The Community for Technology Leaders
RSS Icon
Issue No.04 - July-Aug. (2013 vol.10)
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.
Steady-state, Chemical reactions, Nonlinear systems,minimally parameterized solutions, Chemical reaction networks, cell signaling, VEGF, EGF, linear conservation laws, analytical solution, bilinear systems
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
[1] C. Li, M. Donizelli, N. Rodriguez, H. Dharuri, L. Endler, V. Chelliah, L. Li, E. He, A. Henry, M.I. Stefan, J.L. Snoep, M. Hucka, N. Le Novère, and C. Laibe, "BioModels Database: An Enhanced, Curated and Annotated Resource for Published Quantitative Kinetic Models," BMC Systems Biology, vol. 4, article 92, June 2010.
[2] M. Hucka, A. Finney, H. Sauro, H. Bolouri, J.C. Doyle, and H. Kitano, "The Systems Biology Markup Language (SBML): A Medium for Representation and Exchange of Biochemical Network Models," Bioinformatics, vol. 19, no. 4, pp. 524-531, 2003.
[3] W.S. Hlavacek, J.R. Faeder, M.L. Blinov, R.G. Posner, M. Hucka, and W. Fontana, "Rules for Modeling Signal-Transduction Systems," Science Signaling, vol. 2006, no. 344, article re6, 2006.
[4] K. Radhakrishnan, Á. Halász, D. Vlachos, and J.S. Edwards, "Quantitative Understanding of Cell Signaling: The Importance of Membrane Organization," Current Opinion in Biotechnology, vol. 21, no. 5, pp. 677-682, Oct. 2010.
[5] M.A. Savageau, Biochemical Systems Analysis: A Study of Function and Design in Molecular Biology. Addison-Wesley, 1976.
[6] K.C. Chen, A. Csikász-Nagy, B. Györffy, J. Val, B. Novák, and J.J. Tyson, "Kinetic Analysis of a Molecular Model of the Budding Yeast Cell Cycle," Molecular Biology of the Cell, vol. 11, no. 1, pp. 369-391, 2000.
[7] B.N. Kholodenko, O.V. Demin, G. Moehren, and J.B. Hoek, "Quantification of Short Term Signaling by the Epidermal Growth Factor Receptor," J. Biological Chemistry, vol. 274, no. 42, pp. 30169-30181, Oct. 1999.
[8] E. Sontag, "Monotone and Near-Monotone Biochemical Networks," Systems and Synthetic Biology, vol. 1, pp. 59-87, 2007.
[9] G. Shinar, U. Alon, and M. Feinberg, "Sensitivity and Robustness in Chemical Reaction Networks," SIAM J. Applied Math., vol. 69, no. 4, pp. 977-998, 2009.
[10] G. Craciun, Y. Tang, and M. Feinberg, "Understanding Bistability in Complex Enzyme-Driven Reaction Networks," Proc. Nat'l Academy of Science USA, vol. 103, pp. 8697-8702, 2006.
[11] K. Radhakrishnan, J.S. Edwards, D.S. Lidke, T.M. Jovin, B.S. Wilson, and J.M. Oliver, "Sensitivity Analysis Predicts That the ERK-pMEK Interaction Regulates ERK Nuclear Translocation," IET Systems Biology, vol. 3, pp. 329-341, 2009.
[12] L.B. Kleiman, T. Maiwald, H. Conzelman, D.A. Lauffenburger, and P.K. Sorger, "Rapid Phospho-Turnover by Receptor Tyrosine Kinases Impacts Downstream Signaling and Drug Binding," Molecular Cell, vol. 43, pp. 723-737, 2011.
[13] D.J. Klinke, N. Cheng, and E. Chambers, "Quantifying Crosstalk among Interferon-$\gamma$ , Interleukin-12, and Tumor Necrosis Factor Signaling Pathways within a ${\rm T}_H 1$ Cell Model," Science Signaling, vol. 5, no. 220, article ra32, 2012.
[14] D.J. Klinke, "An Empirical Bayesian Approach for Model-Based Inference of Cellular Signaling Networks," BMC Bioinformatics, vol. 10, article 371, 2009.
[15] J. Gunawardena, "A Linear Framework for Time-Scale Separation in Nonlinear Biochemical Systems," PLoS One, vol. 7, no. 5, article e36321, 2012.
[16] G. Batt, B. Yordanov, R. Weiss, and C. Belta, "Robustness Analysis and Tuning of Synthetic Gene Networks," Bioinformatics, vol. 23, no. 18, pp. 2415-2422, 2007.
[17] Á. Halász, V. Kumar, M. Imielinski, C. Belta, O. Sokolsky, S. Pathak, and H. Rubin, "Analysis of Lactose Metabolism in E. coli Using Reachability Analysis of Hybrid Systems," IET Systems Biology, vol. 1, no. 2, pp. 130-148, 2007.
[18] G. Batt, H. de Jong, M. Page, and J. Geiselmann, "Symbolic Reachability Analysis of Genetic Regulatory Networks Using Discrete Abstractions," Automatica, vol. 44, no. 4, pp. 982-989, 2008.
[19] A. Donzé, E. Fanchon, L.M. Gattepaille, O. Maler, and P. Tracqui, "Robustness Analysis and Behavior Discrimination in Enzymatic Reaction Networks," PLoS One, vol. 6, no. 9, article e24246, 2011.
[20] D. Ropers, V. Baldazzi, and H. de Jong, "Model Reduction Using Piecewise-Linear Approximations Preserves Dynamic Properties of the Carbon Starvation Response in Escherichia coli," IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 8, no. 1, pp. 166-180, 2011.
[21] E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, second ed., no. 6, Springer, 1998.
[22] G.E. Shilov, Linear Algebra. Dover Publications, 1977.
[23] F. MacGabhann and A.S. Popel, "Systems Biology of Vascular Endothelial Growth Factors," Microcirculation, vol. 15, no. 8, pp. 715-738, Jan. 2008.
[24] F. MacGabhann and A.S. Popel, "Model of Competitive Binding of Vascular Endothelial Growth Factor and Placental Growth Factor to VEGF Receptors on Endothelial Cells," AJP: Heart and Circulatory Physiology, vol. 286, no. 1, pp. 153H-164H, Sept. 2003.
[25] F. MacGabhann and A.S. Popel, "Differential Binding of VEGF Isoforms to VEGF Receptor 2 in the Presence of Neuropilin-1: A Computational Model," AJP: Heart and Circulatory Physiology, vol. 288, no. 6, pp. H2851-H2860, June 2005.
[26] F. MacGabhann and A.S. Popel, "Dimerization of VEGF Receptors and Implications for Signal Transduction: A Computational Study," Biophysical Chemistry, vol. 128, nos. 2-3, pp. 125-139, July 2007.
[27] K. Mayawala, D.G. Vlachos, and J.S. Edwards, "Heterogeneities in EGF Receptor Density at the Cell Surface can Lead to Concave up Scatchard Plot of EGF Binding," FEBS Letters, vol. 579, no. 14, pp. 3043-3047, June 2005.
[28] K. Mayawala, D.G. Vlachos, and J.S. Edwards, "Spatial Modeling of Dimerization Reaction Dynamics in the Plasma Membrane: Monte Carlo vs. Continuum Differential Equations," Biophysical Chemistry, vol. 121, no. 3, pp. 194-208, June 2006.
[29] K. Mayawala, D.G. Vlachos, and J.S. Edwards, "Computational Modeling Reveals Molecular Details of Epidermal Growth Factor Binding," BMC Cell Biology, vol. 6, article 41, 2005.
[30] M.-Y. Hsieh, S. Yang, M.A. Raymond-Stinz, S. Steinberg, D.G. Vlachos, W. Shu, B.S. Wilson, and J.S. Edwards, "Stochastic Simulations of ErbB Homo and Heterodimerisation: Potential Impacts of Receptor Conformational State and Spatial Segregation," IET Systems Biology, vol. 2, no. 5, article 256, 2008.
[31] M.-Y. Hsieh, S. Yang, M.A. Raymond-Stinz, J.S. Edwards, and B.S. Wilson, "Spatio-Temporal Modeling of Signaling Protein Recruitment to EGFR," BMC Systems Biology, vol. 4, article 57, 2010.
[32] M.N. Costa, K. Radhakrishnan, B.S. Wilson, D.G. Vlachos, and J.S. Edwards, "Coupled Stochastic Spatial and Non-Spatial Simulations of ErbB1 Signaling Pathways Demonstrate the Importance of Spatial Organization in Signal Transduction," PLoS ONE, vol. 4, no. 7, article e6316, July 2009.
[33] M.N. Costa, K. Radhakrishnan, and J.S. Edwards, "Monte-Carlo Simulations of Plasma Membrane Corral-Induced EGFR Clustering," J. Biotechnology, vol. 151, pp. 261-270, Jan. 2011.
[34] N. Mobilia, A. Donzé, J.M. Moulis, and E. Fanchon, "A Model of the Cellular Iron Homeostasis Network Using Semi-Formal Methods for Parameters Space Exploration,", 2012.
123 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool