Math 8803/4803, Spring 2008:Discrete Mathematical Biology

Prof. Christine Heitsch

School of Mathematics

Georgia Institute of Technology

Lecture 6 – January 18, 2008

Designing DNA codewords

Problem:Minimize all possible mishybridizations for {Wi}, {Ci}.Also satisfy biochemical constraints on melting temperature and free energy.

F F

Complements – Wi and Cj for i 6= j

Reverse-complements – Wi and Wj

(or Ci and Cj) for i 6= j

Inverted repeats – Wi with itself

Previously approaches were either “top-down” or “bottom-up.”

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Previous Solution Approaches

• Top-down;

– Hamming distance and coding theory [2, 5, 4, 3].

– De Bruijn sequences and the 2-4 rule [1].

• Bottom-up:

– Culling according to biochemical constraints [6, 7].

– Stochastic local search [9, 8].

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A “mixed” solution strategy

Top-down: Consider the set of (4!)43−1

4−3 ≈ 1.89× 1020 distinct

De Bruijn sequences with n = 4 and k = 3 having length 43.

F F

0

0

0

1

1

1

00

11

0 1

0

1

01

00010111

)

)10( ,

10( ,

)1( ,

)10( ,

0

Bottom-up: Select B(4, 3) (uniformly at random) which minimize

other mishybrizations and satisfy required biochemical constraints.

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Recall our graph theory definitions

An Euler circuit is a walk in a graph G which begins and ends at

the same vertex and uses every edge exactly once.

A Hamiltonian circuit is a walk in a graph G which begins and

ends at the same vertex and visits every vertex (except the

endpoints) exactly once.

An arborescence is a rooted directed tree with all the edges

pointing in the direction of the root.

A spanning tree of a connected graph G is a subgraph which

contains all the vertices and is a tree.

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Examples from our De Bruijn graph

Vertices v0, v1, v2, v3 where vertex

vi is labeled with integer i in binary.

An Euler circuit:

v3 → v2 → v0 → v0 → v1

→ v2 → v1 → v3 → v3

A Hamiltonian circuit:

v3 → v2 → v0 → v1 → v3

A spanning arborescence rooted at v3:

E = {(v0, v1), (v2, v1), (v1, v3)}

0

0

0

1

1

1

00

11

0 1

0

1

01

00010111

)

)10( ,

10( ,

)1( ,

)10( ,

0

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Combinatorial explosion

2 3 4 5 6

2 1 24 20, 736 995, 318, 000 ≈ 3.87× 1015

3 2 373, 248 ≈ 1.89× 1020 ≈ 7.63× 1049 ·

4 16 ≈ 1.26× 1019 · · ·

5 2048 · · · ·

6 67, 108, 864 · · · ·

7 ≈ 1.44× 1017 · · · ·

How to generate a (uniformly) random B(n, k) De Bruijn sequence?

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Random walks on the De Bruijn graph

0

0

0

1

1

1

00

11

0 1

0

1

01

00010111

)

)10( ,

10( ,

)1( ,

)10( ,

0

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Random walks on the De Bruijn graph

0

0

0

1

1

1

11

0

1

01

00 10111

)

)10( ,

10( ,

)1( ,

)10( ,

00

1

00

0

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Uniformly random De Bruijn sequences

Total of (4!)43−1

4−3 ≈ 1.89× 1020 distinct B(4, 3) with length 43.

Randomized Algorithm.

• For vertex vi, i = 0, . . . , 15, chose a random permutation pi of {0, 1, 2, 3}.

• Beginning at v1, build a sequence by walking around G(4, 3) according to pi.

• Stop when a permutation is exhausted. Accept if sequence has length 64.

The sequence is De Bruijn with probability 1/64;

about 64 trials on average will generate a B(4, 3).(Can be improved slightly to 4/81 and about 20 trials on average.)

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Can we weave a net of DNA strands?

Woman Arranging a Fishing Net – Zanzibar, Tanzania, September 2000

By Martin Wierzbicki from http://photosbymartin.com/.

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Biochemical constraints

Thermodynamics predictions calculated by mfold.

∆G = ∆H − T∆S

Tm =1000∆H

A + ∆S + R ln (C/4)− 273.15 + 16.6 log [Na+]

Free energy G chemical potential of a substance

Enthalpy H heat content of a substance

Entropy S disorder of a substance

Melting temperature Tm 50% duplex

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Predicted properties

The DNA word W from a random De Bruijn sequence B(4, 3):aatgctggagcaaccactcctttcatcgcgtattgttagggtgaaagtctacggcccgacagat

For RCA, there should be no (significant) secondary structures.

Sequence Secondary structure

∆G (kcal/mole) Tm (◦C)

Linear W -7.6 49.9

Circular W 0.48 23.7

RCA average -6.26/repeat 46.7

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Rolling Circle Amplification (RCA)

Enzymatic Ligation

Circular DNA TemplateGuide DNA (20bp)

(64bp)

Circular Template

cDNA Primer

Amplified Sequence

DNA Polymerase

Circular DNA construction Sequence amplification

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RCA product for random De Bruijn sequence

1kb

2kb

10kb23kb

4.4kb

9.4kb6.6kb

2.3kb

564bp

1 2 3 4 5 6 7 8 Lane 1: 1Kb marker

Lane 2: no circular DNA

Lane 3: RCA 5min

Lane 4: RCA 20min

Lane 5: RCA 35min

Lane 6: RCA 1hr

Lane 7: RCA overnight

Lane 8: Lamda marker

Observed on 0.8% agarose gel: significant amplification at 30 ◦C.

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Predicted surface array hybridization

The DNA word W from a random De Bruijn sequence B(4, 3):aatgctggagcaaccactcctttcatcgcgtattgttagggtgaaagtctacggcccgacagat

Parse W into overlapping 16mer targets Ti for i = 1, . . . , 64.

Predict binding of complement Pi with Ti and difference W \ Ti.

Sequence Hybridization with Pi

∆G (kcal/mole) Tm (◦C)Ti ≤ −18.6 ≥ 65.8

W \ Ti ≥ −3.8 ≤ 18.0

Also check hybridization with a second B(4, 3) DNA sequence W2.

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Minimal free energy

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Melting temperature

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Surface Plasmon Resonance (SPR) imaging

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Verification of complementary binding

aatgctggagcaaccactcctttcatcgcgtattgttagggtgaaagtctacggcccgacagat

= W and another random B(4, 3) W2 =

gctgccctaaagagtcggcagcgatcttccaacgtgacactcatttgtagggttatggaatacc

Probes:

P1 = ctaacaatacgcgatg

P2 = agtgtcacgttggaag

P3 = gtgtatccgacatgtg

Targets:

W

G = gggctctctattacgacctc

H = cttccaacgtgacact

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SPR imaging result of complementary binding

P1 P2 P3

Hybridization of W onto DNA array

P1 P2 P3

Hybridization of G onto DNA array

P1 P2 P3

Hybridization of H onto DNA array

P3P2P1

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Proof of principle

W1 = ctaacaatacgcgatg W2 = agtgtcacgttggaag

W3 = gtgtatccgacatgtg C2 = cttccaacgtgacact

P3P2P1 P1 P2 P3

Hybridization of H onto DNA array

“Proof of principle” experimental results support using

random De Bruijn sequences to design DNA codewords.

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Acknowledgments

• Ming Li for his DNA9 PowerPoint slides for “Attomole Detection of DNA inan Array Format with SPR Imaging Using RCA for the ProgrammableNanoscale DNA Biosensors.”

• Ming Li, Prof. Rob Corn, and the 2000 – 2002 Corn Research Group at theUniversity of Wisconsin – Madison. Graduate Students: Greta Wegner, Terry Goodrich, MingLi, Shiping Fang, Yuan Li, Heesuk Kim, Johanna Kwok. Senior Staff Scientist: Hye Jin Lee. PostdoctoralAssociates: Dr. Alastair Wark, Dr. Eric Codner. Visitors: Dr. Tomonori Saeki from Hitachi, Ltd.Undergraduates: Misha Wolfson, Takeyoshi Goto.

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References

[1] A. Ben-Dor, R. M. Karp, B. Schwikowski, and Z. Yakhini. Universal DNA tag systems: a

combinatorial design scheme. In RECOMB, pages 65–75, 2000.

[2] A. G. Frutos, Q. Liu, A. J. Thiel, A. M. W. Sanner, A. E. Condon, L. M. Smith, and R. M.

Corn. Demonstartion of a word design strategy for DNA computing on sur faces. Nucleic

Acids Res., 25(23):4748 – 4757, 1997.

[3] P. Gaborit and O. D. King. Linear constructions for DNA codes. Theoret. Comput. Sci.,

334(1-3):99–113, 2005.

[4] O. D. King. Bounds for DNA codes with constant GC-content. Electron. J. Combin.,

10:Research Paper 33, 13 pp. (electronic), 2003.

[5] M. Li, H. J. Lee, A. E. Condon, and R. M. Corn. DNA word design strategy for creating sets

of non-interacting sets of oligonucleotides for DNA microarrays. Langmuir, 18(3):805–812,

2002.

[6] M. R. Shortreed, S. B. Chang, D. Hong, M. Phillips, B. Campion, D. C. Tulpan,

M. Andronescu, A. Condon, H. H. Hoos, and L. M. Smith. A thermodynamic approach to

designing structure-free combinatorial DNA word sets. Nucleic Acids Res,

33(15):4965–4977, 2005.

[7] D. Tulpan, M. Andronescu, S. B. Chang, M. R. Shortreed, A. Condon, H. H. Hoos, and

L. M. Smith. Thermodynamically based DNA strand design. Nucleic Acids Res,

33(15):4951–4964, 2005.

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[8] D. Tulpan and H. Hoos. Hybrid randomised neighbourhoods improve stochastic local search

for dna code design, 2003.

[9] D. Tulpan, H. Hoos, and A. Condon. Stochastic local search algorithms for dna word design,

2002.

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