Investigation on changes of modularity and robustness by edge-removal mutations in signaling networks

Background Biological networks consisting of molecular components and interactions are represented by a graph model. There have been some studies based on that model to analyze a relationship between structural characteristics and dynamical behaviors in signaling network. However, little attention has been paid to changes of modularity and robustness in mutant networks. Results In this paper, we investigated the changes of modularity and robustness by edge-removal mutations in three signaling networks. We first observed that both the modularity and robustness increased on average in the mutant network by the edge-removal mutations. However, the modularity change was negatively correlated with the robustness change. This implies that it is unlikely that both the modularity and the robustness values simultaneously increase by the edge-removal mutations. Another interesting finding is that the modularity change was positively correlated with the degree, the number of feedback loops, and the edge betweenness of the removed edges whereas the robustness change was negatively correlated with them. We note that these results were consistently observed in randomly structure networks. Additionally, we identified two groups of genes which are incident to the highly-modularity-increasing and the highly-robustness-decreasing edges with respect to the edge-removal mutations, respectively, and observed that they are likely to be central by forming a connected component of a considerably large size. The gene-ontology enrichment of each of these gene groups was significantly different from the rest of genes. Finally, we showed that the highly-robustness-decreasing edges can be promising edgetic drug-targets, which validates the usefulness of our analysis. Conclusions Taken together, the analysis of changes of robustness and modularity against edge-removal mutations can be useful to unravel novel dynamical characteristics underlying in signaling networks. Electronic supplementary material The online version of this article (10.1186/s12918-017-0505-2) contains supplementary material, which is available to authorized users.


Supporting Text. Nested Canalyzing Functions
Given a Boolean network ( , ), the value of each variable at time + 1 is determined by the values of other variables 1 , 2 , … , with a link to at time by the Boolean function : ( + 1) = ( 1 ( ), 2 ( ), … , ( )). The rule is called canalyzing on the input variable if there exist Boolean values, and , such that ( ) = → ( + 1) = .
Then, and are called the canalyzing and canalyzed values for the output variable , respectively.
The notion of nested canalyzing functions (NCFs) was introduced in (Kauffman, et al., 2003), and they are a natural subset of canalyzing rules. It was inspired by the question of what happens in the noncanalyzing case: When a rule is not canalyzed by the value of the first input variable, is it canalyzed by one of the remaining input variables? This consecutive canalization test can be repeated for all inputs and therefore an NCFs to update can be represented as follows: ( + 1) = { 1 1 ( ) = 1 2 1 ( ) ≠ 1 and 2 ( ) = 2 3 1 ( ) ≠ 1 and 2 ( ) ≠ 2 and 3 ( ) = 3 ⋮ 1 ( ) ≠ 1 and ⋯ and −1 ( ) ≠ −1 and ( ) =

− ℎ
Where all and ( = 1, 2, … , ) denote the canalyzing and canalyzed values, respectively, and ≠ . In addition, as in the previous studies (Kauffman, et al., 2003;Kauffman, et al., 2004), we independently and randomly specified 1 , … , values with the probabilities where is a constant. On the other hand, the value of is deterministically specified by the value of and the sign of the interaction from to ( = 1, … , ) as the following table.
Sign of the interaction from to Fig. S1. Analysis of the changes of the modularity and the robustness by edge-removal mutations in STF signaling network. The removal rate of edges was varied from 1% to 5% (More specifically, the numbers of removed edges were 5, 11, 16, 22, and 27, respectively, among a total of 557 edges). For each removal rate, 5,000 trials of edge-removal were examined. (a) Results of average changes of the modularity and the robustness against the removal rate of edges. Y-axis value and error bar represent the average and the standard deviation divided by the square root of the sample size (5000), respectively. Both average values were significantly larger than zero (All P-values <0.0001, using one-sample t-test). The one-sample t-test was valid because the average values were normally distributed (see Additional file 1: Fig. S4) and there were very few outliers (see Additional file 1: Fig. S7). (b)-(c) Relationship between the changes of the modularity and the robustness in the case that the removal rate is 1% and 2%, respectively. A significant negative relationship was observed (Correlation coefficients were -0.05254 and -0.022068, respectively, with all P-values <0.0001). This relationship was consistently observed for larger removal rates (Correlation coefficients when the removal rate of edge is 3%, 4%, and 5% were -0.03272 and, -0.04156, and -0.02795, respectively, with all P-values <0.0001). (d) A trend of correlation coefficients between the changes of the modularity and the robustness against the removal rate of edges.

Fig. S2.
Analysis of the changes of the modularity and the robustness by edge-removal mutations in HIV-1 signaling network. The removal rate of edges was varied from 1% to 5% (More specifically, the numbers of removed edges were 3, 7, 11, 14, and 18, respectively, among a total of 368 edges). For each removal rate, 5,000 trials of edge-removal were examined. (a) Results of average changes of the modularity and the robustness against the removal rate of edges. Y-axis value and error bar represent the average and the standard deviation divided by the square root of the sample size (5000), respectively. Both average values were significantly larger than zero (All P-values <0.0001 using one-sample t-test). The one-sample t-test was valid because the average values were normally distributed (see Additional file 1: Fig. S5) and there were very few outliers (see Additional file 1: Fig. S8). (b)-(c) Relationship between the changes of the modularity and the robustness in the case that the removal rate is 1% and 2%, respectively. A significant negative relationship was observed (Correlation coefficients were -0.03867 and -0.05417, respectively with all P-values <0.0001). This relationship was consistently observed for larger removal rates (Correlation coefficients when the edge-removal rate is 3%, 4%, and 5% were -0.06862 and, -0.05948, and -0.09733, respectively, with all P-values <0.0001). (d) A trend of correlation coefficients between the changes of the modularity and the robustness against the removal rate of edges.

Fig. S3. Analysis of normal distributions of averages of modularity changes and robustness changes in T-LGL network. (a)-(e)
Results of the average of the modularity change with removal rates of 1% to 5%, respectively. (f)-(j) Results of the average of the robustness change with removal rates of 1% to 5%, respectively. In each subfigure, the average of the modularity or the robustness changes over 50 trials is computed, and this process was repeated 100 times to examine the distribution of the average variable. Kolmogorov-Smirnov test was run and all average values were normally distributed (All P-values > 0.10). S4. Analysis of normal distributions of averages of modularity changes and robustness changes in STF network. (a)-(e) Results of the average of the modularity change with removal rates of 1% to 5%, respectively. (f)-(j) Results of the average of the robustness change with removal rates of 1% to 5%, respectively. In each subfigure, the average of the modularity or the robustness changes over 50 trials is computed, and this process was repeated 100 times to examine the distribution of the average variable. Kolmogorov-Smirnov test was run and all average values were normally distributed (All P-values > 0.10 except that P-value = 0.069 in (e)) .

Fig. S5. Analysis of normal distributions of averages of modularity changes and robustness changes in HIV-1 network. (a)-(e)
Results of the average of the modularity change with removal rates of 1% to 5%, respectively. (f)-(j) Results of the average of the robustness change with removal rates of 1% to 5%, respectively. In each subfigure, the average of the modularity or the robustness changes over 50 trials is computed, and this process was repeated 100 times to examine the distribution of the average variable. Kolmogorov-Smirnov test was run and all average values were normally distributed (All P-values > 0.10 except that P-value = 0.053 in (d)). S6. Analysis of outliers of averages of modularity changes and robustness changes in T-LGL network. (a)-(e) Results of the average of the modularity change with removal rates of 1% to 5%, respectively. (f)-(j) Results of the average of the robustness change with removal rates of 1% to 5%, respectively. In each subfigure, the average of the modularity or the robustness changes over 50 trials is computed, and this process was repeated 100 times to examine the distribution of the average variable. We examined outliers by a boxplot inspection and found no significant outliers in all subfigures except for (e), (h) and (j). S7. Analysis of outliers of averages of modularity changes and robustness changes in STF network. (a)-(e) Results of the average of the modularity change with removal rates of 1% to 5%, respectively. (f)-(j) Results of the average of the robustness change with removal rates of 1% to 5%, respectively. In each subfigure, the average of the modularity or the robustness changes over 50 trials is computed, and this process was repeated 100 times to examine the distribution of the average variable. We examined outliers by a boxplot inspection and found no significant outliers in all subfigures except for (c), (i) and (j). S8. Analysis of outliers of averages of modularity changes and robustness changes in HIV-1 network. (a)-(e) Results of the average of the modularity change with removal rates of 1% to 5%, respectively. (f)-(j) Results of the average of the robustness change with removal rates of 1% to 5%, respectively. In each subfigure, the average of the modularity or the robustness changes over 50 trials is computed, and this process was repeated 100 times to examine the distribution of the average variable. We examined outliers by a boxplot inspection and found no significant outliers in all subfigures except for (c) and (i). S9. Relationship between the changes of the modularity and the robustness in T-LGL signaling network. (a)-(c) Results in the cases that the edge-removal rate is 3%, 4%, and 5%, respectively. Each plot is a result of 5,000 trials. Correlation coefficients of (a)-(c) were -0.30652, -0.30684, and -0.28626, respectively (All P-values <0.0001).

Fig. S10.
Relationship between the changes of the modularity and the robustness in random networks. (a)-(c) Results of the random networks shuffled from T-LGL, STF, and HIV-1 signaling networks, respectively. In each subfigure, a set of 100 random networks were generated and 500 trials of edge-removals were tested for each network (Hence, each correlation coefficient was obtained over a total of 50,000 samples). We analyzed the relationship by varying the edge-removal rate from 1% to 5%. All cases showed significantly negative relationship (All P-values <0.0001).    Subgraphs with respect to High-MI-incident and High-RD-incident nodes, respectively. Red link and yellow node represent High-MI edge and High-MI-incident node, respectively, in both (a) and (c), whereas they represent High-RD edge and High-RD-incident node, respectively, in both (b) and (d).

Fig. S18.
Edge-removal analysis for edgetic drug discovery in STF signaling network. The arrows and bar-headed lines represent positive and negative interactions, respectively. Line thickness is proportional to the inclusion frequency of the interaction in top-K edge sets ranked in a decreasing order of the robustness change among 5000 trials of edge-removal mutations with 1% removal rate. The interaction ( 3 1 → ) was observed 17 times in top-K edge sets (K was chosen to 18).

Fig. S19.
Edge-removal analysis for edgetic drug discovery in HIV-1 signaling network. The arrows and bar-headed lines represent positive and negative interactions, respectively. Line thickness is proportional to the inclusion frequency of the interaction in top-K edge sets ranked in a decreasing order of the robustness change among 5000 trials of edge-removal mutations with 1% removal rate. The interactions ( 41 → 28) and ( 3 → 3) were observed 6 and 3 times, respectively, in top-K edge sets (K was chosen to 16).

Type of GO analysis GO term
High-MIincident (%)   Table. S2. GO analysis results between High-MI-incident/High-RD-incident group and the rest of genes in HIV-1 network. All P-values were calculated by using Bonferroni test.