Seeing the results of a mutation with a vertex weighted hierarchical graph

We represent the protein structure of scTIM with a graph-theoretic model. We construct a hierarchical graph with three layers - a top level, a midlevel and a bottom level. The top level graph is a representation of the protein in which its vertices each represent a substructure of the protein. In turn, each substructure of the protein is represented by a graph whose vertices are amino acids. Finally, each amino acid is represented as a graph where the vertices are atoms. We use this representation to model the effects of a mutation on the protein.

There are 19 vertices (substructures) in the top level graph and thus there are 19 distinct graphs at the midlevel. The vertices of each of the 19 graphs at the midlevel represent amino acids. Each amino acid is represented by a graph where the vertices are atoms in the residue structure. All edges are determined by proximity in the protein's 3D structure. The vertices in the bottom level are labelled by the corresponding molecular mass of the atom that it represents. We use graph-theoretic measures that incorporate vertex weights to assign graph based attributes to the amino acid graphs. The attributes of the corresponding amino acids are used as vertex weights for the substructure graphs at the midlevel. Graph-theoretic measures based on vertex weighted graphs are subsequently calculated for each of the midlevel graphs. Finally, the vertices of the top level graph are weighted with attributes of the corresponding substructure graph in the midlevel.

We can visualize which mutations are more influential than others by using properties such as vertex size to correspond with an increase or decrease in a graph-theoretic measure. Global graph-theoretic measures such as the number of triangles or the number of spanning trees can change as the result. Hence this method provides a way to visualize these global changes resulting from a small, seemingly inconsequential local change.

This modelling method provides a novel approach to the visualization of protein structures and the consequences of amino acid deletions, insertions or substitutions and provides a new way to gain insight on the consequences of diseases caused by genetic mutations.