exact and approximation algorithms for dna tag set design ion mandoiu and dragos trinca computer...

Exact and Approximation Algorithms for

DNA Tag Set Design

Ion Mandoiu and Dragos Trinca

Computer Science & Engineering Department

University of Connecticut

Overview

Background on universal tag arrays Tag set design problem

- c-h code formulation- Integer program and LP-approximation- Cycle packing formulation

Experimental results Conclusions

Watson-Crick Complementarity

• Four nucleotide types: A,C,G,T

• A’s paired with T’s (2 hydrogen bonds)

• C’s paired with G’s (3 hydrogen bonds)

DNA Microarrays• Exploit Watson-Crick complementarity to simultaneously

perform a large number of substring tests

• Used in a variety of genomic analyses– Transcription (gene expression) analysis – Single Nucleotide Polymorphism (SNP) genotyping– Genomic-based microorganism identification

• Common microarray formats involve direct hybridization between labeled DNA/RNA sample and DNA probes attached to a glass slide

Direct Hybridization Experiment

Images courtesy of Affymetrix.

Labeled DNA/RNA sample hybridized to array of probes

Laser activation of fluorescent labels

Optical scanning used to identify

probes with complements in the

mixture

Limitations of Common Array Formats

• Arrays of cDNAs– Probes obtained by reverse transcription from Expressed

Sequence Tags (ESTs)

– Inexpensive, but can only be used for transcription analysis

• Oligonucleotide arrays– Short (20-60bp) synthetic DNA probes

– Flexible, but expensive unless produced in large quantities

Universal Tag Arrays

• [Brenner 97, Morris et al. 98] “Programmable” oligonucleotide arrays – Array consists of application independent oligonucleotides called

tags

– Two-part “reporter” probes: aplication specific primers ligated to antitags

– Detection carried by a sequence of reactions separately involving the primer and the antitag part of reporter probes

+

Mix reporter probes with genomic DNA

Universal Tag Array Experiment

Solution phase hybridization

Single-Base Extension

Solid phase hybridization

Universal Tag Array Advantages

• Cost effective– Same array used in many analyses economies of scale

• Easy to customize– Only need to synthesize new set of reporter probes

• Reliable– Solution phase hybridization better understood than

hybridization on solid support

Tag Hybridization Constraints

(H1) Antitags hybridize strongly to complementary tags

(H2) No antitag hybridizes to a non-complementary tag

(H3) Antitags do not cross-hybridize to each other

t1t1 t2t2 t1 t2t1

Tag Set Design Problem: Find a maximum cardinality set of tags satisfying (H1)-(H3)

More Hybridization Constraints…

• Enforced during tag assignment by- Leaving some tags unassigned and distributing primers to multiple arrays [Ben-Dor et al. 03]

- Exploiting availability of multiple primer candidates [MPT05]

t1 t2t1

Hybridization Models: Stability

• Melting temperature models, e.g., nearest neighbor [SantaLucia 96]

• Tag weight h [Ben-Dor et al. 00]– wt(A)=wt(T)=1, wt(C)=wt(G)=2

– Equivalent to “2-4 rule” melting temperature model

• Tag length = l [Affymetrix]– Combined with additional constraints on GC-content, etc.

• Hamming distance model, e.g. [Marathe et al. 01]– Models rigid DNA strands

• LCS/edit distance model, e.g., [Torney et al. 03] – Models infinitely elastic DNA strands

• c-token model [Ben-Dor et al. 00]:– Derived from nucleation complex theory: duplex formation

requires formation of nucleation complex between perfectly complementary substrings

– Nucleation complex must have weight c

Hybridization Models: Non-Interaction

c-h Code Problem

• c-token: left-minimal DNA string of weight c, i.e.,– w(x) c

– w(x’) < c for every proper suffix x’ of x

• A set of tags is called c-h code if(C1) Every tag has weight h

(C2) Every c-token is used at most once

• c-h code problem: given c and h, find maximum cardinality c-h code

Previous Work

• [Ben-Dor et al.00]- Constructive upper-bound on c-h code size based on token tail-weight- Approximation algorithm based on DeBruijn sequences

• [MPT05]- Upper-bound for c-h codes including antitag-to-antitag hybridization constraints- Simple alphabetic tree search heuristic

Token Content of a Tag

c=4 CCAGATT CC CCA CAG AGA GAT GATT

Tag sequence of c-tokensEnd pos: 2 3 4 5 6 7 c-token: CCCCACAGAGAGATGATT

Layered c-token graph

st

c1

cN

ll-1c/2 (c/2)+1 …

Integer Program Formulation

• Maximum integer flow problem w/ set capacity constraints

•O(lN)/O(hN) constraints & variables, where N = #c-tokens

Number of c-tokensc # c-tokens

5 208

6 568

7 1552

8 4240

9 11584

10 31648

Packing ILP Formulation path every for variable0/1 ps-txp

Garg-Konemann Algorithm1. x 0; y // yi are variables of the dual LP

2. Find min weight s-t path p, where weight(v) = yi for every vVi

3. While weight(p) < 1 do

M maxi |p Vi|

xp xp + 1/M

For every i, yi yi( 1 + * |p Vi|/M )

Find min weight s-t path p, where weight(v) = yi for vVi

4. For every p, xp xp / (1 - log1+)

[GK98] The algorithm computes a factor (1- )2 approximation to the optimal LP solution with (N/)* log1+N shortest path computations

LP Based c-h Code Construction

1. Run Garg-Konemann and store the minimum weight paths in a list

2. Traversing the list in reverse order, pick tags corresponding to paths if they are feasible and do not share c-tokens with already selected tags

3. Mark used c-tokens and run the alphabetic tree search algorithm to select additional tags

Experimental Results (h=15)

Periodic Tags

• Key observation: c-token uniqueness constraint in c-h code formulation is too strong– A c-token should not appear in two different tags, but

can be repeated within a tag!

• A tag t is called periodic if it is the prefix of () for some – Periodic strings make better use of c-tokens (t uses at

most || c-tokens)

c-token factor graph, c=4 (incomplete)

CC

AAG AAC

AAAA

AAAT

Cycle Packing Problem

• Vertex-Disjoint Cycle Packing Problem: Given directed graph G, find maximum number of vertex disjoint directed cycles in G

Theorem 1: APX-hard even for regular directed graphs with in-degree and out-degree 2

• [Salavatipour and Verstraete 05] For general graphs:– Quasi-NP-hard to approximate within (log1- n)

– O(n1/2) approximation algorithm

Tag Set Design Algorithm1. Construct c-token factor graph G

2. T{}

3. For all cycles C defining periodic tags, in increasing

order of cycle length, do

• Add to T the tag defined by C

• Remove C from G

4. Perform an alphabetic tree search and add to T tags

with no c-tokens in common with T

5. Return T

Experimental Results

h

Antitag-to-Antitag Hybridization

• Formalization in c-token hybridization model:(C3) No two (anti)tags contain complementary substrings of

weight c

• Cycle packing and tree search extend easily

Results w/ Extended Constraints

Herpes B Gene Expression Assay

Tm # poolsPool size

500 tags 1000 tags 2000 tags

# arrays % Util. # arrays % Util. # arrays % Util.

60 14461 4 94.06 2 97.20 1 72.30

5 4 96.13 2 100.00 1 72.30

67 15601 4 96.53 2 98.70 1 78.00

5 4 98.00 2 99.90 1 78.00

70 15221 4 96.73 2 98.90 1 76.10

5 4 97.80 2 99.80 1 76.10

Tm # poolsPool size

500 tags 1000 tags 2000 tags

# arrays % Util. # arrays % Util. # arrays % Util.

60 14461 4 82.26 3 65.35 2 57.05

5 4 88.26 3 70.95 2 63.55

67 15601 4 86.33 3 69.70 2 61.15

5 4 91.86 3 76.00 2 67.20

70 15221 4 88.46 3 73.65 2 65.40

5 4 92.26 2 91.10 2 70.30

GenFlex Tags

Periodic Tags

Conclusions• Use of periodic tags yields significant increase in tag

set size and enables higher multiplexing rates in tag assignment

• Algorithms extend to more accurate hybridization models– Monotonic melting temperature c-tokens factor graph

• Other applications of non-interacting DNA tag sets – Lab-on-chip, DNA-mediated assembly, DNA computing

[Brenneman&Condon 02]

• Open problems– Settle approximation complexity of disjoint cycle packing– Improved tag set design algorithms?

Acknowledgments

• UCONN Research Foundation