Relations between the dynamics of network systems and their subnetworks

Yunjiao Wang; Kiran Chilakamarri; Demetrios Kazakos; Maria C. Leite; Yunjiao Wang; Kiran Chilakamarri; Demetrios Kazakos; Maria C. Leite

doi:10.3934/Math.2017.2.437

AIMS Mathematics

2017, Volume 2, Issue 3: 437-450. doi: 10.3934/Math.2017.2.437

Previous Article Next Article

Research article

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

Received: 31 January 2017 Accepted: 08 August 2017 Published: 14 August 2017

Statistical analysis of the connectivity of real world networks have revealed interesting features such as community structure, network motif and as on. Such discoveries tempt us to understand the dynamics of a complex network system by studying those of its subnetworks. This approach is feasible only if the dynamics of the subnetwork systems can somehow be preserved or partially preserved in the whole system. Most works studied the connectivity structures of networks while very few considered the possibility of translating the dynamics of a subnetwork system to the whole system. In this paper, we address this issue by focusing on considering the relations between cycles and fixed points of a network system and those of its subnetworks based on Boolean framework. We proved that at a condition we called agreeable, if X0 is a fixed point of the whole system, then X₀ restricted to the phase-space of one of the subnetwork systems must be a fixed point as well. An equivalent statement on cycles follows from this result. In addition, we discussed the relations between the product of the transition diagrams (a representation of trajectories) of subnetwork systems and the transition diagram of the whole system.
- Boolean network systems,
- cycle,
- fixed point,
- subnetwork systems,
- dynamics
Citation: Yunjiao Wang, Kiran Chilakamarri, Demetrios Kazakos, Maria C. Leite. Relations between the dynamics of network systems and their subnetworks[J]. AIMS Mathematics, 2017, 2(3): 437-450. doi: 10.3934/Math.2017.2.437

Related Papers:

Abstract

Statistical analysis of the connectivity of real world networks have revealed interesting features such as community structure, network motif and as on. Such discoveries tempt us to understand the dynamics of a complex network system by studying those of its subnetworks. This approach is feasible only if the dynamics of the subnetwork systems can somehow be preserved or partially preserved in the whole system. Most works studied the connectivity structures of networks while very few considered the possibility of translating the dynamics of a subnetwork system to the whole system. In this paper, we address this issue by focusing on considering the relations between cycles and fixed points of a network system and those of its subnetworks based on Boolean framework. We proved that at a condition we called agreeable, if X0 is a fixed point of the whole system, then X₀ restricted to the phase-space of one of the subnetwork systems must be a fixed point as well. An equivalent statement on cycles follows from this result. In addition, we discussed the relations between the product of the transition diagrams (a representation of trajectories) of subnetwork systems and the transition diagram of the whole system.

References

[1]	W. Abuo-Jaoude, D. Ouattara, and M. Kaufman, From structure to dynamics: frequency tuning in the 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 pattern of 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 Hern ández-Lemus, Community structure reveals biologically functional modules in mef2c transcriptional regulatory network, Front Physiol, 7 (2016), 184.
[5]	B. B. Aldrige, J. M. Burke, D. A. Lauffenburge, and P.K. Sorger, Physicochemical modeling of cell signaling 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 pathways activated by bacteria and allergen, Physical Reveiw. E, Statistical physics, plasmas, fluids and related interdisciplinary topics, 84 (2011), 031929.
[8]	R. Edwards and L. Glass, Combinatorial explosion in model gene networks, Chaos: An inerdisciplinary Journal of Nonlinear Science, 11 (2000), 691-704.
[9]	R. Edwards, H. Siegelmann, K. Aziza, and L. Glass, Symbolic dynamics and computation in model gene 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 model for cell-fate determination during arabidopsis thaliana flower development that is robus and recovers experiental 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 control network, J. Theor. Biol., 39 (1973), 103-139.
[13]	Cantini L., Medico E., Fortunato S., and CaselleMDetection of gene communities in multi-networks reveals cancer drivers., Sci. Rep., 5 (2015), 17386.
[14]	R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon Network motifs: Simple building blocks of complex networks, Science, 298 (2002), 824-827.
[15]	A. Mogilner, R.Wollman, andW. F. Marshall Quantitative modeling in cell biology, Developmental Cell, 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 National Academy 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 proven to 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 cell receptor 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 in networks 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 regulation network 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 immune responses reveals key efectors to single and co-infection dynamics by a respiratory bacterium and 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 stochastic dynamics 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 regulatory and 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 reaction networks, J. Math. Bio., 76 (2014), 2945-2984.
[33]	R. S. Wang and R. Albert, Discrete dynamical modeling of cellular signaling networks, Methods in Enzymology, 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 protein interaction networks, Proteomics, 4 (2004), 928-942.

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)