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. 2016 Sep;25(9):1617-27.
doi: 10.1002/pro.2963. Epub 2016 Jun 19.

Network representation of protein interactions: Theory of graph description and analysis

Dennis Kurzbach  1
Affiliations

Network representation of protein interactions: Theory of graph description and analysis

Dennis Kurzbach. Protein Sci. 2016 Sep.

Abstract

A methodological framework is presented for the graph theoretical interpretation of NMR data of protein interactions. The proposed analysis generalizes the idea of network representations of protein structures by expanding it to protein interactions. This approach is based on regularization of residue-resolved NMR relaxation times and chemical shift data and subsequent construction of an adjacency matrix that represents the underlying protein interaction as a graph or network. The network nodes represent protein residues. Two nodes are connected if two residues are functionally correlated during the protein interaction event. The analysis of the resulting network enables the quantification of the importance of each amino acid of a protein for its interactions. Furthermore, the determination of the pattern of correlations between residues yields insights into the functional architecture of an interaction. This is of special interest for intrinsically disordered proteins, since the structural (three-dimensional) architecture of these proteins and their complexes is difficult to determine. The power of the proposed methodology is demonstrated at the example of the interaction between the intrinsically disordered protein osteopontin and its natural ligand heparin.

Keywords: chemical shift; graph theory; network description; nuclear magnetic resonance; protein interactions; relaxation.

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Figures

Figure 1
Figure 1
Histograms of 15N‐R 2 and η values of protein backbone amides found in the BMRB at 600 (left) and 800 MHz proton Lamor frequency (right). Note that the distributions are not scaled to correlation times or protein sizes. This might transform the distributions into monomodal functions and will be treated elsewhere. For the present purpose, the unscaled distributions yield the desired information.
Figure 2
Figure 2
NMR observables ΔCS(1HN), ΔCS(15N), ΔR 2, and Δη as a function of residue position for the OPN–heparin interaction (left) and normalized NMR observables parameters ΔCS(1HN)*, ΔCS(15N)*, Δ R2*, and Δη * derived according to Eqs. (1), and (2) (right).
Figure 3
Figure 3
(a) Graphical display of the covariance matrix (squared values) corresponding to the OPN/heparin interaction. The yellow plain depicts the noise level, <n>, found for the covariance matrix (see text). (b) Adjacency matrix derived for the OPN/heparin interaction if all values smaller than <n> in (a) are set to 0 and all other values to 1 [blue dots, cf. Eqs. (3), (4)]. The adjacency matrix can be grouped into three clusters of residues as indicated at the bottom of the figure (red: heparin binding site, yellow: compensatory site, purple: residual affected residues).
Figure 4
Figure 4
Display of the correlation network of the OPN/heparin interaction. It is represented with three pronounced hubs as indicated by the blue loops. These hubs correspond to the binding site, the affected site and residual correlated residues Isolated nodes are ignored. Every spot corresponds to a node, that is, to a diagonal element of the adjacency matrix and every line indicates an edge between two nodes, that is, a nonzero off‐diagonal matrix elements of A. The residue patches of the primary sequence corresponding to the nodes are indicated at the bottom; see Figure 3 for the corresponding nodes in the adjacency matrix. The clustering was performed by means of binary hierarchical clustering using an Euclidean distance norm and a predefined number of three clusters. The graphical visualization was done using Mathematica 10's spring electrical embedding method.
Figure 5
Figure 5
A graph with four nodes and four edges. For the node υ x, δx = 3, and Cx = 1/3 [cf. Eqs. (5), (6)].
Figure 6
Figure 6
Residue plots of connectivity and centrality measures W, C, and δ corresponding to the matrix in Figure 3(B) [cf. Eqs. (5)–(7)]. All three parameters show increased values around the heparin binding site (aa 140–160) and the compensatory site (aa 100–140).
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References

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