1. Comp602 Bioinformatics Algorithms -m werner 2011
DNA Microarrays
Comp602 Bioinformatics Algorithms
-m werner 2011

2. Comp602 Bioinformatics Algorithms
Cell Snapshot
What if you could capture what a cell is doing at any given instant?
You could compare diseased cells to normal ones
You could follow changes over the cell’s lifetime
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Comp602 Bioinformatics Algorithms

3. Expressed Proteins Provide a Snapshot
In humans every cell (almost) contains genetic coding for all human functions
i.e. It can express any human protein
But specialized cells only express proteins appropriate to their function
Further – they only express proteins as needed
Knowing which proteins are expressed by a cell at any instant reveals information on what kind of cell it is and what it is currently doing
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Comp602 Bioinformatics Algorithms

4. DNA Microarrays to Measure Protein Expression
Protein expression can be measured efficiently
Given a sample (cytoplasm, bloodstream, etc.)
Can measure mRNA present
A gene is:
On if the cell is currently transcribing its mRNA
Off otherwise
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Comp602 Bioinformatics Algorithms

5. Comp602 Bioinformatics Algorithms
Probes for mRNA
To see if a gene is on use a probe
The probe is the DNA complement of the mRNA we are checking
i.e. if the mRNA has the sequence AGGTATC the probe would have TCCATAG
So the mRNA would stick to the probe
You may not need the entire gene for the probe
A short oligonucleotide of 25 – 40 base pairs may suffice
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Comp602 Bioinformatics Algorithms

6. Comp602 Bioinformatics Algorithms
Probing
To test a sample for the expression of a particular protein
Paint a glass slide with lots of its probes (complements of its mRNA)
Add a dye to the sample
Bathe the slide in the sample and rinse off
If the protein is being expressed the slide glows with the dye color
The intensity of the glow indicates the level of the protein expression
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Comp602 Bioinformatics Algorithms

7. Comp602 Bioinformatics Algorithms
Parallel Probes
A DNA Microarray allows for simultaneous probes for multiple proteins
The slide is divided into small squares
The squares are printed (sometimes with inkjet technology) with the probes
A single slide can test for 1000 or more proteins
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Comp602 Bioinformatics Algorithms

8. Comp602 Bioinformatics Algorithms
Ink Jet Printing
Four cartridges are loaded with the four nucleotides: A, G, C,T
As the printer head moves across the array, the nucleotides are deposited in pixels where they are needed.
This way (many copies of) a base long oligonucleotide are deposited in each pixel.
By Steve Hookway lecture and Sorin Draghici’s book “Data Analysis Tools for DNA Microarrays”
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Comp602 Bioinformatics Algorithms

9. Inkjet Printed Microarrays
Inkjet head, squirting phosphor-ammodites
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Comp602 Bioinformatics Algorithms
From Zohar Yakhini

10. Thermal Ink Jet Arrays, by Agilent Technologies
In-Situ synthesized oligonucleotide array mers.
cDNA array,
Inkjet deposition
Comp602 Bioinformatics Algorithms
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11. Comp602 Bioinformatics Algorithms
Wash the slide with two samples labeled with green and red dye
Results:
Green – Sample 1 expressed
Red – Sample 2 expressed
Yellow – Samples 1 and 2 co-expressed
Flash Demo
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Comp602 Bioinformatics Algorithms

12. Comp602 Bioinformatics Algorithms
Two Channel Detection
Two-channel microarrays are hybridized with cDNA prepared from two samples to be compared (e.g. diseased tissue versus healthy tissue) and that are labeled with two different fluorophores, red and green.
@mw
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Comp602 Bioinformatics Algorithms

13. Comp602 Bioinformatics Algorithms
Log Ratios
For a particular protein (gene) we are interested in the ratio of expression between the malignant sample m and the benign sample b.
The ratio m/b is not symetric. i.e if malignant is more expressed the ratios go from 1 to ∞, but if benign is more expressed the ratios are squeezed between 0 and 1.
To equalize, take log2(b/n). Now, ratios >1 have logs ranging up from 0 and negative ones ranging down from 0.
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Comp602 Bioinformatics Algorithms

14. Using Microarrays (cont’d)
Green: expressed only from control
Red: expressed only from experimental cell
Yellow: equally expressed in both samples
Black: NOT expressed in either control or experimental cells
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Comp602 Bioinformatics Algorithms

15. Comp602 Bioinformatics Algorithms
Microarray Data
Microarray data are usually transformed into an intensity matrix (below)
The intensity matrix allows biologists to make correlations between different genes (even if they are
dissimilar) and to understand how genes functions might be related
Time:
Time X
Time Y
Time Z
Gene 1 10 8
Gene 2 9
Gene 3 4
8.6 3
Gene 4 7
Gene 5 1 2
Intensity (expression level) of gene at measured time
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Comp602 Bioinformatics Algorithms

16. Clustering of Microarray Data
Plot each datum as a point in N-dimensional space
Make a distance matrix for the distance between every two gene points in the N-dimensional space
Genes with a small distance share the same expression characteristics and might be functionally related or similar.
Clustering reveals groups of functionally related genes
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Comp602 Bioinformatics Algorithms

17. Clustering of Microarray Data (cont’d)
Clusters
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Comp602 Bioinformatics Algorithms

18. Homogeneity and Separation Principles
Homogeneity: Elements within a cluster are close to each other
Separation: Elements in different clusters are further apart from each other
…clustering is not an easy task!
Given these points a clustering algorithm might make two distinct clusters as follows
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Comp602 Bioinformatics Algorithms

19. Comp602 Bioinformatics Algorithms
Bad Clustering
This clustering violates both Homogeneity and Separation principles
Close distances from points in separate clusters
Far distances from points in the same cluster
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Comp602 Bioinformatics Algorithms

20. Comp602 Bioinformatics Algorithms
Good Clustering
This clustering satisfies both Homogeneity and Separation principles
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Comp602 Bioinformatics Algorithms

21. Clustering Techniques
Agglomerative: Start with every element in its own cluster, and iteratively join clusters together
Divisive: Start with one cluster and iteratively divide it into smaller clusters
Hierarchical: Organize elements into a tree, leaves represent genes and the length of the pathes between leaves represents the distances between genes. Similar genes lie within the same subtrees
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Comp602 Bioinformatics Algorithms

22. Hierarchical Clustering
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23. Hierarchical Clustering: Example
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24. Hierarchical Clustering: Example
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25. Hierarchical Clustering: Example
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26. Hierarchical Clustering: Example
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27. Hierarchical Clustering: Example
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28. Hierarchical Clustering (cont’d)
Hierarchical Clustering is often used to reveal evolutionary history
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Comp602 Bioinformatics Algorithms

29. Hierarchical Clustering Algorithm
Hierarchical Clustering (d , n)
Form n clusters each with one element
Construct a graph T by assigning one vertex to each cluster
while there is more than one cluster
Find the two closest clusters C1 and C2
Merge C1 and C2 into new cluster C with |C1| +|C2| elements
Compute distance from C to all other clusters
Add a new vertex C to T and connect to vertices C1 and C2
Remove rows and columns of d corresponding to C1 and C2
Add a row and column to d corrsponding to the new cluster C
return T
The algorithm takes a nxn distance matrix d of pairwise distances between points as an input.
4/17/2017
Comp602 Bioinformatics Algorithms

30. Hierarchical Clustering Algorithm
Hierarchical Clustering (d , n)
Form n clusters each with one element
Construct a graph T by assigning one vertex to each cluster
while there is more than one cluster
Find the two closest clusters C1 and C2
Merge C1 and C2 into new cluster C with |C1| +|C2| elements
Compute distance from C to all other clusters
Add a new vertex C to T and connect to vertices C1 and C2
Remove rows and columns of d corresponding to C1 and C2
Add a row and column to d corrsponding to the new cluster C
return T
Different ways to define distances between clusters may lead to different clusterings
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Comp602 Bioinformatics Algorithms

31. Hierarchical Clustering: Recomputing Distances
dmin(C, C*) = min d(x,y)
for all elements x in C and y in C*
Distance between two clusters is the smallest distance between any pair of their elements
davg(C, C*) = (1 / |C*||C|) ∑ d(x,y)
Distance between two clusters is the average distance between all pairs of their elements
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Comp602 Bioinformatics Algorithms

32. Squared Error Distortion
Given a data point v and a set of points X,
define the distance from v to X
d(v, X)
as the (Euclidian) distance from v to the closest point from X.
Given a set of n data points V={v1…vn} and a set of k points X,
define the Squared Error Distortion
d(V,X) = ∑d(vi, X)2 / n < i < n
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Comp602 Bioinformatics Algorithms

33. Comp602 Bioinformatics Algorithms
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Comp602 Bioinformatics Algorithms

34. Comp602 Bioinformatics Algorithms
Clustering a Heat Map
Each sample (column) is a point in n-space where n is the number of proteins tested (rows)
On the top samples are clustered hierarchically into a dendrogram according to some distance measure. Columns have been swapped to show close samples next to each other.
On the right proteins are clustered to show how closely they are co-expressed in the various samples. Rows have been swapped.
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Comp602 Bioinformatics Algorithms

35. Comp602 Bioinformatics Algorithms
A: Hierarchical cluster analysis of gene expression patterns of normal and cancerous prostate specimens. The dendrogram indicates relationships among samples based on gene expression profiles. Cluster analysis is seen to distinguish malignant from normal prostate (pink branches). Cluster analysis also defines three subtypes of prostate cancer (numbered above) not distinguishable histologically.
A Perspective on DNA Microarrays in Pathology Research and Practice, Jonathan R. Pollack
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Comp602 Bioinformatics Algorithms

36. Comp602 Bioinformatics Algorithms
A Data Mining Problem
On a given microarray, we test on the order of 10k elements in one time
Number of microarrays used in typical
experiment is no more than 100.
Insufficient sampling.
Data is obtained faster than it can be processed.
High noise.
Algorithmic approaches to work through this large data set and make sense of the data are desired.
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

37. Comp602 Bioinformatics Algorithms
A Data Mining Problem
On a given Microarray we test on the order of 10k elements at a time
Data is obtained faster than it can be processed
We need some ways to work through this large data set and make sense of the data
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
4/17/2017
Comp602 Bioinformatics Algorithms

38. Grouping and Reduction
Grouping: discovers patterns in the data from a microarray
Reduction: reduces the complexity of data by removing redundant probes (genes) that will be used in subsequent assays
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

39. Unsupervised Grouping: Clustering
Pattern discovery via grouping similarly expressed genes together
Three techniques most often used
k-Means Clustering
Hierarchical Clustering
Kohonen Self Organizing Feature Maps
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

40. Clustering Limitations
Any data can be clustered, therefore we must be careful what conclusions we draw from our results
Clustering is non-deterministic and can and will produce different results on different runs
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
4/17/2017
Comp602 Bioinformatics Algorithms

41. Comp602 Bioinformatics Algorithms
K-means Clustering
Given a set of n data points in d-dimensional space and an integer k
We want to find the set of k points in d-dimensional space that minimizes the mean squared distance from each data point to its nearest center
No exact polynomial-time algorithms are known for this problem
“A Local Search Approximation Algorithm for k-Means Clustering” by Kanungo et. al
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42. K-means Algorithm (Lloyd’s Algorithm)
Has been shown to converge to a locally optimal solution
But can converge to a solution arbitrarily bad compared to the optimal solution
Data Points
Optimal Centers
Heuristic Centers
K=3
“K-means-type algorithms: A generalized convergence theorem and characterization of local optimality” by Selim and Ismail
“A Local Search Approximation Algorithm for k-Means Clustering” by Kanungo et al.
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43. Distance Measures Between Two Datapoints in n-Space
Euclidean Distance
Cosine Coefficient – Compare 2 vectors for direction using the dot product:
x∙y = ∑xi∙yi = |x||y|cos(Ѳ )
Manhattan Distance
Pearson Correlation Coefficient
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44. Pearson Correlation Coefficient, r. values in [-1,1] interval
Gene expression over d experiments is a vector in Rd, e.g. for gene C: (0, 3, 3.58, 4, 3.58, 3)
Given two vectors X and Y that contain N elements, we calculate r as follows:
Cho & Won, 2003
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Comp602 Bioinformatics Algorithms

45. Comp602 Bioinformatics Algorithms
Euclidean Distance
Now to find the distance between two points, say the origin and the point (3,4):
Simple and Fast! Remember this when we consider the complexity!
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
4/17/2017
Comp602 Bioinformatics Algorithms

46. Comp602 Bioinformatics Algorithms
Finding a Centroid
We use the following equation to find the n dimensional centroid point amid k n dimensional points:
Let’s find the midpoint between 3 2D points, say: (2,4) (5,2) (8,9)
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
4/17/2017
Comp602 Bioinformatics Algorithms

47. Comp602 Bioinformatics Algorithms
K-means Algorithm
Choose k initial center points randomly
Cluster data using Euclidean distance (or other distance metric)
Calculate new center points for each cluster using only points within the cluster
Re-Cluster all data using the new center points
This step could cause data points to be placed in a different cluster
Repeat steps 3 & 4 until the center points have moved such that in step 4 no data points are moved from one cluster to another or some other convergence criteria is met
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From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
Comp602 Bioinformatics Algorithms

48. Comp602 Bioinformatics Algorithms
An example with k=2
We Pick k=2 centers at random
We cluster our data around these center points
Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

49. K-means example with k=2
We recalculate centers based on our current clusters
Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

50. K-means example with k=2
We re-cluster our data around our new center points
Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

51. K-means example with k=2
We repeat the last two steps until no more data points are moved into a different cluster
Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

52. Comp602 Bioinformatics Algorithms
Choosing k
Use another clustering method
Run algorithm on data with several different values of k
Use advance knowledge about the characteristics of your test
Cancerous vs Non-Cancerous
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

53. Comp602 Bioinformatics Algorithms
Cluster Quality
Since any data can be clustered, how do we know our clusters are meaningful?
The size (diameter) of the cluster vs. The inter-cluster distance
Distance between the members of a cluster and the cluster’s center
Diameter of the smallest sphere
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
4/17/2017
Comp602 Bioinformatics Algorithms

54. Cluster Quality Continued
distance=5
size=5
distance=20
Quality of cluster assessed by ratio of distance to nearest cluster and cluster diameter
size=5
Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
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Comp602 Bioinformatics Algorithms

55. Cluster Quality Continued
Quality can be assessed simply by looking at the diameter of a cluster
A cluster can be formed even when there is no similarity between clustered patterns. This occurs because the algorithm forces k clusters to be created.
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
4/17/2017
Comp602 Bioinformatics Algorithms

56. Characteristics of k-means Clustering
The random selection of initial center points creates the following properties
Non-Determinism
May produce clusters without patterns
One solution is to choose the centers randomly from existing patterns
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
4/17/2017
Comp602 Bioinformatics Algorithms

57. Squared Error Distortion
Given a data point v and a set of points X,
define the distance from v to X
d(v, X)
as the (Euclidian) distance from v to the closest point from X.
Evaluate the error induced by mapping each data point to one of the k-means centroids
Given a set of n data points V={v1…vn} and a set of k centroids X={c1…ck}, define the Squared Error Distortion
d(V,X) = ∑d(vi, X)2 / n < i < n

58. Comp602 Bioinformatics Algorithms
Algorithm Complexity
Linear in the number of data points, N
Can be shown to have time of cN
c does not depend on N, but rather the number of clusters, k
Low computational complexity
High speed
From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici
4/17/2017
Comp602 Bioinformatics Algorithms

59. K-Means Clustering: Lloyd Algorithm
Arbitrarily assign the k cluster centers
while the cluster centers keep changing
Assign each data point to the cluster Ci corresponding to the closest cluster representative (center) (1 ≤ i ≤ k)
After the assignment of all data points, compute new cluster representatives according to the center of gravity of each cluster, that is, the new cluster representative is
∑v / |C| for all v in C for every cluster C
*This may lead to merely a locally optimal clustering.

60. x1 x2 x3

61. x1 x2 x3

62. x1 x3 x2

63. x1 x2 x3

64. Conservative K-Means Algorithm
Lloyd algorithm is fast but in each iteration it moves many data points, not necessarily causing better convergence.
A more conservative method would be to move one point at a time only if it improves the overall clustering cost
The smaller the clustering cost of a partition of data points is the better that clustering is
Different methods (e.g., the squared error distortion) can be used to measure this clustering cost

65. K-Means “Greedy” Algorithm
ProgressiveGreedyK-Means(k)
Select an arbitrary partition P into k clusters
while forever
bestChange  0
for every cluster C
for every element i not in C
if moving i to cluster C reduces its clustering cost
if (cost(P) – cost(Pi  C) > bestChange
bestChange  cost(P) – cost(Pi  C)
i*  I
C*  C
if bestChange > 0
Change partition P by moving i* to C*
else
return P