December  2011, 4(6): 1565-1575. doi: 10.3934/dcdss.2011.4.1565

Topology and dynamics of boolean networks with strong inhibition

1. 

Department of Mathematics, The George Washington University, Washington, DC 20052, United States, United States

2. 

Department of Physics, The George Washington University, Washington, DC 20052, United States, United States, United States

Received  May 2009 Revised  October 2009 Published  December 2010

A major challenge in systems biology is to understand interactions within biological systems. Such a system often consists of units with various levels of activities that evolve over time, mathematically represented by the dynamics of the system. The interaction between units is mathematically represented by the topology of the system. We carry out some mathematical analysis on the connections between topology and dynamics of such networks. We focus on a specific Boolean network model - the Strong Inhibition Model. This model defines a natural map from the space of all possible topologies on the network to the space of all possible dynamics on the same network. We prove this map is neither surjective nor injective. We introduce the notions of "redundant edges" and "dormant vertices" which capture the non-injectiveness of the map. Using these, we determine exactly when two different topologies yield the same dynamics and we provide an algorithm that determines all possible network solutions given a dynamics.
Citation: Yongwu Rong, Chen Zeng, Christina Evans, Hao Chen, Guanyu Wang. Topology and dynamics of boolean networks with strong inhibition. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1565-1575. doi: 10.3934/dcdss.2011.4.1565
References:
[1]

F. Li, T. Long, Y. Lu, Q. Ouyang and C. Tang, The yeast cell-cycle network is robustly designed, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 4781-4786. doi: 10.1073/pnas.0305937101.

[2]

N. Tan and Q. Ouyang, Design of a network with state stability, J. Theor. Biol., 240 (2006), 592-598. doi: 10.1016/j.jtbi.2005.10.019.

[3]

J. Hopfield, Neural networks and physical systems with emergent collective computational properties, Proc. National Academy of Sciences of the USA, 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554.

[4]

G. Wang, C. Du, H. Chen, R. Simha, Y. Rong, Y. Xiao, and C. Zeng, Process-Based Network Decomposition Reveals Backbone Motif Structure, Proc. National Academy of Sciences, 107 (2010), 10478-10483. doi: 10.1073/pnas.0914180107.

[5]

R. Laubenbacher and B. Stigler, A computational algebra approach to the reverse engineering of gene regulatory networks, Journal of Theoretical Biology, 229 (2004), 523-537. doi: 10.1016/j.jtbi.2004.04.037.

[6]

A. Salam Jarrah, R. Laubenbacher and A. Veliz-Cuba, The dynamics of conjunctive and disjunctive Boolean networks, preprint (2008). arXiv:0805.0275.

show all references

References:
[1]

F. Li, T. Long, Y. Lu, Q. Ouyang and C. Tang, The yeast cell-cycle network is robustly designed, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 4781-4786. doi: 10.1073/pnas.0305937101.

[2]

N. Tan and Q. Ouyang, Design of a network with state stability, J. Theor. Biol., 240 (2006), 592-598. doi: 10.1016/j.jtbi.2005.10.019.

[3]

J. Hopfield, Neural networks and physical systems with emergent collective computational properties, Proc. National Academy of Sciences of the USA, 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554.

[4]

G. Wang, C. Du, H. Chen, R. Simha, Y. Rong, Y. Xiao, and C. Zeng, Process-Based Network Decomposition Reveals Backbone Motif Structure, Proc. National Academy of Sciences, 107 (2010), 10478-10483. doi: 10.1073/pnas.0914180107.

[5]

R. Laubenbacher and B. Stigler, A computational algebra approach to the reverse engineering of gene regulatory networks, Journal of Theoretical Biology, 229 (2004), 523-537. doi: 10.1016/j.jtbi.2004.04.037.

[6]

A. Salam Jarrah, R. Laubenbacher and A. Veliz-Cuba, The dynamics of conjunctive and disjunctive Boolean networks, preprint (2008). arXiv:0805.0275.

[1]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079

[2]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations and Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83

[3]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations and Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59

[4]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443

[5]

Kangkang Deng, Zheng Peng, Jianli Chen. Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1881-1896. doi: 10.3934/jimo.2018127

[6]

P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692

[7]

Rumi Ghosh, Kristina Lerman. Rethinking centrality: The role of dynamical processes in social network analysis. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1355-1372. doi: 10.3934/dcdsb.2014.19.1355

[8]

K. L. Mak, J. G. Peng, Z. B. Xu, K. F. C. Yiu. A novel neural network for associative memory via dynamical systems. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 573-590. doi: 10.3934/dcdsb.2006.6.573

[9]

Tibye Saumtally, Jean-Patrick Lebacque, Habib Haj-Salem. A dynamical two-dimensional traffic model in an anisotropic network. Networks and Heterogeneous Media, 2013, 8 (3) : 663-684. doi: 10.3934/nhm.2013.8.663

[10]

Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004

[11]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701

[12]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure and Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249

[13]

Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001

[14]

Yi Ming Zou. Dynamics of boolean networks. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1629-1640. doi: 10.3934/dcdss.2011.4.1629

[15]

Constanza Riera, Pantelimon Stănică. Landscape Boolean functions. Advances in Mathematics of Communications, 2019, 13 (4) : 613-627. doi: 10.3934/amc.2019038

[16]

Julian Newman. Synchronisation of almost all trajectories of a random dynamical system. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4163-4177. doi: 10.3934/dcds.2020176

[17]

Mika Yoshida, Kinji Fuchikami, Tatsuya Uezu. Realization of immune response features by dynamical system models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 531-552. doi: 10.3934/mbe.2007.4.531

[18]

Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091

[19]

Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507

[20]

Sanjay K. Mazumdar, Cheng-Chew Lim. A neural network based anti-skid brake system. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 321-338. doi: 10.3934/dcds.1999.5.321

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (0)

[Back to Top]