- Open Access
Seeing the results of a mutation with a vertex weighted hierarchical graph
© Knisley and Knisley; licensee BioMed Central Ltd. 2014
- Published: 28 August 2014
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.
- Span Tree
- Cystic Fibrosis Conductance Transmembrane Regulator
- Minimum Span Tree
- Domination Number
- Single Point Mutation
Historically, graphs have been used to represent chemical structures since the inception of graph theory and there is a well-developed field known as chemical graph theory [1, 2]. Whereas in chemical graph theory each vertex represents an atom, the size of a protein molecule does not lend itself well to this representation. Thus, in many cases in the literature where a protein is represented by a graph, each vertex represents an amino acid and therefore each vertex represents ten, more or less, atoms. Two vertices are connected by an edge in the graph if the corresponding amino acid residues are within a specified distance threshold, typically 7 or 8 angstroms. Using this approach, protein structures can be viewed as networks of amino acids [3–6]. Even so, due to the size of many proteins, these graphs still tend to be very large. Since many of the chemical descriptors are defined for small molecules, measures from network science proved to be more informative for macromolecules.
Topological features of protein structures exhibit many desirable network properties such as high clustering coefficients and short average path lengths which shed light on aspects of protein folding [7–9]. A review of the uses of graphs as models of protein structure can be found in , which is a summary of work prior to 2002 for each method of representation - that is, vertices representing atoms versus vertices representing amino acids. However, again due to the size of these graphs, graph-theoretic representations have not provided an effective visualization tool, nor has it been an effective way to determine other properties of the protein's structure, such as the location of binding sites. In order to address the challenge of modelling a molecule at different scales - that is, simultaneously capturing full scale global properties of a large graph while identifying local properties of a small region of the graph - we developed a vertex weighted hierarchical graph model of a protein structure . There are distinct advantages to each of the approaches discussed above, namely to let each vertex represent an atom, or to let each vertex represent an amino acid. To capture the advantages of each, we build a representation that uses both methods. That is, we use the nested graphs concept to integrate the information obtained at each scale with all other scales. To do so we construct a hierarchical graph-theoretic structure to represent the three-dimensional structure of a protein. Since the goal in  was to identify the global effects of a single point mutation, we now address the BioVis Data Challenge with the vertex-weighted, hierarchical graph approach.
Substructure intervals used for the midlevel graph.
D2 - alpha
D3 - beta
D4 - alpha
D5 - loop
D6 - alpha
D7 - beta
D8 - alpha
D9 - alpha
D10 - beta
D11 - alpha
D12 - alpha
D13 - beta
D14 - loop
D15 - alpha
D16 - beta
D17 - alpha
D18 - beta
D19 - alpha
Finally, each amino acid is represented by a graph where each vertex represents an atom and two vertices are adjacent if the corresponding atoms have a bond. We do not consider the hydrogen atoms in this model (this is the commonly known hydrogen suppressed ball-and-stick representation of a molecule). We let a single vertex represent the central carbon in the backbone and thus we obtain a rooted graph where we denote the backbone carbon as the root. There are twenty graphs at this level which we can refer to as the bottom level. More layers are possible and may be desirable for a very large protein or a protein complex. Thus this general method can be applied to a very large structure such as a protein complex or a relatively small protein. We determined that three layers are sufficient for the scTIM model.
We first describe the process for a single point mutation. Associated with each amino acid is a set of descriptors derived from graph-theoretic measures of its graph representation. These values were first defined by the authors in  where a neural network was trained to recognize a change in binding affinity due to mutations. A single point mutation in a protein results in a change in exactly one of the amino acids in the protein's amino acid sequence. Thus, one vertex in exactly one of the midlevel graphs will receive a new set of descriptors that corresponds to the change at the bottom level. This results in a change of attributes of a single vertex at the midlevel. This change of attributes at the midlevel results in a change of graph-theoretic measures of the midlevel graph that utilize vertex weights. Consequently, the vertex at the top level graph that represents the substructure where the single point mutation occurred will receive a new set of attributes. In this way we are able to capture the flow of information from a single point mutation to the entire protein and visualize the effects. The process described above changes the vertex weights, but not the structure itself, of the top level graph. Using graph-theoretic measures that incorporate vertex weights, we obtain a unique set of graph-theoretic values associated with each mutation.
In addition, there are a number of folding algorithms that will provide a predicted structure for a given amino acid sequence. For example, PhYre2  and I-TASSER  have both consistently performed well in the annual protein prediction competition known as CASP- Critical Assessment of Structure Prediction . Consequently, a predicted pdb file can be obtained for a mutated sequence and the process described above can be iterated with the predicted structure. This results in a top-level graph whose structure may differ from the wild type. For the life scientists, we are developing a tool for "virtual mutations". By changing a residue (or a set of residues) in the sequence, this changes a vector of descriptors for a vertex (or vertices) at the midlevel. This in turn changes the values of the Top Level graph that are associated with that mutation. The consequences of the mutation on the structure of the graph can be viewed immediately. We describe the methods in more detail in the methods section and the results are below.
Modelling a single point mutation with a predicted structural change
A vertex set S is said to be a dominating set of vertices if every vertex in the graph is either in the set S or has a neighbor in S. Necessarily then, the entire set of vertices of a graph is a dominating set. The domination number of a graph is the minimum cardinality among all dominating sets. Since the cardinality of a set can be found by assigning a weight of 1 to each element of the set and then summing the weights, we define the weighted-domination number to be the minimum weight among all dominating sets. The maximum degree of a graph is the maximum value among all degree measures in the graph. Whereas the degree of a vertex is a local measure, the maximum degree of a graph is a global measure. Almost all standard graphical measures thus lend themselves to a "weighted" version. Graph-theoretic measures such as the maximum weighted-degree provide a rich source for numerical characterizations for the mid-level graphs which in turn are weights for the vertices of the top level graph. By including the vertex weights, we show the graph for the top level graph below generated by Cytoscape. Figure 2 is the wild type and Figure 3 is the mutant. Substructure D4, where the mutation occurred, is highlighted together with the neighbors of D4. Notice that there are significant structural changes predicted by PhYre2, such as the loss of the edge connecting D1 with D4. However, without the vertex weights, the change would not be nearly as striking.
Using this approach, the effects of a single point mutation on the entire protein can be observed. One would expect that the vertex representing the substructure where the mutation occurred to be affected. However, other consequences can now also be observed, such as the loss of the "heavy" cycle, D1, D4, D5, D9, D13, D16. The differences and similarities between a wild type protein and a given mutation of that protein are difficult to discern in a model where each vertex represents an amino acid. Without the hierarchical structure, a change in a single amino acid (a single vertex) may seem inconsequential, especially among the hundreds of such changes possible. With the hierarchical structure, however, the influence of such mutations can be tracked at each scale of representation.
Modelling mutations without a change in predicted structure
As the designers of this challenge intended, a problem of great interest in the field of molecular biology and biomedical science is how a single point mutation in some instances can have virtually no effect on the structure and function of a protein while in other cases the results can be disastrous. For example, a mutation in the gene for the cystic fibrosis conductance transmembrane regulator causes the protein to misfold and be tagged for degradation . Consequently, people with this mutation do not have this needed membrane protein in their epithelial cells and the result is the disease Cystic Fibrosis. We note that for most people, the mutation is a single point mutation, the deletion of phenylalanine (F) at position 508. The protein has a total of 1482 residues and thus the absence of only one residue out of the nearly 1500 residues has severe consequences. Molecular dynamics has shown the deletion of F at 508 causes very little change at the local level , so there must be some means by which this single deletion, i.e., a minor change at the local level percolates the entire structure.
This is the idea behind the vertex-weighted hierarchical graph model. A change at the amino acid level can be quantified on the bottom level and relayed to the mid-level by a change in vertex weights in the corresponding midlevel graph. This change in turn results in a change in the weights of the Top Level graph. The discussion about CFTR is an illustrative example of the concept and not meant to be restrictive. One could replace "deletion" with "insertion" and the discussion would remain the same in that the corresponding midlevel graph with the insertion would change and consequently the vertex representing that midlevel graph would receive a new set of descriptors, i.e, new weights. In addition, biochemical properties associated with the residues such as ss-stability and Vander Waals are included as vertex weights for the amino acids. For access to the IPython notebook and other materials, the reader may contact the authors of the paper.
Not all graphical invariants are informative for every graph. For example, the connectivity number provides no discerning information on a set of trees since all trees have connectivity number 2. In the same way, it should be noted that not all descriptors can be used to infer the impact of a mutation to the residue sequence. Proteins vary widely in size and structure. Thus, in practice, results and meaningful visualizations require a careful selection and testing of candidate descriptors and vertex weighting methods. The general model however can always be applied. For each application, the size of the structure, the types of the descriptors, and even the number of levels of the hierarchical graph must be determined by the modeller. In addition, the method only works as good as the selected protein prediction software when that software is used to determine the corresponding mutant graph. Our focus here is on visualizing what the software predicts so that, at a glance, one can observe the global structural consequences of a mutation when viewed through the lens of graph theory.
We now discuss some of the specifics in the methods section.
We implemented the hierarchical modelling process as an IPython notebook  running on the Python distribution Anaconda 1.8. This implementation begins by reading in protein three dimensional conformation data in the pdb file format via the module biopython . A single chain in the pdb model must be selected, and then either all or sections of the chain can be used to produce the hierarchical structure. In this way a connected graph is constructed for each chain. These chain graphs can then be connected by edges based on proximity if a protein has more than one chain. Given that the contest designers only provided mutations for one of the chains of TIM, our work was restricted to that chain.
An atom-based contact map is used to generate the lowest level graph. Measures for distances between residues include Cα to Cα, between centroids, and between corresponding centers of mass. Edges are weighted with the number of contacts between two residues, and the distance measures can be all-atom or restricted to side chains. In addition to the pdb file, the notebook uses a file "AADescriptorsRaw.csv" which contains a number of amino acid descriptors and graph-theoretic measures. As described earlier, we modify a number of the standard graph-theoretic measures to incorporate the vertex weights. In particular we find weighted upper domination, weighted lower domination, weighted diameter, circumference, average weighted degree, weighted periphery which we define by generalizing standard graphical invariants. Additionally, we use Plr, Chrg, Hydpthy, stablty, ss-stability, vanderWaal, chargetransf, chargedonar, averhydrophocitiy, coilConformation, IsoElectric, Balaban index, RofGyr, ShapeIndex, EIIP to be the most informative from a long list of highly used amino acid indicies. Many of these can be found in the Amino Acid Index Database .
Thus, the lowest level - the all atom level - is used both to define the structure and the vertex properties of the mid-level graph. A list of ranges defines the vertices (substructures) of the mid-level, contact-map generated graphs. Each of these substructures are in turn the vertices of the top level graph. Once again, edges are defined by the contacts between the structures, with at least two contacts between substructures necessary for an edge in the top level graph. Also, the top level edges are once again weighted by the number of contacts between the substructures (which are the vertices of the top level graph.)
Graph based descriptors are defined for the substructure graphs Di and for the top level graph. For example, the maximum generalized degree of a substructure graph is the largest vertex weighted degree corresponding to a given amino acid descriptor. The vertex weighted degree over a given descriptor is the sum of the descriptor values over the neighborhood of that vertex. A substructure-wide descriptor thus provides vertex weights for the vertices in a top level graph, and these vertex weights can be used to infer properties of the top level graph. The result can be exported to graphml as a vertex weighted graph, after which a visualization tool such as Cytoscape can be used to visualize the impact of sequence level changes on the top level graph of a protein.
There are numerous way to quantify structural aspects of a graph and these quantities are typically called graphical invariants in graph theory. For example, if the edges of the graph are weighted, then the minimum weight of a spanning tree is a quantity that is well known and highly studied. Thus, we use the vertex weights to determine a corresponding scheme for edge weights and then utilize the fact that there exist algorithms to find the minimum spanning trees of (edge) weighted graphs. The minimum weight among all spanning trees is just one of many ways to quantify a graph and these are the quantities that can then be drawn upon as vertex weights for the next level up the hierarchy, or used as part of the final quantification of the graph if one is calculating the minimum spanning tree of the top level graph. We have explored a number of graphical invariants from graph theory and molecular descriptors from computational chemistry. This work illustrates the concept and utility of the vertex-weighted hierarchical graph as an effective modelling and visualization tool for the investigation of the consequences of a mutation.
The data for the Data Contest was provided by Drs. Magliery and Sullivan. We gratefully acknowledge the dataset provided by Drs. Magliery and Sullivan at The Ohio State University for the purposes of the BioVis 2013 Contest. This work was partially funded by an East Tennessee State University Research Development Grant (#E82258).
Publication of this work was supported by an East Tennessee State University Research Development Grant (#E82258).
This article has been published as part of BMC Proceedings Volume 8 Supplement 2, 2014: Proceedings of the 3rd Annual Symposium on Biological Data Visualization: Data Analysis and Redesign Contests. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcproc/supplements/8/S2
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