Starting from K = 2, we apply the method in [14] to this network, and the condition (6) holds until K = 10. To make the solution more accurate, we do a global maximization by changing the module index of boundary nodes starting from the approximate solution. Since the approximate solution selleckchem Alisertib is already good, this step is very fast. The structure of the network is shown in Figure 5(a), with different colors and shapes denoting different modules as described in Table 1. Then we construct the hierarchical modular structure as shown in Figure 5(b). On the lowest level, there are ten modules, while on the highest level, there are four modules.Figure 5Yeast gene coexpression network. (a) The network structure, (b) the hierarchical structure. Table 1GO enrichment analysis results of the gene modules on the lowest level.

Since coexpressed genes tend to be coregulated and possibly have similar functions, genes in the same module are expected to be enriched for some function categories. In order to understand the biological basis of the network modules, we consider each identified module for enrichment of annotations from gene ontology (GO) [25]. In our analysis, the enrichment analysis was performed by GO stats from Bioconductor. For each module, the statistically most significant GO categories are analyzed. Table 1 shows the enrichment results for the ten modules. ��M-size�� and ��G-size�� are the size of both the modules and the GO categories, respectively. ��Overlap�� is the overlap size of the module and the GO category. Table 2 shows the enrichment results for the modules on different levels.

From the tables, it is easy to see that different gene function categories are enriched most on different levels. For example, module M2 enriches the GO category ��translation” most significantly, while the combined module M2, M8 enriches ��Ribonucleoprotein complex biogenesis” most significantly, with M2 containing 42 genes having this function. The combined module M2, M8, M4, and M1 also enriches this function, while M4 itself enriches ��cellular respiration” significantly. On the uppermost level, the module composed of M2, M8, M1, M4, M3, and M7 enriches four GO function categories most significantly, and all the genes are overlapped. Three (��cellular component biogenesis,�� ��cellular component biogenesis at cellular level,�� and ��ribosome biogenesis��) of them are different from the most enriched gene functions for each of these six modules.

All these results indicate that hierarchical modular structure does exist in gene coexpression networks and different gene functions are enriched most on different levels.Table 2GO enrichment analysis results of gene modules on the upper level.We use the software REViGO to check the hierarchical structure of the enriched Entinostat GO categories [26].