Visualization of Biological Information with Circular Drawings

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: meancluster Intra-cluster relation All pairs of genes in cluster i with correlation higher than threshold1* meani 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 visualizationplacing 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 DrawingDetermine 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 DrawingDetermine 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 DrawingDetermine 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 v u w v u w

18. Circular DrawingDetermine 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 DrawingDetermine 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 DrawingDetermine 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 DrawingDetermine 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 DrawingDetermine 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 visualizationreduce 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?