# American Institute of Mathematical Sciences

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.
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