AIMS Mathematics, 2017, 2(3): 437-450. doi: 10.3934/Math.2017.2.437

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Relations between the dynamics of network systems and their subnetworks

1 Department of Mathematics, Texas Southern University, 3100 Cleburne, Houston, TX, 77004, USA
2 Department of Mathematics, University of South Florida at St. Pete, 140 7th Avenue South St.Petersburg, Florida 33701, USA

Statistical analysis of the connectivity of real world networks have revealed interesting featuressuch as community structure, network motif and as on. Such discoveries tempt us to understandthe dynamics of a complex network system by studying those of its subnetworks. This approach isfeasible only if the dynamics of the subnetwork systems can somehow be preserved or partially preservedin the whole system. Most works studied the connectivity structures of networks while very fewconsidered the possibility of translating the dynamics of a subnetwork system to the whole system. Inthis paper, we address this issue by focusing on considering the relations between cycles and fixedpoints of a network system and those of its subnetworks based on Boolean framework. We proved thatat a condition we called agreeable, if X0 is a fixed point of the whole system, then X0 restricted to thephase-space of one of the subnetwork systems must be a fixed point as well. An equivalent statementon cycles follows from this result. In addition, we discussed the relations between the product of thetransition diagrams (a representation of trajectories) of subnetwork systems and the transition diagramof the whole system.
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1. W. Abuo-Jaoude, D. Ouattara, and M. Kaufman, From structure to dynamics: frequency tuning inthe p53-mdm2 network: I. logical approach, Journal of Theoretical Biology, 258 (2009), 561-577.    

2. R. Albert and H. Othmer, The topology of the regulatory interactions perdicts the expression patternof the segment polarity genes in drosophila melanogaster, Journal of Theoretical Biology, 233(2003) , 1-18.

3. Reka Albert, Scale-free networks in cell biology, Journal of Cell Science, 118 (2005), 4947-4957.    

4. Sergio A. Alcalá-Corona, Tadeo E. Velázquez-Caldelas, Jesús Espinal-Enríquez, and Enrique Hernández-Lemus, Community structure reveals biologically functional modules in mef2c transcriptionalregulatory network, Front Physiol, 7 (2016), 184.

5. B. B. Aldrige, J. M. Burke, D. A. Lauffenburge, and P.K. Sorger, Physicochemical modeling of cellsignaling pathways, Nature Cell Biol, 8 (2006).

6. U. Alon, Network motifs: Theory and experimental approaches, Nature Reviews Genetics, 8 (2007),450.

7. C. Campbell, J. Thakar, and R. Albert, Network analysis reveals cross-links of the immune pathwaysactivated by bacteria and allergen, Physical Reveiw. E, Statistical physics, plasmas, fluids andrelated interdisciplinary topics, 84 (2011), 031929.

8. R. Edwards and L. Glass, Combinatorial explosion in model gene networks, Chaos: An inerdisciplinaryJournal of Nonlinear Science, 11 (2000), 691-704.

9. R. Edwards, H. Siegelmann, K. Aziza, and L. Glass, Symbolic dynamics and computation in modelgene models, Chaos: An inerdisciplinary Journal of Nonlinear Science, 11 (2001), 160-169.    

10.C. Espionza-Soto, P. Padilla-Longoria, and E.R. Alvarez-Buylla, A gene regulatory network modelfor cell-fate determination during arabidopsis thaliana flower development that is robus and recoversexperiental gene expression profiles, Plant Cell, 16 (2004), 2923-2939.    

11.Newman M. E and Girvan M., Community structure in social and biological networks, Proc. Natl.Acad. Sci. U.S.A., 99 (2002), 7821-7826.    

12.L. Glass and S. A. Kauffman, The logical analysis of continuous, nonlinear biochemical controlnetwork, J. Theor. Biol., 39 (1973), 103-139.    

13.Cantini L., Medico E., Fortunato S., and CaselleMDetection of gene communities in multi-networksreveals cancer drivers., Sci. Rep., 5 (2015), 17386.

14.R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon Network motifs: Simplebuilding blocks of complex networks, Science, 298 (2002), 824-827.

15.A. Mogilner, R.Wollman, andW. F. Marshall Quantitative modeling in cell biology, DevelopmentalCell, 11 (2006), 279-287.

16.S. Li, S. M. AssMann, and R. Albert Predicting essential components of signal transduction networks:A dynamic model of guard cell abscisic acid signaling, PLoS Biol., 4 (2006), e312.

17.M. E. J. Newman, Modularity and community structure in networks, Proceedings of the NationalAcademy of Sciences, 103 (2006), 8577-8582.

18.M. E. J. Newman, Communities, modules and large scale structure, Nature Physics, 8 (2012), 25-31.

19.A. Saadatpour, R. Albert, and T. C. Reluga, A reduction method for boolean network models provento conserve attractors, SIAM J. Appl. Dyn. Sys., 12 (2013), 1997-2011.    

20.J. Saez-Rodriguez, L. Simeoni, J. A. Lindquist, R. Hemenway, U. Bommhardt, U. U. Haus B. Arndt,R.Weismantel, E. D. Gilles, S. Klamt, and B. Schraven, A logical model provides insights into t cellreceptor signaling, PLoS Computational Biology, 3 (2007), e163.

21.L. Sanchez and D. Thieffry, A logical analysis of the drosophila gap-gene system, J. Theor. Biol.,211 (2001), 115-141.    

22.Adam J. Schwarz, Alessandro Gozzi, and Angelo Bifone, Community structure and modularity innetworks of correlated brain activity, Magnetic Resonance Imaging., 27 (2008), 914-920.

23.S. S. Shen-Orr, R. Milo, S. Mangan, and U. Alon, Network motifs in the transcriptional regulationnetwork of escherichia coli, Nature Genet 31 (2002), 64-68.

24.E. Snoussi, Qualitative dynamics of piecewise differential equations: a discrete mapping approach.,Dynamical and Stability of Systems, 4 (1989), 189-207.

25.R Tanaka, Scale-rich metabolic networks., Phys. Rev. Lett., 94 (2005), 168101.

26.J. Thakar, A. K. Pathak, L. Murphy, R. Albert, I. Cattadori, and R. J. De Boer, Network model of immuneresponses reveals key efectors to single and co-infection dynamics by a respiratory bacteriumand a gatrointestinal helminth, PLoS Computational Biology, 8 (2012), 1.

27.R. Thomas, Biological feedback, CRC, 1990.

28.D. Turner, P. Paszek, D.J.Woodcock, D. E. Nelson, C.A.Horton, Y.Wang, D.G. Spiller, D. A. Rand,M. R. H. White, and C. V. Harper, Physiological levels of tnfalpha stimulation induce stochasticdynamics of nf-kappab responses in single living cells, Journal of Cell Biology, 123 (2010), 2834-2843.

29.J. Tyson and B. Novak Functional motifs in biochemical reaction networks, Annu. Rev. Phys.Chem., 61 (2010), 219-240.

30.J. J. Tyson, K. C. Chen, and B. Novak, Network dynamics and cell physiology, Nature Rev. Mol.Cell Biol, 2 (2001), 908-916.    

31.J. J. Tyson, K. C. Chen, and B. Novak, Sniffers, buzzers, toggles and blinkers: dynamics of regulatoryand signaling pathways in the cell, Curr. Op. Cell Biol, 15 (2003), 221-231.    

32.A. Veliz-Cuba, A. Kumar, and K. Josic, Piecewise linear and boolean models of chemical reactionnetworks, J. Math. Bio., 76 (2014), 2945-2984.    

33.R. S. Wang and R. Albert, Discrete dynamical modeling of cellular signaling networks, Methods inEnzymology, 467 (2009) , 281-306.

34.Sebastian Wernicke, A faster algorithm for detecting network motifs, Proc. 5th WABI-05, 3692(2005).

35.S. H. Yook, Z. N. Oltvai, and A. L. Barabási, Functional and topological characterization of proteininteraction networks, Proteomics, 4 (2004), 928-942.    

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