# Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks
## 4 Result and Discussion  
The cell cycle of eukaryotes consists of four phases generally: G1, S, G2 and M phases. At this point, the controllable cancer network [44] is formed by a portion of key molecules in the G1/S phase as shown in Figure 3. The dotted arrows refer to induced gene expression. The E2F protein, for example, stimulates the expression of its own gene. The expression of suppressor genes is indicated by hammerheads. The proliferation and apoptosis of cells are regulated by transcription factors E2F and Myc in the cancer network. According to the particular expression status of cells, they fulfill the roles of tumor suppressor gene and oncogene. The microRNA, called miR-17-92, regulates E2F and Myc in the process of transcription. Meanwhile, pRb also regulates E2F. On the contrary, transcription of miR-17-92 is also induced by E2F and Myc. Under the regulation of this interaction, miR-17-92 is expressed as a tumor inhibitor or a tumor gene, severally. As a result, the cancer network (Figure 3) was added to a straightforward regulatory network [45], including E2F, Myc, and miR-17-92 (Figure 4). In this way, the PBMCN can simulate the simplified network more effectively, and the resulting network model is shown in Figure 4.  
Figure 4: The PBMCNs structure of the cell cycle of eukaryotes. Figure 3: (a) Cancer network. (b) Myc/E2F/miR-17-92 gene regulatory network. The networks are summarized as follows:  
\[l=1:\left\{ \begin{array}{l}E2F(t+1)=f_{E2F}(E2F(t),miR_{1}(t),pRb(t)),\\ miR_{1}(t+1)=f^{1}_{miR_{1}}(E2F(t),miR_{1}(t),pRb(t))\\ \\ l=2:\left\{\begin{array}{l}miR_{2}(t+1)=f^{2}_{miR_{2}}(miR_{2}(t),Myc(t), pRb(t)),\\ Myc(t+1)=f^{2}_{miR_{2}}(miR_{2}(t),Myc(t),pRb(t))\\ \end{array}\right.\end{array} \right.\]  
For simplicity, E2F, \(miR_{1}\), \(miR_{2}\), Myc and pRb are represented by \(x^{1}_{1},x^{1}_{2},x^{2}_{2},x^{2}_{3}\) and \(u(t)\), respectively. The specific variables involved are defined as follows:  
* \(x^{1}_{1}\): Transcription factor involved in gene transcription;
* \(x^{2}_{3}\): Transcription factor involved in gene transcription;
* \(x^{1}_{2},x^{2}_{3}\): Non-coding RNA cluster;
* \(u(t)\): Retinoblastoma protein;
* \(f^{1}_{11}\): The transcription of miRNA-17-92 is promoted, allowing E2F to enter the cancerous regions under the regulation of retinoblastoma protein;
* \(f^{1}_{21}\): The expression of oncogenes is promoted by E2F;
* \(f^{1}_{22}\): The expression of oncogenes is inhibited by E2F, and tumor suppressor genes are expressed by E2F;
* \(f^{1}_{21}\): The expression of oncogenes is promoted by Myc;
* \(f^{2}_{22}\): The expression of oncogenes is inhibited by Myc, and tumor suppressor genes are expressed by Myc;
* \(f^{2}_{31}\): The transcription of miRNA-17-92 is promoted, allowing Myc to enter the cancerous regions. The logic function \(f^{1}_{11}\) can be selected with the probabilities for \(x^{1}_{1}\). \[x^{1}_{1}(t+1)=f^{1}_{11}(x^{1}_{1}(t),x^{1}_{2}(t),u(t))=x^{1}_{2}(t)\wedge u (t).\]  
The logic function \(f^{1}_{21}\) and \(f^{1}_{22}\) can be selected with the probabilities for \(x^{1}_{2}\). \[x^{1}_{2}(t+1):\left\{ \begin{array}{l}f^{1}_{21}(x^{1}_{1}(t),x^{1}_{2}(t), u(t))=x^{1}_{1}(t),\\ f^{1}_{22}(x^{1}_{1}(t),x^{1}_{2}(t),u(t))=\
eg x^{1}_{1}(t)\end{array} \right.\]  
The corresponding probabilities are given: \(p(f^{1}_{2}=f^{1}_{21})=0.5\) and \(p(f^{1}_{2}=f^{1}_{22})=0.5\), respectively. The logic function \(f^{2}_{21}\) and \(f^{2}_{22}\) can be selected with the probabilities for \(x^{2}_{2}\). \[x^{2}_{2}(t+1):\left\{ \begin{array}{l}f^{2}_{21}(x^{2}_{2}(t),x^{2}_{3}(t), u(t))=x^{2}_{3}(t),\\ f^{2}_{22}(x^{2}_{2}(t),x^{2}_{3}(t),u(t))=\
eg x^{2}_{3}(t)\end{array} \right.\]  
The corresponding probabilities are given: \(p(f^{2}_{2}=f^{2}_{21})=0.5\) and \(p(f^{2}_{2}=f^{2}_{22})=0.5\), respectively. The logic function \(f^{2}_{31}\) and \(f^{2}_{32}\) can be selected with the probabilities for \(x^{2}_{3}\).