Cluster Analysis of Microarray Gene Expression

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Cluster Analysis of Microarray Gene Expression
CSCI5980 Functional Genomics, Systems Biology
and Bioinformtics

• Rui Kuang and Chad Myers
• Department of Computer Science and Engineering
• University of Minnesota
• kuang_at_cs.umn.edu

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Hierarchical Clustering

• Organize the genes in a structure of a
  hierarchical tree
• Initial step each gene is regarded as a cluster
  with one item
• Find the 2 most similar clusters and merge them
  into a common node
• The length of the branch is proportional to the
  distance
• Iterate on merging nodes until all genes are
  contained in one cluster- the root of the tree.

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K-MEANS

• The user sets the number of clusters- k
• Initialization each gene is randomly assigned to
  one of the k clusters
• Average expression vector is calculated for each
  cluster (clusters profile)
• Iterate over the genes
• For each gene- compute its similarity to the
  cluster profiles.
• Move the gene to the cluster it is most similar
  to.
• Recalculated cluster profiles.
• Stop criteria further shuffling of genes results
  in minor improvement in the clustering score

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How Many Clusters?

• Try several parameters and compare the clustering
  solutions
• Mathematical criteria for comparison of
  clustering solutions later in the presentation
• PCA (Principle Component Analysis)
• A technique for projecting the gene expression
  data set onto a reduced (2 or 3 dimensional)
  easily visualized space

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PCA - Example

• Dataset Thousands of genes probed in 5
  conditions (time points relative to treatment)
• The expression profile of each gene is presented
  by the vector of its expression levels X (X1,
  X2, X3, X4, X5)
• Imagine each gene X as a point in a 5-dimentional
  space.
• Each direction/axis corresponds to a specific
  condition
• Genes with similar profiles are close to each
  other in this space
• PCA- Project this dataset to 2 dimensions,
  preserving as much information as possible

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PCA Example
Visual estimation of the number of clusters in
the data
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K-MEANS example 4 clusters
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Cluster 1
Cluster 3
Mis-classified
Cluster 4
Cluster 2
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K-means example 3 clusters
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Too few clusters K2
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Convergence of K-MEANS

• Define goodness measure of cluster k as sum of
  squared distances from cluster centroid.
• Reassignment monotonically decreases G, since
  each vector is assigned to the closest centroid.
• Stop criteria, e.g.,
• A fixed number of iterations.
• Gene partition unchanged.
• Centroid positions dont change.

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Time Complexity

• Computing distance between two genes is O(m),
  where m is the number of experiments.
• Reassigning clusters O(Kn) distance
  computations, or O(Knm).
• Computing centroids Each doc gets added once to
  some centroid O(nm).
• Assume these two steps are each done once for I
  iterations O(IKnm).

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K-MEANS Example

• Tavazoie et al., Nature 1999
• Cell cycles of S. cerevisiae
• Measurements of 6,220 ORFs at 15 time points
• K-means clustering discovers 30 clusters of 3,000
  genes
• Interpretation with MIPS protein functions and
  common transcription factor binding motifs

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K-MEANS Example
MluI cell-cycle box
Swi 4/6 cell-cycle box
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K-means

• Advantages
• Simple and fast!
• Disadvantages
• The number of gene clusters is unknown
• No structure between the clusters
• Not robust to outliers centroid can be greatly
  affected by outliers

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SOMs (Self-Organizing Maps)

• User sets the number of clusters in a form of a
  rectangular grid (e.g., 3x2) map nodes
• Imagine genes as points in (M-dimensional) space
• Initialization map nodes (corresponding to
  clusters) are randomly placed in the data space

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Genes data points
Clusters map nodes
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SOM - Scheme

• Randomly choose a data point (gene).
• Find its closest map node
• Move this map node towards the data point
• Move the neighbor map nodes towards this point,
  but to lesser extent
• Iterate over data points

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• The extent of node displacements is relaxed with
  the iteration number
• After thousands of iterations
• Assign each gene to the map node (cluster) it is
  most similar to

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SOM Example

• Tamayo et al., PNAS 1999
• Cell cycles of S. cerevisiae dataset and human
  hematopoietic differentiation.

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SOM Example

• Two Cell cycles of S. cerevisiae
• 6 X 5 SOM

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How to evaluate clustering solution

• S and T are two clustering solution.
• n11 of pairs that are mates in both S and T
• n10 of pairs that are mates only in S
• n01 of pairs that are mates only in T
• Jaccard coefficient correctly identified mates
  vs all mates in S and T
• Minkowski coefficient disagreements vs true
  mates in S