Visualization of Biological Information with Circular Drawings

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

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

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

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

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

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

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

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

Circular Drawing

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

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

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

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

CIRCULAR BICONNECTED

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

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

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

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

CIRCULAR NON-BICONNECTED

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

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

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

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

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

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