
C2
C3
47
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
1
2
g
S1
0
p
r
S
P
R
g
g
p
g
3
0
2
3
p
r
C1
C2
C3
C0
S3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
S1
0
0
0
1
1
1
2
2
3
3
1
2
2
3
P
R
S
C0
C1
C2
C3
C1
C2
C3
C0
C1
C2
C3
C0
C1
C2
C3
48
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
1
2
g
S1
0
p
r
S
P
R
g
g
p
g
3
0
2
3
p
r
C1
C2
C3
C0
S3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
S1
0
0
0
1
1
1
2
2
3
3
1
2
2
3
P
R

S
C0
C1
C2
C3
C1
C2
C3
C0
C1
C2
C3
C0
C1
C2
C3
C0
C1
C2
49
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
1
2
g
S1
0
p
r
S
P
R
g
g
p
g
3
0
2
3
p
r
C1
C2
C3
C0
S3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1

S1
0
0
0
1
1
1
2
2
3
3
1
2
2
3
P
R
0
0
S
C0
C1
C2
C3
C1
C2
C3
C0
C1
C2
C3
C0
C1
C2
C3
C0
C1
C2
50
Build a 4 x 256 rectangle
t 2
g
g
1
0
0
1
S3
0
p
r
g
S2
0
0
1
2
3
0
0
1
2
g
S1
0
p
r
S
P
R
g
g
p
g
3
0
2
3
p
r
C1
C2
C3
C0
P
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
P
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
3
P
3
3
2
1
2
1
1
0
1
0
0
R
P
3
3
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
N
Tile Complexity
52
2-temperature model
t 4
3
1
3
3
53
2-temperature model
t 4 6
54
2-temperature 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 1 0 2 0 0
1 b 0 0 1 0 1 0 c 0 0 3 0 1
1 d 2 2 2 2 0 1 e 0 0 0 1
2 1 f 1 1 2 2 1 1
a b c d e f a 1 - - -
- - b - 0 - - - - c - -
3 - - - d - - - 2 - - e
- - - - 2 - f - - - -
- 1
59
N x N Square --- Flexible Glue Model
N log N
seed row
log N
60
N x N Square --- Flexible Glue Model
N log N
Complexity
seed row
log N
61
N x N Square --- Flexible Glue Model
goal - seed binary counter to a given
value -
0
1
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
2
log N
62
N x N Square --- Flexible Glue Model
5
3
3
3
4
4
4
4
4
4
5
5
5
5
. . .
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
63
N x N Square --- Flexible Glue Model
key idea
5
0 0 1 1 0 1 1 0 0 1
1 1 0

5
3
3
3
4
4
4
4
4
4
5
5
5
5
. . .
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
64
N x N Square --- Flexible Glue Model
G(b4, p5) 1 G(b4, w5) 0
5
p5
5
5
5
5
w5
b4
4
5
3
2
1
65
N x N Square --- Flexible Glue Model
5

• given B 011011 110101 010111
• encode B into glue function

p5
b4
4
p0 p1 p2 p3 p4 p5 b0 0 1 1
0 1 1 b1 1 1 0 1 0 1 b2
0 1 0 1 1 1 b3 0 0 1
0 1 0 b4 0 0 0 0 0 1 b5
1 1 1 1 1 0
B 011011 110101 010111
66
N x N Square --- Flexible Glue Model

• build block

• Complexity

0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 0 1 0 1
67
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 1 1 1 0
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 1 1 0 1
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 1 1 0 0
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 1 0 1 1
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 1 0 1 0
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 1 0 0 1
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 1 0 0 0
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 0 1 1 1
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 0 1 1 0
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1
1 0 1 1 1 0 0 0 1 0 1
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
log N
Y
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


c2
c1
c3

79
Unique-Shape Model

1
x
x3
x
0
x
x2
x
x1
x


c2
c1
c3

80
Unique-Shape Model

x3
1
x2
1
x1
0


c2
c1
c3

81
Unique-Shape Model

x3
1
x2
1
x1
c1
0


c2
c1
c3

82
Unique-Shape Model

x3
1
x2
ok
1
x1
c1
0


c2
c1
c3

83
Unique-Shape Model

x3
ok
1
x2
ok
1
x1
c1
0


c2
c1
c3

84
Unique-Shape Model

x3
ok
1
x2
ok
1
x1
c2
c1
0


c2
c1
c3

85
Unique-Shape Model

x3
ok
1
x2
ok
c2
1
x1
c2
c1
0


c2
c1
c3

86
Unique-Shape Model

x3
ok
ok
1
x2
ok
c2
1
x1
c2
c1
0


c2
c1
c3

87
Unique-Shape Model

x3
ok
ok
1
x2
ok
c2
1
x1
ok
c2
c1
0


c2
c1
c3

88
Unique-Shape Model

x3
ok
ok
ok
1
x2
ok
ok
c2
1
x1
ok
c2
c1
0


c2
c1
c3

89
Unique-Shape Model


x3
ok
ok
ok
1

x2
ok
ok
c2
1

x1
ok
c2
c1
0



c2
c1
c3

90
Unique-Shape Model


T
x3
ok
ok
ok
1

x2
ok
ok
c2
1

x1
ok
c2
c1
0



c2
c1
c3

91
Unique-Shape Model


T
T
x3
ok
ok
ok
1

x2
ok
ok
c2
1

x1
ok
c2
c1
0



c2
c1
c3

92
Unique-Shape Model


T
T
T
x3
ok
ok
ok
1

x2
ok
ok
c2
1

x1
ok
c2
c1
0



c2
c1
c3

93
Unique-Shape Model


T
T
T
SAT
x3
ok
ok
ok
1

x2
ok
ok
c2
1

x1
ok
c2
c1
0



c2
c1
c3

Satisfied
(LaBean and Lagoudakis, 1999)
94
Unique-Shape Model


T
T
T
SAT


x3
ok
ok
ok

1
x3
ok
c2
ok

0
x2
ok
ok
c2

1
x2
ok
ok
c2

1
x1
ok
c2

c1
0
x1
ok
c2

c1
0


c2
c1
c3



c2
c1
c3

Satisfied
(LaBean and Lagoudakis, 1999)
95
Unique-Shape Model


T
T
T
SAT


T
x3
ok
ok
ok

1
x3
ok
c2
ok

0
x2
ok
ok
c2

1
x2
ok
ok
c2

1
x1
ok
c2

c1
0
x1
ok
c2

c1
0


c2
c1
c3



c2
c1
c3

Satisfied
(LaBean and Lagoudakis, 1999)
96
Unique-Shape Model


T
T
T
SAT


T
F
x3
ok
ok
ok

1
x3
ok
c2
ok

0
x2
ok
ok
c2

1
x2
ok
ok
c2

1
x1
ok
c2

c1
0
x1
ok
c2

c1
0


c2
c1
c3



c2
c1
c3

Satisfied
(LaBean and Lagoudakis, 1999)
97
Unique-Shape Model


T
T
T
SAT


T
F
F
x3
ok
ok
ok

1
x3
ok
c2
ok

0
x2
ok
ok
c2

1
x2
ok
ok
c2

1
x1
ok
c2

c1
0
x1
ok
c2

c1
0


c2
c1
c3



c2
c1
c3

Not Satisfied
Satisfied
(LaBean and Lagoudakis, 1999)
98
Multiple Temperature Model










x3
x3
x2
x2
x1
x1


c1
c2
c3



c1
c2
c3

Not Satisfied
Satisfied
99
Multiple Temperature Model









T
T
T
T
SAT

T
T
F
F
NO
x3
1
ok
ok
ok

x3
0
ok
c2
ok

x2
1
ok
c2
ok

x2
1
ok
c2
ok

x1
0
c1
c2
ok

x1
0
c1
c2
ok



c1
c2
c3



c1
c2
c3

Not Satisfied
Satisfied
100
Multiple Temperature Model









T
T
T
T
SAT

T
T
F
F
NO
x3
1
ok
ok
ok

x3
0
ok
c2
ok

x2
1
ok
c2
ok

x2
1
ok
c2
ok

x1
0
c1
c2
ok

x1
0
c1
c2
ok



c1
c2
c3



c1
c2
c3

Not Satisfied
Satisfied
101
Multiple Temperature Model










x3
x3
x2
x2
x1
x1


Not Satisfied
Satisfied
102
Unique Shape Problem Results
Standard P Flexible Glue P Multiple
Temperature NP-hard Unique Shape Co-NPC Multip
le 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
Error Resistance Insufficient Bindings
t 2
105
Further Research
Error Resistance Insufficient Bindings
t 2
106
Further Research
Error Resistance Insufficient Bindings
t 2
107
Further Research
Error Resistance Insufficient Bindings
t 2
108
Further Research
Error Resistance Insufficient Bindings
t 2
109
Further Research
Error Resistance Insufficient Bindings
t 2
110
Further Research
Error Resistance Insufficient Bindings
t 2
111
Further Research
Error Resistance Insufficient Bindings
Standard
Fluctuating
b
temperature
a
112
Further Research
113
Further Research
114
Further Research
115
Further Research
116
Outline

• Importance of DNA Self-Assembly
• Tile Self-Assembly (Generalized Models)
• DNA Word Design

117
DNA Word Design
5
1
2
3
4
6
7
8
9
118
DNA Word Design
5
1
2
3
4
6
7
8
9
green red yellow blue purple white black te
al
ACCT GAAA GCTA CGTA CTCG CATG ACGA TTTA

• Must be sufficiently
• different
• -Must have similar
• thermodynamic properties
• -Must be short

119
Hamming Constraint (k)
ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTA
GCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGC
CCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTA
GGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCT
AGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATC
AGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC
X GCTTCGTAGCATAG Y
TTAGCCGCGTAGCT
n strings
HAMM(X,Y) 11 gt k
length L 14
120
Free Energy Constraint
A C G T A 2 1 5 3 C 7 2 6 9 G 1 1 3 1 T 8 7 4 2
ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTA
GCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGC
CCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTA
GGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCT
AGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATC
AGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC
Pairwise free energies

n strings
length L 14
121
Free Energy Constraint
A C G T A 2 1 5 3 C 7 2 6 9 G 1 1 3 1 T 8 7 4 2
ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTA
GCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGC
CCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTA
GGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCT
AGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATC
AGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC
Pairwise free energies

n strings
X AGCATTATAGATAC
FE(X) 517...
length L 14
122
Free Energy Constraint
A C G T A 2 1 5 3 C 7 2 6 9 G 1 1 3 1 T 8 7 4 2
ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTA
GCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGC
CCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTA
GGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCT
AGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATC
AGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC
Pairwise free energies

n strings
X AGCATTATAGATAC
FE(X) 517...
For all strings X and Y FE(X) FE(Y) lt 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) gt k
2) FE(X) FE(Y) lt C Minimize L
124
DNA Word Design
Simple Lower Bound
L gt log n L gt k L gt 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
PrFAILURE
All Random
length L 5(klog n)
127
Free Energy Constraint
n
length L O(klog n)
128
Free Energy Constraint
All length L strings
n
length L O(klog n)
129
Free Energy Constraint
Low FE
All length L strings
n
length L O(klog n)
130
Free Energy Constraint
Low FE
All length L strings
n
High FE
length L O(klog n)
131
Free Energy Constraint
Low FE
All length L strings
n
High FE
length L O(klog n)
132
Free Energy Constraint
All length L strings
n
length L O(klog n)
Fact Strings can be chosen to satisfy the Free
Energy Constraint
133
Free Energy Constraint
For each string X a lt FE(X) lt b
n
How do you get these strings?
length L O(klog n)
134
Free Energy Constraint
Given
135
Free Energy Constraint
Given
Find
136
Free Energy Constraint
Given
Find
a lt FE lt b
Problem 4L 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
What if we knew
fLa,b, fL/2a,b, fL/4a,b, , f1a,b for all
a,b in A,C,G,T
140
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
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
c
d
FEc,d
L/2
L
142
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
b
c
d
FEc,d
L/2
L
143
Recursive Property
a
b
c
d
FEc,d
L/2
L
144
Recursive Property
T(L)
a
b
c
d
FEc,d
L/2
L
145
Recursive Property
T(L) T(L/2)
a
b
c
d
FEc,d
L/2
L
146
Recursive Property
T(L) T(L/2) L log L
a
b
c
d
FEc,d
L/2
L
147
Recursive Property
T(L) T(L/2) L log L O(L
log L)
a
b
c
d
FEc,d
L/2
L
148
Summary for Word Design
Hamming Constraint (k) -Randomly generate
words of length L O(k log n)
n
length L O(klog 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(klog 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(klog n)
151
DNA Self-Assembly

• Importance of DNA Self-Assembly
• Tile Self-Assembly
• DNA Word Design

Questions?