# American Institute of Mathematical Sciences

August  2017, 22(6): 2169-2206. doi: 10.3934/dcdsb.2017091

## Domain control of nonlinear networked systems and applications to complex disease networks

 1 School of Mathematics and Statistics, Wuhan University, School of Mathematics and Statistics, Central China Normal University, Wuhan 430072, China 2 Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, S7N 5A9, Canada 3 School of Mathematics and Statistics, Wuhan University, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China

Received  June 2016 Revised  August 2016 Published  March 2017

The control of complex nonlinear dynamical networks is an ongoing challenge in diverse contexts ranging from biology to social sciences. To explore a practical framework for controlling nonlinear dynamical networks based on meaningful physical and experimental considerations, we propose a new concept of the domain control for nonlinear dynamical networks, i.e., the control of a nonlinear network in transition from the domain of attraction of an undesired state (attractor) to the domain of attraction of a desired state. We theoretically prove the existence of a domain control. In particular, we offer an approach for identifying the driver nodes that need to be controlled and design a general form of control functions for realizing domain controllability. In addition, we demonstrate the effectiveness of our theory and approaches in three realistic disease-related networks: the epithelial-mesenchymal transition (EMT) core network, the T helper (Th) differentiation cellular network and the cancer network. Moreover, we reveal certain genes that are critical to phenotype transitions of these systems. Therefore, the approach described here not only offers a practical control scheme for nonlinear dynamical networks but also helps the development of new strategies for the prevention and treatment of complex diseases.

Citation: Suoqin Jin, Fang-Xiang Wu, Xiufen Zou. Domain control of nonlinear networked systems and applications to complex disease networks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2169-2206. doi: 10.3934/dcdsb.2017091
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A schematic illustration of the domain control. (A) The nonlinear networked system can exhibit transitions between the different stable steady states. The red solid, blue solid and green dashed lines represent one stable steady state (attractor), another stable steady (another attractor) state and unstable states, respectively. The green dashed line with an arrow indicates the transition from one attractor to another attractor. (B) and (C) provide illustrations of the concept of domain control, namely the transition from the domain of attraction of one attractor to the domain of attraction of another attractor. (B) The domain of attraction of one attractor (corresponding to the red solid line in (A)). Each state in the domain of attraction is represented by a red ball. The horizontal axis is a projected state-space and the vertical axis is the potential, which indicates the relative instability of individual states. (C) The domain of attraction of another attractor (corresponding to the blue solid line in (A)). Each state in the domain of attraction is represented by a blue ball
The EMT core network and domain control strategies for its phenotype transition. (A) Diagram (redrawn from [41]) of the EMT core network. The arrows and short bars represent activation and inhibition, respectively. For example, SNAI1 activates ZEB1 but inhibits miR-203 expression. (B) The practically required driver nodes (genes) for realizing domain controllability of this network. Mesenchymal cells can be induced from epithelial cells and back by any one of the four genes (SNAI1, ZEB1, ZEB2 and miR-203) and any one of the two genes (SNAI1 and miR-203), respectively
Domain control of the EMT core network transition from the epithelial state to the mesenchymal state. The representative node state trajectories $x(t)$ (A-B), control function $u(t)$ (C) and control error $e(t)$ (D) are depicted when the practically required driver node, namely miR-203, is controlled. The symbols $x_2$ and $x_6$ indicate the activities of SNAI1 and miR-203, respectively. The black dashed and blue solid lines represent the initial epithelial state (i.e., the undesired state with no control) and desired mesenchymal state, respectively
The Th differentiation cellular network and domain control strategies for its phenotype transition. (A) Diagram (redrawn from the reference [39]) of the Th differentiation cellular network. (B) The practically required driver nodes (genes) for realizing domain controllability of this network. Th1 and Th2 cells can be induced from Th0 cells by any one of the five genes (T-bet, IFN-$\gamma$, IFN-$\gamma$R, JAK1 and STAT1) and any one of the four genes (GATA-3, IL-4, IL-4R and STAT6), respectively
Domain control of the Th differentiation cellular network transitions from Th0 to Th1 and Th2. The representative node state trajectories $x(t)$ (A, B), control function $u(t)$ (C) and control error $e(t)$ (D) are depicted. The symbols $x_1$ and $x_{22}$ indicate the activities of GATA-3 and T-bet, respectively. The black dashed line indicates the Th0 phenotype, i.e., the initial state with no control. The red line with circle markers and blue line with square markers represent transitions from Th0 to Th1 and Th0 to Th2, respectively. The driver node is T-bet when Th0 is steered to the Th1 phenotype (Th0$\rightarrow$Th1) and GATA-3 when Th0 is driven to the Th2 phenotype (Th0$\rightarrow$ Th2)
Diagram of the cancer network. This network is redrawn from the reference [33], including 32 nodes (genes) and 111 edges (66 activation interactions and 45 repression interactions). The network mainly includes four types of marker genes: apoptosis marker genes (green rectangles), cancer marker genes (red rectangles), tumor repressor genes (light blue rectangles) and other genes (blue rectangles).The red arrows represent activation and the green short bars represent repression
Domain control of the cancer network transitions from the cancer state to the apoptosis state and normal state. The representative node state trajectories $x(t)$ (A-D), control function $u(t)$ (E) and control error $e(t)$ (F) are depicted when AKT and RB are controlled, respectively. The symbols $x_4$, $x_6$, $x_{10}$ and $x_{25}$ indicate the activities of PTEN, RB, AKT and NF-$\kappa$B, respectively. The black dashed lines with star markers indicate the cancer state, i.e., the undesired state with no control. The blue solid lines with squares and red solid lines with filled circles represent transitions from cancer (C) to apoptosis (A) (denoted by C$\rightarrow$A) and cancer (C) to normal (N) (denoted by C$\rightarrow$N), respectively. The driver node is AKT when cancer is steered to the apoptosis and RB when cancer is driven to the normal state
Domain control strategies for phenotype transitions of the cancer network. The practically required driver nodes (genes) for realizing domain controllability of this network were presented. For example, the normal and apoptosis states can be induced from the cancer state by any one of the two genes (RB and CDK2) and any one of the three genes (AKT, PTEN and NF$\kappa$B), respectively
The gene-regulatory network of the genetic toggle switch
A two-gene regulatory circuit with a self-feedback
A schematic diagram of the relationships between the controlled variables (nodes) and the remaining variables. We relabel the indexes of variables of the system such that the control set $C= \{1, ..., m\}$. Then the ($m+1$)-th equation only contains the variables in the control set $C$. The ($m+2$)-th equation only contains the variables in the union of the control set $C$ and ($m+1$)-th node. By this analogy, the ($n$)-th equation contains the variables in the set $C\bigcup \{m+1, \cdots, n-1\}$
The Goodwin model for a genetic regulatory system involving end-product inhibition
Distributions of the domains of attraction of the EMT core network model. We project the domains into a two-dimensional plane. The light red and light yellow regions represent the domains of attraction of the epithelial attractor and mesenchymal attractor, respectively
Domain control of the EMT core network transition from the epithelial state to the mesenchymal state by the theoretically required driver nodes. The node state trajectories $x(t)$ (A-F), control function $u(t)$ (G) and control error $e(t)$ (H) are depicted when the theoretically required driver nodes (i.e., miR-200 and miR-203) are controlled. The symbols $x_1, x_2, x_3, x_4, x_5$ and $x_6$ indicate the activities of CDH1, SNAI1, ZEB1, ZEB2, miR-200 and miR-203, respectively. $u_5$ and $u_6$ indicate the control functions for miR-200 and miR-203, respectively. The black dashed and blue solid lines represent the initial epithelial state (i.e., the undesired state with no control) and desired mesenchymal state, respectively
Effect of the parameter $\lambda$ in the designed control function on the domain control of the EMT core network. The node state trajectories $x(t)$ (A-F), control function $u(t)$ (G) and control error $e(t)$ (H) are depicted when the practically required driver node, namely miR-203, is controlled. The symbols $x_1, x_2, x_3, x_4, x_5$ and $x_6$ indicate the activities of CDH1, SNAI1, ZEB1, ZEB2, miR-200 and miR-203, respectively. The black dashed and colored solid lines represent the initial epithelial state (i.e., the undesired state with no control) and desired mesenchymal state, respectively. Here, we take the EMT core network as an example to show how the parameter $\lambda$ in our designed control function influences the control process. As shown in (H), to a certain extent, a higher absolute value of λ indicates a less time the system takes to evolve to the desirable attractor. However, there is no significant influence when the absolute value of $\lambda$ is larger than 1
Distributions of the domains of attraction of the Th differentiation cellular network model. We project the domains into a two-dimensional plane. The light red and light yellow regions represent the domains of attraction of the Th1 attractor and Th2 attractor, respectively
Domain control of the T helper differentiation cellular network transition from Th1 to Th2. The representative node state trajectories $x(t)$ (A-K), control function $u(t)$ (L) and control error $e(t)$ (M) are depicted. The symbols $x_1, x_4, x_5, x_6, x_7, x_{12}, x_{13}, x_{17}, x_{19}, x_{21}$ and $x_{22}$ indicate the activities of GATA-3, IFN-$\gamma$, IFN-$\gamma$R, IL-10, IL-10R, IL-4, IL-4R, SOCS1, STAT3, STAT6 and T-bet, respectively. The dashed black line indicates the Th1 phenotype, i.e. the initial state with no control. The solid blue line represents the Th2 phenotype. The driver node is GATA-3 when Th1 is driven to the Th2 phenotype
Distributions of the domains of attraction of the cancer network. We project the domains into a two-dimensional plane. The light red and light yellow regions represent the domains of attraction of the cancer attractor and normal attractor, respectively
Domain control of the cancer network transitions from the normal state to the apoptosis state and cancer state. The representative node state trajectories $x(t)$ (A-L), control function $u(t)$ (M) and control error $e(t)$ (N) are depicted when AKT ($u_{N\rightarrow A}$) and RB ($u_{N\rightarrow C}$) are controlled, respectively. The symbols $x_4, x_5, x_6, x_{10}, x_{12}, x_{13}, x_{14}, x_{16}, x_{18}, x_{21}, x_{22}$ and $x_{25}$ indicate the activities of PTEN, CDH1, RB, AKT, VEGF, HGF, HIF1, MDM2, CDK4, Caspase, BAX and NF$\kappa$B, respectively. The black dashed lines with star markers indicate the normal state, i.e., the initial state with no control. The red solid lines with filled circles and blue solid lines with squares represent transitions from normal (N) to apoptosis (A) (denoted by N$\rightarrow$A) and normal (A) to cancer (C) (denoted by N$\rightarrow$C), respectively. The driver node is AKT when normal is steered to the apoptosis and RB when normal is driven to the cancer state
Domain control of the cancer network transitions from the apoptosis state to the normal state and cancer state. The black dashed lines with star markers indicate the apoptosis state, i.e., the undesired state with no control. The red solid lines with filled circles and blue solid lines with squares represent transitions from the apoptosis state (A) to normal state (N) (denoted by A$\rightarrow$N) and apoptosis state (A) to cancer (C) (denoted by A$\rightarrow$C), respectively. The driver node is NF$\kappa$B when the apoptosis state is steered to the normal state and AKT when apoptosis state is driven to the cancer state
Robustness analysis of the control effectiveness against parameter perturbation. The evolution of the control function $u(t)$ and control error $e(t)$ are depicted when miR-203 is controlled. Each curve represents one simulation for one perturbation of the parameter values. In our numerical simulations, we randomly generate $100$ parameter sets in which every parameter of the system is perturbed within a range of $\pm10\%$. The initial state is the epithelial state, which is set to be $(1, 0, 0, 0, 1, 1)^T$ in all the numerical experiments. We zoom into the curves in a small window
The attractors of the EMT network. The two observed attractors A and B correspond to the epithelial state and mesenchymal state [41], respectively.
 A B A B $x_1$ (CDH1)} 1.7924 0.1111 $x_4$ (ZEB2)} 0.0170 1.8169 $x_2$ (SNAI1)} 0.3359 3.2229 $x_5$ (miR-200)} 1.7924 0.1111 $x_3$ (ZEB1)} 0.0522 1.8224 $x_6$ (miR-203)} 2.9873 0.1851
 A B A B $x_1$ (CDH1)} 1.7924 0.1111 $x_4$ (ZEB2)} 0.0170 1.8169 $x_2$ (SNAI1)} 0.3359 3.2229 $x_5$ (miR-200)} 1.7924 0.1111 $x_3$ (ZEB1)} 0.0522 1.8224 $x_6$ (miR-203)} 2.9873 0.1851
The positively invariant sets of the mathematical models of the three disease-related networks
 Networks Positively invariant sets EMT $\left\{ ({{x}_{1}}, \cdots, {{x}_{6}})\in {{\mathbb{R}}^{6}}\left| 0 < {{x}_{i}} < \frac{1}{{{d}_{i}}}, i=1, \cdots, 6 \right. \right\}$ T helper \left\{x\in {{\mathbb{R}}^{23}}\left| \begin{aligned} & 0\le {{x}_{5}} < \frac{1}{{{d}_{4}}{{d}_{5}}}, 0\le {{x}_{6}} < \frac{1}{{{d}_{1}}{{d}_{6}}}, 0\le {{x}_{7}} < \frac{1}{{{d}_{1}}{{d}_{6}}{{d}_{7}}}, 0\le {{x}_{14}} < \frac{1}{{{d}_{11}}{{d}_{14}}}, \\ & 0\le {{x}_{17}} < ((\frac{1}{{{d}_{3}}}+\frac{1}{{{d}_{15}}})\frac{1}{{{d}_{18}}}+\frac{1}{{{d}_{22}}})\frac{1}{{{d}_{17}}}, 0\le {{x}_{18}} < (\frac{1}{{{d}_{3}}}+\frac{1}{{{d}_{15}}})\frac{1}{{{d}_{18}}}, \\ & 0\le {{x}_{19}} < \frac{1}{{{d}_{1}}{{d}_{6}}{{d}_{7}}{{d}_{19}}}, 0\le {{x}_{20}} < \frac{1}{{{d}_{9}}{{d}_{20}}}, 0\le {{x}_{21}} < \frac{1}{{{d}_{13}}{{d}_{21}}}, \\ & 0\le {{x}_{i}} < \frac{1}{{{d}_{i}}}, 0\le {{x}_{j}} < 1, i=1, 3, 4, 9, 11, 12, 13, 15, 16, j=8, 10, 23 \\ \end{aligned} \right. \right\} Cancer $\left\{ ({{x}_{1}}, \cdots, {{x}_{32}})\in {{\mathbb{R}}^{32}}\left| 0\le {{x}_{i}}\le 1, i=1, \cdots, 32 \right. \right\}$
 Networks Positively invariant sets EMT $\left\{ ({{x}_{1}}, \cdots, {{x}_{6}})\in {{\mathbb{R}}^{6}}\left| 0 < {{x}_{i}} < \frac{1}{{{d}_{i}}}, i=1, \cdots, 6 \right. \right\}$ T helper \left\{x\in {{\mathbb{R}}^{23}}\left| \begin{aligned} & 0\le {{x}_{5}} < \frac{1}{{{d}_{4}}{{d}_{5}}}, 0\le {{x}_{6}} < \frac{1}{{{d}_{1}}{{d}_{6}}}, 0\le {{x}_{7}} < \frac{1}{{{d}_{1}}{{d}_{6}}{{d}_{7}}}, 0\le {{x}_{14}} < \frac{1}{{{d}_{11}}{{d}_{14}}}, \\ & 0\le {{x}_{17}} < ((\frac{1}{{{d}_{3}}}+\frac{1}{{{d}_{15}}})\frac{1}{{{d}_{18}}}+\frac{1}{{{d}_{22}}})\frac{1}{{{d}_{17}}}, 0\le {{x}_{18}} < (\frac{1}{{{d}_{3}}}+\frac{1}{{{d}_{15}}})\frac{1}{{{d}_{18}}}, \\ & 0\le {{x}_{19}} < \frac{1}{{{d}_{1}}{{d}_{6}}{{d}_{7}}{{d}_{19}}}, 0\le {{x}_{20}} < \frac{1}{{{d}_{9}}{{d}_{20}}}, 0\le {{x}_{21}} < \frac{1}{{{d}_{13}}{{d}_{21}}}, \\ & 0\le {{x}_{i}} < \frac{1}{{{d}_{i}}}, 0\le {{x}_{j}} < 1, i=1, 3, 4, 9, 11, 12, 13, 15, 16, j=8, 10, 23 \\ \end{aligned} \right. \right\} Cancer $\left\{ ({{x}_{1}}, \cdots, {{x}_{32}})\in {{\mathbb{R}}^{32}}\left| 0\le {{x}_{i}}\le 1, i=1, \cdots, 32 \right. \right\}$
The theoretically required driver nodes and designed control functions for realizing domain control of the three disease-related networks. $x_{2, i}^{*}$ is the $i$-th component of the state vector of the desired attractor
 Networks Driver nodes Control functions ($u(t)$) EMT miR-200 and miR-203 $\left[\begin{matrix} -\left(\frac{1}{1+{{K}_{52}}x_{2}^{2}+{{K}_{53}}x_{3}^{2}+{{K}_{54}}x_{4}^{2}}-{{d}_{5}}{{x}_{5}} \right)+\lambda \left({{x}_{5}}-x_{2, 5}^{*} \right) \\ -\left(\frac{1}{1+{{K}_{62}}x_{2}^{2}+{{K}_{63}}x_{3}^{2}+{{K}_{64}}x_{4}^{2}}-{{d}_{6}}{{x}_{6}} \right)+\lambda \left({{x}_{6}}-x_{2, 6}^{*} \right) \\ \end{matrix} \right]$ T helper GATA-3 and STAT1 $\left[\begin{matrix} -\left(\frac{{{k}_{1-1}}{{x}_{1}}+{{k}_{1-21}}{{x}_{21}}}{1+{{k}_{1-1}}{{x}_{1}}+{{k}_{1-21}}{{x}_{21}}+{{k}_{1-22}}{{x}_{22}}}-{{d}_{1}}{{x}_{1}} \right)+\lambda \left({{x}_{1}}-x_{2, 1}^{*} \right) \\ -\left({{k}_{18-3}}{{x}_{3}}+{{k}_{18-15}}{{x}_{15}}-{{d}_{18}}{{x}_{18}} \right)+\lambda \left({{x}_{18}}-x_{2, 18}^{*} \right) \\ \end{matrix} \right]$ Cancer P53, RB, AKT, EGFR, HIF1, CDK2, CDK1, BCL2 and NF$\kappa$B $\left[\begin{matrix} -\left(\frac{a\left({{s}_{1}}{{x}_{1}}^{n}+{{s}_{9}}{{x}_{9}}^{n}+{{s}_{14}}{{x}_{14}}^{n} \right)}{3{{S}^{n}}+{{s}_{1}}{{x}_{1}}^{n}+{{s}_{9}}{{x}_{9}}^{n}+{{s}_{14}}{{x}_{14}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+1/2{{s}_{16}}{{x}_{16}}^{n}+1/2{{s}_{31}}{{x}_{31}}^{n}}-k{{x}_{2}} \right)+\lambda \left({{x}_{2}}-x_{2, 2}^{*} \right) \\ -\left(\frac{2b{{S}^{n}}}{{{S}^{n}}+1/2{{s}_{17}}{{x}_{17}}^{n}+1/2{{s}_{18}}{{x}_{18}}^{n}}-k{{x}_{6}} \right)+\lambda \left({{x}_{6}}-x_{2, 6}^{*} \right) \\ -\left(\frac{a\left({{s}_{11}}{{x}_{11}}^{n}+{{s}_{12}}{{x}_{12}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)/3}{{{S}^{n}}+1/3{{s}_{11}}{{x}_{11}}^{n}+1/3{{s}_{12}}{{x}_{12}}^{n}+1/3{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{10}} \right)+\lambda \left({{x}_{10}}-x_{2, 10}^{*} \right) \\ -\left(\frac{a\left({{s}_{9}}{{x}_{9}}^{n}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{11}}{{x}_{11}}^{n}+{{s}_{13}}{{x}_{13}}^{n}+{{s}_{14}}{{x}_{14}}^{n}+{{s}_{15}}{{x}_{15}}^{n} \right)}{6{{S}^{n}}+{{s}_{9}}{{x}_{9}}^{n}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{11}}{{x}_{11}}^{n}+{{s}_{13}}{{x}_{13}}^{n}+{{s}_{14}}{{x}_{14}}^{n}+{{s}_{15}}{{x}_{15}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{2}}{{x}_{2}}^{n}}-k{{x}_{12}} \right)+\lambda \left({{x}_{12}}-x_{2, 12}^{*} \right) \\ -\left(\frac{a\left({{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n}+{{s}_{25}}{{x}_{25}}^{n}+{{s}_{26}}{{x}_{26}}^{n}+{{s}_{28}}{{x}_{28}}^{n} \right)}{5{{S}^{n}}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n}+{{s}_{25}}{{x}_{25}}^{n}+{{s}_{26}}{{x}_{26}}^{n}+{{s}_{28}}{{x}_{28}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{2}}{{x}_{2}}^{n}}-k{{x}_{14}} \right)+\lambda \left({{x}_{14}}-x_{2, 14}^{*} \right) \\ -\left(\frac{a\left({{s}_{9}}{{x}_{9}}^{n}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{20}}{{x}_{20}}^{n} \right)}{3{{S}^{n}}+{{s}_{9}}{{x}_{9}}^{n}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{20}}{{x}_{20}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+1/2{{s}_{3}}{{x}_{3}}^{n}+1/2{{s}_{6}}{{x}_{6}}^{n}}-k{{x}_{17}} \right)+\lambda \left({{x}_{17}}-x_{2, 17}^{*} \right) \\ -\left(\frac{2b{{S}^{n}}}{{{S}^{n}}+1/3{{s}_{3}}{{x}_{3}}^{n}+1/3{{s}_{5}}{{x}_{5}}^{n}+1/3{{s}_{30}}{{x}_{30}}^{n}}-k{{x}_{19}} \right)+\lambda \left({{x}_{19}}-x_{2, 19}^{*} \right) \\ -\left(\frac{a\left({{s}_{22}}{{x}_{22}}^{n}+{{s}_{23}}{{x}_{23}}^{n} \right)/2}{{{S}^{n}}+1/2{{s}_{22}}{{x}_{22}}^{n}+1/2{{s}_{23}}{{x}_{23}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+1/2{{s}_{10}}{{x}_{10}}^{n}+1/2{{s}_{24}}{{x}_{24}}^{n}}-k{{x}_{21}} \right)+\lambda \left({{x}_{21}}-x_{2, 21}^{*} \right) \\ -\left(\frac{a\left({{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)/2}{{{S}^{n}}+1/2{{s}_{10}}{{x}_{10}}^{n}+1/2{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{25}} \right)+\lambda \left({{x}_{25}}-x_{2, 25}^{*} \right) \\ \end{matrix} \right]$
 Networks Driver nodes Control functions ($u(t)$) EMT miR-200 and miR-203 $\left[\begin{matrix} -\left(\frac{1}{1+{{K}_{52}}x_{2}^{2}+{{K}_{53}}x_{3}^{2}+{{K}_{54}}x_{4}^{2}}-{{d}_{5}}{{x}_{5}} \right)+\lambda \left({{x}_{5}}-x_{2, 5}^{*} \right) \\ -\left(\frac{1}{1+{{K}_{62}}x_{2}^{2}+{{K}_{63}}x_{3}^{2}+{{K}_{64}}x_{4}^{2}}-{{d}_{6}}{{x}_{6}} \right)+\lambda \left({{x}_{6}}-x_{2, 6}^{*} \right) \\ \end{matrix} \right]$ T helper GATA-3 and STAT1 $\left[\begin{matrix} -\left(\frac{{{k}_{1-1}}{{x}_{1}}+{{k}_{1-21}}{{x}_{21}}}{1+{{k}_{1-1}}{{x}_{1}}+{{k}_{1-21}}{{x}_{21}}+{{k}_{1-22}}{{x}_{22}}}-{{d}_{1}}{{x}_{1}} \right)+\lambda \left({{x}_{1}}-x_{2, 1}^{*} \right) \\ -\left({{k}_{18-3}}{{x}_{3}}+{{k}_{18-15}}{{x}_{15}}-{{d}_{18}}{{x}_{18}} \right)+\lambda \left({{x}_{18}}-x_{2, 18}^{*} \right) \\ \end{matrix} \right]$ Cancer P53, RB, AKT, EGFR, HIF1, CDK2, CDK1, BCL2 and NF$\kappa$B $\left[\begin{matrix} -\left(\frac{a\left({{s}_{1}}{{x}_{1}}^{n}+{{s}_{9}}{{x}_{9}}^{n}+{{s}_{14}}{{x}_{14}}^{n} \right)}{3{{S}^{n}}+{{s}_{1}}{{x}_{1}}^{n}+{{s}_{9}}{{x}_{9}}^{n}+{{s}_{14}}{{x}_{14}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+1/2{{s}_{16}}{{x}_{16}}^{n}+1/2{{s}_{31}}{{x}_{31}}^{n}}-k{{x}_{2}} \right)+\lambda \left({{x}_{2}}-x_{2, 2}^{*} \right) \\ -\left(\frac{2b{{S}^{n}}}{{{S}^{n}}+1/2{{s}_{17}}{{x}_{17}}^{n}+1/2{{s}_{18}}{{x}_{18}}^{n}}-k{{x}_{6}} \right)+\lambda \left({{x}_{6}}-x_{2, 6}^{*} \right) \\ -\left(\frac{a\left({{s}_{11}}{{x}_{11}}^{n}+{{s}_{12}}{{x}_{12}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)/3}{{{S}^{n}}+1/3{{s}_{11}}{{x}_{11}}^{n}+1/3{{s}_{12}}{{x}_{12}}^{n}+1/3{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{10}} \right)+\lambda \left({{x}_{10}}-x_{2, 10}^{*} \right) \\ -\left(\frac{a\left({{s}_{9}}{{x}_{9}}^{n}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{11}}{{x}_{11}}^{n}+{{s}_{13}}{{x}_{13}}^{n}+{{s}_{14}}{{x}_{14}}^{n}+{{s}_{15}}{{x}_{15}}^{n} \right)}{6{{S}^{n}}+{{s}_{9}}{{x}_{9}}^{n}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{11}}{{x}_{11}}^{n}+{{s}_{13}}{{x}_{13}}^{n}+{{s}_{14}}{{x}_{14}}^{n}+{{s}_{15}}{{x}_{15}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{2}}{{x}_{2}}^{n}}-k{{x}_{12}} \right)+\lambda \left({{x}_{12}}-x_{2, 12}^{*} \right) \\ -\left(\frac{a\left({{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n}+{{s}_{25}}{{x}_{25}}^{n}+{{s}_{26}}{{x}_{26}}^{n}+{{s}_{28}}{{x}_{28}}^{n} \right)}{5{{S}^{n}}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n}+{{s}_{25}}{{x}_{25}}^{n}+{{s}_{26}}{{x}_{26}}^{n}+{{s}_{28}}{{x}_{28}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{2}}{{x}_{2}}^{n}}-k{{x}_{14}} \right)+\lambda \left({{x}_{14}}-x_{2, 14}^{*} \right) \\ -\left(\frac{a\left({{s}_{9}}{{x}_{9}}^{n}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{20}}{{x}_{20}}^{n} \right)}{3{{S}^{n}}+{{s}_{9}}{{x}_{9}}^{n}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{20}}{{x}_{20}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+1/2{{s}_{3}}{{x}_{3}}^{n}+1/2{{s}_{6}}{{x}_{6}}^{n}}-k{{x}_{17}} \right)+\lambda \left({{x}_{17}}-x_{2, 17}^{*} \right) \\ -\left(\frac{2b{{S}^{n}}}{{{S}^{n}}+1/3{{s}_{3}}{{x}_{3}}^{n}+1/3{{s}_{5}}{{x}_{5}}^{n}+1/3{{s}_{30}}{{x}_{30}}^{n}}-k{{x}_{19}} \right)+\lambda \left({{x}_{19}}-x_{2, 19}^{*} \right) \\ -\left(\frac{a\left({{s}_{22}}{{x}_{22}}^{n}+{{s}_{23}}{{x}_{23}}^{n} \right)/2}{{{S}^{n}}+1/2{{s}_{22}}{{x}_{22}}^{n}+1/2{{s}_{23}}{{x}_{23}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+1/2{{s}_{10}}{{x}_{10}}^{n}+1/2{{s}_{24}}{{x}_{24}}^{n}}-k{{x}_{21}} \right)+\lambda \left({{x}_{21}}-x_{2, 21}^{*} \right) \\ -\left(\frac{a\left({{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)/2}{{{S}^{n}}+1/2{{s}_{10}}{{x}_{10}}^{n}+1/2{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{25}} \right)+\lambda \left({{x}_{25}}-x_{2, 25}^{*} \right) \\ \end{matrix} \right]$
An example of the practically required driver nodes and designed control functions for realizing domain control of the three disease-related networks. E and M represent the epithelial state and mesenchymal state, respectively. C, A and N represent the cancer, apoptosis and normal states, respectively. $x_{2, i}^{*}$ is the $i$-th component of the state vector of the desired attractor
 Transitions Drivers Control functions ($u(t)$) E$\leftrightarrow$M miR-203 ${-\left(\frac{1}{1+{{K}_{62}}x_{2}^{2}+{{K}_{63}}x_{3}^{2}+{{K}_{64}}x_{4}^{2}}-{{d}_{6}}{{x}_{6}} \right)+\lambda \left({{x}_{6}}-x_{2, 6}^{*} \right)}$ Th0$\leftrightarrow$Th1 T-bet ${-\left(\frac{{{k}_{22-18}}{{x}_{18}}+{{k}_{22-22}}{{x}_{22}}}{1+{{k}_{22-1}}{{x}_{1}}+{{k}_{22-22}}{{x}_{22}}+{{k}_{22-18}}{{x}_{18}}}-{{d}_{22}}{{x}_{22}} \right)+\lambda \left({{x}_{22}}-x_{2, 22}^{*} \right)}$ Th0$\leftrightarrow$Th2 GATA-3 ${-\left(\frac{{{k}_{1-1}}{{x}_{1}}+{{k}_{1-21}}{{x}_{21}}}{1+{{k}_{1-1}}{{x}_{1}}+{{k}_{1-21}}{{x}_{21}}+{{k}_{1-22}}{{x}_{22}}}-{{d}_{1}}{{x}_{1}} \right)+\lambda \left({{x}_{1}}-x_{2, 1}^{*} \right)}$ C$\leftrightarrow$A AKT ${-\left(\frac{a\left({{s}_{11}}{{x}_{11}}^{n}+{{s}_{12}}{{x}_{12}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)}{3{{S}^{n}}+{{s}_{11}}{{x}_{11}}^{n}+{{s}_{12}}{{x}_{12}}^{n}+{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{10}} \right)+\lambda \left({{x}_{10}}-x_{2, 10}^{*} \right)}$ C$\leftrightarrow$N RB ${-\left(\frac{4b{{S}^{n}}}{2{{S}^{n}}+{{s}_{17}}{{x}_{17}}^{n}+{{s}_{18}}{{x}_{18}}^{n}}-k{{x}_{6}} \right)+\lambda \left({{x}_{6}}-x_{2, 6}^{*} \right)}$ A$\leftrightarrow$N NF$\kappa$B ${-\left(\frac{a\left({{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)}{2{{S}^{n}}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{25}} \right)+\lambda \left({{x}_{25}}-x_{2, 25}^{*} \right)}$
 Transitions Drivers Control functions ($u(t)$) E$\leftrightarrow$M miR-203 ${-\left(\frac{1}{1+{{K}_{62}}x_{2}^{2}+{{K}_{63}}x_{3}^{2}+{{K}_{64}}x_{4}^{2}}-{{d}_{6}}{{x}_{6}} \right)+\lambda \left({{x}_{6}}-x_{2, 6}^{*} \right)}$ Th0$\leftrightarrow$Th1 T-bet ${-\left(\frac{{{k}_{22-18}}{{x}_{18}}+{{k}_{22-22}}{{x}_{22}}}{1+{{k}_{22-1}}{{x}_{1}}+{{k}_{22-22}}{{x}_{22}}+{{k}_{22-18}}{{x}_{18}}}-{{d}_{22}}{{x}_{22}} \right)+\lambda \left({{x}_{22}}-x_{2, 22}^{*} \right)}$ Th0$\leftrightarrow$Th2 GATA-3 ${-\left(\frac{{{k}_{1-1}}{{x}_{1}}+{{k}_{1-21}}{{x}_{21}}}{1+{{k}_{1-1}}{{x}_{1}}+{{k}_{1-21}}{{x}_{21}}+{{k}_{1-22}}{{x}_{22}}}-{{d}_{1}}{{x}_{1}} \right)+\lambda \left({{x}_{1}}-x_{2, 1}^{*} \right)}$ C$\leftrightarrow$A AKT ${-\left(\frac{a\left({{s}_{11}}{{x}_{11}}^{n}+{{s}_{12}}{{x}_{12}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)}{3{{S}^{n}}+{{s}_{11}}{{x}_{11}}^{n}+{{s}_{12}}{{x}_{12}}^{n}+{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{10}} \right)+\lambda \left({{x}_{10}}-x_{2, 10}^{*} \right)}$ C$\leftrightarrow$N RB ${-\left(\frac{4b{{S}^{n}}}{2{{S}^{n}}+{{s}_{17}}{{x}_{17}}^{n}+{{s}_{18}}{{x}_{18}}^{n}}-k{{x}_{6}} \right)+\lambda \left({{x}_{6}}-x_{2, 6}^{*} \right)}$ A$\leftrightarrow$N NF$\kappa$B ${-\left(\frac{a\left({{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)}{2{{S}^{n}}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{25}} \right)+\lambda \left({{x}_{25}}-x_{2, 25}^{*} \right)}$
The attractors of the Th differentiation cellular network. The three observed attractors (A, B and C) correspond to the Th0, Th1 and Th2 phenotypes [17], respectively
 A B C A B C $x_{1}$ (GATA-3) 0.0000 0.0000 2.0238 $x_{13}$ (IL-4R) 0.0000 0.0000 1.3358 $x_{2}$ (IFN-$\beta$) 0.0000 0.0000 0.0000 $x_{14}$ (IRAK) 0.0000 0.0000 0.0000 $x_{3}$ (IFN-$\beta$R) 0.0000 0.0000 0.0000 $x_{15}$ (JAK1) 0.0000 0.4406 0.0000 $x_{4}$ (IFN-$\gamma$) 0.0000 2.1012 0.0000 $x_{16}$ (NFAT) 0.0000 0.0000 0.0000 $x_{5}$ (IFN-$\gamma$R) 0.0000 2.1012 0.0000 $x_{17}$ (SOCS1) 0.0000 2.1459 0.0000 $x_{6}$ (IL-10) 0.0000 0.0000 2.0238 $x_{18}$ (STAT1) 0.0000 0.4406 0.0000 $x_{7}$ (IL-10R) 0.0000 0.0000 2.0238 $x_{19}$ (STAT3) 0.0000 0.0000 2.0238 $x_{8}$ (IL-12) 0.0000 0.0000 0.0000 $x_{20}$ (STAT4) 0.0000 0.0000 0.0000 $x_{9}$ (IL-12R) 0.0000 0.0000 0.0000 $x_{21}$ (STAT6) 0.0000 0.0000 2.2264 $x_{10}$ (IL-18) 0.0000 0.0000 0.0000 $x_{22}$ (T-bet) 0.0000 1.7053 0.0000 $x_{11}$ (IL-18R) 0.0000 0.0000 0.0000 $x_{23}$ (TCR) 0.0000 0.0000 0.0000 $x_{12}$ (IL-4) 0.0000 0.0000 2.0094
 A B C A B C $x_{1}$ (GATA-3) 0.0000 0.0000 2.0238 $x_{13}$ (IL-4R) 0.0000 0.0000 1.3358 $x_{2}$ (IFN-$\beta$) 0.0000 0.0000 0.0000 $x_{14}$ (IRAK) 0.0000 0.0000 0.0000 $x_{3}$ (IFN-$\beta$R) 0.0000 0.0000 0.0000 $x_{15}$ (JAK1) 0.0000 0.4406 0.0000 $x_{4}$ (IFN-$\gamma$) 0.0000 2.1012 0.0000 $x_{16}$ (NFAT) 0.0000 0.0000 0.0000 $x_{5}$ (IFN-$\gamma$R) 0.0000 2.1012 0.0000 $x_{17}$ (SOCS1) 0.0000 2.1459 0.0000 $x_{6}$ (IL-10) 0.0000 0.0000 2.0238 $x_{18}$ (STAT1) 0.0000 0.4406 0.0000 $x_{7}$ (IL-10R) 0.0000 0.0000 2.0238 $x_{19}$ (STAT3) 0.0000 0.0000 2.0238 $x_{8}$ (IL-12) 0.0000 0.0000 0.0000 $x_{20}$ (STAT4) 0.0000 0.0000 0.0000 $x_{9}$ (IL-12R) 0.0000 0.0000 0.0000 $x_{21}$ (STAT6) 0.0000 0.0000 2.2264 $x_{10}$ (IL-18) 0.0000 0.0000 0.0000 $x_{22}$ (T-bet) 0.0000 1.7053 0.0000 $x_{11}$ (IL-18R) 0.0000 0.0000 0.0000 $x_{23}$ (TCR) 0.0000 0.0000 0.0000 $x_{12}$ (IL-4) 0.0000 0.0000 2.0094
The attractors of the cancer network. The three observed attractors (A, B and C) correspond to the apoptosis, normal and cancer states [33], respectively.
 A B C A B C $x_1$ (ATM) 0.4165 0.4277 0.4712 $x_{17}$ (CDK2) 0.1336 0.4613 0.8022 $x_2$ (P53) 0.4668 0.4642 0.4545 $x_{18}$ (CDK4) 0.1342 0.4273 0.8620 $x_3$ (P21) 0.5700 0.4511 0.4378 $x_{19}$ (CDK1) 0.5788 0.4853 0.5550 $x_4$ (PTEN) 0.7438 0.2973 0.2836 $x_{20}$ (E2F1) 0.1922 0.2620 0.3441 $x_5$ (CDH1) 0.3215 0.6263 0.5298 $x_{21}$ (Caspase) 0.8766 0.0688 0.0621 $x_6$ (RB) 0.9970 0.7105 0.1921 $x_{22}$ (BAX) 0.7159 0.2641 0.2498 $x_7$ (ARF) 0.2756 0.2645 0.3087 $x_{23}$ (BAD) 0.8486 0.1024 0.0923 $x_8$ (AR) 0.4134 0.2439 0.1955 $x_{24}$ (BCL2) 0.1740 0.7533 0.7705 $x_9$ (MYC) 0.6647 0.4760 0.4758 $x_{25}$ (NF$\kappa$B) 0.1433 0.8853 0.9007 $x_{10}$ (AKT) 0.3044 0.8058 0.8294 $x_{26}$ (RAS) 0.4089 0.5158 0.5427 $x_{11}$ (EGFR) 0.5262 0.4606 0.4636 $x_{27}$ (TGF$\alpha$) 0.0000 0.0000 0.0000 $x_{12}$ (VEGF) 0.4145 0.6239 0.6464 $x_{28}$ (TNF$\alpha$) 0.0000 0.0000 0.0000 $x_{13}$ (HGF) 0.1484 0.5908 0.6214 $x_{29}$ (TGF$\beta$) 0.0806 0.6772 0.7016 $x_{14}$ (HIF1) 0.2998 0.6632 0.6823 $x_{30}$ (Wee1) 0.6282 0.4550 0.5882 $x_{15}$ (hTERT) 0.3765 0.4680 0.4702 $x_{31}$ (MdmX) 0.6915 0.8148 0.6413 $x_{16}$ (MDM2) 0.2471 0.4911 0.7550 $x_{32}$ (Wip1) 0.4933 0.4877 0.4666
 A B C A B C $x_1$ (ATM) 0.4165 0.4277 0.4712 $x_{17}$ (CDK2) 0.1336 0.4613 0.8022 $x_2$ (P53) 0.4668 0.4642 0.4545 $x_{18}$ (CDK4) 0.1342 0.4273 0.8620 $x_3$ (P21) 0.5700 0.4511 0.4378 $x_{19}$ (CDK1) 0.5788 0.4853 0.5550 $x_4$ (PTEN) 0.7438 0.2973 0.2836 $x_{20}$ (E2F1) 0.1922 0.2620 0.3441 $x_5$ (CDH1) 0.3215 0.6263 0.5298 $x_{21}$ (Caspase) 0.8766 0.0688 0.0621 $x_6$ (RB) 0.9970 0.7105 0.1921 $x_{22}$ (BAX) 0.7159 0.2641 0.2498 $x_7$ (ARF) 0.2756 0.2645 0.3087 $x_{23}$ (BAD) 0.8486 0.1024 0.0923 $x_8$ (AR) 0.4134 0.2439 0.1955 $x_{24}$ (BCL2) 0.1740 0.7533 0.7705 $x_9$ (MYC) 0.6647 0.4760 0.4758 $x_{25}$ (NF$\kappa$B) 0.1433 0.8853 0.9007 $x_{10}$ (AKT) 0.3044 0.8058 0.8294 $x_{26}$ (RAS) 0.4089 0.5158 0.5427 $x_{11}$ (EGFR) 0.5262 0.4606 0.4636 $x_{27}$ (TGF$\alpha$) 0.0000 0.0000 0.0000 $x_{12}$ (VEGF) 0.4145 0.6239 0.6464 $x_{28}$ (TNF$\alpha$) 0.0000 0.0000 0.0000 $x_{13}$ (HGF) 0.1484 0.5908 0.6214 $x_{29}$ (TGF$\beta$) 0.0806 0.6772 0.7016 $x_{14}$ (HIF1) 0.2998 0.6632 0.6823 $x_{30}$ (Wee1) 0.6282 0.4550 0.5882 $x_{15}$ (hTERT) 0.3765 0.4680 0.4702 $x_{31}$ (MdmX) 0.6915 0.8148 0.6413 $x_{16}$ (MDM2) 0.2471 0.4911 0.7550 $x_{32}$ (Wip1) 0.4933 0.4877 0.4666
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