Co-expression networks: graph properties and topological comparisons

Abstract

Motivation: Microarray-based gene expression data have been generated widely to study different biological processes and systems. Gene co-expression networks are often used to extract information about groups of genes that are 'functionally' related or co-regulated. However, the structural properties of such co-expression networks have not been rigorously studied and fully compared with known biological networks. In this article, we aim at investigating the structural properties of co-expression networks inferred for the species Saccharomyces Cerevisiae and comparing them with the topological properties of the known, well-established transcriptional network, MIPS physical network and protein-protein interaction (PPI) network of yeast.

Results: These topological comparisons indicate that co-expression networks are not distinctly related with either the PPI or the MIPS physical interaction networks, showing important structural differences between them. When focusing on a more literal comparison, vertex by vertex and edge by edge, the conclusion is the same: the fact that two genes exhibit a high gene expression correlation degree does not seem to obviously correlate with the existence of a physical binding between the proteins produced by these genes or the existence of a MIPS physical interaction between the genes. The comparison of the yeast regulatory network with inferred yeast co-expression networks would suggest, however, that they could somehow be related.

Conclusions: We conclude that the gene expression-based co-expression networks reflect more on the gene regulatory networks but less on the PPI or MIPS physical interaction networks.

Supplementary information: Supplementary data are available at Bioinformatics online.

Fig. 1.

Degree distribution, P(k), of the true PPI network (cycles) and the corresponding co-expression network inferred by network inference procedure I (squares) as a function of k. The inset panel displays P(k) in double logarithmic scales, showing that P(k) approximately decays as a power law. These results indicate that, in spite of being both networks approximately scale free (at least, for all k : k>1), their degree distribution is different. For the co-expression network, the standard errors are plotted based on 10⁴ bootstrap samples.

Fig. 2.

Nearest neighbors average degree function, , of the true MIPS network (cycles) and the corresponding co-expression network inferred by network inference procedure I (squares), indicating that the MIPS network is dissortative while the co-expression network is assortative. For the association network, the standard errors are also plotted based on 10⁴ bootstrap samples.

Fig. 3.

Degree-dependent clustering coefficient, C(k), of the true PPI network (cycles) and the corresponding co-expression network inferred by network inference procedure I (squares). The picture shows that both C(k) functions are different, being the local clustering coefficient of the co-expression network always substantially larger than that of the yeast PPI network. Note that C(k) does not decay as C(k) ∼ k^−β (β a constant, usually close to 1). For the co-expression network, the standard errors are also plotted based on 10⁴ bootstrap samples.

Fig. 4.

Panels (1a–c): degree-dependent clustering coefficient, C(k), of the true PPI networks (cycles) and the corresponding co-expression networks inferred by procedure I (squares). Panels (2a–c): degree distribution, P(k), of the true PPI networks (cycles) and the corresponding co-expression networks inferred by procedure I (squares). Panels (3a–c): nearest neighbors average degree function, , of the true protein-protein interaction networks (cycles) and the corresponding co-expression networks inferred by procedure I (squares). Note that panels (a) correspond to the direct physical BioGRID PPI network, panels (b) correspond to the entire BioGRID PPI network, and panels (c) correspond to the STRING PPI network (see text for details). For the co-expression networks, the standard errors are plotted based on 2000 bootstrap samples. The plots show (i) that the direct physical BioGRID PPI network is topologically different from its associated co-expression network (they show, for example, different types of degree-degree correlation and a big difference in C(k)), (ii) the topological differences between the entire BioGRID network and its associated co-expression networks are sensibly smaller and (iii) those differences, although still appreciable, begin to disappear when the compared networks are the STRING PPI and its associated co-expression network.