2.4. Stratification of genes

The genes were stratified by time-lag extent, as defined by the method in the previous section. For genome-wide analyses, they were further stratified by clustering on the basis of the time-course distance of mRNA and protein abundance changes between genes. For that purpose, we used the E matrix (explained in the previous section) to define the distance between genes (g and g′):

Using this distance, we conducted hierarchical gene clustering using the Ward's method.

We evaluated the confidence of each gene cluster using bootstrap probability (BP) [12]. In addition, we calculated the approximately unbiased (AU) p-value [13] developed for reducing the known bias of the BP test. If a cluster has an AU p-value of >0.95, then the hypothesis that “the cluster does not exist” is rejected with a significance level of 0.05. In this study, we focused on clusters of genes with BP >0.80 and AU >0.95 as being reliable clusters. We used BP in addition to AU because we sometimes saw considerable differences between these two values in a cluster, which seemed to be unreliable.

We calculated BP and AU using the R ver. 3.2.2 (2015-08-14) pvclust package (ver. 2.0–0).