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```					Complexities for Generalized
Models of Self-Assembly

Gagan Aggarwal               Stanford University
Michael H. Goldwasser        St. Louis University
Ming-Yang Kao                Northwestern University
Robert T. Schweller          Northwestern University

Some results were obtained independantly
by Cheng, Espanes 2003
Outline
• Importance of DNA Self-Assembly
– Synthesis of Nanostructures
– DNA Computing

• Tile Self-Assembly

• DNA Word Design
TILE
TILE

GCATCG

CGTAGC
TILE

GCATCG

CGTAGC
TILE
Super Small Circuits, Built Autonomously
Molecular-scale pattern for a RAM memory with demultiplexed addressing
(Winfree, 2003)
DNA Computers

+               Output!

Computer Program       Input
DNA Computers

+               Output!

Computer Program       Input

Program
DNA Computers

+               Output!

Computer Program       Input

+

Program              Input
DNA Computers

+               Output!

Computer Program       Input

+

Program              Input       Output!
Outline
• Importance of DNA Self-Assembly

• Tile Self-Assembly (Generalized Models)
– Tile Complexity
– Shape Verification
– Error Resistance

• DNA Word Design
Tile Model of Self-Assembly
(Rothemund, Winfree STOC 2000)

Tile System:    {t , G, T , s}
t : temperature, positive integer

G: glue function G :     {0,1,...,t}

r           p               r

T: tileset      {     b

g
,
y   y

r
w
,   b

r
b
, ... }
s: seed tile
How a tile system self assembles

S     a     b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x     c     d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2

S
How a tile system self assembles

S     a     b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x     c     d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2

S     a
How a tile system self assembles

S     a     b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x     c     d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2

c

S     a
How a tile system self assembles

S     a     b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x     c     d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2
d

c

S     a
How a tile system self assembles

S       a   b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x       c   d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2
d

c

S     a   b
How a tile system self assembles

S       a   b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x       c   d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2
d

c     x

S     a   b
How a tile system self assembles

S       a   b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x       c   d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2
d

c     x   x

S     a   b
How a tile system self assembles

S       a   b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x       c   d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2
d     x

c     x   x

S     a   b
How a tile system self assembles

S       a   b
G(y,y) = 2
T=
G(g,g) = 2
G(r, r) = 2
x       c   d             G(b,b) = 2
G(p,p) = 1
G(w,w) = 1

t=2
d     x   x

c     x   x

S     a   b
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
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
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
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
Focus
• Multiple Temperature Model
• Flexible Glue Model
– Remove the restriction that G(x, y) = 0 for x!=y

Goal:
Reduce Tile Complexity
Our Tile Complexity Results
Multiple temperature model:
log N
k x N rectangles:    (           )           (our paper)
log log N

N 1/ k
beats standard model:   (        )          (our paper)
k

Flexible Glue:
N x N squares:          ( log N )           (our paper)

log N
beats standard model: (
)
log log N     Goel, Huang STOC 2001)
Building k x N Rectangles

k-digit, base N(1/k) counter:
0    0   0   0   0            2   2   2

kk
0    0   0   0   0
...      2   2   2

0    0   0   1   1            2   2   2

S0    1   2   0   1            0   1   2

N
N
Building k x N Rectangles

k-digit, base N(1/k) counter:
0    0   0   0   0                         2   2   2

k   k
0    0   0   0   0
...                 2   2   2

0    0   0   1   1                         2   2   2

S0    1   2   0   1                         0   1   2

N
N

Tile Complexity:        O (k  N   1/ k
)
Build a 4 x 256 rectangle:   t=2
S3 0

S2 0
S1 0
S   g g g p
C0 C1 C2 C3

S
Build a 4 x 256 rectangle:   t=2
S3 0
g
S2 0    0 0 0   1    2   3
S1 0      g

S     g g g p
C0 C1 C2 C3

S3 0
0
S2 0
S1 0 g g p
S C1 C2 C3
Build a 4 x 256 rectangle:    t=2     g   g
S3 0                               0 0   1 1
g                          p   r
S2 0    0 0 0   1    2   3
S1 0      g

S      g g g p
C0 C1 C2 C3

S3 0 0
S2 0 0
S1 0 0 p
S C1 C2 C3
Build a 4 x 256 rectangle:    t=2     g   g
S3 0                               0 0   1 1
g                          p   r
S2 0    0 0 0   1    2   3
S1 0      g

S      g g g p
C0 C1 C2 C3

S3 0 0
S2 0 0 g g
S1 0 0 0 1
S C1 C2 C3
Build a 4 x 256 rectangle:    t=2     g   g
S3 0                               0 0   1 1
g                          p   r
S2 0    0 0 0   1    2   3
S1 0      g

S      g g g p
C0 C1 C2 C3

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

S      g g g p                      2    3
C0 C1 C2 C3

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

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

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

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

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

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

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

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 0 0 0 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
Building k x N Rectangles

k-digit, base N(1/k) counter:
0    0   0   0   0                         2   2   2

k   k
0    0   0   0   0
...                 2   2   2

0    0   0   1   1                         2   2   2

S0    1   2   0   1                         0   1   2

N
N

Tile Complexity:        O (k  N   1/ k
)
2-temperature model
t= 4

3
1
j
3
k                            3

N
2-temperature model
t=4   6

j

k

N
2-temperature model
log N
O(           )
O( j  N   1/ j
)      log log N
(our paper)

log N
Kolmogorov Complexity     (           )   (Rothemund,
log log N     Winfree STOC 2000)

N 1/ k
Beats Standard Model      (        )        (our paper)
k

k

N
Assembly of N x N Squares
Assembly of N x N Squares

N-k

k
N-k
k
Assembly of N x N Squares

Complexity:
N-k                             log N
(           )
log log N
X
Goel, Huang
STOC 2001)
k
N-k
Y     k
N x N Squares --- Flexible Glue Model
Kolmogorov lower bounds:
log N
Standard                     (           )       (Rothemund, Winfree
log log N          STOC 2000)

Flexible                  ( log N )

Standard Glue Function                     Flexible Glue Function
a   b   c    d   e   f                         a   b   c   d   e   f
a   1   -   -    -   -   -                     a   1   0   2   0   0   1
b   -   0    -   -   -    -                    b   0   0   1   0   1   0
c   -   -   3    -   -   -                     c   0   0   3   0   1   1
d   -   -    -   2   -    -                    d   2   2   2   2   0   1
e   -   -   -    -   2   -                     e   0   0   0   1   2   1
f   -   -    -   -   -   1                     f   1   1   2   2   1   1
N x N Square --- Flexible Glue Model

N – log N

seed row

log N
N x N Square --- Flexible Glue Model

N – log N

Complexity:
seed row                  k  log N

log N
N 1/ k  2
O (k  N 1/ k )  O (log N )
N x N Square --- Flexible Glue Model

goal:
- seed binary counter to a given value

- O( log N )

0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 1
2

log N
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
N x N Square --- Flexible Glue Model

key idea:

0 0   1    1   0    1   1   0   0   1    1   1   0   5
| |   |    |    |   |   |   |   |    |   |   |   |       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
N x N Square --- Flexible Glue Model
G(b4, p5) = 1
G(b4, w5) = 0
5
p5

5
w5
5 5 5 b          4

1 2 3 4 5
N x N Square --- Flexible Glue Model

• given B = 011011 110101 010111 …
5
p5
• encode B into glue function                   b4
4

p0 p1 p2 p3 p4 p5
b0    0 1 1 0 1 1
b1    1 1 0 1 0 1
B = 011011 110101 010111 …       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
N x N Square --- Flexible Glue Model

• build   2 log N block

• Complexity:      O( log N )

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
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
N – log N

2 x log N block

log N
N – log N

N – log N
log N

log N
N – log N          X
Complexity:
( log N )

N – log N
log N

Y      log N
Our Tile Complexity Results
Multiple temperature model:
log N
k x N rectangles:    (           )           (our paper)
log log N

N 1/ k
beats standard model:   (        )          (our paper)
k

Flexible Glue:
N x N squares:          ( log N )           (our paper)

log N
beats standard model: (
)
log log N     Goel, Huang STOC 2001)
Molecular-scale pattern for a RAM memory with demultiplexed addressing
(Winfree, 2003)
Outline
• Importance of DNA Self-Assembly

• Tile Self-Assembly (Generalized Models)
– Tile Complexity
– Shape Verification
– Error Resistance

• DNA Word Design
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:            Co-NPC       (our paper)
Unique-Shape Model

*    T   T    T    T SAT       *    T   T    F    F

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   *

Satisfied                     Not Satisfied

(LaBean and Lagoudakis, 1999)
Multiple Temperature Model
*                           *

*                           *

*                           *

*                           *
*                           *

x3                          x3

x2                          x2

x1                          x1

*    *    c1 c2 c3   *      *    *   c1 c2 c3   *

Satisfied                  Not Satisfied
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   *

Satisfied                   Not Satisfied
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   *

Satisfied                  Not Satisfied
Multiple Temperature Model
*                        *

*                        *

*                        *

*                        *
*                        *

x3                       x3

x2                       x2

x1                       x1

*                        *

Satisfied                     Not Satisfied
Unique Shape Problem Results

Goel, Huang, Kempe,
Espanes, Rothemund,
STOC 2002)
Flexible Glue          P

Multiple Temperature   NP-hard    (our paper)

Unique Shape           Co-NPC     (our paper)

Multiple Tile          NP-hard    (our paper)
Outline
• Importance of DNA Self-Assembly

• Tile Self-Assembly (Generalized Models)
– Tile Complexity
– Shape Verification
– Error Resistance

• DNA Word Design
Further Research
Error Resistance: Insufficient Bindings

t=2
Further Research
Error Resistance: Insufficient Bindings

t=2
Further Research
Error Resistance: Insufficient Bindings

t=2
Further Research
Error Resistance: Insufficient Bindings

t=2
Further Research
Error Resistance: Insufficient Bindings

t=2
Further Research
Error Resistance: Insufficient Bindings

t=2
Further Research
Error Resistance: Insufficient Bindings

t=2
Further Research
Error Resistance: Insufficient Bindings

Standard                     Fluctuating

b
temperature

a
Outline
• Importance of DNA Self-Assembly

• Tile Self-Assembly (Generalized Models)

• DNA Word Design
DNA Word Design

1   2       3          4       5      6       7   8   9

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

green: ACCT
GAAA       -Must be sufficiently
red:
different
yellow: GCTA
blue: CGTA         -Must have similar
purple: CTCG        thermodynamic properties
white: CATG
black: ACGA        -Must be short
teal:   TTTA
Hamming Constraint (k)
ACCTGAGAGAGCTC
GCGCAGCTGGCTCA
TTAGCAGACTGACA
GCTTCGTAGCATAG
ATAGCTGCATCGAT
TGCTAGCGTCAAGC
AGCATTATAGATAC
GCCCGTAGACTCGA
TCGAGTAGATCGAT   X=   GCTTCGTAGCATAG
CGACGTAGGCTTTG
n strings   CTGATGATTAGGCG
|   | |
TTCAGCTGCGGCTA   Y=   TTAGCCGCGTAGCT
TCGATGCGTAGCTA
GAGTGCTGCTAGCT
AGCTAGTCACTCGA          HAMM(X,Y) = 11 > k
TCGACTAGCTTCGA
TTAGCCGCGTAGCT
GACTAGTCGATCAG
TCGCGCTTATATAT
ATCGTAGTCTAGTC
TACGATCGCTAGTC

length L = 14
Free Energy Constraint
ACCTGAGAGAGCTC                         A   C   G   T
GCGCAGCTGGCTCA                     A   2   1   5   3
TTAGCAGACTGACA
GCTTCGTAGCATAG
Pairwise      =   C   7   2   6   9
free energies
ATAGCTGCATCGAT                     G   1   1   3   1
TGCTAGCGTCAAGC
AGCATTATAGATAC                     T   8   7   4   2
GCCCGTAGACTCGA
TCGAGTAGATCGAT
CGACGTAGGCTTTG
n strings   CTGATGATTAGGCG
TTCAGCTGCGGCTA
TCGATGCGTAGCTA
GAGTGCTGCTAGCT
AGCTAGTCACTCGA
TCGACTAGCTTCGA
TTAGCCGCGTAGCT
GACTAGTCGATCAG
TCGCGCTTATATAT
ATCGTAGTCTAGTC
TACGATCGCTAGTC

length L = 14
Free Energy Constraint
ACCTGAGAGAGCTC                           A   C   G   T
GCGCAGCTGGCTCA                       A   2   1   5   3
TTAGCAGACTGACA
GCTTCGTAGCATAG
Pairwise      =     C   7   2   6   9
free energies
ATAGCTGCATCGAT                       G   1   1   3   1
TGCTAGCGTCAAGC
AGCATTATAGATAC                       T   8   7   4   2
GCCCGTAGACTCGA
TCGAGTAGATCGAT
CGACGTAGGCTTTG
n strings   CTGATGATTAGGCG
TTCAGCTGCGGCTA   X= AGCATTATAGATAC
TCGATGCGTAGCTA
GAGTGCTGCTAGCT   FE(X) = 5+1+7+...
AGCTAGTCACTCGA
TCGACTAGCTTCGA
TTAGCCGCGTAGCT
GACTAGTCGATCAG
TCGCGCTTATATAT
ATCGTAGTCTAGTC
TACGATCGCTAGTC

length L = 14
Free Energy Constraint
ACCTGAGAGAGCTC                           A   C   G   T
GCGCAGCTGGCTCA                       A   2   1   5   3
TTAGCAGACTGACA
GCTTCGTAGCATAG
Pairwise      =     C   7   2   6   9
free energies
ATAGCTGCATCGAT                       G   1   1   3   1
TGCTAGCGTCAAGC
AGCATTATAGATAC                       T   8   7   4   2
GCCCGTAGACTCGA
TCGAGTAGATCGAT
CGACGTAGGCTTTG
n strings   CTGATGATTAGGCG
TTCAGCTGCGGCTA   X= AGCATTATAGATAC
TCGATGCGTAGCTA
GAGTGCTGCTAGCT   FE(X) = 5+1+7+...
AGCTAGTCACTCGA
TCGACTAGCTTCGA
TTAGCCGCGTAGCT
GACTAGTCGATCAG
TCGCGCTTATATAT
For all strings X and Y:
ATCGTAGTCTAGTC   |FE(X) – FE(Y)| < C
TACGATCGCTAGTC

length L = 14
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
DNA Word Design

Simple Lower Bound:        Hamming Constraint:

L > log n                Set L = 5*(k + log n)

L>k                      Generate n strings of
length L uniformly at
L > ½(k + log n)          random.

L  (k  log n)          -satisfies hamming
constraint with high
probability.

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