1. Visualization of Biological Information with Circular Drawings

2. Outline Preliminaries Gene clustering Graph extraction from biological data Graph visualization Circular Drawings Conclusions and Discussion

3. Preliminaries Graph G(V,E) : set of vertices V, set of edges E joining vertices Each vertex represents an entity (e.g., gene) Each edge represents a strong correlation between the genes Several clustering algorithms give groups of vertices

4. Preliminaries Correlation: Compute Pearson's correlation coefficient for every pair of genes Select only the genes with the highest signal – to – noise ratio

5. Gene clustering Select an unclustered gene Add all genes with Pearson coef>threshold in the same cluster Repeat until no new cluster can be found For the unclustered genes, repeat the procedure, with decreased threshold value new_threshold=threshold*threshold

6. Preliminaries Correlation: Compute Pearson's correlation coefficient for every pair of genes

7. Graph extraction from biological data(1) Genes are represented as vertices Clusters are represented as groups Edges represent a relationship- correlation between genes

8. Graph extraction from biological data(2) Compute mean value of correlation co-efficients for all genes in a cluster: mean cluster Intra-cluster relation All pairs of genes in cluster i with correlation higher than threshold1* mean i are considered highly correlated Inter-cluster relation For every pair of genes dis=distance between clustering levels thres= The threshold used for the lowest level All pairs of genes with correlation higher than threshold2* (dis+1)(thres) are considered highly correlated

9. Graph visualization Gene → Vertex → circle The brightness of the color reflects the level in which the gene has been clustered High correlation → Edge → line The brightness of the color reflects the value of the Pearson coefficient Cluster → Group → Circle with respective genes-vertices on its periphery

10. Circular Drawing

11. Graph visualization Place groups in an aesthetic and comprehensive manner Determine ordering of vertices in group such that there are as few intra-edge crossings as possible Further reduce overall number of crossings using heuristics

12. Graph visualization placing groups Force - directed method over groups Groups are represented as electric loads and inter- group edges as springs Allow the system to converge

13. Graph visualization Place groups in an aesthetic and comprehensive manner ۷ Determine ordering of vertices in group such that there are as few intra-edge crossings as possible Further reduce overall number of crossings using heuristics

14. Circular Drawing Determine ordering of vertices in group-TREE The ordering is determined by the discovery time of a depth-first search A cross-free result is guaranteed

15. CIRCULAR BICONNECTED

16. Circular Drawing Determine ordering -BICONNECTED GRAPH Biconnected graph: A graph that remains connected after removing any (one) vertex/edge Find cross free embedding Can find this it if such an embedding exists Minimize number of crossings: NP-complete problem

17. Circular Drawing Determine ordering -BICONNECTED GRAPH Decompose the graph For some lowest degree node u Identify / create triangles with neighbors v, w store edge (v, w) remove u Repeat until only three vertices are left u v w u v w

18. Circular Drawing Determine ordering -BICONNECTED GRAPH Restore graph Remove all stored edges Perform depth-first search, compute longest path and place it on the circle Place any remaining vertices next to as many neighbors as possible between 2 neighbors next to 1neighbor next to 0 neighbors

19. Circular Drawing Determine ordering -BICONNECTED GRAPH Time requirement: O(|E|) If a cross-free result can be obtained the algorithm achieves this in O(|V|) Very good results in all cases compared to other circular drawing techniques

20. CIRCULAR NON-BICONNECTED

21. Circular Drawing Determine ordering -non BICON. GRAPH Obtain block cut point tree: Find articulation points: all vertices responsible for non-biconnectivity Find all biconnected components Combined they give the block cut point tree

22. Circular Drawing Determine ordering -non BICON. GRAPH ● Place block-cutpoint tree on embedding circle ● Layout each component with variant of CIRCULAR-BICONNECTED ● Circular drawing of trees ● Articulation points ● Transform component layout for arc

23. Circular Drawing Determine ordering -non BICON. GRAPH ● O(|E|) time requirement Dominated by the block-cut point tree construction ● Biconnectivity structure is clearly displayed ● Low number of crossings

24. Graph visualization Place groups in an aesthetic and comprehensive manner ۷ Determine ordering of vertices in group such that there are as few intra-edge crossings as possible ۷ Further reduce overall number of crossings using heuristics

25. Graph visualization reduce crossings Rotate groups trying to minimize energy, total edge length e.g for edge(9,20) reduce from 9->2cros

26. Conclusions and discussion We presented an algorithm for the visualization of biological data Other visualization techniques? Other types of applications?