Design and Optimization of Universal DNA Arrays

1
Design and Optimization of Universal DNA Arrays

• Ion Mandoiu
• Computer Science Engineering Department
• University of Connecticut
• http//www.engr.uconn.edu/ion/

2
DNA Arrays

• Exploit Watson-Crick complementarity to
  simultaneously perform a large number of
  substring tests
• Used in a variety of high-throughput genomic
  analyses
• Transcription (gene expression) analysis
• Single Nucleotide Polymorphism (SNP) genotyping
• Alternative splicing, ChIP-on-chip, tiling
  arrays, genomic-based species identification,
  point-of-service diagnosis,
• Common array formats involve direct hybridization
  between labeled DNA/RNA sample and DNA probes
  attached to a glass slide

3
SNP Genotyping

• Genome variation 0.1 of the DNA different from
  one individual to another
• 80 of the variation is represented by Single
  Nucleotide Polymorphisms (SNPs)
• 2 possible nucleotides (alleles) for each SNP
• SNP genotyping determining the alleles present
  at the SNP sites
• Highest throughput for SNP genotyping is achieved
  by high-density DNA microarrays based on direct
  hybridization

4
SNP genotyping via direct hybridization
Labeled sample

• SNP1 with alleles T/G
• SNP2 with alleles A/G

Array with 2 probes/SNP
Hybridization
5
Universal DNA Arrays

• Programable arrays
• Array consists of application independent
  oligonucleotides
• Analysis carried by a sequence of reactions
  involving application specific primers
• Flexible AND cost effective
• Universal array architectures tag arrays, APEX
  arrays, SBE/SBH arrays,

6
Overview

• Tag Array Design
• - Tag Set Design
• - Tag Assignment Algorithms
• SBE/SBH Assays
• - Decoding and Multiplexing Algorithms
• Conclusions

7
SNP Genotyping with Tag Arrays
Tag
Primer
G

A
G
2. Solution phase hybridization

• Mix reporter probes with unlabeled genomic DNA

C
antitag
4. Solid phase hybridization
3. Single-Base Extension (SBE)
8
Tag Array Advantages

• Cost effective
• Same array used in many analyses ? can be mass
  produced
• Easy to customize
• Only need to synthesize new set of reporter
  probes
• Reliable
• Solution phase hybridization better understood
  than hybridization on solid support

9
Tag Set Design Problem
t1
t1
t2
t2
t1
t2
t1

• (H1) Tags hybridize strongly to complementary
  antitags
• (H2) No tag hybridizes to a non-complementary
  antitag
• (H3) Tags do not cross-hybridize to each other

Tag Set Design Problem Find a maximum
cardinality set of tags satisfying (H1)-(H3)
10
Hybridization Models

• Melting temperature Tm temperature at which 50
  of duplexes are in hybridized state
• 2-4 rule
• Tm 2 (As and Ts) 4 (Cs and Gs)
• More accurate models exist, e.g., the
  near-neighbor model

11
Hybridization Models (contd.)

• 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
• Duplex formation requires formation of
  nucleation complex between perfectly
  complementary substrings
• Nucleation complex must have weight ? c, where
  wt(A)wt(T)1, wt(C)wt(G)2 (2-4 rule)

12
c-h Code Problem

• c-token left-minimal DNA string of weight ? c,
  i.e.,
• w(x) ? c
• w(x) lt c for every proper suffix x of x
• A set of tags is a c-h code if
• (C1) Every tag has weight ? h
• (C2) Every c-token is used at most once

c-h Code Problem Ben-Dor et al.00 Given c and
h, find maximum cardinality c-h code
13
Algorithms for c-h Code Problem

• Ben-Dor et al.00 approximation algorithm based
  on DeBruijn sequences
• Alphabetic tree search algorithm
• Enumerate candidate tags in lexicographic order,
  save tags whose c-tokens are not used by
  previously selected tags
• Easily modified to handle various combinations
  of constraints
• MT 05, 06 Optimum c-h codes can be computed in
  practical time for small values of c by using
  integer programming
• Practical runtime using Garg-Koneman
  approximation and LP-rounding

14
Token Content of a Tag

• c4
• CCAGATT
• CC
• CCA
• CAG
• AGA
• GAT
• GATT

Tag ? sequence of c-tokens End pos 2
3 4 5 6 7
c-token CC?CCA?CAG?AGA?GAT?GATT
15
Layered c-token graph for length-l tags
l
l-1
c/2
(c/2)1

c1
t
s
cN
16
Integer Program Formulation MPT05

• Maximum integer flow problem w/ set capacity
  constraints
• O(hN) constraints variables, where N c-tokens

17
Number of c-tokens

• WA or T, SC or G
• Gn strings of weight n
• ? G1 2 G2 6 Gn 2Gn-2 2Gn-1

18
Number of c-tokens
19
Packing LP Formulation
20
Garg-Konemann Algorithm

• x ? 0 y ? ? // yi are variables of the dual LP
• Find min weight s-t path p, where weight(v) yi
  for every v?Vi
• While weight(p) lt 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 v?Vi
• 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
21
LP Based Tag Set Design

• Run Garg-Konemann and store the minimum weight
  paths in a list
• 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
• Mark used c-tokens and run the alphabetic tree
  search algorithm to select additional tags

22
Periodic Tags MT05

• 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 in a tag
• A tag t is called periodic if it is the prefix of
  (?)? for some period ?
• Periodic strings make best use of c-tokens

23
c-token factor graph, c4 (incomplete)
CC AAG
AAC
AAAA AAAT
24
Vertex-disjoint Cycle Packing Problem

• Given directed graph G, find maximum number of
  vertex disjoint directed cycles in G
• MT 05 APX-hard even for regular directed graphs
  with in-degree and out-degree 2
• h-c/21 approximation factor for tag set design
  problem
• Salavatipour and Verstraete 05
• Quasi-NP-hard to approximate within ?(log1-? n)
• O(n1/2) approximation algorithm

25
Cycle Packing Algorithm

• Construct c-token factor graph G
• T?
• For all cycles C defining periodic tags, in
  increasing order of cycle length,
• Add to T the tag defined by C
• Remove C from G
• Perform an alphabetic tree search and add to T
  tags consisting of unused c-tokens
• Return T

• Gives an increase of over 40 in the number of
  tags compared to previous methods

26
Experimental Results
27
Antitag-to-Antitag Hybridization

• Additional constraint antitags do not
  cross-hybridize, including self
• Formalization in c-token hybridization model
• (C3) No two tags contain complementary substrings
  of weight ? c
• Cycle packing and tree search extend easily

28
Results w/ Extended Constraints
29
More Hybridization Constraints
t1
t1
t2

• Enforced during tag assignment by
• - Leaving some tags unassigned and distributing
  primers across multiple arrays Ben-Dor et al.
  03
• - Exploiting availability of multiple primer
  candidates MPT05

30
Assignable Primers

• If primer p hybridizes to the complement of tag
  t, at most one of the assignments (p,t), (p,t)
  and (p,t) can be made

• Set P of primers is assignable to a set T of tags
  if the condition above is satisfied for every
  p,p and t,t

31
Characterization of Assignable Sets

• conflict graph
• G(T ? P,E), where (t,p) ? E if t and p
  hybridize
• X number of primers adjacent to a degree 1 tag
• Y number of degree 0 tags

X1
Y2

• Ben-Dor 04 Set P is assignable to T iff
• XY ? P

32
Finding Assignable Primer Sets
Multiplexing Problem given primer set P and tag
set T, find partition of P into minimum number of
assignable sets
Maximum Assignable Primer Set Problem given
primer set P and tag set T, find a maximum size
assignable subset of P

• Both problems are NP-hard Ben-Dor 04

33
Integration with Primer Selection

• In practice, several primer candidates with
  equivalent functionality
• In SNP genotyping, can pick primer from either
  forward and reverse strand
• In gene expression/identification applications,
  many primers have desired length, Tm, etc.

34
Pooled Array Multiplexing Problem
Pooled Multiplexing Problem Given set of primer
pools P and tag set T, find a primer from each
pool and a partition of selected primers into
minimum number of assignable sets
35
XY Characterization Fails for Pools
36
Pooled Multiplexing Algorithms

• Primer-Del greedy deletion for pools similar to
  Ben-Dor et al 04

• Repeatedly delete primer of maximum potential
  until XY ? pools, where
• Potential of tag t is 2-deg(t)
• Potential of primer p is sum of potentials of
  conflicting tags
• Subtract ½ if primer adjacent to a tag of degree 1

37
Pooled Multiplexing Algorithms

• Primer-Del greedy deletion for pools similar to
  Ben-Dor et al 04
• Primer-Del same but never delete last primer
  from pool unless no other choice
• Min-Pot select primer with min potential from
  each pool, then run Primer-Del
• Min-Deg select primer with min degree, then run
  Primer-Del
• Iterative ILP iteratively find a maximum
  assignable pool set using integer linear program

38
Integer Linear Program for MAPS

• where zpt 1 iff primer p is assigned to tag t

39
Results GenFlex Tags, c8
40
Results GenFlex Tags, c7
41
Results 213 MPT05 Tags, c7
42
Herpes B Gene Expression Assay
GenFlex Tags
Periodic Tags
43
Overview

• Tag Array Design
• - Tag Set Design
• - Tag Assignment Algorithms
• SBE/SBH Assays
• - Decoding and Multiplexing Algorithms
• Conclusions

44
SBE/SBH Assay MP 06
Primers
T
T
A
A
T
T
TTGCA
T
CCATT
A
GATAA
T
hybridization to k-mer array (SBH)
single-base extension (SBE)
45
Some notations

• P set of primers, X set of probes
• Ep ? A,C,T,G the set of possible extensions for
  primer p
• The spectrum of primer p, SpecX(p), is the set of
  probes hybridizing with p
• The extended spectrum of primer p with extension
  set Ep,

46
Decodable primer sets

• Four parallel single-color SBE/SBH experiments ?
  one type of extension in each SBE experiment
• P is weakly decodable with respect to extension e
  if for every primer p
• One SBE/SBH experiment with 4 colors (4
  extensions)
• P is weakly decodable if for every primer p and
  every extension e ? Ep

47
Strongly r-decodable primer sets

• Hybridization involving labeled nucleotide is
  less predictable
• ?Informative probes should not rely on it
• Signal from one SNP may obscure signal from
  another when read at the same probe due to
  differences in DNA amplification efficiency
• ?Informative probes cannot be shared between SNPs
• P is strongly r-decodable if for every primer p
• where r redundancy parameter

48
MPPP

• A set of primer pools P P1,,Pn is strongly
  r-decodable iff there is a primer pi in each pool
  Pi such that p1,,pn is strongly r-decodable.

• Minimum Pool Partitioning Problem (MPPP)
• Given
• primer pools set P and extensions sets Ep, for
  every primer p
• probe set X
• redundancy r
• Find
• partition of P into the min number of strongly
  r-decodable subsets

49
MDPSP

• Maximum r-Decodable Pool Subset Problem (MDPSP)
• Given
• primer pools set P and extensions sets Ep, for
  every primer p
• probe set X
• redundancy r
• Find
• strongly r-decodable subset of P of maximum size

50
Min-Greedy Algorithm for Maximum Induced Matching
in General Graphs

• Pick a vertex u of min degree
• Pick a vertex v of min degree from among us
  neighbors
• Add edge (u,v) to the matching
• Delete all neighbors of u and v from the graph
• Repeat the above steps until the graph becomes
  empty
• Duckworth 05 d-1 approximation factor for
  d-regular graphs

51
Min-Greedy Algorithms for MDPSP

• Bipartite hybridization graph G
• Primers in left side, probes in right side
• Two types of edges
• N(p)SpecX(p)
• N-(p)SpecX(p,Ep) \ SpecX(p)
• Two algorithm variants
• MinPrimerGreedy pick primer first
• MinProbeGreedy pick probe first
• Delete primer/probe if N degree drops below r/1

52
Experimental results for k-mers (Ep4, primer
length20)
53
MDPSP Size vs Primer Length
k10
54
Experimental results for c-tokens (Ep4, primer
length20)
55
MDPSP Size vs Primer Length
c13
56
Overview

• Tag Array Design
• - Tag Set Design
• - Tag Assignment Algorithms
• SBE/SBH Assays
• - Decoding and Multiplexing Algorithms
• Conclusions

57
Conclusions and Ongoing Work

• Combinatorial algorithms yield significant
  increases in multiplexing rates of universal
  arrays
• New SBE/SBH architecture particularly promising
  based on preliminary simulation results
• Ongoing work
• Extend methods to more accurate hybridization
  models, e.g., use NN melting temperature models
• More complex (e.g., temperature dependent) DNA
  tag set non-interaction requirements for DNA
  self/mediated assembly
• Probabilistic decoding in presence of
  hybridization errors
• Application to novel domains, e.g., DNA barcoding

58
Acknowledgments

• Claudia Prajescu and Dragos Trinca
• Funding from NSF (Awards 0546457 and 0543365) and
  UCONN Research Foundation