Analysis and Studying of Cascading Failures in Gene Networks Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 163-172 ISSN 1078-6236 International Institute for General Systems Studies, Inc. Analysis and Study of Cascading Failures in Gene Network * Wang Shudong 1 , Shi Songtao 2 , Ge Yanru 3 , Sun Longxiao 1 , Xu Dashun 4 , Meng Dazhi 1,2 1 College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266510, China 2 School of Software Engineering; College of Applied Science, Beijing University of Technology, Beijing 100124, China 3 Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 4 Department of Mathematics, Southern Illinois University Carbondale, Makanida 62958, USA Email: wangshd2008@yahoo.com.cn, dzhmeng07@yahoo.com.cn Abstract Genome forms gene networks in terms of complicated interactions to realize its functions. Further research for gene networks can help to comprehend and predict many unknown functions of genome. In this work, cascading failure models of weighted gene networks are built and the robustness of the models is also analyzed and discussed. Based on the data of normal and lung adenocarcinoma stages, by simulating and analysing the cascading failures of the two gene network models, that cascading failures occur more likely in the networks for adenocarcinoma experimental groups than the one of normal control group is discovered. In the numerical experiments, we notice that nine genes of experimental group and eight genes of control group are of very strong destructibility for the robustness of experimental and control networks respectively. The failures of these genes can lead to the collapse or paralysis of the whole network. Therefore, we conclude that these genes might play important roles in keeping normal level or developing lung adenocarcinoma of organisms. When applying the methods of modeling and analysing cascading failures in gene network to other diseases’ data, biomedical scientists can be enlightened for understanding the mechanism of diseases and predicting the functions of significant genes. Keywords Systems biology Gene network Cascading failure Network statistics 1. Introduction Research on the biological functions of genome is a major issue in life science [1-3] . The study for gene networks describing the complicated genetic interactions in genome is an important way to understand biological functions [4-9] . So far, many methods have been proposed to build and analyze gene networks [4] . Amy Hin Yan Tong et al. [5] investigated the correspondence between the dense local neighborhoods in gene regulatory network and biological functions of yeasts. Mark Kittisopikul and Gürol M. Süel [6] studied the biological significance of feed-forward loop motifs in gene network of Escherichia coli and discovered that most * This work was supported by the National Natural Science Foundation of China (Grant Nos. 60874036, 60503002) and SDUST Research Fund. mailto:wangshd2008@yahoo.com.cn http://www.pnas.org/search?author1=Mark+Kittisopikul&sortspec=date&submit=Submit http://www.pnas.org/search?author1=G%C3%BCrol+M.+S%C3%BCel&sortspec=date&submit=Submit 164 Wang: Analysis and Study of Cascading Failures in Gene Network feed-forward loops have two kinds of regulatory functions. Hallinan J. S. et al. [7] analyzed the relationships among the motifs, feedback loops and their dynamical features in gene regulatory network. Bowers et al. [8] proposed a computational approach-logic analysis of phylogenetic profiles to identify detailed relationships among genes or proteins on the basis of genomic data, which was applied into 4873 distinct orthologous protein families of 67 fully sequenced organisms, and identified 750, 000 triplets previously unknown logic regulatory relationships. Shudong Wang et al. [9] constructed a logical network with 16 active genes of shoot in different external stimuli, and analyzed the dynamics of the logical network. Now, the theoretical models of cascading failures and their mechanisms, prevention and control for various actual complex networks have been relatively deeply studied [10-17] . For instance, R. Kinney et al. [16] analyzed the cascading failures in the North American power grid. The results show that deliberate attacks can lead to a substantial decline in the transmission efficiencies of power grid, while random failures have nearly no influence. Ashley G. Smart et al. [17] investigated the relationships between structure and robustness in the metabolic networks of Escherichia coli, Methanosarcina barkeri, Staphylococcus aureus, and Saccharomyces cerevisiae using a cascading failure model based on a topological flux balance criterion and found that metabolic networks are exceptionally robust compared to appropriate null models. But the reports are rare about cascading failures in gene network. This work investigates the influences of cascading failures on gene networks. Based on the documental data, by comparing cascading failures of gene networks for normal and lung adenocarcinoma groups, we discover control networks are quite robuster than lung adenocarcinoma experimental ones. This indicates the change from normal organisms into lung adenocarcinoma ones may result from dysfunctions or gene mutations (considered as failures) of some genes. Through numerical experiments, we notice that failures of some genes in experimental and control groups can lead to collapse or paralysis of the whole network. These genes might play important roles in keeping normal level or developing lung adenocarcinoma in organisms. This paper is organized as follows: the work background and existing methods for studying gene networks are introduced in the first part. The cascading failure model and its algorithm used in this work are presented in detail in the second part. The methods and main results of numerical experiments of cascading failure model are described based on two data sets of (normal) control group and (lung adenocarcinoma) experimental group in the third part. The obtained results are analyzed and discussed, and the possible corresponding biological significances are also pointed out in the fourth part. 2. The Model and Algorithm of Cascading Failure In this research, we consider cascading failures of complex gene networks, so we treat genes no different from nodes of complex networks. We use the capacity-load in cascading failure model. Let  WEVG ,, be a complex (directed or undirected) gene network with node-set  1,2, ,V N  , edge-set E and weight-set W . Suppose ijw is the weight from node i to j in complex gene network G . Then the edge-length from node i to j is defined as the reciprocal value ijw 1 of ijw . If 0ijw , then the edge-length from node i to j is  . The greater the weight between two nodes is, the lesser the edge-length is; the lesser the weight is, the greater the edge-length is, and vice versa. The shortest paths from node i to j are these paths corresponding to the smallest sum of edge-length in all the paths from node i to j . Obviously, the shortest paths from node i to j are not always unique. Suppose there exist app:ds:have app:ds:describe app:ds:always Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 165 p shortest paths from node i to j . Then the load of any shortest path R is defined as p 1 of the product of all the weights in R , i.e.   p w Rji ij , . The load jL of node j is defined as the sum of the loads of all the shortest paths passing through node j . The capacity jC of node j is proportional to its initial load 0 jL , i.e.   01 jj LC  , 1,2, ,j N  , where constant 0  is a tolerance factor. If the load of a node is greater than its capacity, then it is called a failure node. After deleting node i , and causing is failure nodes (including node i ), then is is defined as the size of cascading failure of node i and N s d i i  as the size-ratio of cascading failure. If cfi td  , then the network breaks down, otherwise, the network doesn’t have failure. This is a criterion of network failure, where cft is the threshold of network failure. Let       cfi cfi td td isign ,0 ,1 )(1 . Then the percentage of failure nodes of the network N isign P N i   1 )(1 ; the largest size-ratio of cascading failure  max max , 1,2, ,iR d i N   ; the average size-ratio of cascading failure 1 N i i d R N   . Let 1, 2( ) 0, i i d d sign i d d     ( d is a variable parameter). Then the cumulative probability of size-ratio of cascading failure N isign ddP N i   1' )(2 )( , which indicates the probability of size-ratio id of cascading failure greater than d . Obviously, maxR , R and '( )P d d are the important parameters measuring the robustness or fragility of network. Based on the above mentioned definitions and symbols, we present the algorithm （CFA）of cascading failure model as follows: ① Input the weight matrix of complex gene network  , ,G V E W . ② Calculate initial load 0 jL of node j and its capacity   01 jj LC  , Nj ,,2,1  . 1i . ③ Delete node i and its incident edges in the network. ④ Calculate the load of every node in the present network and compare the capacity with the load of every node. If the load is lesser than the capacity for every node in the present network, then go to ⑤, otherwise, delete every node and its incident edges whose load is greater than its capacity, go to ④. ⑤ If the size-ratio of cascading failure after deleting node i is greater than or equal to the threshold cft of network failure, then the network breaks down. ⑥ 1 ii . If Ni  , then go to ③. ⑦ Calculate the largest size-ratio of cascading failure maxR , average size-ratio of cascading 166 Wang: Analysis and Study of Cascading Failures in Gene Network failure R , and the cumulative probability of size-ratio of cascading failure '( )P d d . 3. The Methods and R esults of Numerical Experiments 3.1 Data Sources Data used in this work are from the results of lung adenocarcinoma network studied by Yuanyuan Zhang et al. (the detailed data sources can be seen in). We use the mutual information network and directed (or 1-order logic) network for control group (abbreviated as N) and lung adenocarcinoma experimental group (abbreviated as AC). In the mutual information network (directed weighted network), mutual information value (U value) is the weight of networks. We denote the weight matrices of mutual information (directed-weighted) network for control and experimental groups by NM ( NL ) and ACM ( ACL ), respectively. Table 1 The list of  ddP ' along with the change of the network threshold nett in control and experimental networks respectively. When 34.0d ,  ddP ' is equal to zero nett 0.52 0.5 0.45 0.4 d AC N AC N AC N AC N 0.01 1 1 1 1 1 1 1 1 0.02 1 1 1 1 1 0.0794 1 0.0714 0.03 1 0.1053 1 0.0870 0.1351 0.0794 0.1667 0.0714 0.04 0.2069 0.1053 0.1875 0.0870 0.1351 0.0794 0.1667 0.0714 0.05 0.2069 0.1053 0.1875 0.0870 0.1351 0.0794 0.1667 0.0714 0.06 0.2069 0.1053 0.1875 0.0870 0.1351 0.0794 0.1667 0.0714 0.07 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.08 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.09 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.1 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.11 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.12 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.13 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.14 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.15 0.1724 0.1053 0.1563 0.0870 0.1351 0 0.1667 0.0714 0.16 0.1724 0.1053 0.1563 0.0870 0.1351 0 0.1667 0.0286 0.17 0.1724 0.1053 0.1563 0.0870 0.1351 0 0.1667 0.0286 0.18 0.1724 0.1053 0.1563 0 0.1351 0 0.1667 0 0.19 0.1724 0.1053 0.1563 0 0.1351 0 0.1667 0 0.2 0.1724 0.1053 0.1563 0 0.1351 0 0.1667 0 0.21 0.1724 0.1053 0.1563 0 0.1351 0 0.1667 0 0.22 0.1724 0.0790 0.1563 0 0.1351 0 0.1667 0 0.23 0.1724 0.0790 0.1563 0 0.1351 0 0.1667 0 0.24 0.1724 0 0.1563 0 0.1351 0 0.1667 0 0.25 0.1724 0 0.1563 0 0.1351 0 0.1667 0 0.26 0.1724 0 0.1563 0 0.1351 0 0.1667 0 0.27 0.1724 0 0.1563 0 0.1351 0 0.1667 0 Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 167 0.28 0.1724 0 0.1563 0 0.1351 0 0.1667 0 0.29 0.1724 0 0.0625 0 0.1351 0 0.1667 0 0.3 0.1724 0 0.0625 0 0 0 0.1667 0 0.31 0.1724 0 0.0625 0 0 0 0 0 0.32 0.069 0 0 0 0 0 0 0 0.33 0.069 0 0 0 0 0 0 0 0.34 0.069 0 0 0 0 0 0 0 3.2 The Methods and Results 3.2.1 The Results of Mutual Information Gene Network In order to highlight the characteristics of the network structure, we analyze the changes of '( )P d d along with the network threshold nett . When taking different network thresholds, we can obtain the mutual information networks with different coarse granularities. The corresponding weight matrices NM and ACM are the inputs in the above CFA algorithm. Obviously, the greater the network threshold nett is, the coarser the granularity is, the more the lost information is and the computational complexity is relatively low; on the contrary, the lesser the network threshold nett is, the finer the granularity is, the less the lost information is, but the computational complexity is relatively high. The detailed data and the changing curves are in Table 1 and Fig. 1. From Table 1 and Fig. 1, it is obvious that '( )P d d of experimental network is clearly higher than that of control one under any network threshold. This indicates the ratio of nodes in experimental network which can result in cascading failures is much greater than the one of control group. With the increasing of d ,  ddP ' in control network reduces to zero earlier than in experimental one. In other words, taking certain appropriate d , control network has no failure while experimental network has more failures. Moreover, with the increasing of network threshold nett , the platform value of  ddP ' of control network is 0.0714, 0.0794, 0.0870 and 0.1053 respectively, showing gradually increasing tendency. This indicates that with the decreasing of the numbers of nodes and edges, cascading failures are more likely to occur in the gene networks, namely: the robustness goes worse. The genes resulting in the cascading failures of control and experimental groups under all four network thresholds are NRAS, PIK3CA, MAPK9, TOP2A and FGF1, RET, WT1, TCL1A, HRK, respectively. The detailed situations can be seen in Table 2 and 3. Fig. 1 Taking the network threshold nett as 0.4, 0.45, 0.5, 0.52 respectively, the changing curves of )( ' ddP  along with d in control and experimental networks. Table 2 The list of the relative greater size-ratio *d of cascading failure of control network under different network thresholds nett , where * denotes the gene. The following presentation is similar. 0.52 0.5 0.45 0.4 app:ds:clearly app:ds:appropriate app:ds:situation app:ds:relative 168 Wang: Analysis and Study of Cascading Failures in Gene Network gene *d (%) gene *d (%) gene *d (%) gene *d (%) NRAS 23.68 NRAS 17.39 NRAS 14.2857 NRAS 15.71 PIK3CA 23.68 PIK3CA 17.39 PIK3CA 14.2857 PIK3CA 17.14 MAPK9 21.05 MAPK9 17.39 MAPK9 14.2857 MAPK9 17.14 TOP2A 23.68 TOP2A 17.39 RBL1 14.2857 RBL1 15.71 TOP2A 14.2857 TOP2A 15.71 Table 3 The list of the relative greater size-ratio *d of cascading failure of experimental network under different network thresholds nett . 0.52 0.5 0.45 0.4 gene *d (%) gene *d (%) gene *d (%) gene *d (%) FGF1 34.48 FGF1 31.25 FGF1 29.73 FGF1 30.95 RET 31.03 RET 28.125 RET 29.73 FGF2 30.95 WT1 31.03 WT1 28.125 WT1 29.73 HSPB2 30.95 TCL1A 34.48 TCL1A 31.25 TCL1A 29.73 RET 30.95 HRK 31.03 HRK 28.125 HRK 29.73 WT1 30.95 TCL1A 30.95 HRK 30.95 3.2.2 The Results of Directed Gene Network To comprehensively measure the robustness and fragility of directed weighted gene network, we analyze the situations of R , maxR and '( )P d d with the changes of network thresholds nett (Table 4 and Table 5). From Table 4 and 5, we discover that R , maxR and '( )P d d of experimental network are clearly greater than the ones of control network under any network threshold. This shows that cascading failures occur in experimental network more easily than in control one. The genes resulting in cascading failures of control and experimental networks under five network thresholds are BAD, ING1, RAF1, TRAF3 and ESR2, HSPB2, NOV, TAL1 respectively. The detailed situations can be seen in Table 6 and 7. app:ds:relative app:ds:situation app:ds:clearly app:ds:situation Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 169 Fig. 2 Taking the network threshold nett as 0.1, 0.125, 0.15, 0.175, 0.2 respectively, the changing curves of )( ' ddP  along with d in control and experimental networks. Table 4 The list of R , maxR along with the change of network threshold nett in control and experimental networks respectively. nett Stage No. of nodes No. of edges R maxR 0.100 AC 60 487 0.1355 0.2167 N 98 1124 0.0858 0.1531 0.125 AC 60 392 0.1385 0.2000 N 95 887 0.0756 0.1158 0.150 AC 59 338 0.1390 0.2034 N 90 700 0.0635 0.1000 0.175 AC 58 285 0.1281 0.1897 N 86 560 0.0686 0.1163 0.200 AC 58 240 0.0888 0.1552 N 77 446 0.0727 0.1169 Table 5 The list of '( )P d d along with the change of network threshold nett in control and experimental networks respectively. When 22.0d ,  ddP ' is equal to zero. nett 0.1 0.125 0.15 0.175 0.2 d AC N AC N AC N AC N AC N 0 1 1 1 1 1 1 1 1 1 1 0.01 1 1 1 1 1 1 1 1 1 1 0.02 0.5167 0.3469 0.4833 0.2947 0.4237 0.2333 0.3621 0.2326 0.3448 0.2597 0.03 0.5167 0.2959 0.4833 0.2947 0.4237 0.2000 0.3621 0.1977 0.3448 0.2208 0.04 0.5167 0.2449 0.4833 0.2316 0.4237 0.1889 0.3621 0.1744 0.2931 0.1818 0.05 0.5167 0.2449 0.4833 0.2211 0.4237 0.1556 0.3621 0.1512 0.2931 0.1818 170 Wang: Analysis and Study of Cascading Failures in Gene Network 0.06 0.5167 0.2347 0.4667 0.2211 0.4068 0.1222 0.3448 0.1512 0.2759 0.1558 0.07 0.4833 0.2143 0.4667 0.2000 0.3898 0.0889 0.3103 0.1163 0.2069 0.1299 0.08 0.4833 0.2143 0.4667 0.1684 0.3898 0.0778 0.3103 0.1163 0.2069 0.1169 0.09 0.4500 0.2143 0.4167 0.1158 0.3390 0.0333 0.2414 0.0814 0.1552 0.1169 0.10 0.4500 0.1939 0.4167 0.0632 0.3390 0.0333 0.2414 0.0233 0.1552 0.0909 0.11 0.3667 0.1837 0.3833 0.0105 0.3051 0 0.2241 0.0116 0.0517 0.0390 0.12 0.3000 0.1122 0.3000 0 0.2712 0 0.2241 0 0.0517 0 0.13 0.3000 0.0510 0.3000 0 0.2712 0 0.2069 0 0.0517 0 0.14 0.2333 0.0102 0.2333 0 0.2542 0 0.1552 0 0.0517 0 0.15 0.2333 0.0102 0.2333 0 0.2542 0 0.1552 0 0.0517 0 0.16 0.1333 0 0.1833 0 0.1864 0 0.0690 0 0 0 0.17 0.0833 0 0.0833 0 0.0339 0 0.0690 0 0 0 0.18 0.0833 0 0.0833 0 0.0339 0 0.0517 0 0 0 0.19 0.0500 0 0.0333 0 0.0169 0 0 0 0 0 0.20 0.0500 0 0.0333 0 0.0169 0 0 0 0 0 0.21 0.0333 0 0 0 0 0 0 0 0 0 0.22 0 0 0 0 0 0 0 0 0 0 Table 6 The list of the relative greater size-ratio *d of cascading failure of control network under different network thresholds nett , where * denotes the gene. 0.1 0.125 0.15 0.175 0.2 gene *d (%) gene *d (%) gene *d (%) gene *d (%) gene *d (%) APC 15.31 ING1 11.58 ELK1 10 TRAF3 11.63 BAD 11.69 AKT1 13.27 APC 10.53 RAF1 10 BAD 10.47 ATF2 11.69 AXL 13.27 BAD 10.53 TRAF3 10 FAS 9.30 ING1 11.69 FOSL2 13.27 MLL 10.53 BAD 8.89 HCK 9.30 APC 10.39 GRB2 13.27 PML 10.53 HCK 8.89 ING1 9.30 HCK 10.39 BAD 12.24 RAF1 10.53 ING1 8.89 NRAS 9.30 MLL 10.39 ING1 12.24 AKT1 9.47 MLL 8.89 RAF1 9.30 TRAF3 10.39 NRAS 12.24 AXL 9.47 AKT1 7.78 AKT1 8.14 CXCL2 9.09 SELL 12.24 ELK1 9.47 AXL 8.14 RAF1 9.09 TP53 12.24 GRB2 9.47 CXCL2 8.14 SFRS3 7.79 TRAF3 12.24 NRAS 9.47 MCC 11.22 BCL2 8.42 MLL 11.22 CXCL2 8.42 NOTCH1 11.22 MAPK9 8.42 MAPK3 11.22 TRAF3 8.42 RAF1 11.22 AVEN 8.42 RARA 11.22 SUPT4H1 11.22 ELK1 10.2 HCK 9.18 app:ds:relative Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 171 MAPK9 9.18 Table 7 The list of the relative greater size-ratio *d of cascading failure of the experimental network under different network thresholds nett , where * denotes the gene. 0.1 0.125 0.15 0.175 0.2 gene *d (%) gene *d (%) gene *d (%) gene *d (%) gene *d (%) GLI2 21.67 FES 20.00 HSPB2 20.34 NOV 18.97 EXTL3 15.52 TP63 21.67 ROS1 20.00 NOV 18.64 WNT3 18.97 NOV 15.52 RET 20.00 HSPB2 18.33 ERG 16.95 TCL1A 18.97 ROS1 15.52 FES 18.33 WNT3 18.33 ESR2 16.95 EXTL3 17.24 ESR2 10.34 TAL1 18.33 TCL1A 18.33 EXTL3 16.95 ESR2 15.52 GLI2 10.34 ESR2 16.67 ESR2 16.67 FES 16.95 HSPB2 15.52 HSPB2 10.34 EXTL3 16.67 GLI2 16.67 CXCL3 16.95 TAL1 15.52 IL1A 10.34 NOV 16.67 CXCL3 16.67 TAL1 16.95 TP63 15.52 TAL1 10.34 E2F1 15.00 IL1A 16.67 WNT3 16.95 HRK 15.52 TCL1A 10.34 CXCL3 15.00 NOV 16.67 TP63 16.95 HSPB2 15.00 TAL1 16.67 HRK 16.95 ROS1 15.00 WNT3 15.00 TCL1A 15.00 4. Conclusion and Analysis In this research, we analyze and investigate the cascading failures in control and experimental networks. Through numerical experiments, we discover: under all the network thresholds, cascading failures occur in experimental networks more easily than in control ones for undirected and directed weighted gene networks. This indicates that the normal organisms are quite robust while diseased organisms are more fragile. In Table1, we notice that: with the increasing of network thresholds, the platform values of  ddP ' of control network are gradually increasing. This shows that with the decreasing of the numbers of nodes and edges, the robustness goes worse. In other words, with the increasing of the numbers of nodes and edges, the robustness goes better. This indicates the intrinsic reason of organisms functioning normally and stably maybe is that most genes play their own roles in organisms. In the process of numerical experiments, we notice that failures of genes BAD, ING1, RAF1, TRAF3, NRAS, PIK3CA, MAPK9, TOP2A of control group and ESR2, HSPB2, NOV, TAL1, FGF1, RET, WT1, TCL1A, HRK of experimental group under all the network thresholds result in collapse or paralysis of the whole network. This provides some useful reference informations for the normal or dieased organisms. For example, activation of gene Bad may induce apoptosis in human lung adenocarcinoma cells [18] . In other words, the failure of gene Bad leads to the defunctionalization of inducing the apoptosis of human lung adenocarcinoma cells and the organism might suffer from lung adenocarcinoma. Gene MAPK9 may enhance the stability of tumor suppressor p53 and its failure can reduce the stability of p53. Thus the organism might develop into cancer. 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