American Institute of Mathematical Sciences

Advanced
June  2008, 3(2): 295-322. doi: 10.3934/nhm.2008.3.295

Graph theory and qualitative analysis of reaction networks

Mirela Domijan 1, and  Markus Kirkilionis 1,

1. 

Zeeman Building, Mathematics Institute, University of Warwick, CV4 7AL Coventry, United Kingdom, United Kingdom

Received  December 2007 Revised  February 2008 Published  March 2008

Different types of macroscopic reaction kinetics can be derived from microscopic molecular interactions, with the law of mass action being the most widely used one in standard situations. After such a modeling step, where primarily the types of reactions are identified, it becomes a problem to analyse qualitative properties of complete regulatory networks. This problem has to be tackled, because chemical reaction networks play a part in some of the most fundamental cellular processes such as cell metabolism and regulation of cell signalling processes. This paper discusses how reaction networks can be described and analysed by graph theoretic means. Graph theory is a useful analysis tool for complex reaction networks, in situations where there is parameter uncertainty or modeling information is incomplete. Graphs are very robust tools, in the sense that whole classes of network topologies will show similar behaviour, independently of precise information that is available about the reaction constants. Nevertheless, one still has to take care to incorporate sufficient dynamical information in the network structure, in order to obtain meaningful results.
Keywords: Graph Theory, Qualitative Behaviour, Reaction Networks.
Mathematics Subject Classification: Primary: 92C45, 37G15; Secondary: 14A1.
Citation: Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295
[1]

M. D. König, Stefano Battiston, M. Napoletano, F. Schweitzer. On algebraic graph theory and the dynamics of innovation networks. Networks & Heterogeneous Media, 2008, 3 (2) : 201-219. doi: 10.3934/nhm.2008.3.201
[2]

Maya Mincheva, Gheorghe Craciun. Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1207-1226. doi: 10.3934/mbe.2013.10.1207
[3]

Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks & Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197
[4]

Anirban Banerjee, Jürgen Jost. Spectral plot properties: Towards a qualitative classification of networks. Networks & Heterogeneous Media, 2008, 3 (2) : 395-411. doi: 10.3934/nhm.2008.3.395
[5]

Barton E. Lee. Consensus and voting on large graphs: An application of graph limit theory. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1719-1744. doi: 10.3934/dcds.2018071
[6]

Anne Shiu, Timo de Wolff. Nondegenerate multistationarity in small reaction networks. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2683-2700. doi: 10.3934/dcdsb.2018270
[7]

Susana Merchán, Luigi Montoro, I. Peral. Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 245-268. doi: 10.3934/cpaa.2015.14.245
[8]

Yunfeng Jia, Yi Li, Jianhua Wu. Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4785-4813. doi: 10.3934/dcds.2017206
[9]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 183-219. doi: 10.3934/naco.2016008
[10]

Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257
[11]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427
[12]

D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 205-217. doi: 10.3934/dcdss.2020346
[13]

A. C. Eberhard, J-P. Crouzeix. Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory. Journal of Industrial & Management Optimization, 2007, 3 (2) : 233-255. doi: 10.3934/jimo.2007.3.233
[14]

Shuichi Jimbo, Yoshihisa Morita. Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4013-4039. doi: 10.3934/dcds.2021026
[15]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173
[16]

Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1927-1941. doi: 10.3934/jimo.2019036
[17]

Ivanka Stamova, Gani Stamov. On the stability of sets for reaction–diffusion Cohen–Grossberg delayed neural networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1429-1446. doi: 10.3934/dcdss.2020370
[18]

Murat Arcak, Eduardo D. Sontag. A passivity-based stability criterion for a class of biochemical reaction networks. Mathematical Biosciences & Engineering, 2008, 5 (1) : 1-19. doi: 10.3934/mbe.2008.5.1
[19]

Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042
[20]

Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences