DNA copy numbers and the Circular Binary Segmentation Algorithm

DNA copy numbers and the
Circular Binary Segmentation Algorithm
Venkatraman E. Seshan
Department of Biostatistics / HICCC
Columbia University
June 15, 2009
Statistical Genomics
Institute of Mathematical Sciences
National University of Singapore
Joint work with Adam B. Olshen, Richard A. Olshen

CBS
Background
• The DNA sequence copy number at any locus in a genome
is the number of copies of genomic DNA.
• The normal copy number is two for human autosomes.
• Copy number alterations are gains and losses of DNA
* they modify the function and/or expression of genes
* they are common in cancer: copy number
- Increased at sites of oncogenes;
- Decreased at sites of tumor suppressor genes.
1

CBS
Background
Comparative genomic hybridization
• first to scan entire genome for copy number variation
Kallioniemi, et al (1992), du Manoir et al (1993)
• metaphase chromosomes from test and reference samples
• imaged separately using fluorescence microscope
• quantified using image analysis software
• smooth signal – “continuous curve”, low resolution
http://amba.charite.de/cgh/cgh01.html
2

CBS
Background
Array based methods
• DNA extracted, fragmented, labeled and hybridized
• different probes
* Bacterial artificial clones (BACs) (1-3k orig, 31k tiling)
* Long oligo arrays: Agilent (44k, 244k),
Nimblegen (350k), ROMA (85k+)
* Short oligos – Affymetrix SNP arrays (100k, 500k, 1.8m)
• higher resolutions – noisier
• Review: Pinkel & Albertson (2005, Nature Genetics)
3

CBS
Example
Lung cancer BAC array with 3160 clones
log−ratio
−2 −1 0 1 2
0 500 1000 1500 2000 2500 3000
Genomic Position
4

CBS
Example
Breast cancer ROMA array with 9820 probes
log−ratio
−2 −1 0 1 2
0 500 1000 1500 2000 2500 3000
Genomic Position
5

CBS
Analysis
Overall plan
In sample identify regions of abnormal copy number
Region(s) repeatedly deleted or amplified across samples
How to identify locations of aberrations?
Thresholds: 3 SDs above 0 is a gain, 3 below is a loss
SDs estimated from probe level data of normal/normal experiments
Alteration calls can oscillate due to data overlap
Smoothing techniques (lowess, quantreg, wavelets etc.)
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CBS
Analysis
Copy number alterations are discrete events
affecting contiguous regions of the genome
• Hodgson et al. (2001) - 3-component Gaussian mixture
• Autio et al. (2003) - CGH-Plotter: 3-means clustering
combined with dynamic programming
• Fridlyand et al. (2004) - Gaussian hidden Markov model
• Wang et al. (2005) - Clustering along chromosomes
• Tibshirani and Wang (2008) - cghFlasso
Our algorithm based on a change-point model
Olshen, Venkatraman, Lucito & Wigler (2004)
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CBS Change-points Methods
Let Z 1 , Z 2 , . . . , Z n be the data ordered by an indexing set
If Z 1 , . . . , Z ν ∼ F 0 and Z ν+1 , . . . , Z n ∼ F 1 ,
then ν is a change-point (Page, 1954).
For our problem
• the data are the log-ratio measurements
• ordered by the location of a probe on a chromosome
• a change-point corresponds to where
the copy number changed on a chromosome
• There may be multiple changes.
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CBS Change-points Methods
Binary segmentation (Sen and Srivastava, 1975; Vostrikova, 1981)
1. Partial sums: S i = Z 1 + · · · + Z i , i = 1, . . . , n
2. Test the hypothesis H 0 : no change against H 1 : ν = i
Statistic if Zs are normal is S i
i − S n−S i
n−i
3. Unknown ν - maximize the t-statistic over all i
It is a recursive procedure - segments the data at a
change-point and tests for change-points in the segments
Tail probability approximation by Siegmund (1986)
9

CBS Binary Segmentation Fails Methods
Chromosome 12 Max t-statistic: 4.242/3.462, p-value: 0.221/0.254
log−ratio
−2 −1 0 1 2
1940 1960 1980 2000 2020 2040 2060 2080
Genomic Position
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CBS Binary Segmentation Fails Methods
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CBS Circular Binary Segmentation Methods
View the data as if on a circle and segment into two arcs.
Partial sums: S i = Z 1 + · · · + Z i , i = 1, . . . , n
Test statistic: T =
max
1 ≤ i < j ≤ N
|T ij |, where
⌢
Z ij − ⌣ Z ij
T ij =
s √ (j − i) −1 + (i + n − j) −1,
⌢
Z ij = (S j − S i )/(j − i)
and
⌣
Z ij = {S i + (S n − S j )}/(i + n − j)
Hence named circular binary segmentation (CBS).
Either one change-point (j = n) or two (j < n).
Levin and Kline (1985): Similar statistic for epidemic alternative
12

CBS More on CBS Methods
• Recursively split the data if p-value below threshold
• Tail probability approximation for Normal data
– Siegmund (1988); Yao (1989)
• Calculate reference distribution by permuting
• If ternary split, the first and third segment are tested for
a binary split with the middle segment (edge effect)
• Algorithmic tweaks:
* Maximize difference in means for fixed segment length
* Stop permutations once p-value exceeds threshold
* Moving windows for large data sets (deprecated)
13

CBS More on CBS Example
−2 −1 0 1 2
xx2$log2R
log−ratio
log−ratio
−2 −1 0 1 2
xx2$genomic.pos
0 500 1000 1500 2000 2500 3000
Genomic Position
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CBS Faster CBS Example
CBS performed consistently well (Lai et al., 2005) but slow
has superior performance (Willenbrock and Fridlyand, 2005)
Why is CBS slow?
• Test statistic: require n(n − 1)/2 computations
• Reference distribution is based on permutation
• Not a problem for 3k BAC arrays but larger arrays?
Especially 350k, 500k arrays (let alone 1.8m or 4m)
Improvements: hybrid p-value & a stopping rule for change
Venkatraman & Olshen (2007)
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CBS Hybrid p-value Faster CBS
Test statistic T = max{T 1 , T 2 } where
T 1 = max
A 1
|T ij | and T 2 = max
A 2
|T ij |
A 1 = {i, j : j − i ≤ k or > n − k} (small arcs)
A 2 = {i, j : k + 1 ≤ j − i ≤ n − k} (non-small arcs)
Choose k to call an arc small
Split data if P (T > T obs ) < α
Bound P (T > T obs ) ≤ P (T 1 > T obs ) + P (T 2 > T obs )
Use permutation for P (T 1 > T obs ) and
Approximate P (T 2 > T obs ) (Siegmund (1988), Yao (1993))
16

CBS p-value Approximation Faster CBS
Let ˜T 2 = max
A 2
T ij . If data are normal, then
P ( ˜T 2 > b) = 1 4 b3 φ(b)
∫ 1−δ
1/2
ν 2 (b/[nt(1 − t)] 1/2 )
t 2 (1 − t) 2 dt,
where δ = k/n, φ is the normal density and ν is
⎧
⎨ ∞
ν(x) = 2x −2 exp
⎩ −2 ∑
r −1 Φ
(− 1 ) ⎫ ⎬
2 xr1/2 ⎭ .
By symmetry P (T 2 > b) ≈ 2P ( ˜T 2 > b).
If data are “regular” and k sufficiently large,
random field approximately Gaussian
1
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CBS Stopping rule Faster CBS
• data segmented if p-value ≤ α
• p-value computed from P permuted statistics
• all P permutations conducted when change detected
• what if none of the first 1000 T perm > T obs
Number T perm > T obs
0 20 40 60 80 100
0 2000 4000 6000 8000 10000
Number of permutations
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CBS Stopping rule Faster CBS
Let e 1 , . . . , e P be the indicators of T perm > T obs
i∑
Let r(i) = e j . Declare no change if r(i) = αP = m
1
• Stopping rule to declare change is a boundary b 1 , . . . , b m
• Stop and declare change detected if r(b i ) < i
• Choosing the boundary? – Repeated significance test
• Control Prob{r(b i ) < i for some i|r(P) = m} at η
• b i is the smallest j such that Prob{r(j) < i} = η ⋆
• obtained using the hypergeometric distribution
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CBS Performance gain Faster CBS
100k SNP array data for 3 breast cancer cell lines
data from http://pevsnerlab.kennedykrieger.org/snpscan.htm
Cell line Time in
MCF7 SKBR3 ZR75 minutes
P 220 242 270 7085.8
H 217 242 271 100.2
H+ES 216 243 269 25.0
Identical 213 242 263
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CBS Lung project Example
• Two broad categories of lung cancer: small cell (SCLC)
and non-small cell (NSCLC)
• Adenocarcinoma is a histological subtype of NSCLC
• Recent advance: EGFR mutation and effective TKI drugs
• Prognosis is in general poor especially KRAS mutation
Can we learn more by profiling DNA copy number?
≈ 250 samples using Agilent 44k arrays.
Also gene expressions using Affy U133 arrays
21

CBS Lung Project Example
22

CBS Lung Project Example
23

CBS Lung Project Example
24

CBS Lung Project Example
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25

CBS Lung Project Example
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26

CBS Lung Project Example
• No obvious focal gain or loss stood out
• Clusters had weak association with EGFR but not KRAS
• Incorporated information from U133 expression arrays
• Potentially interesting gene DUSP4
Acknowledgements
Marc Ladanyi, William Gerald, Valerie Rusch, Stephen Broderick,
Cameron Brennan, Dhananjay Chitale, Bhuvanesh Singh and others.
27

CBS Copy Number Variation TCGA
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CBS Multi platform data wCBS
• Method to merge data from multiple platforms
• Platform specific probe noise - weights w i
• Partial sums: S i = w 1 Z 1 + · · · + w i Z i ,
W i = w 1 + · · · + w i
i = 1, . . . , n
• Test statistic: T =
max
1 ≤ i < j ≤ N
|T ij |, where
T ij =
⌢
Z ij − ⌣ Z ij
s
√(W j − W i ) −1 + (W i + W n − W j ) −1,
⌢
Z ij = (S j − S i )/(W j − W i )
and
⌣
Z ij = {S i + (S n − S j )}/(W i + W n − W j )
29

CBS Multi platform data wCBS
Weighted CBS uses the same hybrid approach.
Bengtsson, et al (2009)
HaasSeg is a Haar wavelt based segmentation algorithm
30

CBS
ASCN
Allele Specific Copy Number
• Traditional methods measure the sum of copy numbers of
the parental chromosomes.
• SNP arrays can be used for allele specific copy numbers.
• Total copy number of 2 can be from AA, AB or BB.
• Data from Molecular Inversion Probes (MIPs) SNP genotyping/copy
number technology from Affymetrix.
• Intensities for the two alleles converted to copy numbers.
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CBS MIPs Example ASCN
Allele A copy number
0 1 2 3 4 5 6
Allele B copy number
0 1 2 3 4 5 6
0 500 1000 1500 2000 2500 3000
Genomic Position (Mb)
0 500 1000 1500 2000 2500 3000
Genomic Position (Mb)
Sum of copy numbers
0 1 2 3 4 5 6
0 500 1000 1500 2000 2500 3000
Genomic Position (Mb)
32

CBS Analysis plan ASCN
• How to estimate parental chromosome copy numbers?
• Phase ambiguity causes problem.
parental origin of heterozygous SNPs unknown.
Proposed plan:
Begin by segmenting total copy number using CBS.
change-points in total present in one of the parent
analyze allele specific CN within each segment
33

CBS Allele A vs B ASCN
Homozygotes are non informative for parental CN
Phase ambiguity can lead to 2 heterozygous blobs
Chromosome 1p
Chromosome 1q
Allee B copy number
0 1 2 3 4 5
Allee B copy number
0 1 2 3 4 5
0 1 2 3 4 5
Allee A copy number
0 1 2 3 4 5
Allee A copy number
34

CBS Allele A vs B ASCN
Step 2: Determine the homozygous locations.
Minimum of A & B
−0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 500 1000 1500 2000 2500 3000
Genomic Position (Mb)
35

CBS Allele A vs B ASCN
Step 3: Test for additional change-points
Why? TCN = 4 = 3 + 1 = 2 + 2 ASCN
Step 4: Estimate difference in parental copy numbers
mean absolute difference of heterozygous ASCNs
Step 5: Combine parental copy numbers
adjacent segment copy number transitions
copy numbers 1 + 2 → 2 + 3
36

CBS Reconstruction ASCN
Copy numbers
0 1 2 3 4 5 6
0 500 1000 1500 2000 2500 3000
Genomic Position (Mb)
Total High copy Low copy
37

CBS Reconstruction ASCN
Allele A copy number
0 1 2 3 4 5 6
Allele B copy number
0 1 2 3 4 5 6
0 500 1000 1500 2000 2500 3000
Genomic Position (Mb)
0 500 1000 1500 2000 2500 3000
Genomic Position (Mb)
Sum of copy numbers
0 1 2 3 4 5 6
0 500 1000 1500 2000 2500 3000
Genomic Position (Mb)
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CBS Equality of parental CN ASCN
Chromosome 1p
Chromosome 1q
Chromosome 6p
Allee B copy number
0 1 2 3 4 5
0 1 2 3 4 5
Allee A copy number
Chromosome 1p
0 1 2 3 4 5
Allee A copy number
Density
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Density
0.0 0.1 0.2 0.3 0.4 0.5
Allee B copy number
0 1 2 3 4 5
Allee B copy number
0 1 2 3 4 5
Chromosome 1q
0 1 2 3 4 5
Allee A copy number
Chromosome 6p
Density
0.0 0.5 1.0 1.5 2.0
−2 −1 0 1 2
N = 678 Bandwidth = 0.09884
−3 −2 −1 0 1 2 3
N = 604 Bandwidth = 0.2596
−2 −1 0 1 2
N = 290 Bandwidth = 0.05112
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CBS Equality of parental CN ASCN
Chromosome 6q
Chromosome 14
Chromosome 15
Allee B copy number
0 1 2 3 4 5
0 1 2 3 4 5
Allee A copy number
Chromosome 6q
0 1 2 3 4 5
Allee A copy number
Density
0.0 0.2 0.4 0.6
Density
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Allee B copy number
0 1 2 3 4 5
Allee B copy number
0 1 2 3 4 5
Chromosome 14
0 1 2 3 4 5
Allee A copy number
Chromosome 15
Density
0.0 0.5 1.0 1.5 2.0
−2 −1 0 1 2
N = 468 Bandwidth = 0.2054
−1.0 −0.5 0.0 0.5 1.0
N = 414 Bandwidth = 0.09307
−1.0 −0.5 0.0 0.5 1.0
N = 408 Bandwidth = 0.04398
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CBS
Summary
• New technologies for studying biological process present
interesting applications of statistical techniques
• The CBS algorithm is a practical solution which is widely
used to study DNA copy number data
• Improvements make it highly attractive for larger arrays
• Software is available as an open source R (www.r-project.
org) library called DNAcopy that is part of Bioconductor
(www.bioconductor.org).
41

CBS
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CBS
References
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