Exact and Approximation Algorithms for DNA Tag Set Design Ion Mandoiu and Dragos Trinca Computer...
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Transcript of 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