2. Introduction to Bioinformatics Lecture 19 Intracellular Networks Graph theory C E N T R F O R I N T E G R A T I V E B I O I N F O R M A T I C S V U E

3. High-throughput Biological Data Enormous amounts of biological data are being generated by high-throughput capabilities; even more are coming –genomic sequences –gene expression data –mass spectrometry data –protein-protein interaction data –protein structures –...... Hidden in these data is information that reflects –existence, organization, activity, functionality …… of biological machineries at different levels in living organisms

4. Bio-Data Analysis and Data Mining Existing/emerging bio-data analysis and mining tools for –DNA sequence assembly –Genetic map construction –Sequence comparison and database search –Gene finding –…. –Gene expression data analysis –Phylogenetic tree analysis to infer horizontally-transferred genes –Mass spec. data analysis for protein complex characterization –…… Current prevailing mode of work Developing ad hoc tools for each individual application

5. Bio-Data Analysis and Data Mining As the amount and types of data and the needs to establish connections across multi-data sources increase rapidly, the number of analysis tools needed will go up “exponentially” –blast, blastp, blastx, blastn, … from BLAST family of tools –gene finding tools for human, mouse, fly, rice, cyanobacteria, ….. –tools for finding various signals in genomic sequences, protein-binding sites, splice junction sites, translation start sites, ….. Many of these data analysis problems are fundamentally the same problem(s) and can be solved using the same set of tools Developing ad hoc tools for each application problem (by each group of individual researchers) may soon become inadequate as bio-data production capabilities further ramp up

6. Data Clustering Many biological data analysis problems can be formulated as clustering problems –microarray gene expression data analysis –arrayCGH data (chromosomal gains and losses) –identification of regulatory binding sites (similarly, splice junction sites, translation start sites,......) –(yeast) two-hybrid data analysis (for inference of protein complexes) –phylogenetic tree clustering (for inference of horizontally transferred genes) –protein domain identification –identification of structural motifs –prediction reliability assessment of protein structures –NMR peak assignments –......

7. Data Clustering: an example Regulatory binding-sites are short conserved sequence fragments in promoter regions Solving binding-site identification as a clustering problem –Project all fragments into Euclidean space so that similar fragments are projected to nearby positions and dissimilar fragments to far positions –Observation: conserved fragments form “clusters” in a noisy background........acgtttataatggcg..............ggctttatattcgtc..............ccgaatataatcta.......

8. Data Clustering Problems Clustering: partition a data set into clusters so that data points of the same cluster are “similar” and points of different clusters are “dissimilar” Cluster identification -- identifying clusters with significantly different features than the background

9. Multivariate statistics – Cluster analysis 1234512345 C1 C2 C3 C4 C5 C6.. Raw table Any set of numbers per column Multi-dimensional problems Objects can be viewed as a cloud of points in a multidimensional space Need ways to group the data

10. Multivariate statistics – Cluster analysis Dendrogram Scores Similarity matrix 5×5 1234512345 C1 C2 C3 C4 C5 C6.. Raw table Similarity criterion Cluster criterion Any set of numbers per column

11. Cluster analysis – data normalisation/weighting 1234512345 C1 C2 C3 C4 C5 C6.. Raw table Normalisation criterion 1234512345 C1 C2 C3 C4 C5 C6.. Normalised table Column normalisationx/max Column range normalisation(x-min)/(max-min)

12. Cluster analysis – (dis)similarity matrix Scores Similarity matrix 5×5 1234512345 C1 C2 C3 C4 C5 C6.. Raw table Similarity criterion D i,j = (  k | x ik – x jk | r ) 1/r Minkowski metrics r = 2 Euclidean distance r = 1 City block distance

13. Cluster analysis – Clustering criteria Dendrogram (tree) Scores Similarity matrix 5×5 Cluster criterion Single linkage - Nearest neighbour Complete linkage – Furthest neighbour Group averaging – UPGMA (phylogeny) Ward Neighbour joining – global measure (phylogeny)

14. Cluster analysis – Clustering criteria 1.Start with N clusters of 1 object each 2.Apply clustering distance criterion iteratively until you have 1 cluster of N objects 3.Most interesting clustering somewhere in between Dendrogram (tree) distance N clusters1 cluster

15. Single linkage clustering (nearest neighbour) Char 1 Char 2

16. Single linkage clustering (nearest neighbour) Char 1 Char 2

17. Single linkage clustering (nearest neighbour) Char 1 Char 2

18. Single linkage clustering (nearest neighbour) Char 1 Char 2

19. Single linkage clustering (nearest neighbour) Char 1 Char 2

20. Single linkage clustering (nearest neighbour) Char 1 Char 2 Distance from point to cluster is defined as the smallest distance between that point and any point in the cluster

21. Single linkage clustering (nearest neighbour) Single linkage dendrograms typically show chaining behaviour (i.e., all the time a single object is added to existing cluster) Let C i and C j be two disjoint clusters: d i,j = Min(d p,q ), where p  C i and q  C j

22. Complete linkage clustering (furthest neighbour) Char 1 Char 2

23. Complete linkage clustering (furthest neighbour) Char 1 Char 2

24. Complete linkage clustering (furthest neighbour) Char 1 Char 2

25. Complete linkage clustering (furthest neighbour) Char 1 Char 2

26. Complete linkage clustering (furthest neighbour) Char 1 Char 2

27. Complete linkage clustering (furthest neighbour) Char 1 Char 2

28. Complete linkage clustering (furthest neighbour) Char 1 Char 2

29. Complete linkage clustering (furthest neighbour) Char 1 Char 2 Distance from point to cluster is defined as the largest distance between that point and any point in the cluster

30. Complete linkage clustering (furthest neighbour) More ‘structured’ clusters than with single linkage clustering Let C i and C j be two disjoint clusters: d i,j = Max(d p,q ), where p  C i and q  C j

31. Clustering algorithm 1. Initialise (dis)similarity matrix 2. Take two points with smallest distance as first cluster 3. Merge corresponding rows/columns in (dis)similarity matrix 4. Repeat steps 2. and 3. using appropriate cluster measure until last two clusters are merged

32. Average linkage clustering ( Unweighted Pair Group Mean Averaging -UPGMA) Char 1 Char 2 Distance from cluster to cluster is defined as the average distance over all within-cluster distances

33. UPGMA Let C i and C j be two disjoint clusters: 1 d i,j = ————————  p  q d p,q, where p  C i and q  C j |Ci| × |Cj| In words: calculate the average over all pairwise inter-cluster distances CiCi CjCj

34. Multivariate statistics – Cluster analysis Phylogenetic tree Scores Similarity matrix 5×5 1234512345 C1 C2 C3 C4 C5 C6.. Data table Similarity criterion Cluster criterion

35. Multivariate statistics – Cluster analysis Scores 5×5 1234512345 C1 C2 C3 C4 C5 C6 Similarity criterion Cluster criterion Scores 6×6 Cluster criterion Make two-way ordered table using dendrograms

36. Multivariate statistics – Two-way cluster analysis 1425314253 C4 C3 C6 C1 C2 C5 Make two-way (rows, columns) ordered table using dendrograms; This shows ‘blocks’ of numbers that are similar

37. Multivariate statistics – Two-way cluster analysis

38. Graph theory The river Pregal in Königsberg – the Königsberg bridge problem and Euler’s graph Can you start at some land area (S 1, S 2, I 1, I 2 ) and walk each bridge exactly once returning to the starting land area?

40. Graphs - definition Digraphs: Directed graphs Complete graphs: have all possible edges Planar graphs: can be presented in 2D and have no crossing edges (e.g. chip design)

41. 0 1 1.5 2 5 6 7 9 1 0 2 1 6.5 6 8 8 1.5 2 0 1 4 4 6 5.5. Graph Adjacency matrix Graphs - definition An undirected graph has a symmetric adjacency matrix A digraph typically has a non-symmetric adjacency matrix

42. Example application – OBSTRUCT: creating non-redundant datasets of protein structures Based on all-against-all global sequence alignment Create all-against-all sequence similarity matrix Filter matrix based on desired similarity range (convert to ‘0’ and ‘1’ values) Form maximal clique (largest complete subgraph) by ordering rows and columns This is an NP-complete problem (NP = non- polynomial) and thus problem scales exponentially with number of vertices (proteins)

43. Example application 1 – OBSTRUCT: creating non-redundant datasets of protein structures Statistical research on protein structures typically requires a database of a maximum number of non- redundant (i.e. non-homologous) structures Often, two structures that have a sequence identity of less than 25% are taken as non-redundant Given an initial set of N structures (with corresponding sequences) and all-against-all pair-wise alignments: Find the largest possible subset where each sequence has <25% sequence identity with any other sequence Heringa, J., Sommerfeldt, H., Higgins, D., and Argos, P. (1992). Obstruct: a program to obtain largest cliques from a protein sequence set according to structural resolution and sequence similarity. Comp. Appl. Biosci. (CABIOS) 8, 599-600.

44. Example application 1 – OBSTRUCT: creating non-redundant datasets of protein structures (Cnt.) The problem now can be formalised as follows: Make a graph containing all sequences as vertices (nodes) Connect two nodes with an edge if their sequence identity < 25% Make an adjacency matrix following the above rules Heringa, J., Sommerfeldt, H., Higgins, D., and Argos, P. (1992). Obstruct: a program to obtain largest cliques from a protein sequence set according to structural resolution and sequence similarity. Comp. Appl. Biosci. (CABIOS) 8, 599-600.

45. Example application 1 – OBSTRUCT: creating non-redundant datasets of protein structures (Cnt.) The algorithm: Now try and reorder the rows (and columns in the same way) such that we get a square only consisting of 1’s in the upper left corner This corresponds to a complete graph (also called clique) containing a set of non-redundant proteins Heringa, J., Sommerfeldt, H., Higgins, D., and Argos, P. (1992). Obstruct: a program to obtain largest cliques from a protein sequence set according to structural resolution and sequence similarity. Comp. Appl. Biosci. (CABIOS) 8, 599-600.

46. Example application 1 – OBSTRUCT: creating non-redundant datasets of protein structures (Cnt.)  1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1....... 5 4 6 4.. Adjacency matrix 1.Order sum array and reorder rows and columns accordingly… 2.Estimate largest possible clique and take subset of adj. matrix containing only rows with enough 1s 3.For a clique of size N, a subset of M rows (and columns), where M  N, with at least N 1s is selected. 4.Go to step 1. Heringa, J., Sommerfeldt, H., Higgins, D., and Argos, P. (1992). Obstruct: a program to obtain largest cliques from a protein sequence set according to structural resolution and sequence similarity. Comp. Appl. Biosci. (CABIOS) 8, 599-600.

48. Some books call graphs containing multiple edges or loops a multigraph, and those without a graph. Other books allow multiple edges or loops in a graph, but then talk about a graph without multiple edges and loops as a simple graph.

49. Remarks A multigraph might have no multiple edges or loops. Every (simple) graph is a multigraph, but not every multigraph is a (simple) graph. Every graph is finite Sometimes even “multigraph” folks talk about a “simple graph” to emphasize that there are no multiple edges and loops.

50. Further definitions K 3,3

51. Further definitions K 3,3 bipartite A graph is bipartite if its vertices can be partitioned into two disjoint subsets U and V such that each edge connects a vertex from U to one from V. A bipartite graph is a complete bipartite graph if every vertex in U is connected to every vertex in V. If U has n elements and V has m, then we denote the resulting complete bipartite graph by K n,m.

52. The Stable Marriage Algorithm Given two non-overlapping equally sized graphs of men (A, B, C,..) and women (a, b, c, …), where each man and woman has a preference list about persons of the opposite sex A pairing denotes a 1-to-1 correspondence between men and women (each man marries one woman) A pairing is unstable if there are couples X-x and Y-y such that X prefers y to x and y prefers X to Y –if this happens, pair X-y is called unsatisfied A pairing in which there are no unsatisfied couples is called a stable pairing or stable marriage The Stable Marriage Algorithm forms a bipartite graph that is stable

53. A: abcd denotes the preferences of A (likes a the most, then b, then c, while d is liked least)

57. 0 1 1.5 2 5 6 7 9 1 0 2 1 6.5 6 8 8 1.5 2 0 1 4 4 6 5.5. Graph Adjacency matrix Graphs - definition An undirected graph has a symmetric adjacency matrix A digraph typically has a non-symmetric adjacency matrix

58. A Theoretical Framework Representation of a set of n-dimensional (n-D) points as a graph –each data point represented as a node –each pair of points represented as an edge with a weight defined by the “distance” between the two points 0 1 1.5 2 5 6 7 9 1 0 2 1 6.5 6 8 8 1.5 2 0 1 4 4 6 5.5. n-D data points graph representation distance matrix

59. A Theoretical Framework Spanning tree: a sub-graph that has all nodes connected and has no cycles Minimum spanning tree: a spanning tree with the minimum total distance (a) (b) (c)

60. Spanning tree Prim’s algorithm (graph, tree) –step 1: select an arbitrary node as the current tree –step 2: find an external node that is closest to the tree, and add it with its corresponding edge into tree –step 3: continue steps 1 and 2 till all nodes are connected in tree. 4 10 6 7 3 5 8 (e) 4 7 3 5 (b) 44 (c) 7 4 3 (d) 7 (a)

61. Kruskal’s algorithm –step 1: consider edges in non-decreasing order –step 2: if edge selected does not form cycle, then add it into tree; otherwise reject –step 3: continue steps 1 and 2 till all nodes are connected in tree. (f) 4 7 3 5 4 10 6 7 3 5 8 (a) (b) 3 4 (c) 3 Spanning tree 4 3 (d) 5 4 3 (e) 5 6 reject 4 3 (e) 5 6 4 3 5 6

62. A Theoretical Framework A formal definition of a cluster: –C forms a cluster in D only if for any partition C = C1 U C2, the closest point, from D-C1, to C1 is from C2. Key results c1 c2 For any data set D, any of its cluster is represented by a sub-tree of its MST

63. A Theoretical Framework The selection order of nodes by PRIM’s algorithm defines a linear representation, L(D), of a data set D Any contiguous block in L(D) represents a cluster if and only if its elements form a sub-tree of the MST, plus some minor additional conditions (each cluster forms a valley) Valleys correspond to clusters (red bars)

64. Application Examples Regulatory binding site identification: cAMP-receptor protein (CRP) binding site Two hybrid data analysis l Gene expression data analysis Are all solvable by the same algorithm

65. More Application Examples Phylogenetic tree clustering analysis Protein sidechain packing prediction Assessment of prediction reliability of protein structures Protein secondary structures NMR peak assignments ……

66. Example 2: Graph-based clustering: REPRO Heringa, J., and Argos P. (1993). A method to recognize distant repeats in protein sequences. Proteins Struct. Func. Genet. 17, 391-411. Non-supervised algorithm for finding repeats in protein sequences, where Repeats can be evolutionary distant (low sequence similarity) Multiple sets of repeats can be recognised

67. Fibronectin repeat example

68. Graph-based clustering: Repro 1. Calculate top-scoring non-overlapping local alignments 2. Stacking of local alignments 3. Make graph with N-termini of top- alignments as nodes 4. Perform graph-based clustering Heringa, J., and Argos P. (1993). A method to recognize distant repeats in protein sequences. Proteins Struct. Func. Genet. 17, 391-411.

69. TFIIIA: seven top-scoring non-overlapping local alignments

70. TFIIIA: Stacking of local alignments

71. TFIIIA: Graph-based clustering