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DNA Self-Assembly

Robert Schweller Northwestern University

Speaking of Science talk Buena Vista

University February 28, 2005

Outline

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

Smart Bricks

Wang Tiles

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G C A T C G

C G T A G C

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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

Input

Program

DNA Computers

Output!

Computer Program

Input

Output!

Input

Program

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 temperature, positive integer

G glue function

T tileset

s seed tile

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

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

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

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

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

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

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

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

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

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
- Adjust temperature during assembly
- 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

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

Building k x N Rectangles

k-digit, base N(1/k) counter

k

N

Building k x N Rectangles

k-digit, base N(1/k) counter

k

N

Tile Complexity

Build a 4 x 256 rectangle

t 2

S3

0

S2

0

S1

0

S

g

g

p

g

C1

C2

C3

C0

S

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

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

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

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

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

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

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

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

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

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

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

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

Building k x N Rectangles

k-digit, base N(1/k) counter

k

N

Tile Complexity

2-temperature model

t 4

3

1

3

3

2-temperature model

t 4 6

2-temperature model

(our paper)

Kolmogorov Complexity

(Rothemund, Winfree STOC 2000)

Beats Standard Model

(our paper)

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

X

(Adleman, Cheng, Goel, Huang STOC 2001)

k

N - k

Y

k

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

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

log N

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

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

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

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

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

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

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

X

N log N

Complexity

N log N

log N

log N

Y

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

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 NP-hard (our paper)

3-SAT Problem

Clause 1 Clause 2 Clause 3

Unique-Shape Model

Unique-Shape Model

x3

x2

x1

Unique-Shape Model

x3

x2

x1

c2

c1

c3

Unique-Shape Model

1

x

x3

x

0

x

x2

x

x1

x

c2

c1

c3

Unique-Shape Model

x3

1

x2

1

x1

0

c2

c1

c3

Unique-Shape Model

x3

1

x2

1

x1

c1

0

c2

c1

c3

Unique-Shape Model

x3

1

x2

ok

1

x1

c1

0

c2

c1

c3

Unique-Shape Model

x3

ok

1

x2

ok

1

x1

c1

0

c2

c1

c3

Unique-Shape Model

x3

ok

1

x2

ok

1

x1

c2

c1

0

c2

c1

c3

Unique-Shape Model

x3

ok

1

x2

ok

c2

1

x1

c2

c1

0

c2

c1

c3

Unique-Shape Model

x3

ok

ok

1

x2

ok

c2

1

x1

c2

c1

0

c2

c1

c3

Unique-Shape Model

x3

ok

ok

1

x2

ok

c2

1

x1

ok

c2

c1

0

c2

c1

c3

Unique-Shape Model

x3

ok

ok

ok

1

x2

ok

ok

c2

1

x1

ok

c2

c1

0

c2

c1

c3

Unique-Shape Model

x3

ok

ok

ok

1

x2

ok

ok

c2

1

x1

ok

c2

c1

0

c2

c1

c3

Unique-Shape Model

T

x3

ok

ok

ok

1

x2

ok

ok

c2

1

x1

ok

c2

c1

0

c2

c1

c3

Unique-Shape Model

T

T

x3

ok

ok

ok

1

x2

ok

ok

c2

1

x1

ok

c2

c1

0

c2

c1

c3

Unique-Shape Model

T

T

T

x3

ok

ok

ok

1

x2

ok

ok

c2

1

x1

ok

c2

c1

0

c2

c1

c3

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)

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)

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)

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)

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)

Multiple Temperature Model

x3

x3

x2

x2

x1

x1

c1

c2

c3

c1

c2

c3

Not Satisfied

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

Not Satisfied

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

Not Satisfied

Satisfied

Multiple Temperature Model

x3

x3

x2

x2

x1

x1

Not Satisfied

Satisfied

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)

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

Further Research

Further Research

Further Research

Further Research

Outline

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

DNA Word Design

5

1

2

3

4

6

7

8

9

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

Hamming Constraint (k)

ACCTGAGAGAGCTCGCGCAGCTGGCTCATTAGCAGACTGACAGCTTCGTA

GCATAGATAGCTGCATCGATTGCTAGCGTCAAGCAGCATTATAGATACGC

CCGTAGACTCGATCGAGTAGATCGATCGACGTAGGCTTTGCTGATGATTA

GGCGTTCAGCTGCGGCTATCGATGCGTAGCTAGAGTGCTGCTAGCTAGCT

AGTCACTCGATCGACTAGCTTCGATTAGCCGCGTAGCTGACTAGTCGATC

AGTCGCGCTTATATATATCGTAGTCTAGTCTACGATCGCTAGTC

X GCTTCGTAGCATAG Y

TTAGCCGCGTAGCT

n strings

HAMM(X,Y) 11 gt k

length L 14

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

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

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

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

DNA Word Design

Simple Lower Bound

L gt log n L gt k L gt ½(k log n)

DNA Word Design

Word Length

Run-Time

DNA Word Design

Hamming Constraint k

-Set L 5(k log n) -Generate all random

strings

PrFAILURE

All Random

length L 5(klog n)

Free Energy Constraint

n

length L O(klog n)

Free Energy Constraint

All length L strings

n

length L O(klog n)

Free Energy Constraint

Low FE

All length L strings

n

length L O(klog n)

Free Energy Constraint

Low FE

All length L strings

n

High FE

length L O(klog n)

Free Energy Constraint

Low FE

All length L strings

n

High FE

length L O(klog n)

Free Energy Constraint

All length L strings

n

length L O(klog n)

Fact Strings can be chosen to satisfy the Free

Energy Constraint

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)

Free Energy Constraint

Given

Free Energy Constraint

Given

Find

Free Energy Constraint

Given

Find

a lt FE lt b

Problem 4L length L strings

Free Energy Constraint

Fixed Energy String Problem Input

Length L, Energy E

Output a string with 1) length L 2) free

energy E

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

What if we knew

fLa,b, fL/2a,b, fL/4a,b, , f1a,b for all

a,b in A,C,G,T

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

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

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

Recursive Property

a

b

c

d

FEc,d

L/2

L

Recursive Property

T(L)

a

b

c

d

FEc,d

L/2

L

Recursive Property

T(L) T(L/2)

a

b

c

d

FEc,d

L/2

L

Recursive Property

T(L) T(L/2) L log L

a

b

c

d

FEc,d

L/2

L

Recursive Property

T(L) T(L/2) L log L O(L

log L)

a

b

c

d

FEc,d

L/2

L

Summary for Word Design

Hamming Constraint (k) -Randomly generate

words of length L O(k log n)

n

length L O(klog n)

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)

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)

DNA Self-Assembly

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

Questions?