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Techcnics in DNA: copy number
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What is DNA copy number?
Normally, each somatic cell contains 2 copies of everychromosome.
What is DNA copy number?
One of the earliest observed “copy number” changes is trisomyof chromosome 21 in Down’s Syndrome.
What is DNA copy number?
In fact, it became apparent later that chromosome aberrationscome in all forms and sizes.
High density DNA copy number data
Array-based Comparative Genomic Hybridization
Figures from Garnis et al. (2004)
DNA Copy Number Data from Different Platforms
Why analyze DNA copy number?Cancer genomics
Why analyze DNA copy number?Douglas et al. (2004), colorectal cancer.
Why analyze DNA copy number?Copy number polymorphisms in Hapmap samples
CNV in natural population may be risk factors for diseases.
Statistical methods for single sample, total copynumber segmentation
1. Circular Binary Segmentation algorithm of Olshen et al.(2004)
2. HMM based methods (Fridlyand et al. (2004), Lai et al.(2007))
3. Wavlet based methods of Hsu et al. (2005)4. Cluster ALong Chromosomes method of Wang et al.
(2005)5. Many others: CBS, HMM, GLAD, CNV, CGHseg,
Quantreg,Wavelet, Lowess, ChARM, GA, L1Regularizaiton, ACE...
Statistical methods for single sample, total copynumber segmentation
1. Circular Binary Segmentation algorithm of Olshen et al.(2004)
2. HMM based methods (Fridlyand et al. (2004), Lai et al.(2007))
3. Wavlet based methods of Hsu et al. (2005)4. Cluster ALong Chromosomes method of Wang et al.
(2005)5. Many others: CBS, HMM, GLAD, CNV, CGHseg,
Quantreg,Wavelet, Lowess, ChARM, GA, L1Regularizaiton, ACE...
Statistical methods for single sample, total copynumber segmentation
1. Circular Binary Segmentation algorithm of Olshen et al.(2004)
2. HMM based methods (Fridlyand et al. (2004), Lai et al.(2007))
3. Wavlet based methods of Hsu et al. (2005)
4. Cluster ALong Chromosomes method of Wang et al.(2005)
5. Many others: CBS, HMM, GLAD, CNV, CGHseg,Quantreg,Wavelet, Lowess, ChARM, GA, L1Regularizaiton, ACE...
Statistical methods for single sample, total copynumber segmentation
1. Circular Binary Segmentation algorithm of Olshen et al.(2004)
2. HMM based methods (Fridlyand et al. (2004), Lai et al.(2007))
3. Wavlet based methods of Hsu et al. (2005)4. Cluster ALong Chromosomes method of Wang et al.
(2005)
5. Many others: CBS, HMM, GLAD, CNV, CGHseg,Quantreg,Wavelet, Lowess, ChARM, GA, L1Regularizaiton, ACE...
Statistical methods for single sample, total copynumber segmentation
1. Circular Binary Segmentation algorithm of Olshen et al.(2004)
2. HMM based methods (Fridlyand et al. (2004), Lai et al.(2007))
3. Wavlet based methods of Hsu et al. (2005)4. Cluster ALong Chromosomes method of Wang et al.
(2005)5. Many others: CBS, HMM, GLAD, CNV, CGHseg,
Quantreg,Wavelet, Lowess, ChARM, GA, L1Regularizaiton, ACE...
HMM Model of Fridlyand et al. (2004)This is a classic application of hidden Markov models:
▶ The underlying states 1, . . . ,K represent the “true” copynumber.
▶ Given state k , the observed intensity levels are N(�k , �2).
▶ The transition matrices and emission parameters areestimated by EM.
▶ The AIC or BIC criterion is used to choose K .
A Bayesian Model for Inference
When we estimate model parameters,confidence intervals are desirable!
1. Confidence bands on estimated copy number.2. How certain are we that [i , j] contains a CNV?3. Confidence intervals on the aberration boundaries.4. Confidence intervals on global measures of “complexity",
such as total number of aberrations.
A Bayesian Model for Inference
When we estimate model parameters,confidence intervals are desirable!
1. Confidence bands on estimated copy number.
2. How certain are we that [i , j] contains a CNV?3. Confidence intervals on the aberration boundaries.4. Confidence intervals on global measures of “complexity",
such as total number of aberrations.
A Bayesian Model for Inference
When we estimate model parameters,confidence intervals are desirable!
1. Confidence bands on estimated copy number.2. How certain are we that [i , j] contains a CNV?
3. Confidence intervals on the aberration boundaries.4. Confidence intervals on global measures of “complexity",
such as total number of aberrations.
A Bayesian Model for Inference
When we estimate model parameters,confidence intervals are desirable!
1. Confidence bands on estimated copy number.2. How certain are we that [i , j] contains a CNV?3. Confidence intervals on the aberration boundaries.
4. Confidence intervals on global measures of “complexity",such as total number of aberrations.
A Bayesian Model for Inference
When we estimate model parameters,confidence intervals are desirable!
1. Confidence bands on estimated copy number.2. How certain are we that [i , j] contains a CNV?3. Confidence intervals on the aberration boundaries.4. Confidence intervals on global measures of “complexity",
such as total number of aberrations.
Observations
1. For array-CGH data, there is a known baseline at 0.
2. Due to mosaicism, the data is drawn from mixtures ofdiscrete copy number levels, and thus is continuous.
3. In some tumors the number of distinct levels is very high.
Observations
1. For array-CGH data, there is a known baseline at 0.2. Due to mosaicism, the data is drawn from mixtures of
discrete copy number levels, and thus is continuous.
3. In some tumors the number of distinct levels is very high.
Observations
1. For array-CGH data, there is a known baseline at 0.2. Due to mosaicism, the data is drawn from mixtures of
discrete copy number levels, and thus is continuous.3. In some tumors the number of distinct levels is very high.
Fitted Levels
Heterogeneity of cancer samples
Image from: http://science.kennesaw.edu/ mhermes/cisplat/cisplat19.htm
Stochastic Change Model
St ∈ {baseline, changed}
Stochastic Change Model
St ∈ {baseline, changed}
baseline state: �t = 0, changed state: �t ∼ N(�, v).
If St jumps, �t takes on new value. Otherwise �t = �t − 1.
Stochastic Change Model
St ∈ {baseline, changed}
baseline state: �t = 0, changed state: �t ∼ N(�, v).
If St jumps, �t takes on new value. Otherwise �t = �t − 1.
yt = �t + ��t , �t ∼ N(0,1)
Stochastic Change Model
P(St = changed ∣ St−1 = baseline) = p
P(St = different changed state ∣ St−1 = changed) = b
P(St = baseline ∣ St−1 = changed) = c
Stochastic Change Model
P(St = changed ∣ St−1 = baseline) = p
P(St = different changed state ∣ St−1 = changed) = b
P(St = baseline ∣ St−1 = changed) = c
This can be modeled with a 3-state Markov model with transitionmatrix:
P =
⎛⎝ 1− p 12p 1
2pc a bc b a
⎞⎠ .
Estimating �t , St
We can compute:
E(�t ∣ y1:n) “smoothed" estimate of meanP(St = changed ∣ y1:n) probability of CNV at t
P(CNV at [i,j] ∣ y1:n) probability of aberration at [i , j]
Estimating �t , St
The posterior distribution of �t given Yn (1 ≤ t ≤ n), which is amixture of normal distributions and a point mass at 0:
�t ∣Yn ∼ �t�0 +∑
1≤i≤t≤j≤n
�ijtN(�ij , vij).
The parameters of this distribution can be computed byrecursive formulas.
E(�t ∣ y1:n) =∑
1≤i≤t≤j≤n
�ijt�ij ,
P(St = changed ∣ y1:n) = �t
P(CNV at [i,j] ∣ y1:n) = �ijt ,
where�t = �
∗t/
At , �ijt = �∗ijt/
At , At = �∗t +
∑1≤i≤t≤j≤n
�∗ijt ,
�∗t = pt [(1− p)p̃t+1 + cq̃t+1]
/c,
�∗ijt =
{qi,t (pp̃t+1 + bq̃t+1)
/p, i ≤ t = j,
aqi,t q̃j,t+1 i,t t+1,j/(p i,j ), i ≤ t < j.
Estimating �t , St
The posterior distribution of �t given Yn (1 ≤ t ≤ n), which is amixture of normal distributions and a point mass at 0:
�t ∣Yn ∼ �t�0 +∑
1≤i≤t≤j≤n
�ijtN(�ij , vij).
The parameters of this distribution can be computed byrecursive formulas.
E(�t ∣ y1:n) =∑
1≤i≤t≤j≤n
�ijt�ij ,
P(St = changed ∣ y1:n) = �t
P(CNV at [i,j] ∣ y1:n) = �ijt ,
where�t = �
∗t/
At , �ijt = �∗ijt/
At , At = �∗t +
∑1≤i≤t≤j≤n
�∗ijt ,
�∗t = pt [(1− p)p̃t+1 + cq̃t+1]
/c,
�∗ijt =
{qi,t (pp̃t+1 + bq̃t+1)
/p, i ≤ t = j,
aqi,t q̃j,t+1 i,t t+1,j/(p i,j ), i ≤ t < j.
Hyperparameter Estimation
The model was defined as:
yt = �t + ��t , �t ∼ N(0,1)
baseline state: �t = 0, changed state: �t ∼ N(�, v).St modeled by a 3-state Markov model with transition matrix:
P =
⎛⎝ 1− p 12p 1
2pc a bc b a
⎞⎠ .
The hyperparameters of this model are �, �, v , a, b, c , p.
Likelihood of the data as a function of these hyperparameterscan be expressed by recursive formulas. Maximum-likelihoodvalues, computed by the EM algorithm, are used.
Hyperparameter Estimation
The model was defined as:
yt = �t + ��t , �t ∼ N(0,1)
baseline state: �t = 0, changed state: �t ∼ N(�, v).St modeled by a 3-state Markov model with transition matrix:
P =
⎛⎝ 1− p 12p 1
2pc a bc b a
⎞⎠ .
The hyperparameters of this model are �, �, v , a, b, c , p.
Likelihood of the data as a function of these hyperparameterscan be expressed by recursive formulas. Maximum-likelihoodvalues, computed by the EM algorithm, are used.
Confidence Bands for BT474
Inference on Measures of Genome Complexity
