Cluster Analysis of Gene Expression Profiles

1
Cluster Analysis of Gene Expression Profiles

• Identifying groups of genes that exhibit a
  similar expression "behavior" across a number of
  experimental conditions
• Assuming that such "co-expression" will tell us
  something about these genes are regulated or even
  possibly something about their function
  (Functional Genomics)
• Using information from multiple genes at a time -
  as opposed to the single gene at a time analysis
  we did so far
• We can also cluster biological samples based on
  the expression of some or all of the genes
• Example Identifying groups of molecularly
  similar tumor
• "Molecular phenotyping"
• "Unsupervised learning" in the computer science
  lingo

2
Cluster analysissource("http//eh3.uc.edu/teachin
g/cfg/2006/R/ClusterAnalysis.R")

• Often a large portion of genes are "not
  interesting"
• The meaning of the "not interesting" depends on
  the context
• Possibly we are interested in genes that whose
  expression is not constant across all
  experimental conditions. To remove
  "non-interesting" genes one can apply a
  "variation filter".
• Various sorts of "filtering" of "non-interesting"
  genes generally amounts to performing some kind
  of informal statistical testing with a very low
  confidence.
• For now, we will just play with our data with
  some more exciting examples to follow
• We have six measurements for each gene and will
  try to cluster genes and experimental conditions
  using this data

3
Cluster analysis

• gt load(url("http//eh3.uc.edu/teaching/cfg/2006/da
  ta/SimpleData.RData"))
• gt Niclt-grep("Nic",dimnames(SimpleData)2)
• gt Ctllt-grep("Ctl",dimnames(SimpleData)2)
• gt MNiclt-apply(SimpleData,Nic,1,mean,na.rmTRUE)
• gt VNiclt-apply(SimpleData,Nic,1,var,na.rmTRUE)
• gt MCtllt-apply(SimpleData,Ctl,1,mean,na.rmTRUE)
• gt VCtllt-apply(SimpleData,Ctl,1,var,na.rmTRUE)
• gt NNiclt-apply(!is.na(SimpleData,Nic),1,sum,na.rm
  TRUE)
• gt NCtllt-apply(!is.na(SimpleData,Ctl),1,sum,na.rm
  TRUE)
• gt VNicCtllt-(((NNic-1)VNic)((NCtl-1)VCtl))/(NCtl
  NNic-2)
• gt DFlt-NNicNCtl-2
• gt TStatlt-abs(MNic-MCtl)/((VNicCtl((1/NNic)(1/NCt
  l)))0.5)
• gt TPvaluelt-2pt(TStat,DF,lower.tailFALSE)
• gt SigGeneslt-(TPvaluelt0.001)
• gt sum(SigGenes)
• 1 7

4
Cluster analysis

• gt library(marray)
• gt library(mclust)
• gt pallt-maPalette(low"green", high"red",
  mid"black")
• gt MinExplt-min(SimpleDataSigGenes,27)
• gt MaxExplt-max(SimpleDataSigGenes,27)
• gt heatmap(data.matrix(SimpleDataSigGenes,27),Co
  lvNA,RowvNA,colpal,labRowas.character(SimpleDa
  taSigGenes,1),scale"none")
• gt maColorBar(seq(MinExp,MaxExp,(MaxExp-MinExp)/5)
  , colpal, horizontalFALSE, k5)

5
Cluster analysis
gt heatmap(data.matrix(SimpleDataSigGenes,27),co
lpal,labRowas.character(SimpleDataSigGenes,1),
scale"none")

• Genes were selected based on their differences
  between Nic and Ctl treatments - not obvsious
  except for one gene

6
Cluster analysis - centered data

• gt CenteredDatalt-SimpleData,27-apply(SimpleData
  ,27,1,mean,na.rmT)
• gt heatmap(data.matrix(CenteredDataSigGenes,),col
  pal,labRowas.character(SimpleDataSigGenes,1),s
  cale"none")
• gt heatmap(data.matrix(SimpleDataSigGenes,27),co
  lpal,labRowas.character(SimpleDataSigGenes,1))

7
Hierarchical Clustering

• Calculating the "distance" or "similarity between
  each pair of expression profiles
• Merging two "closest" profiles, forming a "node"
  in the clustering tree and re-calculating the
  "distance between such a "sub-cluster" and rest
  of the profiles or sub-clusters using on of the
  "linkage" principles. Again merge two closest
  sub-clusters
• Complete linkage - define the distance/similarity
  between the two clusters as the maximum/minimum
  distance/similarity between pairs of profiles in
  which one profile is from the first sub-cluster
  and the other profile is from the second
  sub-cluster
• Average linkage - define the distance/similarity
  between the two clusters as the average
  distance/similarity between pairs of profiles in
  which one profile is from the first sub-cluster
  and the other profile is from the second
  sub-cluster
• Single linkage - define the distance/similarity
  between the two clusters as the minimum/maximum
  distance/similarity between pairs of profiles in
  which one profile is from the first sub-cluster
  and the other profile is from the second
  sub-cluster

8
Euclidian Distance

• R actually operates on distances, so similarities
  have to be transformed into distances - usually
  straightforward
• Euclidian distance

• In 2 and 3 dimensions, this is our usual, every
  day's distance
• gt EDistanceslt-dist(CenteredDataSigGenes,,method
  "euclidean", diag T, upper T)
• gt print(EDistances,digits2)

9
Distance Matrix

• Distance Matrix - whole
• 34 440 596 2797 4466 4512 7651
• 34 0.00 8.55 5.64 5.46 8.15 8.03 9.14
• 440 8.55 0.00 3.01 3.19 0.82 0.82 1.13
• 596 5.64 3.01 0.00 0.33 2.53 2.48 3.59
• 2797 5.46 3.19 0.33 0.00 2.71 2.62 3.72
• 4466 8.15 0.82 2.53 2.71 0.00 0.47 1.18
• 4512 8.03 0.82 2.48 2.62 0.47 0.00 1.14
• 7651 9.14 1.13 3.59 3.72 1.18 1.14 0.00
• Distance Matrix - lower triangular
• gt EDistanceslt-dist(CenteredDataSigGenes,,method
  "euclidean")
• gt print(EDistances,digits2)
• 34 440 596 2797 4466 4512
• 440 8.55
• 596 5.64 3.01
• 2797 5.46 3.19 0.33
• 4466 8.15 0.82 2.53 2.71
• 4512 8.03 0.82 2.48 2.62 0.47

10
Dendrograms - Complete Linkage

• gt Clusteringlt-hclust(EDistances,method"complete")
  
• gt plot(Clustering)

Distance Matrix - lower triangular 34
440 596 2797 4466 4512 440 8.55
596 5.64 3.01 2797
5.46 3.19 0.33 4466 8.15 0.82 2.53
2.71 4512 8.03 0.82 2.48 2.62 0.47
7651 9.14 1.13 3.59 3.72 1.18 1.14
11
Clustering genes and samples

• gt EDistancesSlt-dist(t(CenteredDataSigGenes,),met
  hod "euclidean")
• gt ClusteringSlt-hclust(EDistancesS,method"complete
  ")
• gt heatmap(data.matrix(CenteredDataSigGenes,),Col
  vas.dendrogram(ClusteringS),Rowvas.dendrogram(Cl
  ustering),
• colpal,scale"none")

gt TwoClusterslt-cutree(ClusteringS,k 2, h
NULL) gt TwoClusters Ctl Nic Nic.1 Nic.2 Ctl.1
Ctl.2 1 2 2 2 1 1
12
Clustering by partitioning K-means algorithm

• For a pre-specified number of clusters iterate
  between calculating cluster "centroides" (i.e.
  cluster means) and re-assigning each profile to
  the cluster with the closest "centroid"

• t1st iteration

• iterate until ct1ct

13
Clustering k-means

• gt TwoCKmeanslt-kmeans(t(CenteredDataSigGenes,),
  2, iter.max 10)
• gt TwoCKmeans
• K-means clustering with 2 clusters of sizes 3, 3
• Cluster means
• 34 440 596 2797
  4466 4512 7651
• 1 2.510742 -0.9565299 0.2554164 0.3246475
  -0.770937 -0.7398173 -1.181848
• 2 -2.510742 0.9565299 -0.2554164 -0.3246475
  0.770937 0.7398173 1.181848
• Clustering vector
• Ctl Nic Nic.1 Nic.2 Ctl.1 Ctl.2
• 1 2 2 2 1 1
• Within cluster sum of squares by cluster
• 1 1.0805679 0.9474704
• Available components
• 1 "cluster" "centers" "withinss" "size"

14
Questions

• How many clusters there are in the data?
• What is the statistical significance of a
  clustering?
• What is a confidence in assigning any particular
  expression profile to any particular cluster?
• Difficult questions, particularly difficult to
  answer when using heuristic methods like
  hierarchical clustering and k-means
• Need statistical models

15
Two genes at a time

• Are these two genes co-expressed?
• By looking at their expression patterns alone,
  combined with the null distribution of the
  similarity measure in non-co-expressed genes, we
  could conclude that this is the case.

16
Another look

• What if we knew that there are two and only two
  distinct patterns in the data and we know how
  they look (thick dashed lines)?
• Given this additional information we are likely
  to conclude that our two genes actually have
  different patterns of expression.

17
Many genes at a time

• Simultaneous detection of patterns of
  expression defined by groups of expression
  profiles and assignment of individual expression
  profiles to appropriate patterns.
• By looking at all genes at the same time, we
  came up with a completely different conclusion
  than when looking at only two of them.
• Questions How many clusters? How confident are
  we in the number of clusters in the data? How
  confident are we that our two genes belong to two
  different clusters? Is such a confidence
  statement taking into account the uncertainty
  about the true number of clusters?

18
Gene-specific normalization of the data
19
Clustering using non-normalized data
K-means
Euclidian Distance
Pearson's correlation
20
Clustering using normalized data
K-means
Euclidian Distance
Pearson's correlation
21
Why do we cluster?

• Guilt by association

Co-expression
Co-regulation
Functional relationship
Assigning function to genes
22
Why do we cluster - Functional Annotation?
23
Dissecting the gene expression regulatory
mechanisms
S.Tavazoie, J.D.Hughes, M.J.Campbell, R.J.Cho,
G.M.Church. Systematic determination of genetic
network architecture, Nat.Genet., 22, (1999)
281-285.