Quality and Error Control
Coding for DNA Microarrays
Olgica Milenkovic
ECE Department
University of Colorado, Boulder
IEEE Denver ComSoc
Outline
• DNA Microarrays
• VLSIPS (Very Large Scale Immobilized Polymer Synthesis)
• Production of DNA Microarrays (http://www.affymetrix.com/)
– Base Scheduling
– Mask Design
– Quality-Control Coding
• Error-Correcting DNA Microarrays (Multiplexed Arrays)
• Production of Multiplexed DNA Microarrays
– Base/Color Scheduling
– Mask Design
– Quality-Control Coding
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DNA microarrays I
Slide #1
Goal: Determining which genes are expressed (active) and which are
unexpressed (inactive)
Comparative gene expression study of multiple cells
Transcription
Translation
Control of
Transcription &
Translation
Transcription
Translation
Protein Coding
Sequence
Protein Coding
Sequence
Protein
Protein
Gene expression and co-regulation
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DNA microarrays II
Slide #2
Creating the `cell cultures’ to be compared…
`Green’ Cell Culture :
`Red’ Cell Culture:
DNA
Subsequence
3’- AATTT CGC… - 5’
DNA
Subsequence
3’- AATTT CGC… - 5’
mRNA
5’ - UUAAAGCG… - 3’
mRNA
3’ - UUAAAGCG… - 5’
cDNA
3’ - AATTTCGC… - 5’
cDNA
3’- AATTT CGC… - 5’
“Color Coding”
3’ - AATTTCGC… - 5’
“Color Coding”
3’- AATTT CGC… - 5’
Creation of tagged cDNA sequences
from first cell type
Creation of tagged cDNA sequences
from first cell type
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DNA microarrays III
Spots
Slide #3
Gene Probes
Complementary sequences
hybridize with each other, forming
stable double-helices
Hybridization:
3’-AAGCT-5’
5’-TTCGA-3’
DNA microarray is scanned by laser light of different wave-lengths
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Probe synthesis in microarrays I
VLSIPS (Gene Chip, AFFYMETRIX, Array
Manufacturing Manual)
Linkers
Mask
Quartz Wafer
Linker Activation
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Slide #4
Probe synthesis in microarrays II
VLSIPS (Gene Chip, AFFYMETRIX, Array
Manufacturing Manual)
Solution of one DNA base
(A or T or G or C)
Solution of one DNA base
(A)
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Slide #5
Slide #6
Base scheduling I
Fixed probe length: N
Production steps
ATGC ATGC ATGC ATGC
Spots
Synchronous
1
schedule
2
(length 4N)
3
4
CTGA
5
ACAA
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Slide #7
Base scheduling II
Production steps
AGGC TTGC TTGC CCGC
Spots
1
2
Asynchronous
schedule
3
4
5
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Base Scheduling III
Slide #8
• Shortest asynchronous base schedule
– Shortest common super-sequence of set of M sequences (NP-hard)
ESN(M,k) – expected length of a longest common subsequence of M
randomly chosen sequences of length N over an alphabet of size k
 k( l )  lim
N 
 k( M ) 
ES N ( M , k )
N
M log(z0 k )
log(k(( 1  z0 ) M  1 ))
 z0  21 / M  1
No significant gain for N≈20-30
Periodic schedule used instead (length 4N)
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Mask Design
Slide #9
Border-length minimization
Feldman and Pevzner, 1994
Hannehalli et.al., 2002
Kahng et.al. 2003, 2004
Key idea:
Arrange the probes on the array in such a way that the border-length of all
masks is minimal
Border-length graph: complete graph on M vertices, weight of edges equal to
the Hamming distance between probes
Greedy traveling salesman algorithm+ threading (discrete space-filing curve)
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Quality Control
Slide #10
Hubbell and Pevzner, 1999
Sengupta and Tompa, 2002
Colbourn et.al., 2002
Quality control (fidelity)
spots
Manufacture identical probes at several
quality-control spots in order to test precision
of production steps
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Relevant coding-theoretic ideas
Slide #11
Balanced code (Sengupta and Tompa, 2002):
An b×v binary matrix of zeros and ones with
• each row has weight k;
• each column has weight bounded between l and b-l, for some constant l;
• any pair of columns is at least at Hamming distance d apart;
Superimposed designs in Renyi’s search model (Kautz and Singleton, 1964,
Dyachkov and Rykov, 1983):
An b×v binary matrix of zeros and ones with
• all Boolean sums composed of no more than s columns are distinct;
•each row has weight exactly t;
Additional constraints: the Boolean sums form an error-correcting code with
prescribed minimum distance d;
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Slide #12
Error-correcting microarray design
• Probe multiplexing (Khan et.al, 2003)
Probes
Excluding hybridization effects, spot formation quality and
under iid measurement noise,
min tr(G* T G* )
G*  ( G T G ) 1 G T
n  k 0  1 G matrix, n  k
rank( G )  k
0
1

0
G
0
1

1
1
0
0
1
0
1
1
1
1
0
0
0
0
0

1

1
1

0
 G( i , j )  c  const.
j
X – vector of RNA levels corresponding to N genes
Y – total concentration of RNA at all spots
Y  TGS X
S - hybridization affinity matrix, T - spot quality matrix
Decoding algorithm: numerical optimization
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s
p
o
t
s
VLSIPS/analysis for multiplexed arrays
Slide #13
Features:
• Multiple polymer synthesis at one given spot (for simplicity, will consider only two
probes per spot)
• Can use two different classes of linkers sensitive to different wavelengths so to select
probes for extension (say, `blue’ and `green’ and `cyan’)
Spots
A T G C
A T G C
g b
c c
A T G C
A T G C
g g c
b g b g
1
2
3
4
5
6
g
b
b c
b
Slide #14
VLSIPS/analysis for multiplexed arrays
Scheduling: shortest schedule of bases/colors
(Using results from V. Dancık, Expected Length of Longest Common Subsequences, 1994)
Set-up: two identical sets of M `blue’ and M `green’ randomly and uniformly
chosen sequences of length N over the alphabet of size four
Length of shortest schedule
lim  2 4( M )   4( M ) ( 2 4( M )  1 )
N 
Chvatal-Sankoff
constants
Synchronous schedule, no `cyan’ colored steps: 8N
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Mask
design:
A
C
G
T
A
C
G
Slide #15
T
S1
AT,CA
S2
AC,CC
S3
GT,GA
S4
TT,TA
b
g
c
c
g
c
c
b
s1 s4
s3 s2
L(M)=4, L(M)=4,
L(M)=2,
L(M)=2, L(M)=2, L(M)=2,
L(M)=2
L(M)=2, L(M)=2,
L(M)=2,
L(M)=2, L(M)=2, L(M)=2,
L(M)=2
s1 s3
s2 s4
Mask Design / Scheduling
Slide #16
Neighborhood graph: complete graph with M vertices labeled by two distinct
sequences ( p ( ), p ( ))
1
2
No `cyan’ steps: weight of edge between two vertices
( p1 (1 ), p2 (1 )), ( p1 (2 ), p2 (2 ))
sums of Hamming distances
Issues: For reasons of controlled hybridization, different probes (blue and
green) at the same spot should have fairly large Hamming distance
(Milenkovic and Kashyap, 2005)
Border-length minimization becomes less effective
With cyan colored steps involved, the distance measure also depends on the
longest common subsequence of the probes at the same spot
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Quality Control Coding
Slide #17
Theorem: Assume that there exists a linear error-control code with parameters
[n,k,d] containing the all-ones codeword. Then one can construct a quality control
array for a multiplexed DNA chip with 2(2k-2) disjoint blue and green production
steps and M probes such that the length of each quality control probe is 2(k-1)-1,
and that the weights w of the columns in the quality control array satisfy
d wnd
Furthermore, with such an array any collection of less than n/(n-d) failed blue or
green steps, respectively, can be uniquely identified.
Open question: how does one extend this result for schedules involving `cyan’
colored production steps, and under `spot’ failures.
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