Evaluation of Two Methods to
Cluster Gene Expression Data
Odisse Azizgolshani
Adam Wadsworth
Protein Pathways
SoCalBSI
Overview:






Background information
Statement of the project
Materials and methods
Results
Discussion and conclusion
Acknowledgements
Microarray Data

Transcriptional response of genes to variations in
cellular states

Cellular States: Mutations, Compound-Treated

State 1
State 2
State 3
…
State Y
Gene 1
0.054
…
…
…
…
Gene 2
…
…
…
…
…
Gene 3
…
…
…
…
…
…
…
…
…
…
…
Gene X
…
…
…
…
…
The data values are the log ratios of the level of gene
expression in the mutant or compound-treated state
over the level of expression in the wild-type state
Clustering

Clustering: Organizing into groups genes with similar
expression profiles

Correlation Coefficient: The metric used to determine the
similarity between two expression profiles

Hierarchical Clustering: A way of forming a multi-level
hierarchy of gene expression profiles, which can be cut off at
certain places to form gene clusters

Project: Evaluating two different methods of hierarchically
clustering expression data
Hierarchical Clustering
Method 1
EXPRESSION PROFILES
State 1
State 2
…
State Y
Gene 1
0.054
…
…
…
Gene 2
…
…
…
…
…
…
…
…
…
Gene X
…
…
…
…
Correlation
Calculations
GENE CORRELATIONS
Gene 1
Gene 2
…
Gene X
Gene 1
0
1.242
…
…
Gene 2
1.242
0
…
…
…
…
…
0
…
Gene X
…
…
…
0
Linking genes by
expression similarity
DENDROGRAM
gene 1 gene 2
gene 3 gene 4
gene 5
Method 1 Example
Hughes, T.R., et al. (2000). Functional Discovery via a Compendium of
Expression Profiles. Cell 102, 109-126.
Hierarchical Clustering
Method 2
EXPRESSION PROFILES
State 1
State 2
…
State Y
Gene 1
0.054
…
…
…
Gene 2
…
…
…
…
…
…
…
…
…
Gene X
…
…
…
…
Correlation
Calculations
GENE CORRELATIONS 1
GENE CORRELATIONS 2
Gene 1
Gene 2
…
Gene X
Gene 1
0
1.242
…
…
Gene 2
1.242
0
…
…
…
…
…
0
…
Gene X
…
…
…
0
Correlation of
Correlations
Gene 1
Gene 2
…
Gene X
Gene 1
0
0.766
…
…
Gene 2
0.766
0
…
…
…
…
…
0
…
Gene X
…
…
…
0
Linking genes by
correlation similarity
DENDROGRAM
gene 1 gene 2
gene 3 gene 4
gene 5
Method 2 Example
Provided by Matteo Pelligrini, Protein Pathways.
Applications of Clustering

Functional Genomics: Gaining information about the
possible function of genes with unknown function
Looking at the function of genes that cluster together with genes
of unknown function

Diagnostics: Tissues from clinical samples can be
clustered together to determine disease subtypes (e.g.
tumor classification)
Project Details

Project Question: In the process of hierarchically
clustering gene expression data, which metric generates
better clusters:
1. The correlation of gene expression ratios (Method 1)
2. The correlation of the correlations (Method 2)

Dataset: Yeast microarray gene expression data (6317
genes, 300 strains)*

Programming Environment: MATLAB v 6.5
*Hughes, T.R., et al. (2000). Functional Discovery via a Compendium of
Expression Profiles. Cell 102, 109-126.
Two Approaches

Problem: Determining the quality of clusters formed so as
to evaluate the two clustering methods

Approach I: Determine the quality of the clusters by
seeing if genes with the same function have clustered
together more often in one method over the other method

Approach II: Determine the quality of the clusters by
analyzing the variances of the clusters and seeing if there is
a difference between clustering methods
Approach I

Gene Function Analysis

If clusters contain lots of genes with the same function (i.e.
transcription, then the clustering method is good.

Two Function Annotation Options

2221 annotated genes with 318 different functions obtained
from http://mips.gsf.de/genre/proj/yeast/index.jsp

1155 annotated genes with 99 different functions obtained
from http://genome.ad.jp/kegg/kegg2.html
6317 genes
ANNOTATE
Approach I Steps
2000 genes
For both annotation options…




Out of the 6317 yeast genes,
select only those genes that
have known functions
Cluster the genes according to
the two methods
For each cluster, compare each
gene to every other gene in that
cluster and see how many pairs
have the same function
If a cluster contains n genes,
then there are (n)(n-1) / 2 gene
pairs to compare
CLUSTER
Cluster 1
Cluster 2
Cluster 3
1000 genes
600 genes
400 genes
NUMBER OF PAIRS TO COMPARE
499500 pairs
179700 pairs
79800 pairs
759000 pairs total
NUMBER OF PAIRS THAT HAVE SAME FUNCTION
150000 pairs
40200 pairs
3204 pairs
193404 pairs same
193404 / 759000 = 0.25
When the genes are partitioned into three clusters,
25% of the gene pairs have the same function
Approach I Results
Avg % Same vs. No. Clusters
1
0.9
Average % Pairs w/ Same Function
0.8
0.7
0.6
Annotations I, Method I
0.5
Annotations I, Method II
Annotations II, Method I
Annotations II, Method II
0.4
0.3
0.2
0.1
0
0
200
400
600
800
1000
1200
1400
Number of Clusters
1600
1800
2000
2200
2400
Approach II (1):


Determining the quality of the clusters
based on their volume.
Comparing the average volume of the
clusters generated by method 1 and
method 2.
Approach II (2):

If there are M genes in each cluster and for each gene, N
experiments are chosen:
We’ll have:




M vectors in a N-dimensional space that can be visualized as M
points.
The M points generate an ellipsoid if M > dimensionality of
the space.
The closer the points to each other, the more correlated they
are together, and the smaller the volume of the cluster.
The smaller the volume of the ellipsoid (cluster), the better the
quality of the cluster.
?
?
M original points in the 3-D space
?
j
i
k
?
D2
Centered ellipsoid with known axes
Approach II (3):

To compute the volume of the cluster, we first
compute its covariance matrix.

We then use Principal Components Analysis
(PCA) to estimate the dimensions of the cluster.

PCA will construct a new space using N
orthogonal linear combinations of old vectors of
the space. (Each linear combination is a
principal component.)
Approach II (4):

In the new space, the ellipsoid is transformed into a
centered ellipsoid, and the covariance matrix is
diagonalized.

The axes of the centered ellipsoid are the elements on the
diagonal of the diagonalized matrix, which are the variance
of the data points in the new space along the principal
components.

The volume of the ellipsoid = 4/3 x  x D1xD2 x…x DN
M original points in the 3-D space
?
?
Covariance matrix
?
j
i
k
0.0058
0.0015
0.0057
0.0015
0.0068
0.0023
0.0057
0.0023
0.0232
v2
v1
v3
PCA and
diagonalizing
the covariance
matrix
V1 = c1i + c2j + c3k
Diagonalized
Covariance Matrix
D2
D3
D1
0.0039
0
0
0
0.0066
0
0
0
V2 = c4i + c5j + c6k
V3 = c7i + c8j + c9k
0.0253
D1
M points in the new 3-D space
Approach II (5):
Question:
 Is one of the methods systematically
generating smaller ellipsoids?
Volume Calculation Results:
Average log cluster volume vs. number of clusters
-45
Average log cluster volume
0
5
10
15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110
-50
avelogvolume
method 1
-55
avelogvolume
method 2
-60
-65
Number of clusters
Similarity of the Clusters from Two
Methods:

Make a KxK matrix (K: the number of clusters in
each method) whose elements are:
aij = Difference (cluster i in method I , cluster j in method II)
N((A,B))

Difference (A,B) =
N(AB) + N(AB)
where:






A and B are two sets (Here: cluster i from method I and
cluster j from method II)
(A,B): symmetrical difference of A and B
N((A,B)) = N(A-B) + N(B-A)
AB: Union of the two sets
AB: Intersect of the two sets
0 < Difference(A,B) < 1
An Example:
A
B
B
A
1 43
89 22
16 73
1 2 6
32 21
7 89 43
26
94
10
11
A-B = {1,2,32,7,89}
N(A-B) = 5
B-A = {26,94,10,11}
N(B-A) = 4
B
A
1 3 54
76 98 6
11 88
23 13
45
N(A-B) = A
N(AB) = N(A-B) + N(B-A) = 5 + 4 = 9
N(B-A) = B
A=B
AB = {1,2,6,32,21,7,89,43,26,94,10,11}
N(AB) = N(A) + N(B)
A–B=B –A=Ø
N(AB) = 12
N(AB) = N(A) + N(B)
N(AB) = 0
AB = {6,21,43}
Dissimilarity score(A,B) = 0
Dissimilarity score (A,B) = 9/(12 + 3 ) = 0.6
A: cluster i from method I
B: cluster j from method II
N(AB) = 3
N(AB) = 0
Dissimilarity score = 1
Results: Dissimilarity Matrix
Dissimilarity score
for cluster 1 from
method 1 and
cluster 2 from
method 2
K: the number of clusters generated for each method
Discussion and Conclusion (1):

Conclusions:

Neither approach can favor one method over the other
with certainty; however,




Approach I favors method I when the number of
clusters is small.
In the range of 1-100 clusters, while approach I favors
method I, approach II fluctuates in choosing the better
method or the other.
The efficiency of both approaches in clustering genes is
dependant on the number of clusters.
The similarity of the clusters from method I and method
II decreases as the number of clusters increases; in fact,
the two methods generate very different clusters.
Discussion and Conclusion (2):

Problems faced and future questions:
What is the best cutoff value for clustering?
 In approach I, not all genes were annotated, so
around 2/3 of the dataset was ignored.
 Gene annotations are somewhat arbitrary.
 What are other ways to quantify the quality of
clusters?
 Memory problem: We couldn’t include all the genes
and all the experiments at the same time to
analyze the quality of clusters.

Acknowledgments:
Special thanks to:
 Our mentor: Dr. Matteo Pellegrini
 Protein Pathways team:
Dr. Darin Taverna
Dr. Peter Bowers
Dr. Mike Thompson
Leon Kopelevich
 SoCalBSI faculty:
Dr. Jamil Momand
Dr. Silvia Heubach
Dr. Sandra Sharp
Dr. Elizabeth Torres
Dr. Wendie Johnston
Dr. Jennifer Faust
Dr. Nancy Warter-Perez
Dr. Beverly Krilowicz
 NIH and NSF : whose funding made this internship possible.
Appendix I: Covariance (1)

The covariance of two features is the measure of
how the two features vary together.



If they both have an increasing or decreasing trend, c ij> 0.
If one decreases while the other one increases, c ij < 0.
If the changes of one is independent of the changes of the
other, c ij = 0.
*
*: http://www.engr.sjsu.edu/~knapp/HCIRODPR/PR_Mahal/cov.htm
Appendix I: Covariance (2)
If we have M variables and each variable has N measurements,
the covariance matrix can be obtained as below:
 (xi - x) (yi - y )
cij =
M
Where: cij ( i  j ) is the covariance of (measurement i and
measurement j) for all M variables.



The diagonals are the variances of each measurement.
Variance: A measure of how much the points vary around the
mean.
Appendix I: Diagonalizing The
Covariance Matrix
data =
cij =
-0.0300
0.2500
0.0340
0.1430
0.2230
0.1900
-0.0230
0.0410 -0.0110
-0.0060
0.1780
-0.0400
0.2120 -0.0560
 (xi - x) (yi - y )
M
covariance_data = cov(data);
0.3100
Covariance matrix
covariance_data =
0.0058
0.0015
0.0057
0.0015
0.0068
0.0023
0.0057
0.0023
0.0232
Diagonalize the
Covariance matrix
[V, D] = eig (covariance_data);
V=
D=
-0.9285 -0.2370
0.2859
0.0039
0.2856 -0.9478
0.1417
0.2374
0.9477
0.2133
0
0
0
0.0066
0
0
0
0.0253
Appendix II: Eigenvalues and
Eigenvectors
An eigenvector of an nxn matrix is a
nonzero vector x such that Ax = x for
some scalar . A scalar  is called an
eigenvalue of A if there is a nontrivial
solution x of Ax = x; such an x is called
an eigenvector corresponding to .*
*: Lay, C. David. Linear Algebra and Its Applications. 3rd ed. P. 303
Appendix III: Principal Components
Analysis (PCA)
data =
[ pcs , newdata , variances , t2 ] = princomp (data) ;
-0.0300
0.2500
0.0340
0.1430
0.2230
0.1900
-0.0230
0.0410 -0.0110
-0.0060
0.1780
-0.0400
0.2120 -0.0560
PCA
0.3100
newdata =
t2 =
variances =
pcs =
-0.0576 -0.0691 -0.0417
0.2859 -0.2370
0.9285
0.1417 -0.9478 -0.2856
0.9477
0.2133 -0.2374
0.1359 -0.0512
-0.1278
0.2006
0.1178
0.0896
0.0352
0.0524 -0.0644
-0.1511 -0.0499 -0.0188
1.2996
0.0253
0.0066
0.0039
3.1981
3.0600
3.0737
1.3686
Volume Calculation Results (20
random experiments, 10-20 clusters:
Volume Calculation Results:
(300 experiments chosen first, then the
dimensionality of the space was reduced to 20):
Average log of cluster volumes vs. number of clusters
-15
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
-16
ave log of clusters
-17
avelogvolume
method 2
-18
avelogvolume
method1
-19
-20
-21
-22
-23
-24
-25
number of clusters
110
Method 1 vs. Method 2 in a More
Conceptual View:


Method 1 links together the two genes
that have the most similar expression
patterns.
Method 2 links together the two genes
whose correlation with all other genes is
most similar; i.e. it looks at a genes in a
more global view (in a context of all
other genes).
Appendix IV:
1
4
4
2
3 4
5
1
2
2
1
5
3
3
5
33
2
1
2
2
5
3
3
4
5
4
5
1
1
4