1. DB Seminar Series: Biclustering Methods for Microarray Data Analysis By: Kevin Yip 10 Sep 2003

2. 2 Outline Introduction Overview of the algorithms Some details of each algorithm Summary Research opportunities

3. 3 Introduction Microarray data can be viewed as an NM matrix: Each of the N rows represents a gene (or a clone, ORF, etc.). Each of the M columns represents a condition (a sample, a time point, etc.). Each entry represents the expression level of a gene under a condition. It can either be an absolute value (e.g. Affymetrix GeneChip) or a relative expression ratio (e.g. cDNA microarrays). A row/column is sometimes referred to as the “expression profile” of the gene/condition.

4. 4 Introduction It is common to visualize a gene expression datasets by a color plot: Red spots: high expression values (the genes have produced many copies of the mRNA). Green spots: low expression values. Gray spots: missing values. N genes M conditions

5. 5 Introduction If two genes are related (have similar functions or are co-regulated), their expression profiles should be similar (e.g. low Euclidean distance or high correlation). However, they can have similar expression patterns only under some conditions (e.g. they have similar response to a certain external stimulus, but each of them has some distinct functions at other time). Similarly, for two related conditions, some genes may exhibit different expression patterns (e.g. two tumor samples of different sub-types).

6. 6 Introduction As a result, each cluster may involve only a subset of genes and a subset of conditions, which form a “checkerboard” structure: In reality, each gene/condition may participate in multiple clusters.

7. 7 Introduction To discover such data patterns, some “biclustering” methods have been proposed to cluster both genes and conditions simultaneously. Differences with projected clustering (by observation, not be definition): Projected clustering has a primary clustering target, biclustering usually treats rows and columns equally. Most projected clustering methods define attribute relevance based on value distances, most biclustering methods define biclusters based on other measures. Some biclustering methods do not have the concept of irrelevant attributes.

8. 8 Overview of the Biclustering Methods MethodPublishCluster ModelGoal Cheng & ChurchISMB 2000Background + row effect + column effect Minimize mean squared residue of biclusters Getz et al. (CTWC) PNAS 2000Depending on plugin clustering algorithm Lazzeroni & Owen (Plaid Models) Bioinformatics 2000 Background + row effect + column effect Minimize modeling error Ben-Dor et al. (OPSM) RECOMB 2002All genes have the same order of expression values Minimize the p-values of biclusters Tanay et al. (SAMBA) Bioinformatics 2002 Maximum bounded bipartite subgraph Minimize the p-values of biclusters Yang et al. (FLOC) BIBE 2003Background + row effect + column effect Minimize mean squared residue of biclusters Kluger et al. (Spectral) Genome Res. 2003 Background  row effect  column effect Finding checkerboard structures

9. 9 Overview of the Biclustering Methods MethodAllow overlap? Bicluster Discovery ComplexityTesting Data Cheng & ChurchYes (rare in reality) One at a time O(MN) or O(MlogN) Yeast (288417), lymphoma (402696) Getz et al. (CTWC) YesOne set at a time Exponential Leukemia (175372), colon cancer (200062) Lazzeroni & Owen (Plaid Models) YesOne at a time Polynomial Food (9616), forex (27618), yeast (246779) Ben-Dor et al. (OPSM) YesAll at the same time O(NM 3 l) Breast tumor (322622) Tanay et al. (SAMBA) YesAll at the same time O((N2 d+1 ) log (r +1) /r(rd) ) Lymphoma (402696), yeast (6200515) Yang et al. (FLOC) YesAll at the same time O((N+M) 2 kp) Yeast (288417) Kluger et al. (Spectral) NoAll at the same time PolynomialLymphoma (1 rel., 1 abs.), leukemia, breast cell line, CNS embryonal tumor

10. 10 Cheng and Church Model: A bicluster is represented the submatrix A of the whole expression matrix (the involved rows and columns need not be contiguous in the original matrix). Each entry A ij in the bicluster is the superposition (summation) of: 1.The background level 2.The row (gene) effect 3.The column (condition) effect A dataset contains a number of biclusters, which are not necessarily disjoint.

11. 11 Cheng and Church Example: Correlation between any two columns = correlation between any two rows = 1. a ij = a iJ + a Ij – a IJ, where a iJ = mean of row i, a Ij = mean of column j, a IJ = mean of A. Biological meaning: the genes have the same (amount of) response to the conditions. Back.: 5Col 0: 1Col 1: 3Col 2: 2 Row 0: 28109 Row 1: 4101211 Row 2: 1798

12. 12 Cheng and Church Goal: to find biclusters with minimum squared residue: For an ideal bicluster, H(I, J) = 0. adding a constant to all entries of a row or column yields an ideal bicluster. multiplying all entries in the bicluster by a constant yields an ideal bicluster.

13. 13 Cheng and Church Constraints: 1M and N1 matrixes always give zero residue. => Find biclusters with maximum sizes, with residues not more than a threshold  (largest -biclusters). Constant matrixes always give zero residue. => Use average row variance to evaluate the “interestingness” of a bicluster. Biologically, it represents genes that have large change in expression values over different conditions.

14. 14 Cheng and Church Finding the largest -bicluster: The problem of finding the largest square - bicluster (|I| = |J|) is NP-hard. Objective function for heuristic methods (to minimize): => sum of the components from each row and column, which suggests simple greedy algorithms to evaluate each row and column independently.

15. 15 Cheng and Church Greedy methods: Algorithm 0: Brute-force deletion (skipped) Algorithm 1: Single node deletion Parameter(s):  (maximum squared residue). Initialization: the bicluster contains all rows and columns. Iteration: 1.Compute all a Ij, a iJ, a IJ and H(I, J) for reuse. 2.Remove a row or column that gives the maximum decrease of H. Termination: when no action will decrease H or H <= . Time complexity: O(MN)

16. 16 Cheng and Church Greedy methods: Algorithm 2: Multiple node deletion (take one more parameter . In iteration step 2, delete all rows and columns with row/column residue > H(I, J)). Algorithm 3: Node addition (allow both additions and deletions of rows/columns).

17. 17 Cheng and Church Handling missing values and masking discovered biclusters: replace by random numbers so that no recognizable structures will be introduced. Data preprocessing: Yeast: x  100log(10 5 x) Lymphoma: x  100x (original data is already log-transformed)

18. 18 Cheng and Church Some results on yeast cell cycle data (288417):

19. 19 Cheng and Church Some results on lymphoma data (402696): No. of genes, no. of conditions 4, 9610, 2911, 25 103, 25127, 1313, 21 10, 572, 9625, 12 9, 513, 962, 96

20. 20 Cheng and Church Discussion: Biological validation: comparing with the clusters in previously published results. No evaluation of the statistical significance of the clusters. Both the model and the algorithm are not tailored for discovering multiple non-disjoint clusters. Normalization is of utmost importance for the model, but this issue is not well-discussed.

21. 21 Yang et al. (FLOC) Model: based on Cheng and Church, but allows missing values. Volume of a bicluster: number of non-missing entries in the submatrix. Goals: Not to introduce random interference. Discover k possibly overlapping clusters simultaneously. Support additional features (e.g. limit the maximum amount of overlapping) using virtually zero additional cost. FLOC: FLexible Overlapped biClustering

22. 22 Yang et al. (FLOC) Missing values handling: Introducing a parameter  (a fraction), so that in a bicluster, all rows and columns must not contain more than  missing values. If =0.6, When calculating the row/column/matrix averages, missing values are not counted. Invalid bicluster: 13 45 34 Valid bicluster: 133 345 344

23. 23 Yang et al. (FLOC) Algorithm: Parameter(s): k (no. of clusters),  (cluster size parameter),  (missing value threshold), r (residue threshold, i.e.,  in Cheng and Church’s notation). Phase 1: create k random biclusters (for each bicluster, each row/column is randomly added with a probability ). Phase 2: repeatedly For each row/column, determine the changes of squared residue if it is selected/deselected from each of the k biclusters. Perform the best actions of the m+n rows and columns.

24. 24 Yang et al. (FLOC) Example (before): Remove from red Remove from green Add to red Remove from green Add to red Remove from green Remove from red Add to green Remove from red Remove from green

25. 25 Yang et al. (FLOC) Example (decisions): Remove from red Remove from green Add to red Remove from green Add to red Remove from green Remove from red Add to green Remove from red Remove from green

26. 26 Yang et al. (FLOC) Example (after, if all actions are performed): Actual algorithm: execute the actions sequentially, keep only the best cluster set out of the M+N potential sets.

27. 27 Yang et al. (FLOC) How to compare different actions? Suppose an action is performed on row/column x in cluster c to form cluster c’, the gain of the action is defined as A +ve gain indicates an improvement of bicluster quality. r c’ > r c  first term is –ve: favor smaller residue. v c’ > v c  second term is +ve: favor larger volume. r c << r  second term dominates: when residue is small, the major goal is to increase volume.

28. 28 Yang et al. (FLOC) What is the execution order of the M+N actions? Based on the gain values, with some probability of swapping the order in order to overcome local optimums. Termination criteria: If none of the M+N new bicluster sets contains only r-biclusters and the aggregated volume is larger than the previous best set. Time complexity of FLOC: O((N+M) 2 kp).

29. 29 Yang et al. (FLOC) Additional features: Limit the maximum amount of bicluster overlapping. Limit the minimum amount of coverage (fraction of entries covered by at least one bicluster). Limit the ratio between the number of genes and conditions in each bicluster. Limit the minimum volume of the biclusters. How? Not to perform any actions that will violate the constraints.

30. 30 Yang et al. (FLOC) Some results on the yeast cell cycle data: Avg. residue Avg. volume Avg. gene num. Avg. cond. num. Time CC algorithm 204.2931576.981671212min FLOC algorithm 187.5431825.7819512.86.7min

31. 31 Yang et al. (FLOC) Some results on Yeast cell cycle data: 1 more gene 2 more conditions, 6 more genes

32. 32 Yang et al. (FLOC) Discussion: The model is still not suitable for non-disjoint clusters. There are more user parameters, including the number of biclusters. There is no justification of having one action per row/column in each iteration. Gain values are based on the biclusters before any of the M+N actions. The additional features can have negative impacts to the clustering process.

33. 33 Lazzeroni and Owen (Plaid Models) Model: Each entry Y ij in the bicluster is the superposition of: 1.The global background level 2.The background level of the layers (biclusters) 3.The row (gene) effect of the layers 4.The column (condition) effect of the layers 1234 1 if bicluster k contains row i 0 otherwise 1 if bicluster k contains column j 0 otherwise

34. 34 Lazzeroni and Owen (Plaid Models) Example: Layer 0:  0 =10. Layer 1:  1 =5,  1 ={2,3,4},  1 ={1,2,3},  1 ={1,1,0}, 1 ={1,1,0}. Layer 2:  2 =2,  2 ={3,3,5},  2 ={4,2,1},  2 ={1,0,0}, 2 ={1,1,1}. ++ =

35. 35 Lazzeroni and Owen (Plaid Models) The model is more suitable for overlapping biclusters. Goal: to find model parameters (K,  0,  k,  ik,  jk,  ik and  jk ) such that the squared error is minimized. For simplicity, call the parameters for cluster k ( k,  ik and  jk )  ijk. Objective function (to minimize):

36. 36 Lazzeroni and Owen (Plaid Models) Algorithm to find 1 layer: 1. Determine initial memberships  (0) and  (0). 2. For (i=0; i<s; i++) 1.Determine cluster parameters  (i+1) from  (i) and  (i). 2.Determine row memberships  (i+1) from  (i+1) and  (i). 3.Determine column memberships  (i+1) from  (i+1) and  (i).

37. 37 Lazzeroni and Owen (Plaid Models) Determining initial memberships  (0) and  (0) (some attempts): All parameters set to 0.5 All parameters set to random values near 0.5 More complicated heuristics: Fix all  ijk to 1. Perform several iterations that update  and  only. Scale  and  so that they sum to N/2 and M/2 respectively.

38. 38 Lazzeroni and Owen (Plaid Models) Determining  (k) from  (k-1) and  (k-1) : deduce the best fit of the models, subject to the condition that every row and column has a zero mean. Solutions (using Lagrange multiplier): where

39. 39 Lazzeroni and Owen (Plaid Models) Similarly, the membership parameters can be determined by: Stopping rule: if a layer has a smaller size than expected (found by random permutation of data) or a K max (a user parameter) layers have been found.

40. 40 Lazzeroni and Owen (Plaid Models) Some results on yeast stress data (246779 ): 34 layers, 5568 parameters (<3% of all observations) No. of layersGenesConditionsObservations 070322170703 11031522872 257921307 314211 4-1812390 Total246779194893

41. 41 Lazzeroni and Owen (Plaid Models) Some results on yeast stress data: Layer 1: includes many genes involved in the cell cycle. Layer 3: includes many genes involved in glycolysis.

42. 42 Lazzeroni and Owen (Plaid Models) Discussion: The model may still be too restrictive for gene expression data in which co-regulated genes may have different magnitudes of response to a stimulus. Again, normalization issues are critical but not addressed.

43. 43 Kluger et al. (spectral) All the previous approaches define NP- hard problems and provide heuristic solutions. This study adopts a model where optimal solution can be found in polynomial time.

44. 44 Kluger et al. (spectral) Model: Each entry in the dataset is the product of: 1.The hidden base expression level 2.The tendency of gene i to be expressed in all conditions 3.The tendency of all genes to be expressed in condition j A normalized dataset should contain a checkerboard structure. Within each block, all row tendencies are equal and all column tendencies are equal.

45. 45 Kluger et al. (spectral) Illustration of the model: Suppose x’= 2 x ( 2 is a scalar), then A T Ax= 2 x  an eigenproblem.

46. 46 Kluger et al. (spectral) Idea of the method: The input gene expression profiles form a non-normalized, non-ordered matrix. Suppose there are ways to normalize the data (discussed later). Call the resulting matrix A. Solve the eigenproblem A T Ax= 2 x and examine the eigenvectors x. If the constants in an eigenvector can be sorted to produce a step-like structure, the condition clusters can be identified accordingly. The gene clusters are found similarly from y.

47. 47 Kluger et al. (spectral) Illustration of the idea: A: The 1 st eigenvector: The corresponding y: By sorting the constants, it can be seen that there are two row clusters and two column clusters.

48. 48 Kluger et al. (spectral) Problem 1: non-normalized data E.g. some rows are multiplied by a scalar. The eigenproblem cannot be formulated.

49. 49 Kluger et al. (spectral) Normalization method 1: independent rescaling of genes and conditions Assume the non-normalized matrix is obtained by multiplying each row i by scalar r i and each column j by scalar c j, then r i1 /r i2 = mean of row i1 / mean of row i2. Let R be a diagonal matrix with entries r i at the diagonal and C is a diagonal matrix defined similarly, then the eigenproblem can be formulated by rescaling the data matrix:

50. 50 Kluger et al. (spectral) Method 2: bi-stochastization By repeating the independent scaling of genes and conditions until stable, the final matrix will have all rows sum to a constant and all columns sum to a different constant. Method 3: log-interactions If the original rows/columns are differed by multiplicative constants, then after taking log, they differ by additive constants. Further, we want each row and each column to have zero mean. This can be achieved by transforming each entry as follows: A’ ij = A ij – A Ij – A iJ + A IJ.

51. 51 Kluger et al. (spectral) Problem 2: when the number of genes/conditions are large, and the input data does not 100% fit the model, it is not easy to find the clusters. Our previous example (0.54, 0.24, 0.54, 0.54, 0.24) obviously contains 2 clusters. But what about (0.07, 0.09, 0.11, 0.11, 0.16, 0.24, 0.31, 0.36, 0.43, 0.45, 0.48, 0.5, 0.53, 0.56, 0.59, 0.65, 0.73, 0.81, 0.83, 0.97)? In such cases, standard one-way clustering techniques (e.g. k-means) can be used to cluster the constant terms in the eigenvectors.

52. 52 Kluger et al. (spectral) Results on lymphoma Affymetrix data:

53. 53 Kluger et al. (spectral) Results on leukemia data:

54. 54 Kluger et al. (spectral) Discussion: Real datasets may deviate from the ideal checkerboard structure. The model does not assume any irrelevant rows/columns, which is different from most biclustering, subspace clustering and projected clustering approaches. The “clusters” are disjoint.

55. 55 3 more approaches to go… In the previous models, every gene in a bicluster has the same amount of response to the conditions. The following three approaches define biclusters in less stringent ways.

56. 56 Ben-Dor et al. (OPSM) Model: For a condition set T and a gene g, the conditions in T can be ordered in a way so that the expression values are sorted in ascending order (suppose the values are all unique). Suppose a submatrix A contains genes G and conditions T. A is a bicluster if there is an ordering (permutation) of T such that the expression values of all genes in G are sorted in ascending order. OPSM: Order-Preserving SubMatrixes.

57. 57 Ben-Dor et al. (OPSM) Example: Valid bicluster: Invalid bicluster: t1t1 t2t2 t3t3 t4t4 t5t5 g1g1 71319250 g2g2 192339642 g3g3 468210 Induced permutation 23415 t1t1 t2t2 t3t3 t4t4 t5t5 g1g1 71319250 g2g2 192339642 g3g3 46827

58. 58 Ben-Dor et al. (OPSM) Goal: to find OPSMs of maximum statistical significance (stochastic model: each row has an independent permutation). Fact: given an NM matrix, the problem of finding an ks OPSM is NP-complete.

59. 59 Ben-Dor et al. (OPSM) Some terms: Complete model (T, ): T is a set of conditions (columns)  is an ordering of the conditions in T. Partial model (,, s): The first a and last b conditions are specified, but not the remaining s-a-b conditions. A row “supports” a model if applying the permutation to the row results in a set of monotonically increasing values.

60. 60 Ben-Dor et al. (OPSM) Idea of algorithm: to grow partial models until they become complete models. Algorithm: Evaluate all (1, 1) partial models (there are O(m 2 ) possible models), keep the best l of them. Expand them to (2, 1) models (there are O(ml) possible models), keep the best l of them. Expand them to (2, 2) models, keep the best l of them. Expand them to (3, 2) models, keep the best l of them. … Until getting l (s/2, s/2) models, which are complete models. Output the best one.

61. 61 Ben-Dor et al. (OPSM) Assume evaluating each model takes O(ns) time, then the whole algorithm requires O(nm 3 l). Evaluating a partial model (idea): A model is more favorable if there are more rows that support it. A row is more likely to support a partial model if there is a large “gap”. t1t1 t2t2 t3t3 t4t4 t5t5 g1g1 12345 g2g2 23415 14 A larger gap A smaller gap

62. 62 Ben-Dor et al. (OPSM) Some results on breast tumor data (322622 (8 with brcal mutations, 8 with brca2 mutations and 6 sporadic breast tumors)): A 3474 bicluster with the first three tissues with brca2 mutations and the last one sporadic. A 426 bicluster with five brca2 mutations followed by one brcal mutation. A 78 bicluster with four brca2 mutations followed by three brcal mutation, followed by a sporadic cancer sample.

63. 63 Ben-Dor et al. (OPSM) The 3474 bicluster:

64. 64 Ben-Dor et al. (OPSM) Discussion: Although the model concerns only the order of values instead of value distance or correlation, the use of total ordering still makes the model quite restrictive (the paper suggests some possible model extensions with no corresponding algorithms). Comparing to previous models, OPSM seems less biologically-intuitive. The algorithm does not prevent the final models from being highly similar to each other.

65. 65 Tanay et al. (SAMBA) SAMBA: Statistical-Algorithmic Method for Bicluster Analysis) Model: The whole dataset forms a bipartite graph G=(U, V, E): U is the set of conditions. V is the set of genes. (u, v)  E iff v responds in condition u (i.e., the expression level of v changes significantly in u). A bicluster: a subgraph of the bipartite graph.

66. 66 Tanay et al. (SAMBA) Example: t1t1 t2t2 t3t3 g1g1 0.81.52.6 g2g2 0.40.73.2 t1t1 t2t2 t3t3 g1g1 g2g2

67. 67 Tanay et al. (SAMBA) Goal: to find the maximum weighted subgraph Assume edges occur independently and equiprobably with density p = |E| / (|U||V|). Denote BP(k, p, n) as the binomial tail, i.e., the probability of observing k or more successes in n trails, then the probability of obtaining a bicluster H=(U’, V’, E’), p(H) is BP(|E’|, |E|/(|U|+|V|), |U’||V’|). Assume p < ½, then the problem can be transformed to finding a maximum weight subgraph of G where each edge has +ve weight (-1-log p) and each non- edge has -ve weight (-1-log(1-p)) (details skipped). A refined model that does not assume independent edges can also be defined.

68. 68 Tanay et al. (SAMBA) Assume gene vertices have d-bounded degree (no more than d edges incident on each gene vertex). Rationale: genes that constantly have abnormal expression are not interesting. Define the neighborhood of a vertex v, N(v) be the set of vertices adjacent to v in G. An O(|V|2 d )-time algorithm to find the maximum weight biclique:

69. 69 Tanay et al. (SAMBA) Based on the algorithm, the maximum weight subgraph can be found in O((n2 d ) log(2d) ) time. The model can also be extended to take into account the sign of expression values (“overexpress” or “underexpress”).

70. 70 Tanay et al. (SAMBA) The SAMBA algorithm: 1. Form the bipartite graph and calculate vertex pair weights. A gene is defined as up regulated (or down regulated) in a condition if its standardized level with mean 0 and variance 1 is above 1 (or below -1). 2. Apply the hashing technique to find the k heaviest bicliques in the graph. 3. Perform greedy addition/removal of vertices and filter biclusters that are too similar.

71. 71 Tanay et al. (SAMBA) Experiments on yeast data (6200515): Use the fourth level GO annotation as class labels. Hide the labels of 30% of the genes. Form biclusters. For biclusters with 60% labeled genes belonging to the same class, all genes with hidden labels are assumed to belong to that class. Compare the assumed and actual class labels to get the accuracy. Repeat for 100 times.

72. 72 Tanay et al. (SAMBA) Some results on yeast data: Actual SAMBA Read: 15% of genes classified as “AA Met” by SAMBA actually belong to class “Pro Met”.

73. 73 Tanay et al. (SAMBA) Discussion: Although the paper reports reasonable running time (a few minutes for 15000500, d set to 40), the exponential time complexity of SAMBA is daunting. It is not easy to define abnormal expression. Performing row standardization is not always appropriate.

74. 74 Getz et al. (CTWC) CTWC: Coupled Two-Way Clustering Goal: to find subsets of genes and conditions such that a single process is the main contributor to the expression of the gene subset over the condition subset. Idea: repeatedly perform one-way clustering on genes/conditions. Stable clusters of genes are used as the attributes for condition clustering, and vice versa. Allow the input of domain knowledge by adding initial gene/condition clusters.

75. 75 Getz et al. (CTWC) Illustration of the idea (assume a 43 dataset): Gene clusters Condition clusters g 1, g 2, g 3, g 4 g 1, g 3 t 1, t 2, t 3 All genes Domain knowledge

76. 76 Getz et al. (CTWC) Illustration of the idea (assume a 43 dataset): Gene clusters Condition clusters g 1, g 2, g 3, g 4 g 1, g 3 t 1, t 2, t 3

77. 77 Getz et al. (CTWC) Illustration of the idea (assume a 43 dataset): Gene clusters Condition clusters Clustering: Rows: g 1, g 2, g 3, g 4 Columns: t 1, t 2, t 3 Clustering: Rows: t 1, t 2, t 3 Columns: g 1, g 2, g 3, g 4 Clustering: Rows: g 1, g 3 Columns: t 1, t 2, t 3 Clustering: Rows: t 1, t 2, t 3 Columns: g 1, g 3 g 1, g 2, g 3, g 4 g 1, g 3 t 1, t 2, t 3

78. 78 Getz et al. (CTWC) Illustration of the idea (assume a 43 dataset): Gene clusters Condition clusters Clustering: Rows: g 1, g 2, g 3, g 4 Columns: t 1, t 2, t 3 Cluster 1: g 1, g 3, g 4 Cluster 2: g 2 g 1, g 2, g 3, g 4 g 1, g 3 t 1, t 2, t 3

79. 79 Getz et al. (CTWC) Illustration of the idea (assume a 43 dataset): Gene clusters Condition clusters Clustering: Rows: g 1, g 2, g 3, g 4 Columns: t 1, t 2, t 3 Cluster 1: g 1, g 3, g 4 Cluster 2: g 2 g 1, g 3, g 4 g2g2 g 1, g 2, g 3, g 4 g 1, g 3 t 1, t 2, t 3

80. 80 Getz et al. (CTWC) Termination: all stable clusters have already been added to the pools. 1-way clustering algorithm used in experiments: super-paramagnetic clustering (SPC). A hierarchical clustering method. Based on an analogy to the physics of inhomogeneous ferromagnets: clusters are broken up due to an increase of temperature. Normalization: 1. Divide by column mean. 2. Standardize each row. Distance function: Euclidean distance.

81. 81 Getz et al. (CTWC) Some results on leukemia data (175372 (47 ALL, 25 AML)): After two iterations, the algorithm formed 49 stable gene clusters and 35 stable sample clusters. One sample cluster contains 37 samples, and is stable when either a cluster of 27 genes or another unrelated cluster of 36 genes was used as the attributes. The latter contains many genes that participate in the glycolysis pathway.

82. 82 Getz et al. (CTWC) Some results on Leukemia data (175372 (47 ALL, 25 AML)): When the AML samples were clustered using a 28-gene cluster as attributes, a stable cluster was found that contains most of the samples (14/15) that were taken from patients that underwent treatment and whose results were known.

83. 83 Getz et al. (CTWC) Discussion: The number of clusters in the pools can be numerous. Only a specific set of one-way clustering algorithms can be used as the “plugin”. They should be able to determine the number of clusters. There should be ways to evaluate the stability of clusters. The meaning of the biclusters is not very intuitive.

84. 84 Summary Definition of (bi-)clusters: Same trend (background +/ row effect +/ column effect). Same ordering of values. Simultaneous abnormal expression (no direction, same direction, same or opposite direction). Depending on plugin algorithm. (Projected clustering): similar values. (Other works): similar shape (e.g. only considers the trend across adjacent time points).

85. 85 Summary The general research approach: 1. Define bicluster model and the clustering goal. 2. Determine if the problem is NP-hard (usually true). 3. Construct a statistical test for evaluating the significance/goodness of a bicluster/a set of biclusters. 4. Sketch the algorithm (usually greedy). 5. If the algorithm has a high complexity, try to speed up by applying reasonable heuristics.

86. 86 Summary The general research approach: 5. Test on synthetic data, validate by statistical tests and known bicluster structures. 6. Test on real data. Validate by Statistical tests. Comparing with previously published results. Using condition types. Using gene annotations. Visualization.

87. 87 Research Opportunities Propose other bicluster models. Based on the current models, propose new algorithms that improve bicluster quality (validated statistically or biologically) and/or time complexity. Combine the strength of multiple studies (e.g. plaid models + graph theory + statistical testing). Investigate the effects of normalization to the models/algorithms. Compare the different methods on some other real datasets. Make better use of domain knowledge.

88. 88 References Yizong Cheng and George M. Church, Biclustering of Expression Data, ISMB 2000. G. Getz, E. Levine and E. Domany, Coupled Two- Way Clustering Analysis of Gene Microarray Data, Proc. Natl. Acad. Sci. USA, 2000. Laura Lazzeroni and Art Owen, Plaid Models for Gene Expression Data, Statistica Sinica, 2002. Amir Ben-Dor, Benny Chor, Richard Karp and Zohar Yakhini, Discovering Local Structure in Gene Expression Data: The Order-Preserving Submatrix Problem, RECOMB 2002.

89. 89 References Amos Tanay, Roded Sharan and Ron Shamir, Discovering Statistically Significant Biclusters in Gene Expression Data, Bioinformatics 2002. Jiong Yang, Haixun Wang, Wei Wang and Philip Yu, Enhanced Biclustering on Expression Data, BIBE 2003. Yuval Kluger, Ronen Basri, Joseph T. Chang and Mark Gerstein, Spectral Biclustering of Microarray Cancer Data: Co-clustering Genes and Conditions, Genome Res., 2003.