
-
Previous Article
Bility and traveling wavefronts for a convolution model of mistletoes and birds with nonlocal diffusion
- DCDS-B Home
- This Issue
-
Next Article
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[5] |
Y. Bar-Yam, D. Harmon and B. de Bivort, Attractors and democratic dynamics, Science, 323 (2009), 1016-1017. Google Scholar |
[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. Google Scholar |
[7] |
R. G. Bartle, The Elements of Integration and Lebesgue Measure, (John Wiley & Sons), 2011. Google Scholar |
[8] |
B. Barzel and A. -L. Barabási, Universality in network dynamics, Nat. Phys., 9 (2013), 673-681. Google Scholar |
[9] |
Y. Ben-Neriah and M. Karin, Inflammation meets cancer, with nf-κb as the matchmaker, Nat.Immunol., 12 (2011), 715-723. Google Scholar |
[10] |
G. Chen and X. Yu, Chaos Control: Theory and Applications, Springer-Verlag Berlin Heidelberg, Berlin, 2003. Google Scholar |
[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. Google Scholar |
[12] |
S. P. Cornelius, W. L. Kath and A. E. Motter, Realistic control of network dynamics, Nat. Commun., 4 (2013), 1942. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[15] |
B. De Craene and G. Berx, Regulatory networks defining EMT during cancer initiation and progression, Nat. Rev. Cancer, 13 (2013), 97-110. Google Scholar |
[16] |
H. De Jong, Modeling and simulation of genetic regulatory systems: A literature review, J. Comput. Biol., 9 (2002), 67-103. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[20] |
J. Gao, Y. -Y. Liu, R. M. D'Souza and A. -L. Barabási, Target control of complex networks, Nat. Commun., 5 (2014), p5415. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[27] |
J. D. Jordan, E. M. Landau and R. Iyengar, Signaling networks: The origins of cellular multitasking, Cell, 103 (2000), 193-200. Google Scholar |
[28] |
R. Kalluri and R. A. Weinberg, The basics of epithelial-mesenchymal transition, J. Clin. Invest., 119 (2009), 1420-1428. Google Scholar |
[29] |
S. Karl and T. Dandekar, Convergence behaviour and control in non-linear biological networks, Sci. Rep., 5 (2015), p9746. Google Scholar |
[30] |
H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, New Jersey, 2002. Google Scholar |
[31] |
Y. C. Lai, Controlling complex, non-linear dynamical networks, National Science Review, 1 (2014), 339-341. Google Scholar |
[32] |
S. Lamouille, J. Xu and R. Derynck, Molecular mechanisms of epithelial-mesenchymal transition, Nat. Rev. Mol. Cell Biol., 15 (2014), 178-196. Google Scholar |
[33] |
C. Li and J. Wang, Quantifying the underlying landscape and paths of cancer, J. R. Soc. Interface, 11 (2014), 20140774. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[36] |
Y. Y. Liu and A. L. Barabasi, Control principles of complex systems, Rev. Mod. Phys., (2016), 88. Google Scholar |
[37] |
Y. Y. Liu, J. J. Slotine and A. L. Barabasi, Controllability of complex networks, Nature, 473 (2011), 167-173. Google Scholar |
[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. Google Scholar |
[39] |
L. Mendoza, A network model for the control of the differentiation process in Th cells, Biosystems, 84 (2006), 101-114. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[42] |
J. F.-Müller and A. Schuppert, Few inputs can reprogram biological networks, Nature, 478 (2011), E4-E4. Google Scholar |
[43] |
T. Nepusz and T. Vicsek, Controlling edge dynamics in complex networks, Nat. Phys., 8 (2012), 568-573. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[46] |
M. Pósfai and P. Hövel, Structural controllability of temporal networks, New J. Phys., 16 (2014), 123055. Google Scholar |
[47] |
J. Ruths and D. Ruths, Control profiles of complex networks, Science, 343 (2014), 1373-1376. Google Scholar |
[48] |
J. -J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, New Jersey, 1991. Google Scholar |
[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. Google Scholar |
[50] |
F. Sorrentino, M. di Bernardo, F. Garofalo and G. R. Chen, Controllability of complex networks via pinning, Phys. Rev. E, 75 (2007), 046103. Google Scholar |
[51] | G. Strang, Calculus, Wellesley-Cambridge Press, Massachusetts, 1991. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[56] |
D. Wang, S. Jin and X. Zou, Crosstalk between pathways enhances the controllability of signalling networks, IET Syst. Biol., 10 (2016), 2-9. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[59] |
R. Weinberg, The Biology of Cancer, 2nd edition, Garland Science, New York, 2013. Google Scholar |
[60] |
U. Wellner, J. Schubert, U. C. Burk, O. Schmalhofer, F. Zhu, A. Sonntag, B. Waldvogel, C. Vannier, D. Darling and A. zur Hausen, The EMT-activator ZEB1 promotes tumorigenicity by repressing stemness-inhibiting micrornas, Nat. Cell Biol., 11 (2009), 1487-1495. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[64] |
S. Wuchty, Controllability in protein interaction networks, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 7156-7160. Google Scholar |
[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. Google Scholar |
[66] |
Z. Yuan, C. Zhao, Z. Di, W. -X. Wang and Y. -C. Lai, Exact controllability of complex networks, Nat. Commun., 4 (2013), p2447. Google Scholar |
[67] |
J. G. Zañudo, G. Yang and R. Albert, Structure-based control of complex networks with nonlinear dynamics, arXiv: 1605.08415v2. Google Scholar |
[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. Google Scholar |
[69] |
N. Zhong, Computational unsolvability of domains of attraction of nonlinear systems, Proc. Amer. Math. Soc., 137 (2009), 2773-2783. Google Scholar |
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[5] |
Y. Bar-Yam, D. Harmon and B. de Bivort, Attractors and democratic dynamics, Science, 323 (2009), 1016-1017. Google Scholar |
[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. Google Scholar |
[7] |
R. G. Bartle, The Elements of Integration and Lebesgue Measure, (John Wiley & Sons), 2011. Google Scholar |
[8] |
B. Barzel and A. -L. Barabási, Universality in network dynamics, Nat. Phys., 9 (2013), 673-681. Google Scholar |
[9] |
Y. Ben-Neriah and M. Karin, Inflammation meets cancer, with nf-κb as the matchmaker, Nat.Immunol., 12 (2011), 715-723. Google Scholar |
[10] |
G. Chen and X. Yu, Chaos Control: Theory and Applications, Springer-Verlag Berlin Heidelberg, Berlin, 2003. Google Scholar |
[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. Google Scholar |
[12] |
S. P. Cornelius, W. L. Kath and A. E. Motter, Realistic control of network dynamics, Nat. Commun., 4 (2013), 1942. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[15] |
B. De Craene and G. Berx, Regulatory networks defining EMT during cancer initiation and progression, Nat. Rev. Cancer, 13 (2013), 97-110. Google Scholar |
[16] |
H. De Jong, Modeling and simulation of genetic regulatory systems: A literature review, J. Comput. Biol., 9 (2002), 67-103. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[20] |
J. Gao, Y. -Y. Liu, R. M. D'Souza and A. -L. Barabási, Target control of complex networks, Nat. Commun., 5 (2014), p5415. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[27] |
J. D. Jordan, E. M. Landau and R. Iyengar, Signaling networks: The origins of cellular multitasking, Cell, 103 (2000), 193-200. Google Scholar |
[28] |
R. Kalluri and R. A. Weinberg, The basics of epithelial-mesenchymal transition, J. Clin. Invest., 119 (2009), 1420-1428. Google Scholar |
[29] |
S. Karl and T. Dandekar, Convergence behaviour and control in non-linear biological networks, Sci. Rep., 5 (2015), p9746. Google Scholar |
[30] |
H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, New Jersey, 2002. Google Scholar |
[31] |
Y. C. Lai, Controlling complex, non-linear dynamical networks, National Science Review, 1 (2014), 339-341. Google Scholar |
[32] |
S. Lamouille, J. Xu and R. Derynck, Molecular mechanisms of epithelial-mesenchymal transition, Nat. Rev. Mol. Cell Biol., 15 (2014), 178-196. Google Scholar |
[33] |
C. Li and J. Wang, Quantifying the underlying landscape and paths of cancer, J. R. Soc. Interface, 11 (2014), 20140774. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[36] |
Y. Y. Liu and A. L. Barabasi, Control principles of complex systems, Rev. Mod. Phys., (2016), 88. Google Scholar |
[37] |
Y. Y. Liu, J. J. Slotine and A. L. Barabasi, Controllability of complex networks, Nature, 473 (2011), 167-173. Google Scholar |
[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. Google Scholar |
[39] |
L. Mendoza, A network model for the control of the differentiation process in Th cells, Biosystems, 84 (2006), 101-114. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[42] |
J. F.-Müller and A. Schuppert, Few inputs can reprogram biological networks, Nature, 478 (2011), E4-E4. Google Scholar |
[43] |
T. Nepusz and T. Vicsek, Controlling edge dynamics in complex networks, Nat. Phys., 8 (2012), 568-573. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[46] |
M. Pósfai and P. Hövel, Structural controllability of temporal networks, New J. Phys., 16 (2014), 123055. Google Scholar |
[47] |
J. Ruths and D. Ruths, Control profiles of complex networks, Science, 343 (2014), 1373-1376. Google Scholar |
[48] |
J. -J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, New Jersey, 1991. Google Scholar |
[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. Google Scholar |
[50] |
F. Sorrentino, M. di Bernardo, F. Garofalo and G. R. Chen, Controllability of complex networks via pinning, Phys. Rev. E, 75 (2007), 046103. Google Scholar |
[51] | G. Strang, Calculus, Wellesley-Cambridge Press, Massachusetts, 1991. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[56] |
D. Wang, S. Jin and X. Zou, Crosstalk between pathways enhances the controllability of signalling networks, IET Syst. Biol., 10 (2016), 2-9. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[59] |
R. Weinberg, The Biology of Cancer, 2nd edition, Garland Science, New York, 2013. Google Scholar |
[60] |
U. Wellner, J. Schubert, U. C. Burk, O. Schmalhofer, F. Zhu, A. Sonntag, B. Waldvogel, C. Vannier, D. Darling and A. zur Hausen, The EMT-activator ZEB1 promotes tumorigenicity by repressing stemness-inhibiting micrornas, Nat. Cell Biol., 11 (2009), 1487-1495. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[64] |
S. Wuchty, Controllability in protein interaction networks, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 7156-7160. Google Scholar |
[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. Google Scholar |
[66] |
Z. Yuan, C. Zhao, Z. Di, W. -X. Wang and Y. -C. Lai, Exact controllability of complex networks, Nat. Commun., 4 (2013), p2447. Google Scholar |
[67] |
J. G. Zañudo, G. Yang and R. Albert, Structure-based control of complex networks with nonlinear dynamics, arXiv: 1605.08415v2. Google Scholar |
[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. Google Scholar |
[69] |
N. Zhong, Computational unsolvability of domains of attraction of nonlinear systems, Proc. Amer. Math. Soc., 137 (2009), 2773-2783. Google Scholar |





















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 |
[1] |
N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079 |
[2] |
Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072 |
[3] |
Péter Koltai. A stochastic approach for computing the domain of attraction without trajectory simulation. Conference Publications, 2011, 2011 (Special) : 854-863. doi: 10.3934/proc.2011.2011.854 |
[4] |
Steven G. Krantz and Marco M. Peloso. New results on the Bergman kernel of the worm domain in complex space. Electronic Research Announcements, 2007, 14: 35-41. doi: 10.3934/era.2007.14.35 |
[5] |
S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593 |
[6] |
Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135 |
[7] |
Jean-Luc Chabert, Ai-Hua Fan, Youssef Fares. Minimal dynamical systems on a discrete valuation domain. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 777-795. doi: 10.3934/dcds.2009.25.777 |
[8] |
João Borges de Sousa, Bernardo Maciel, Fernando Lobo Pereira. Sensor systems on networked vehicles. Networks & Heterogeneous Media, 2009, 4 (2) : 223-247. doi: 10.3934/nhm.2009.4.223 |
[9] |
Ying Wu, Zhaohui Yuan, Yanpeng Wu. Optimal tracking control for networked control systems with random time delays and packet dropouts. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1343-1354. doi: 10.3934/jimo.2015.11.1343 |
[10] |
Ciro D’Apice, Umberto De Maio, Peter I. Kogut. Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 283-314. doi: 10.3934/dcdsb.2009.11.283 |
[11] |
Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022 |
[12] |
Luigi Fontana, Steven G. Krantz and Marco M. Peloso. Hodge theory in the Sobolev topology for the de Rham complex on a smoothly bounded domain in Euclidean space. Electronic Research Announcements, 1995, 1: 103-107. |
[13] |
Rong Liu, Saini Jonathan Tishari. Automatic tracking and positioning algorithm for moving targets in complex environment. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1251-1264. doi: 10.3934/dcdss.2019086 |
[14] |
Said Hadd, Rosanna Manzo, Abdelaziz Rhandi. Unbounded perturbations of the generator domain. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 703-723. doi: 10.3934/dcds.2015.35.703 |
[15] |
Shigeki Akiyama, Edmund Harriss. Pentagonal domain exchange. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4375-4400. doi: 10.3934/dcds.2013.33.4375 |
[16] |
Gleb G. Doronin, Nikolai A. Larkin. Kawahara equation in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 783-799. doi: 10.3934/dcdsb.2008.10.783 |
[17] |
Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333 |
[18] |
Luis A. Caffarelli, Fang Hua Lin. Analysis on the junctions of domain walls. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 915-929. doi: 10.3934/dcds.2010.28.915 |
[19] |
Nicolás Carreño. Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain. Mathematical Control & Related Fields, 2012, 2 (4) : 361-382. doi: 10.3934/mcrf.2012.2.361 |
[20] |
Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure & Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031 |
2018 Impact Factor: 1.008
Tools
Article outline
Figures and Tables
[Back to Top]