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Averaging principle for the Schrödinger equations†
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 |
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.
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K. Axelrod, A. Sanchez and J. Gore, Phenotypic states become increasingly sensitive to perturbations near a bifurcation in a synthetic gene network, eLife, 4 (2015), e07935. |
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S. Balint, E. Kaslik, A. M. Balint and A. Grigis, Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems, Adv. Differ. Equ., 2006 (2006), Art. 23939, 1-15. |
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Y. Bar-Yam, D. Harmon and B. de Bivort, Attractors and democratic dynamics, Science, 323 (2009), 1016-1017. |
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A.-L. Barabási, N. Gulbahce and J. Loscalzo, Network medicine: A network-based approach to human disease, Nat. Rev. Genet., 12 (2011), 56-68. |
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R. G. Bartle, The Elements of Integration and Lebesgue Measure, (John Wiley & Sons), 2011. |
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B. Barzel and A.-L. Barabási, Universality in network dynamics, Nat. Phys., 9 (2013), 673-681. |
[9] |
Y. Ben-Neriah and M. Karin, Inflammation meets cancer, with nf-κb as the matchmaker, Nat.Immunol., 12 (2011), 715-723. |
[10] |
G. Chen and X. Yu, Chaos Control: Theory and Applications, Springer-Verlag Berlin Heidelberg, Berlin, 2003. |
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Y. -Z. Chen, L. Wang, W. Wang and Y. -C. Lai, The paradox of controlling complex networks: Control inputs versus energy requirement, preprint, arXiv: 1509.03196. |
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S. P. Cornelius, W. L. Kath and A. E. Motter, Realistic control of network dynamics, Nat. Commun., 4 (2013), 1942. |
[13] |
P. Creixell, E. M. Schoof, J. T. Erler and R. Linding, Navigating cancer network attractors for tumor-specific therapy, Nat. Biotechnol., 30 (2012), 842-848. |
[14] |
P. Csermely, T. Korcsmáros, H. J. Kiss, G. London and R. Nussinov, Structure and dynamics of molecular networks: A novel paradigm of drug discovery: A comprehensive review, Pharmacol & Therapeut, 138 (2013), 333-408. |
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A. Di Cara, A. Garg, G. De Micheli, I. Xenarios and L. Mendoza, Dynamic simulation of regulatory networks using SQUAD, BMC Bioinformatics, 8 (2007), p462. |
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B. Fiedler, A. Mochizuki, G. Kurosawa and D. Saito, Dynamics and control at feedback vertex sets. Ⅰ: informative and determining nodes in regulatory networks, J. Dyn. Differ. Equ., 25 (2013), 563-604. |
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T. S. Gardner, C. R. Cantor and J. J. Collins, Construction of a genetic toggle switch in Escherichia coli, Nature, 403 (2000), 339-342. |
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J. Gao, Y.-Y. Liu, R. M. D'Souza and A.-L. Barabási, Target control of complex networks, Nat. Commun., 5 (2014), p5415. |
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S. Huang, I. Ernberg and S. Kauffman, Cancer attractors: a systems view of tumors from a gene network dynamics and developmental perspective, Semin. Cell Dev. Biol., 20 (2009), 869-876. |
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E. S. Hwang, S. J. Szabo, P. L. Schwartzberg and L. H. Glimcher, T helper cell fate specified by kinase-mediated interaction of T-bet with GATA-3, Science, 307 (2005), 430-433. |
[24] |
R. G. Jenner, M. J. Townsend, I. Jackson, K. Sun, R. D. Bouwman, R. A. Young, L. H. Glimcher and G. M. Lord, The transcription factors T-bet and GATA-3 control alternative pathways of T-cell differentiation through a shared set of target genes, Proc. Natl. Acad. Sci. U. S. A., 106 (2009), 17876-17881. |
[25] |
S. Jin, Y. Li, R. Pan and X. Zou, Characterizing and controlling the inflammatory network during influenza a virus infection, Sci. Rep., 4 (2014), p3799. |
[26] |
S. Jin, L. Niu, G. Wang and X. Zou, Mathematical modeling and nonlinear dynamical analysis of cell growth in response to antibiotics, Int. J. Bifurcat. Chaos, 25 (2015), 1540007, 12pp. |
[27] |
J. D. Jordan, E. M. Landau and R. Iyengar, Signaling networks: The origins of cellular multitasking, Cell, 103 (2000), 193-200. |
[28] |
R. Kalluri and R. A. Weinberg, The basics of epithelial-mesenchymal transition, J. Clin. Invest., 119 (2009), 1420-1428. |
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S. Karl and T. Dandekar, Convergence behaviour and control in non-linear biological networks, Sci. Rep., 5 (2015), p9746. |
[30] |
H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, New Jersey, 2002. |
[31] |
Y. C. Lai, Controlling complex, non-linear dynamical networks, National Science Review, 1 (2014), 339-341. |
[32] |
S. Lamouille, J. Xu and R. Derynck, Molecular mechanisms of epithelial-mesenchymal transition, Nat. Rev. Mol. Cell Biol., 15 (2014), 178-196. |
[33] |
C. Li and J. Wang, Quantifying the underlying landscape and paths of cancer, J. R. Soc. Interface, 11 (2014), 20140774. |
[34] |
Y. Li, S. Jin, L. Lei, Z. Pan and X. Zou, Deciphering deterioration mechanisms of complex diseases based on the construction of dynamic networks and systems analysis, Sci. Rep., 5 (2015), p9283. |
[35] |
Y. Li, M. Yi and X. Zou, The linear interplay of intrinsic and extrinsic noises ensures a high accuracy of cell fate selection in budding yeast, Sci. Rep., 4 (2014), p5764. |
[36] |
Y. Y. Liu and A. L. Barabasi, Control principles of complex systems, Rev. Mod. Phys., 88 (2016), p035006. |
[37] |
Y. Y. Liu, J. J. Slotine and A. L. Barabasi, Controllability of complex networks, Nature, 473 (2011), 167-173. |
[38] |
L. G. Matallana, A. M. Blanco and J. A. Bandoni, Estimation of domains of attraction: A global optimization approach, Math. Comput. Model., 52 (2010), 574-585. |
[39] |
L. Mendoza, A network model for the control of the differentiation process in Th cells, Biosystems, 84 (2006), 101-114. |
[40] |
A. Mochizuki, B. Fiedler, G. Kurosawa and D. Saito, Dynamics and control at feedback vertex sets. Ⅱ: A faithful monitor to determine the diversity of molecular activities in regulatory networks, J. Theor. Biol., 335 (2013), 130-146. |
[41] |
M. Moes, A. Le Béchec, I. Crespo, C. Laurini, A. Halavatyi, G. Vetter, A. Del Sol and E. Friederich, A novel network integrating a miRNA-203/SNAI1 feedback loop which regulates epithelial to mesenchymal transition, PloS One, 7 (2012), e35440. |
[42] |
F.-J. Müller and A. Schuppert, Few inputs can reprogram biological networks, Nature, 478 (2011), E4-E4. |
[43] |
T. Nepusz and T. Vicsek, Controlling edge dynamics in complex networks, Nat. Phys., 8 (2012), 568-573. |
[44] |
J. A. Papin, T. Hunter, B. O. Palsson and S. Subramaniam, Reconstruction of cellular signalling networks and analysis of their properties, Nat. Rev. Mol. Cell Biol., 6 (2005), 99-111. |
[45] |
J. Pei, N. Yin, X. Ma and L. Lai, Systems biology brings new dimensions for structure-based drug design, J. Am. Chem. Soc., 136 (2014), 11556-11565. |
[46] |
M. Pósfai and P. Hövel, Structural controllability of temporal networks, New J. Phys., 16 (2014), 123055. |
[47] |
J. Ruths and D. Ruths, Control profiles of complex networks, Science, 343 (2014), 1373-1376. |
[48] |
J. -J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, New Jersey, 1991. |
[49] |
M. S. Song, L. Salmena and P. P. Pandolfi, The functions and regulation of the pten tumour suppressor, Nat. Rev. Mol. Cell Biol., 13 (2012), 283-296. |
[50] |
F. Sorrentino, M. di Bernardo, F. Garofalo and G. R. Chen, Controllability of complex networks via pinning, Phys. Rev. E, 75 (2007), 046103. |
[51] |
G. Strang, Calculus, Wellesley-Cambridge Press, Massachusetts, 1991.
![]() |
[52] |
J. Tan and X. Zou, Complex dynamical analysis of a coupled network from innate immune responses, Int. J. Bifurcat. Chaos, 23 (2013), 1350180, 26pp. |
[53] |
J. Tan and X. Zou, Optimal control strategy for abnormal innate immune response, Comput. Math. Method M., 2015 (2015), Art. ID 386235, 16 pp. |
[54] |
A. Vinayagam, T. E. Gibson, H.-J. Lee, B. Yilmazel, C. Roesel, Y. Hu, Y. Kwon, A. Sharma, Y.-Y. Liu, N. Perrimon and A.-L. Barabási, Controllability analysis of the directed human protein inteorkraction netw identifies disease genes and drug targets, Proc. Natl. Acad. Sci. U. S. A., 113 (2016), 4976-4981. |
[55] |
D. Wang, S. Jin, F. X. Wu and X. Zou, Estimation of control energy and control strategies for complex networks, Adv. Complex Syst., 18 (2015), 1550018, 23pp. |
[56] |
D. Wang, S. Jin and X. Zou, Crosstalk between pathways enhances the controllability of signalling networks, IET Syst. Biol., 10 (2016), 2-9. |
[57] |
L. Z. Wang, R. Q. Su, Z. G. Huang, X. Wang, W. X. Wang, C. Grebogi and Y. C. Lai, A geometrical approach to control and controllability of nonlinear dynamical networks, Nat. Commun., 7 (2016), p11323. |
[58] |
Y. Wang, J. Tan, F. Sadre-Marandi, J. Liu and X. Zou, Mathematical modeling for intracellular transport and binding of HIV-1 gag proteins, Math. Biosci., 262 (2015), 198-205. |
[59] |
R. Weinberg, The Biology of Cancer, 2nd edition, Garland Science, New York, 2013. |
[60] |
U. Wellner, J. Schubert, U. C. Burk, O. Schmalhofer, F. Zhu, A. Sonntag, B. Waldvogel, C. Vannier, D. Darling, A. zur Hausen et al., The EMT-activator ZEB1 promotes tumorigenicity by repressing stemness-inhibiting micrornas, Nat. Cell Biol., 11 (2009), 1487-1495. |
[61] |
D. K. Wells, W. L. Kath and A. E. Motter, Control of stochastic and induced switching in biophysical networks, Phys. Rev. X, 5 (2015), 031036. |
[62] |
A. J. Whalen, S. N. Brennan, T. D. Sauer and S. J. Schiff, Observability and controllability of nonlinear networks: The role of symmetry, Phys. Rev. X, 5 (2015), 011005. |
[63] |
F.-X. Wu, L. Wu, J. Wang, J. Liu and L. Chen, Transittability of complex networks and its applications to regulatory biomolecular networks, Sci. Rep., 4 (2014), p4819. |
[64] |
S. Wuchty, Controllability in protein interaction networks, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 7156-7160. |
[65] |
G. Yan, G. Tsekenis, B. Barzel, J.-J. Slotine, Y.-Y. Liu and A.-L. Barabási, Spectrum of controlling and observing complex networks, Nat. Phys., 11 (2015), 779-786. |
[66] |
Z. Yuan, C. Zhao, Z. Di, W.-X. Wang and Y.-C. Lai, Exact controllability of complex networks, Nat. Commun., 4 (2013), p2447. |
[67] |
J. G. Zañudo, G. Yang and R. Albert, Structure-based control of complex networks with nonlinear dynamics, arXiv: 1605.08415v2. |
[68] |
J. Zhang, Z. Yuan, H. X. Li and T. Zhou, Architecture-dependent robustness and bistability in a class of genetic circuits, Biophys. J., 99 (2010), 1034-1042. |
[69] |
N. Zhong, Computational unsolvability of domains of attraction of nonlinear systems, Proc. Amer. Math. Soc., 137 (2009), 2773-2783. |
show all references
References:
[1] |
P. Ao, D. Galas, L. Hood and X. M. Zhu, Cancer as robust intrinsic state of endogenous molecular-cellular network shaped by evolution, Med. Hypotheses, 70 (2008), 678-684. |
[2] |
R. P. Araujo, L. A. Liotta and E. F. Petricoin, Proteins, drug targets and the mechanisms they control: The simple truth about complex networks, Nat. Rev. Drug Discov., 6 (2007), 871-880. |
[3] |
K. Axelrod, A. Sanchez and J. Gore, Phenotypic states become increasingly sensitive to perturbations near a bifurcation in a synthetic gene network, eLife, 4 (2015), e07935. |
[4] |
S. Balint, E. Kaslik, A. M. Balint and A. Grigis, Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems, Adv. Differ. Equ., 2006 (2006), Art. 23939, 1-15. |
[5] |
Y. Bar-Yam, D. Harmon and B. de Bivort, Attractors and democratic dynamics, Science, 323 (2009), 1016-1017. |
[6] |
A.-L. Barabási, N. Gulbahce and J. Loscalzo, Network medicine: A network-based approach to human disease, Nat. Rev. Genet., 12 (2011), 56-68. |
[7] |
R. G. Bartle, The Elements of Integration and Lebesgue Measure, (John Wiley & Sons), 2011. |
[8] |
B. Barzel and A.-L. Barabási, Universality in network dynamics, Nat. Phys., 9 (2013), 673-681. |
[9] |
Y. Ben-Neriah and M. Karin, Inflammation meets cancer, with nf-κb as the matchmaker, Nat.Immunol., 12 (2011), 715-723. |
[10] |
G. Chen and X. Yu, Chaos Control: Theory and Applications, Springer-Verlag Berlin Heidelberg, Berlin, 2003. |
[11] |
Y. -Z. Chen, L. Wang, W. Wang and Y. -C. Lai, The paradox of controlling complex networks: Control inputs versus energy requirement, preprint, arXiv: 1509.03196. |
[12] |
S. P. Cornelius, W. L. Kath and A. E. Motter, Realistic control of network dynamics, Nat. Commun., 4 (2013), 1942. |
[13] |
P. Creixell, E. M. Schoof, J. T. Erler and R. Linding, Navigating cancer network attractors for tumor-specific therapy, Nat. Biotechnol., 30 (2012), 842-848. |
[14] |
P. Csermely, T. Korcsmáros, H. J. Kiss, G. London and R. Nussinov, Structure and dynamics of molecular networks: A novel paradigm of drug discovery: A comprehensive review, Pharmacol & Therapeut, 138 (2013), 333-408. |
[15] |
B. De Craene and G. Berx, Regulatory networks defining EMT during cancer initiation and progression, Nat. Rev. Cancer, 13 (2013), 97-110. |
[16] |
H. De Jong, Modeling and simulation of genetic regulatory systems: A literature review, J. Comput. Biol., 9 (2002), 67-103. |
[17] |
A. Di Cara, A. Garg, G. De Micheli, I. Xenarios and L. Mendoza, Dynamic simulation of regulatory networks using SQUAD, BMC Bioinformatics, 8 (2007), p462. |
[18] |
B. Fiedler, A. Mochizuki, G. Kurosawa and D. Saito, Dynamics and control at feedback vertex sets. Ⅰ: informative and determining nodes in regulatory networks, J. Dyn. Differ. Equ., 25 (2013), 563-604. |
[19] |
T. S. Gardner, C. R. Cantor and J. J. Collins, Construction of a genetic toggle switch in Escherichia coli, Nature, 403 (2000), 339-342. |
[20] |
J. Gao, Y.-Y. Liu, R. M. D'Souza and A.-L. Barabási, Target control of complex networks, Nat. Commun., 5 (2014), p5415. |
[21] |
B. T. Hennessy, D. L. Smith, P. T. Ram, Y. Lu and G. B. Mills, Exploiting the PI3K/AKT pathway for cancer drug discovery, Nat. Rev. Drug Discov., 4 (2005), 988-1004. |
[22] |
S. Huang, I. Ernberg and S. Kauffman, Cancer attractors: a systems view of tumors from a gene network dynamics and developmental perspective, Semin. Cell Dev. Biol., 20 (2009), 869-876. |
[23] |
E. S. Hwang, S. J. Szabo, P. L. Schwartzberg and L. H. Glimcher, T helper cell fate specified by kinase-mediated interaction of T-bet with GATA-3, Science, 307 (2005), 430-433. |
[24] |
R. G. Jenner, M. J. Townsend, I. Jackson, K. Sun, R. D. Bouwman, R. A. Young, L. H. Glimcher and G. M. Lord, The transcription factors T-bet and GATA-3 control alternative pathways of T-cell differentiation through a shared set of target genes, Proc. Natl. Acad. Sci. U. S. A., 106 (2009), 17876-17881. |
[25] |
S. Jin, Y. Li, R. Pan and X. Zou, Characterizing and controlling the inflammatory network during influenza a virus infection, Sci. Rep., 4 (2014), p3799. |
[26] |
S. Jin, L. Niu, G. Wang and X. Zou, Mathematical modeling and nonlinear dynamical analysis of cell growth in response to antibiotics, Int. J. Bifurcat. Chaos, 25 (2015), 1540007, 12pp. |
[27] |
J. D. Jordan, E. M. Landau and R. Iyengar, Signaling networks: The origins of cellular multitasking, Cell, 103 (2000), 193-200. |
[28] |
R. Kalluri and R. A. Weinberg, The basics of epithelial-mesenchymal transition, J. Clin. Invest., 119 (2009), 1420-1428. |
[29] |
S. Karl and T. Dandekar, Convergence behaviour and control in non-linear biological networks, Sci. Rep., 5 (2015), p9746. |
[30] |
H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, New Jersey, 2002. |
[31] |
Y. C. Lai, Controlling complex, non-linear dynamical networks, National Science Review, 1 (2014), 339-341. |
[32] |
S. Lamouille, J. Xu and R. Derynck, Molecular mechanisms of epithelial-mesenchymal transition, Nat. Rev. Mol. Cell Biol., 15 (2014), 178-196. |
[33] |
C. Li and J. Wang, Quantifying the underlying landscape and paths of cancer, J. R. Soc. Interface, 11 (2014), 20140774. |
[34] |
Y. Li, S. Jin, L. Lei, Z. Pan and X. Zou, Deciphering deterioration mechanisms of complex diseases based on the construction of dynamic networks and systems analysis, Sci. Rep., 5 (2015), p9283. |
[35] |
Y. Li, M. Yi and X. Zou, The linear interplay of intrinsic and extrinsic noises ensures a high accuracy of cell fate selection in budding yeast, Sci. Rep., 4 (2014), p5764. |
[36] |
Y. Y. Liu and A. L. Barabasi, Control principles of complex systems, Rev. Mod. Phys., 88 (2016), p035006. |
[37] |
Y. Y. Liu, J. J. Slotine and A. L. Barabasi, Controllability of complex networks, Nature, 473 (2011), 167-173. |
[38] |
L. G. Matallana, A. M. Blanco and J. A. Bandoni, Estimation of domains of attraction: A global optimization approach, Math. Comput. Model., 52 (2010), 574-585. |
[39] |
L. Mendoza, A network model for the control of the differentiation process in Th cells, Biosystems, 84 (2006), 101-114. |
[40] |
A. Mochizuki, B. Fiedler, G. Kurosawa and D. Saito, Dynamics and control at feedback vertex sets. Ⅱ: A faithful monitor to determine the diversity of molecular activities in regulatory networks, J. Theor. Biol., 335 (2013), 130-146. |
[41] |
M. Moes, A. Le Béchec, I. Crespo, C. Laurini, A. Halavatyi, G. Vetter, A. Del Sol and E. Friederich, A novel network integrating a miRNA-203/SNAI1 feedback loop which regulates epithelial to mesenchymal transition, PloS One, 7 (2012), e35440. |
[42] |
F.-J. Müller and A. Schuppert, Few inputs can reprogram biological networks, Nature, 478 (2011), E4-E4. |
[43] |
T. Nepusz and T. Vicsek, Controlling edge dynamics in complex networks, Nat. Phys., 8 (2012), 568-573. |
[44] |
J. A. Papin, T. Hunter, B. O. Palsson and S. Subramaniam, Reconstruction of cellular signalling networks and analysis of their properties, Nat. Rev. Mol. Cell Biol., 6 (2005), 99-111. |
[45] |
J. Pei, N. Yin, X. Ma and L. Lai, Systems biology brings new dimensions for structure-based drug design, J. Am. Chem. Soc., 136 (2014), 11556-11565. |
[46] |
M. Pósfai and P. Hövel, Structural controllability of temporal networks, New J. Phys., 16 (2014), 123055. |
[47] |
J. Ruths and D. Ruths, Control profiles of complex networks, Science, 343 (2014), 1373-1376. |
[48] |
J. -J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, New Jersey, 1991. |
[49] |
M. S. Song, L. Salmena and P. P. Pandolfi, The functions and regulation of the pten tumour suppressor, Nat. Rev. Mol. Cell Biol., 13 (2012), 283-296. |
[50] |
F. Sorrentino, M. di Bernardo, F. Garofalo and G. R. Chen, Controllability of complex networks via pinning, Phys. Rev. E, 75 (2007), 046103. |
[51] |
G. Strang, Calculus, Wellesley-Cambridge Press, Massachusetts, 1991.
![]() |
[52] |
J. Tan and X. Zou, Complex dynamical analysis of a coupled network from innate immune responses, Int. J. Bifurcat. Chaos, 23 (2013), 1350180, 26pp. |
[53] |
J. Tan and X. Zou, Optimal control strategy for abnormal innate immune response, Comput. Math. Method M., 2015 (2015), Art. ID 386235, 16 pp. |
[54] |
A. Vinayagam, T. E. Gibson, H.-J. Lee, B. Yilmazel, C. Roesel, Y. Hu, Y. Kwon, A. Sharma, Y.-Y. Liu, N. Perrimon and A.-L. Barabási, Controllability analysis of the directed human protein inteorkraction netw identifies disease genes and drug targets, Proc. Natl. Acad. Sci. U. S. A., 113 (2016), 4976-4981. |
[55] |
D. Wang, S. Jin, F. X. Wu and X. Zou, Estimation of control energy and control strategies for complex networks, Adv. Complex Syst., 18 (2015), 1550018, 23pp. |
[56] |
D. Wang, S. Jin and X. Zou, Crosstalk between pathways enhances the controllability of signalling networks, IET Syst. Biol., 10 (2016), 2-9. |
[57] |
L. Z. Wang, R. Q. Su, Z. G. Huang, X. Wang, W. X. Wang, C. Grebogi and Y. C. Lai, A geometrical approach to control and controllability of nonlinear dynamical networks, Nat. Commun., 7 (2016), p11323. |
[58] |
Y. Wang, J. Tan, F. Sadre-Marandi, J. Liu and X. Zou, Mathematical modeling for intracellular transport and binding of HIV-1 gag proteins, Math. Biosci., 262 (2015), 198-205. |
[59] |
R. Weinberg, The Biology of Cancer, 2nd edition, Garland Science, New York, 2013. |
[60] |
U. Wellner, J. Schubert, U. C. Burk, O. Schmalhofer, F. Zhu, A. Sonntag, B. Waldvogel, C. Vannier, D. Darling, A. zur Hausen et al., The EMT-activator ZEB1 promotes tumorigenicity by repressing stemness-inhibiting micrornas, Nat. Cell Biol., 11 (2009), 1487-1495. |
[61] |
D. K. Wells, W. L. Kath and A. E. Motter, Control of stochastic and induced switching in biophysical networks, Phys. Rev. X, 5 (2015), 031036. |
[62] |
A. J. Whalen, S. N. Brennan, T. D. Sauer and S. J. Schiff, Observability and controllability of nonlinear networks: The role of symmetry, Phys. Rev. X, 5 (2015), 011005. |
[63] |
F.-X. Wu, L. Wu, J. Wang, J. Liu and L. Chen, Transittability of complex networks and its applications to regulatory biomolecular networks, Sci. Rep., 4 (2014), p4819. |
[64] |
S. Wuchty, Controllability in protein interaction networks, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 7156-7160. |
[65] |
G. Yan, G. Tsekenis, B. Barzel, J.-J. Slotine, Y.-Y. Liu and A.-L. Barabási, Spectrum of controlling and observing complex networks, Nat. Phys., 11 (2015), 779-786. |
[66] |
Z. Yuan, C. Zhao, Z. Di, W.-X. Wang and Y.-C. Lai, Exact controllability of complex networks, Nat. Commun., 4 (2013), p2447. |
[67] |
J. G. Zañudo, G. Yang and R. Albert, Structure-based control of complex networks with nonlinear dynamics, arXiv: 1605.08415v2. |
[68] |
J. Zhang, Z. Yuan, H. X. Li and T. Zhou, Architecture-dependent robustness and bistability in a class of genetic circuits, Biophys. J., 99 (2010), 1034-1042. |
[69] |
N. Zhong, Computational unsolvability of domains of attraction of nonlinear systems, Proc. Amer. Math. Soc., 137 (2009), 2773-2783. |





















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 |
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 | $\left\{ ({{x}_{1}}, \cdots, {{x}_{32}})\in {{\mathbb{R}}^{32}}\left| 0\le {{x}_{i}}\le 1, i=1, \cdots, 32 \right. \right\}$ |
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]$ |
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 | ${-\left(\frac{a\left({{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n} \right)}{2{{S}^{n}}+{{s}_{10}}{{x}_{10}}^{n}+{{s}_{13}}{{x}_{13}}^{n}}+\frac{b{{S}^{n}}}{{{S}^{n}}+{{s}_{4}}{{x}_{4}}^{n}}-k{{x}_{25}} \right)+\lambda \left({{x}_{25}}-x_{2, 25}^{*} \right)}$ |
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 |
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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