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Analytical Solution of Steady-State Equations for Chemical Reaction Networks with Bilinear Rate Laws
July-Aug. 2013 (vol. 10 no. 4)
pp. 957-969
Adam M. Halasz, Dept. of Math., West Virginia Univ., Morgantown, WV, USA
Hong-Jian Lai, Dept. of Math., West Virginia Univ., Morgantown, WV, USA
Meghan McCabe Pryor, Dept. of Chem. & Nucl. Eng., Univ. of New Mexico, Albuquerque, NM, USA
Krishnan Radhakrishnan, Dept. of Preventive Med. & Environ. Health, Univ. of Kentucky, Lexington, KY, USA
Jeremy S. Edwards, Dept. of Mol. Genetics & Microbiol., Univ. of New Mexico Health Sci. Center, NM, USA
True steady states are a rare occurrence in living organisms, yet their knowledge is essential for quasi-steady-state approximations, multistability analysis, and other important tools in the investigation of chemical reaction networks (CRN) used to describe molecular processes on the cellular level. Here, we present an approach that can provide closed form steady-state solutions to complex systems, resulting from CRN with binary reactions and mass-action rate laws. We map the nonlinear algebraic problem of finding steady states onto a linear problem in a higher-dimensional space. We show that the linearized version of the steady-state equations obeys the linear conservation laws of the original CRN. We identify two classes of problems for which complete, minimally parameterized solutions may be obtained using only the machinery of linear systems and a judicious choice of the variables used as free parameters. We exemplify our method, providing explicit formulae, on CRN describing signal initiation of two important types of RTK receptor-ligand systems, VEGF and EGF-ErbB1.
Index Terms:
chemical reactions,algebra,biochemistry,cellular biophysics,EGF-ErbB1,steady-state equations,chemical reaction networks,bilinear rate laws,living organisms,quasisteady-state approximations,multistability analysis,CRN,molecular processes,cellular level,binary reactions,mass-action rate laws,nonlinear algebraic problem,linear conservation laws,RTK receptor-ligand systems,VEGF,Steady-state,Chemical reactions,Nonlinear systems,minimally parameterized solutions,Chemical reaction networks,cell signaling,VEGF,EGF,linear conservation laws,analytical solution,bilinear systems
Citation:
Adam M. Halasz, Hong-Jian Lai, Meghan McCabe Pryor, Krishnan Radhakrishnan, Jeremy S. Edwards, "Analytical Solution of Steady-State Equations for Chemical Reaction Networks with Bilinear Rate Laws," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 10, no. 4, pp. 957-969, July-Aug. 2013, doi:10.1109/TCBB.2013.41
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