Joint Analysis of DNA Copy Numbers and Gene …cs.brown.edu/courses/csci2950-c/Fall2008/LectureSlides/DerekAguiar...Gene 4 0 0 1.5 1 -0.5. GENERAL TERMINOLOGY C(i,j) DNA copy number

SEPARATING-CROSS CORRELATION

Threshold-based analysis of the dependence of u

and v ndash highly dependant on threshold selections

We view the two vectors u and v as n points (uivi)

in the plain

We place an axis parallel

cross centered at (xy)

partitioning the plain into

four quadrants At Bt Ct Dt

Let at bt ct dt equal

the quadrant counts

SEPARATING-CROSS CORRELATION

They are interested in the separating cross score

function

Let π τ denote the samples permutation induced

by u and v respectively Our cross and score

functions now depend only on quadrant counts

and no longer on location of points In other

words we have F(uv) = F(πτ)

SEPARATING-CROSS CORRELATION

We can now compute F(πτ) for every function

f(πτt) by examining at most all (n-1)2 possible

crosses

0

05

1

15

2

25

3

0 05 1 15 2 25 3

0

05

1

15

2

25

3

0 05 1 15 2 25 3

0

05

1

15

2

25

3

0 05 1 15 2 25 3

At=2

At=2

At=2

Dt=1Dt=2 Dt=3

Ct=0 Ct=0

Ct=0

Bt=2

Bt=1

Bt=0

SEPARATING-CROSS CORRELATION

The Diagonal Product DP(πτt) = at dt

One cross score function that they are particular

interested in is the Maximal Diagonal Product

Distinguishes between

samples that contribute

to the MDP (points in

quadrants AtDt) and

those that do not

(points in quadrants BtCt)0

05

1

15

2

25

3

0 05 1 15 2 25 3

At=2

Dt=3Ct=0

Bt=0

OUTLINE

Background and Introduction

Methods

Correlation Scoring Methods

Regional Analysis

Results

Genomic-Continuous Submatrices

Max-hypergeometric Algorithm

CCA ndash Consistent Correlation Algorithm

Results

Conclusion and Discussion

REGIONAL ANALYSIS

Previous studiesrsquo models only consider correlation

between GE levels of single genes and their

respective DNA copy number

This paperrsquos model also considers regional effects

Genes that are close together are likely to be

perturbed in the same fashion

Since DCN events are likely to span more than one

gene

REGIONAL ANALYSIS

Given a gene gi we define its k-neighborhood as

the continuous sequence of genes surrounding it

Straightforward method of quantifying

correlation between GE and DCN of a given gene

is to calculate average correlation

r(ij) is a correlation measure between E(i) and

C(j)

REGIONAL ANALYSIS - EXAMPLE

1-neighborhood around gene 3

We get the average correlation of gene irsquos

expression vector with neighboring genes DCN

vectors

)432()3(1

4333233

133 1 rrrr

REGIONAL ANALYSIS - RESULTS

They test whether or not regional effects are dependent on the original chromosomal ordering using simulations from data in JR Pollack et al PNAS 99(20)12963ndash8 20022

Null hypothesis neighboring genes are independent of each other

p-values are computed as follows Let L be the size of the simulation

Randomly draw L-1 expression vectors E[i1]hellipE[iL-1]

For each E[i1]hellipE[iL-1] compute rl = r(ilΓk(i))

We now compute r= r(iΓk(i))

We assign the rank of r amongst r1r2helliprL-1r the value ρ

The p-value for the region correlation observed at i is pV(i)=ρL

REGIONAL ANALYSIS - RESULTS

Key notes

Randomly permuting the dataset yields a straight line

Significant correlations for single genes and regional areas

Using region-based analysis delivers almost 80 more loci

where DCN-GE correlation may be identified with high

confidence

OUTLINE

Background and Introduction

Methods

Correlation Scoring Methods

Regional Analysis

Results

Genomic-Continuous Submatrices

Max-hypergeometric Algorithm

CCA ndash Consistent Correlation Algorithm

Results

Conclusion and Discussion

GENOMIC-CONTINUOUS SUBMATRICES

This part of the paper is concerned with detecting

the genomic segments in which an aberration has

occurred the affected samples and the

transcriptional effect

Genomic alterations

are often localized to

a subset of samples

as well as specific

chromosome

segments

GENOMIC-CONTINUOUS SUBMATRICES

DEFINITIONS

C and E are DCN and GE matrices (rows = genes columns = samples)

G is the ordered set of genes

Grsquo is a continuous segment of genes

S is the set of samples

Srsquo is a subset of samples

The genomic-continuous submatrix (GCSM) is M=Grsquo x Srsquo

M=Grsquo x S-Srsquo

C(M) and E(M) are the projections of matrices C and E on the subsets Grsquo and Srsquo In other words the submatrices formed by Grsquo and Srsquo of

DNA copy number and gene expression

GENOMIC-CONTINUOUS SUBMATRICES

Remember that the model states a genomic

alteration in a given sample will effect DCN

measurements in the segment but only some of

the GE measurements

A GCSM matrix is significantly amplified if

Most DNA copy values in the set C(M) are positive

Some genes gi in Grsquo have higher expression values in

the samples in Srsquo compared to the samples not in Srsquo

GENOMIC-CONTINUOUS SUBMATRICES

Formallyhellip

Define the score F(MC) that measures

overabundance of positive values in C(M) compared

to C(M)

They use the hypergeometric distribution to test for

significance

Definitions

N=number of values in C(M U M)

n=number of values in C(M)

K=number of positive values in C(M U M)

k=number of positive values in C(M)

GENOMIC-CONTINUOUS SUBMATRICES ndash

HYPERGEOMETRIC SCORE

Then the probability of finding k or more positive

values in C(M) is

Choose i values from number of values in Grsquo

Choose from the complement of Grsquo

the remaining number of positive values

Total ways we can pick positive values

GENOMIC-CONTINUOUS SUBMATRICES ndash

HYPERGEOMETRIC SCORE

Positive DCN

Non-positive DCN

C(M) C(M)N=10

n=6

K=6

k=5

012

6

10

0

4

6

6

6

10

1

4

5

6

CMF

Thus the probability of selecting 5 or 6 positive DCN is 012

GENOMIC-CONTINUOUS SUBMATRICES

A threshold number of misclassifications score

may be assigned to each gene according to its

performance as a Srsquo versus S ndash Srsquo classifier

If the probability for a single gene of obtaining a

score of s or better under the null model is p(s)

then the number of genes

with scores s or better

amongst the |Grsquo|

genes examined is

Binomial(np(s))

distributed

GENOMIC-CONTINUOUS SUBMATRICES

Under the null model DCN and GE vectors are

completely uncorrelated

A total score for an amplification is

where F(ME) is defined to be the maximum

ndashlog(σ(s)) for 0lesleSrsquo and σ(s) is the tail probability

of B(np(s))

The basic idea here is F(MCE) reflects the

overabundance of genes in Grsquo that are

significantly expressed (comparing Srsquo with S-Srsquo)

MAX-HYPERGEOMETRIC ALGORITHM

Heuristic approach to find Srsquo that maximizes

F((Grsquo x Srsquo)CE) for a given Grsquo

Note Since Grsquo is continuous the total number of

permutations = )( 2

1

nin

i

MAX-HYPERGEOMETRIC ALGORITHM

Iterate through all permutations of Grsquo of length le l

For each sample pi = of positive entries in

C(Grsquosample)

Order samples decreasing order

Find the maximum score out of a prefix of ordered

samples

If the maximum score is above threshold add M=(GrsquoArsquo)

to L

CONSISTENT CORRELATION ALGORITHM

Attempts to account for DCN and GE patterns to

locate partitions of Srsquo

Uses

CONSISTENT CORRELATION ALGORITHM

The sample MDP or SMDP is defined as follows

for all input genes

for each sample calculate pi=SMDP(samplei)

order samples decreasing order

For all segments in G of length less than l

Calculate maximum score over all prefixes of samples in the

ordered set ndash F((GrsquoSamplePrefix)CE)

if the maximum score gt threshold then add M=(GrsquoSrsquo) to L

Main difference here is the calculation of pi

ANALYSIS

MHA will produce good fitting GCSMs when

F(MC) is the dominant factor

CCA will produce good fitting GCSMs when there

is a strong correlation between E(M) and C(M)

When F(ME) is the dominant factor neither

algorithm performs well

Not within the scope of this study