DNA Chips and Their Analysis Comp. Genomics: Lectures based on many sources, primarily Zohar Yakhini

DNA Microarras: Basics What are they. Types of arrays (cDNA arrays, oligo arrays). What is measured using DNA microarrays. How are the measurements done?

DNA Microarras: Computational Questions Design of arrays. Techniques for analyzing experiments. Detecting differential expression. Similar expression: Clustering. Other analysis techniques (mmmmmany). Machine learning techniques, and applications for advanced diagnosis.

What is a DNA Microarray (I) A surface (nylon, glass, or plastic). Containing hundreds to thousand pixels. Each pixel has copies of a sequence of single stranded DNA (ssDNA). Each such sequence is called a probe.

What is a DNA Microarray (II) An experiment with k elements. Way to concurrently explore the function of multiple genes. A snapshot of the expression level of k genes under given test conditions

Some Microarray Terminology Probe: ssDNA printed on the solid substrate (nylon or glass). These are short substrings of the genes we are going to be testing Target: cDNA which has been labeled and is to be washed over the probe

Back to Basics: Watson and Crick James Watson and Francis Crick discovered, in 1953, the double helix structure of DNA. From Zohar Yakhini

Watson-Crick Complimentarity A binds to T C binds to G AATGCTTAGTC TTACGAATCAG Perfect match AATGCGTAGTC TTACGAATCAG One-base mismatch From Zohar Yakhini

Array Based Hybridization Assays (DNA Chips) Unknown sequence or mixture (target). Many copies. Array of probes Thousands to millions of different probe sequences per array. From Zohar Yakhini

Array Based Hyb Assays Target hybs to WC complimentary probes only Therefore – the fluorescence pattern is indicative of the target sequence. From Zohar Yakhini

DNA Sequencing Sanger Method Generate all A,C,G,T – terminated prefixes of the sequence, by a polymerase reaction with terminating corresponding bases. Run in four different gel lanes. Reconstruct sequence from the information on the lengths of all A,C,G,T – terminated prefixes. The need for 4 different reactions is avoided by using differentially dye labeled terminating bases. From Zohar Yakhini

Central Dogma of Molecular Biology (reminder) Transcription mRNA Cells express different subset of the genes in different tissues and under different conditions Gene (DNA) Translation Protein From Zohar Yakhini

Expression Profiling on MicroArrays Differentially label the query sample and the control (1-3). Mix and hybridize to an array. Analyze the image to obtain expression levels information. From Zohar Yakhini

Microarray: 2 Types of Fabrication 1.cDNA Arrays: Deposition of DNA fragments –Deposition of PCR-amplified cDNA clones –Printing of already synthesized oligonucleotieds 2.Oligo Arrays: In Situ synthesis –Photolithography –Ink Jet Printing –Electrochemical Synthesis By Steve Hookway lecture and Sorin Draghici’s book “Data Analysis Tools for DNA Microarrays”

cDNA Microarrays vs. Oligonucleotide Probes and Cost cDNA ArraysOligonucleotide Arrays Long Sequences Spot Unknown Sequences More variability Arrays cheaper Short Sequences Spot Known Sequences More reliable data Arrays typically more expensive By Steve Hookway lecture and Sorin Draghici’s book “Data Analysis Tools for DNA Microarrays”

Photolithography (Affymetrix) Similar to process used to generate VLSI circuits Photolithographic masks are used to add each base If base is present, there will be a “hole” in the corresponding mask Can create high density arrays, but sequence length is limited From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici Photodeprotection mask C

Photolithography (Affymetrix) From Zohar Yakhini

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 oligo is deposited in each pixel. By Steve Hookway lecture and Sorin Draghici’s book “Data Analysis Tools for DNA Microarrays”

AG TC … Ink Jet Printing (Agilent) The array is a stack of images in the colors A, C, G, T. From Zohar Yakhini

Inkjet Printed Microarrays Inkjet head, squirting phosphor-ammodites From Zohar Yakhini

Electrochemical Synthesis Electrodes are embedded in the substrate to manage individual reaction sites Electrodes are activated in necessary positions in a predetermined sequence that allows the sequences to be constructed base by base Solutions containing specific bases are washed over the substrate while the electrodes are activated From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

Preparation of Samples Use oligo(dT) on a separation column to extract mRNA from total cell populations. Use olig(dT) initiated polymerase to reverse transcribe RNA into fluorescence labeled cDNA. RNA is unstable because of environment RNA- digesting enzymes. Alternatively – use random priming for this purpose, generating a population of transcript subsequences From Zohar Yakhini

Expression Profiling on MicroArrays Differentially label the query sample and the control (1-3). Mix and hybridize to an array. Analyze the image to obtain expression levels information. From Zohar Yakhini

Expression Profiling: a FLASH Demo URL:

Expression Profiling – Probe Design Issues Probe specificity and sensitivity. Special designs for splice variations or other custom purposes. Flat thermodynamics. Generic and universal systems From Zohar Yakhini

Hybridization Probes Sensitivity: Strong interaction between the probe and its intended target, under the assay's conditions. How much target is needed for the reaction to be detectable or quantifiable? Specificity: No potential cross hybridization. From Zohar Yakhini

Specificity Symbolic specificity Statistical protection in the unknown part of the genome. Methods, software and application in collaboration with Peter Webb, Doron Lipson. From Zohar Yakhini

Reading Results: Color Coding Numeric tables are difficult to read Data is presented with a color scale Coding scheme: –Green = repressed (less mRNA) gene in experiment –Red = induced (more mRNA) gene in experiment –Black = no change (1:1 ratio) Or –Green = control condition (e.g. aerobic) –Red = experimental condition (e.g. anaerobic) We usually use ratio Campbell & Heyer, 2003

cDNA array, Inkjet deposition In-Situ synthesized oligonucleotide array mers. Thermal Ink Jet Arrays, by Agilent Technologies

Application of Microarrays We only know the function of just about 30% of the 30,000 genes in the Human Genome –Gene exploration –Functional Genomics DNA microarrays are just the first among many high throughput genomic devices (first used approx. 1996) By Steve Hookway lecture and Sorin Draghici’s book “Data Analysis Tools for DNA Microarrays”

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.

Informative Genes in a Two Classes Experiment Differentially expressed in the two classes. Identifying (statistically significant) informative genes - Provides biological insight - Indicate promising research directions - Reduce data dimensionality - Diagnostic assay From Zohar Yakhini

Expression pattern and pathological diagnosis information (annotation), for a single gene a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 Permute the annotation by sorting the expression pattern (ascending, say). Informative genes etc Non-informative genes etc Scoring Genes From Zohar Yakhini

Separation Score Compute a Gaussian fit for each class  (  1,  1 ), (  2,  2 ). The Separation Score is (  1 -  2 )/(  1 +  2 )

Threshold Error Rate (TNoM) Score Find the threshold that best separates tumors from normals, count the number of errors committed there # of errors = min(7,8) = 7. Not informative 6 7 Ex 1: Ex 2: A perfect single gene classifier gets a score of 0. Very informative From Zohar Yakhini

p-Values Relevance scores are more useful when we can compute their significance: –p-value: The probability of finding a gene with a given score if the labeling is random p-Values allow for higher level statistical assessment of data quality. p-Values provide a uniform platform for comparing relevance, across data sets. p-Values enable class discovery From Zohar Yakhini

BRCA1 Wildtype Genes over-expressed in BRCA1 mutants Genes under-expressed in BRCA1 mutants BRCA1 mutants Sporadic sample s14321 With BRCA1-mutant expression profile BRCA1 Differential Expression Collab with NIH NEJM 2001 From Zohar Yakhini

Data Analysis: Leave One Out Cross Validation (LOOCV) Repeat, for each tissue (tumor/normal) “Hide” the label of the test tissue Diagnose the test tissue based on the remaining data Compare the diagnosis to the hidden label Perform this using different choices of genes subsets sizes Small, efficient diagnostic assays From Zohar Yakhini

BRCA1 LOOCV Results 95% success rate (21/22) Sporadic tissue (14321) consistently classified as BRCA1 BRCA1 gene is normal, but silenced in the patient’s DNA From Zohar Yakhini

Lung Cancer Informative Genes Data from Naftali Kaminski’s lab, at Sheba. 24 tumors (various types and origins) 10 normals (normal edges and normal lung pools) From Zohar Yakhini

And Now: Global Analysis of Gene Expression Data Most common tasks: 1.Construct gene network from experiments. 2.Cluster - either genes, or experiments

And Now: Global Analysis of Gene Expression Data Most common tasks: 1.Construct gene network from experiments. 2.Cluster - either genes, or experiments

And Now: Global Analysis of Gene Expression Data Most common tasks: 1.Construct gene network from experiments. 2.Cluster - either genes, or experiments

Pearson Correlation Coefficient, r. Values are in [-1,1] interval Gene expression over d experiments is a vector in R d, 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

Intuition for Pearson Correlation Coefficient r(v1,v2) close to 1: v1, v2 highly correlated. r(v1,v2) close to -1: v1, v2 anti correlated. r(v1,v2) close to 0: v1, v2 not correlated.

Pearson Correlation and p-Values When entries in v1,v2 are distributed according to normal distribution, can assign (and efficiently compute) p-Values for a given result. These p-Values are determined by the Pearson correlation coefficient, r, and the dimension, d, of the vectors. For same r, vectors of higher dimension will be assigned more significant (smaller) p-Value.

Replace each entry x i by its rank in vector x. Then compute Pearson correlation coefficients of rank vectors. Example: X = Gene C = (0, 3.00, 3.41, 4, 3.58, 3.01) Y = Gene D = (0, 1.51, 2.00, 2.32, 1.58, 1) Ranks(X)= (1,2,4,6,5,3) Ranks(Y)= (1,3,5,6,4,2) Ties should be taken care of, but: (1) rare (2) can randomize (small effect) Spearman Rank Order Coefficient (a close relative of Pearson, non parametric)

From Pearson Correlation Coefficients to a Gene Network Compute correlation coefficient for all pairs of genes (what about missing data?) Choose p-Value threshold. Put an edge between gene i and gene j iff p-Value exceeds threshold.

Things May Get Messy What to do with significant yet negative correlation coefficients? Usually care only about the p-value and put a “normal edge” Cases composed of multiple experiments where distribution is far from normal.

Things Do Get Messy

What to Do when Things Get Messy?

What to do when things Get Messy

1)Create a single vector of all experiments per gene. Compute correlations based on these vectors. This is the common approach. Disadvantage: Outcome is dominated by the larger experiments.

What to do when things Get Messy 2) For each edge, count the no. of experiments where it appears significantly. Take edges exceeding some threshold. Disadvantage: Outcome is somewhat dominated by experiments with many significant correlations.

What to do when things Get Messy 3) For each edge, make a weighted count the of experiments where it appears significantly. Weights are higher if experiment has few significant correlations. Take edges exceeding some threshold. Disadvantage: No solid mathematical justification.

Public microarray data sets Pearson Correlation Genes Samples Genes Pair-wise gene co- expression matrices Gene pair score - a gene pair n - number of datasets x k i,j - 1 if g i and g j are significantly correlated in dataset k, 0 otherwise p k - proportion of significantly correlated gene pairs in dataset k Summary of the procedure Network of conserved co-expression links Nodes represent genes Edges represent highly correlated expressions Cluster Detection Highly inter-connected clusters

The Outcome (Whole Network)

Node Score Cutoff Node Score Cutoff Ribosome-relatedChloroplast-related ER and mitochondrion- related A B Node Score Cutoff Ribosome-related Chloroplast and Ribosome- related Chloroplast-related Chloroplast and ER- related * (1) * (2) * (3)* (4) + Outcome after Clustering

But what is Clustering?

Example data: fold change (ratios) Name0 hours2 hours4 hours6 hours8 hours10 hours Gene C Gene D Gene E Gene F Gene G Gene H Gene I Gene J Gene K Gene L Gene M Gene N Campbell & Heyer, 2003

Example data 2 Name0 hours2 hours4 hours6 hours8 hours10 hours Gene C Gene D Gene E Gene F Gene G Gene H Gene I02320 Gene J Gene K Gene L Gene M Gene N Campbell & Heyer, 2003

Example data: Pearson correlation coefficients Gene CGene DGene EGene FGene GGene HGene IGene JGene KGene LGene MGene N Gene C Gene D Gene E Gene F Gene G Gene H Gene I Gene J Gene K Gene L Gene M Gene N Campbell & Heyer, 2003

Example: Reorganization of data Campbell & Heyer, 2003 Name0 hours2 hours4 hours6 hours8 hours10 hours Gene M Gene N Gene H Gene K Gene J Gene E Gene C Gene L Gene G Gene D Gene I Gene F

Grouping and Reduction Grouping: Partition items into groups. Items in same group should be similar. Items in different groups should be dissimilar. Grouping may help discover patterns in the data. Reduction: reduce the complexity of data by removing redundant probes (genes).

Unsupervised Grouping: Clustering Pattern discovery via clustering similarly entities together Techniques most often used: k-Means Clustering Hierarchical Clustering Biclustering Alternative Methods: Self Organizing Maps (SOMS), plaid models, singular value decomposition (SVD), order preserving submatrices (OPSM),……

Clustering Overview Different similarity measures in use: –Pearson Correlation Coefficient –Cosine Coefficient –Euclidean Distance –Information Gain –Mutual Information –Signal to noise ratio –Simple Matching for Nominal – –

Clustering Overview (cont.) Different Clustering Methods –Unsupervised k-means Clustering (k nearest neighbors) Hierarchical Clustering Self-organizing map –Supervised Support vector machine Ensemble classifier  Data Mining

Clustering Limitations Any data can be clustered, therefore we must be careful what conclusions we draw from our results Clustering is often randomized. It can, and will, produce different results for different runs on same data

k-means Clustering Given a set of m data points in d-dimensional space and an integer k. We want to find the set of k “centers” in d-dimensional space that minimizes the Euclidean (mean square) distance from each data point to its nearest center. No exact polynomial-time algorithms are known for this problem (no wonder, NP-hard!). “A Local Search Approximation Algorithm for k-Means Clustering” by Kanungo et. al

K-means Heuristic (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 “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. K=3 Data Points Optimal Centers Heuristic Centers

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!

Finding a Centroid We use the following equation to find the n dimensional centroid point (center of mass) amid k (n dimensional) points: Example: The midpoint between three 2D points, say: (2,4) (5,2) (8,9)

K-means Iterative Heuristic 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 some data points to be placed in a different cluster) Repeat steps 3 & 4 until no data points are moved from one cluster to another (stabilization), or till some other convergence criteria is met From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

An example with 2 clusters 1.We Pick 2 centers at random 2.We cluster our data around these center points Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

K-means example with k=2 3.We recalculate centers based on our current clusters Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

K-means example with k=2 4.We re-cluster our data around our new center points Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

k-means example with k=2 5. 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

Choosing k Run algorithm on data with several different values of k Use prior knowledge about the characteristics of your test (e.g. cancerous vs non-cancerous tissues, in case it is the experiments that are being clustered)

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 containing the cluster From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

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

Cluster Quality Continued Quality can be assessed simply by looking at the diameter of a cluster (alone????) Warning: A cluster can be formed by the heuristic 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

Properties of k-means Clustering The random selection of initial center points implies the following properties –Non-Determinism / Randomized –May produce incoherent clusters One solution is to choose the centers randomly from existing points From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

Heuristic’s Complexity Linear in the number of data points, N Can be shown to have run time cN, where c does not depend on N, but rather the number of clusters, k (not sure about dependence on dimension, d?)  efficient From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

Hierarchical Clustering -a different clustering paradigm Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

Hierarchical Clustering (cont.) Gene CGene DGene EGene FGene GGene HGene IGene JGene KGene LGene MGene N Gene C Gene D Gene E Gene F Gene G Gene H Gene I Gene J Gene K Gene L Gene M0.94 Gene N Campbell & Heyer, 2003

Hierarchical Clustering (cont.) F C G D E Gene CGene DGene EGene FGene G Gene C Gene D Gene E Gene F-0.35 Gene G CE 1 1Gene DGene FGene G Gene D Gene F-0.35 Gene G Average “similarity” to Gene D: ( )/2 = 0.89 Gene F: (-0.40+(-0.57))/2 = Gene G: ( )/2 =

Hierarchical Clustering (cont.) F G D CE 1 1Gene DGene FGene G Gene D Gene F-0.35 Gene G GD 2

Hierarchical Clustering (cont.) F CE 1 GD 2 12Gene F Gene F 3

Hierarchical Clustering (cont.) F CE 1 GD 2 3 3Gene F Gene F 4 F

Hierarchical Clustering (cont.) F CE 1 GD algorithm looks familiar?

Clustering of entire yeast genome Campbell & Heyer, 2003

Hierarchical Clustering: Yeast Gene Expression Data Eisen et al., 1998

A SOFM Example With Yeast “Interpresting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation” by Tamayo et al.

SOM Description Each unit of the SOM has a weighted connection to all inputs As the algorithm progresses, neighboring units are grouped by similarity Input Layer Output Layer From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

An Example Using Color Each color in the map is associated with a weight From

Cluster Analysis of Microarray Expression Data Matrices Application of cluster analysis techniques in the elucidation gene expression data. Important paradigm: Guilt by association

The features of a living organism are governed principally by its genes. If we want to fully understand living systems we must know the function of each gene. Once we know a gene’s sequence we can design experiments to find its function: However this approach is too slow to handle all the gene sequence information we have today (HGSP). Function of Genes Delete Gene X Gene X The Classical Approach of Assigning a function to a Gene Conclusion: Gene X = left eye gene. ("זבוב בלי רגליים – חרש")

Microarray Analysis Microarray analysis allows the monitoring of the activities of many genes over many different conditions. Experiments are carried out on a Physical Matrix like the one below: To facilitate computational analysis the physical matrix which may contain 1000’s of gene’s is converted into a numerical matrix using image analysis equipment. G1 G2 G3 G4 G5 G6 G7 G6 G7 C1 C2 C3 C4 C5 C6 C7Low Zero High Conditions Genes Possible inference: If Gene X’s activity (expression) is affected by Condition Y (Extreme Heat), then Gene X may be involved in protecting the cellular components from extreme heat. Each Gene has its corresponding Expression Profile for a set of conditions. This Expression Profile may be thought of as a feature profile for that gene for that set of conditions (A condition feature profile).

Cluster Analysis Cluster Analysis is an unsupervised procedure which involves grouping of objects based on their similarity in feature space. In the Gene Expression context Genes are grouped based on the similarity of their Condition feature profile. Cluster analysis was first applied to Gene Expression data from Brewer’s Yeast (Saccharomyces cerevisiae) by Eisen et al. (1998). Two general conclusions can be drawn from these clusters: Genes clustered together may be related within a biological module/system. If there are genes of known function within a cluster these may help to class this biological/module system. X Y A B C Z Clusters A,B and C represent groups of related genes. Clustering Conditions Genes

From Data to Biological Hypothesis System C Cluster C with four Genes may represent System C Relating these genes aids in elucidation of this System C Gene Expression Microarray Cluster Set Conditions (A-Z) Gene 1 Gene 2 Gene 3 Gene 4 Gene 5 Gene 6 Gene 7 X Y A B C External Stimulus( Condition X) Regulator Protein Toxin DNA Gene a Gene b Gene c Gene d Gene Expression Toxin Pump Cell Membrane

Some Drawbacks of Clustering Biological Data 1.Clustering works well over small numbers of conditions but a typical Microarray may have hundreds of experimental conditions. A global clustering may not offer sufficient resolution with so many features. 2.As with other clustering applications, it may be difficult to cluster noisy expression data. 3.Biological Systems tend to be inter-related and may share numerous factors (Genes) – Clustering enforces partitions which may not accurately represent these intimacies. 4.Clustering Genes over all Conditions only finds the strongest signals in the dataset as a whole. More ‘local’ signals within the data matrix may be missed. X Y A B C Z Inter-related(3) Local Signals(4) May represent more complex system such as:

How do we better model more complex systems? One technique that allows detection of all signals in the data is biclustering. Instead of clustering genes over all conditions biclustering clusters genes with respect to subsets of conditions. -interrelated clusters (genes may belong more than one bicluster). -local signals (genes correlated over only a few conditions). -noisy data (allows erratic genes to belong to no cluster). This enables better representation of:

Biclustering Technique first described by J.A. Hartigan in 1972 and termed ‘Direct Clustering’. First Introduced to Microarray expression data by Cheng and Church(2000) Gene 1 Gene 2 Gene 3 Gene 4 Gene 5 Gene 6 Gene 7 Gene 8 Gene 9 A B C D E F G H Gene 1 Gene 4 Gene 6 Gene 7 Gene 9 B E F Biclustering (of genes AND conditions) A B D E F G H Gene 1 Gene 4 Gene 9 Clustering misses local signal {(B,E,F),(1,4,6,7,9)} present over subset of conditions. Gene 1 Gene 4 Gene 9 A B C D E F G H Clustering (of genes) Biclustering discovers local coherences over a subset of conditions Conditions

Approaches to Biclustering Microarray Gene Expression First applied to Gene Expression Data by Cheng and Church(2000). –Used a sub-matrix scoring technique to locate biclusters. Tanay et al.(2000) –Modelled the expression data on Bipartite graphs and used graph techniques to find ‘complete graphs’ or biclusters. Lazzeroni and Owen –Used matrix reordering to represent different ‘layers’ of signals (biclusters) ‘Plaid Models’ to represent multiple signals within data. Ben-Dor et al. (2002) –“Biclusters” depending on order relations (OPSM).

Bipartite Graph Modelling First proposed in: “Discovering statically significant biclusters in gene expressing data” Tanay et al. Bioinformatics 2000 Within the graph modelling paradigm biclusters are equivalent to complete bipartite sub-graphs. Tanay and colleagues used probabilistic models to determine the least probable sub-graphs (those showing most order and consequently most surprising) to identify biclusters.

The Cheng and Church Approach The core element in this approach is the development of a scoring to prioritise sub-matrices. This scoring is based on the concept of the residue of an entry in a matrix. In the Matrix (I,J) the residue score of element is given by: a i j I J In words, the residue of an entry is the value of the entry minus the row average, minus the column average, plus the average value in the matrix. This score gives an idea of how the value fits into the data in the surrounding matrix.

The mean squared residue score (H) for a matrix (I,J) is then calculated : This Global H score gives an indication of how the data fits together within that matrix- whether it has some coherence or is random. The Cheng and Church Approach(2) A low H score means that there is a correlation in the matrix - a score of H(I,J)= 0 would mean that the data in the matrix fluctuates in unison i.e. the sub-matrix is a bicluster A high H value signifies that the data is uncorrelated. - a matrix of equally spread random values over the range [a,b], has an expected H score of (b-a)/12. range = [0,800] then H(I,J) = 53,333

Worked example of H score: R(1) = = 0.1 R(2) = = 0.1:: R(12) = = 0.1 Col Avg Row Avg Matrix (M) Avg. = 6.5 H (M) = (0.01x12)/12 = 0.01 If 5 was replaced with 3 then the score would changed to: H(M2) = 2.06 If the matrix was reshuffled randomly the score would be around: H(M3) = sqr(12-1)/12 = 10.08

In order to find all possible biclusters in an Expression Matrix all sub- matrices must be tested using the H score. The Cheng and Church Approach: Node Deletion Biclustering Algorithm In a node deletion algorithm all columns and rows are tested for deletion. If removing a row or column decreases the H score of the Matrix than it is removed. This continues until it is not possible to decrease the H score further. This low H score coherent sub-matrix (bicluster) is then returned. The process then masks this located bicluster by inserting random numbers in place of it. And reiterates the process. R R R R Node Deletion

The Cheng and Church Approach: No. of genes, no. of conditions 4, 9610, 2911, , 25127, 1313, 21 10, 572, 9625, 12 9, 513, 962, 96 Some results on lymphoma data (4026  96):

Conclusions: High throughput Functional Genomics (Microarrays) requires Data Mining Applications. Biclustering resolves Expression Data more effectively than single dimensional Cluster Analysis. Future Research/Question’s: Implement a simple H score program to facilitate study if H score concept. Are there other alternative scorings which would better apply to gene expression data? Do un-biclustered genes have any significance? Horizontally transferred genes? Implement full scale biclustering program and look at better adaptation to expression data sets and the biological context.

References Basic microarray analysis: grouping and feature reduction by Soumya Raychaudhuri, Patrick D. Sutphin, Jeffery T. Chang and Russ B. Altman; Trends in Biotechnology Vol. 19 No. 5 May 2001 Self Organizing Maps, Tom Germano, “Data Analysis Tools for DNA Microarrays” by Sorin Draghici; Chapman & Hall/CRC 2003 Self-Organizing-Feature-Maps versus Statistical Clustering Methods: A Benchmark by A. Ultsh, C. Vetter; FG Neuroinformatik & Kunstliche Intelligenz Research Report 0994

References Interpreting patterns of gene expression with self- organizing maps: Methods and application to hematopoietic differentiation by Tamayo et al. A Local Search Approximation Algorithm for k-Means Clustering by Kanungo et al. K-means-type algorithms: A generalized convergence theorem and characterization of local optimality by Selim and Ismail