American Institute of Mathematical Sciences

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August  2012, 32(8): 2937-2950. doi: 10.3934/dcds.2012.32.2937

Why optimal states recruit fewer reactions in metabolic networks

Joo Sang Lee 1, , Takashi Nishikawa 2, and  Adilson E. Motter 3,

1. 

Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, United States
2. 

Department of Mathematics, Clarkson University, Potsdam, NY 13699, United States
3. 

Department of Physics & Astronomy and Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL 60208, United States

Received  May 2011 Revised  August 2011 Published  March 2012

The metabolic network of a living cell involves several hundreds or thousands of interconnected biochemical reactions. Previous research has shown that under realistic conditions only a fraction of these reactions is concurrently active in any given cell. This is partially determined by nutrient availability, but is also strongly dependent on the metabolic function and network structure. Here, we establish rigorous bounds showing that the fraction of active reactions is smaller (rather than larger) in metabolic networks evolved or engineered to optimize a specific metabolic task, and we show that this is largely determined by the presence of thermodynamically irreversible reactions in the network. We also show that the inactivation of a certain number of reactions determined by irreversibility can generate a cascade of secondary reaction inactivations that propagates through the network. The mathematical results are complemented with numerical simulations of the metabolic networks of the bacterium Escherichia coli and of human cells, which show, counterintuitively, that even the maximization of the total reaction flux in the network leads to a reduced number of active reactions.
Keywords: optimization., Network modeling, duality principle, flux balance analysis, cellular metabolism.
Mathematics Subject Classification: Primary: 92C42; Secondary: 90C3.
Citation: Joo Sang Lee, Takashi Nishikawa, Adilson E. Motter. Why optimal states recruit fewer reactions in metabolic networks. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2937-2950. doi: 10.3934/dcds.2012.32.2937
References:
[1]

U. Alon, "An Introduction to Systems Biology: Design Principles of Biological Circuits,", Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007). Google Scholar

[2]

A.-L. Barabási and Z. N. Oltvai, Network biology: Understanding the cell's functional organization,, Nat. Rev. Genet., 5 (2004), 101. Google Scholar

[3]

S. D. Becker, et al., Quantitative prediction of cellular metabolism with constraint-based models: The COBRA Toolbox,, Nat. Protoc., 2 (2007), 727. doi: 10.1038/nprot.2007.99. Google Scholar

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M. J. Best and K. Ritter, "Linear Programming: Active Set Analysis and Computer Programs,", Prentice-Hall, (1985). Google Scholar

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L. M. Blank, L. Kuepfer and U. Sauer, Large-scale 13C-flux analysis reveals mechanistic principles of metabolic network robustness to null mutations in yeast,, Genome Biol., 6 (2005). doi: 10.1186/gb-2005-6-6-r49. Google Scholar

[6]

H. P. J. Bonarius, G. Schmid and J. Tramper, Flux analysis of underdetermined metabolic networks: The quest for the missing constraints,, Trends Biotechnol., 15 (1997), 308. doi: 10.1016/S0167-7799(97)01067-6. Google Scholar

[7]

S. P. Cornelius, J. S. Lee and A. E. Motter, Dispensability of Escherichia coli's latent pathways,, Proc. Natl. Acad. Sci. USA, 108 (2011), 3124. doi: 10.1073/pnas.1009772108. Google Scholar

[8]

N. C. Duarte, et al., Global reconstruction of the human metabolic network based on genomic and bibliomic data,, Proc. Natl. Acad. Sci. USA, 104 (2007), 1777. doi: 10.1073/pnas.0610772104. Google Scholar

[9]

S. S. Fong, A. R. Joyce and B. Ø. Palsson, Parallel adaptive evolution cultures of Escherichia coli lead to convergent growth phenotypes with different gene expression states,, Genome. Res., 15 (2005), 1365. doi: 10.1101/gr.3832305. Google Scholar

[10]

S. S. Fong, A. Nanchen, B. Ø. Palsson and U. Sauer, Latent pathway activation and increased pathway capacity enable Escherichia coli adaptation to loss of key metabolic enzymes,, J. Biol. Chem., 281 (2006), 8024. doi: 10.1074/jbc.M510016200. Google Scholar

[11]

, ILOG CPLEX (Version 10.2.0)., Available from: \url{http://www.ilog.com/products/cplex/}., (). Google Scholar

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D. E. Kaufman and R. L. Smith, Direction choice for accelerated convergence in hit-and-run sampling,, Oper. Res., 46 (1998), 84. doi: 10.1287/opre.46.1.84. Google Scholar

[13]

D.-H. Kim and A. E. Motter, Slave nodes and the controllability of metabolic networks,, New J. Phys., 11 (2009). doi: 10.1088/1367-2630/11/11/113047. Google Scholar

[14]

M. V. Kritz, M. T. dos Santos, S. Urrutia and J.-M. Schwartz, Organising metabolic networks: Cycles in flux distributions,, J. Theo. Biol., 265 (2010), 250. doi: 10.1016/j.jtbi.2010.04.026. Google Scholar

[15]

O. L. Mangasarian, Uniqueness of solution in linear programming,, Linear Algebra Appl., 25 (1979), 151. doi: 10.1016/0024-3795(79)90014-4. Google Scholar

[16]

A. E. Motter, Improved network performance via antagonism: From synthetic rescues to multi-drug combinations,, BioEssays, 32 (2010), 236. doi: 10.1002/bies.200900128. Google Scholar

[17]

A. E. Motter, N. Gulbahce, E. Almaas and A.-L. Barabási, Predicting synthetic rescues in metabolic networks,, Mol. Syst. Biol., 4 (2008). doi: 10.1038/msb.2008.1. Google Scholar

[18]

T. Nishikawa, N. Gulbahce and A. E. Motter, Spontaneous reaction silencing in metabolic optimization,, PLoS Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000236. Google Scholar

[19]

B. Papp, C. Pál and L. D. Hurst, Metabolic network analysis of the causes and evolution of enzyme dispensability in yeast,, Nature, 429 (2004), 661. doi: 10.1038/nature02636. Google Scholar

[20]

B. Ø. Palsson, "Systems Biology: Properties of Reconstructed Networks,", Cambridge University Press, (2006). Google Scholar

[21]

E. Ravasz, A. Somera, D. Mongru, Z. Oltvai and A.-L. Barabási, Hierarchical organization of modularity in metabolic networks,, Science, 297 (2002), 1551. doi: 10.1126/science.1073374. Google Scholar

[22]

J. L. Reed, T. D. Vo, C. H. Schilling and B. Ø. Palsson, An expanded genome-scale model of Escherichia coli K-12 (iJR904 GSM/GPR),, Genome Biol., 4 (2003). doi: 10.1186/gb-2003-4-9-r54. Google Scholar

[23]

W. Rudin, "Real and Complex Analysis,'', Third edition, (1987). Google Scholar

[24]

R. Schuetz, L. Kuepfer and U. Sauer, Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli,, Mol. Syst. Biol., 3 (2007). doi: 10.1038/msb4100162. Google Scholar

[25]

T. Shlomi, T. Benyamini, E. Gottlieb, R. Sharan and E. Ruppin, Genome-scale metabolic modeling elucidates the role of proliferative adaptation in causing the Warburg effect,, PLoS Comput. Biol., 7 (2011). doi: 10.1371/journal.pcbi.1002018. Google Scholar

[26]

R. L. Smith, Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions,, Oper. Res., 32 (1984), 1296. doi: 10.1287/opre.32.6.1296. Google Scholar

[27]

V. Spirin and L. A. Mirny, Protein complexes and functional modules in molecular networks,, Proc. Natl. Acad. Sci. USA, 100 (2003), 12123. doi: 10.1073/pnas.2032324100. Google Scholar

[28]

P. Szilágyi, On the uniqueness of the optimal solution in linear programming,, Rev. Anal. Numér. Théor. Approx., 35 (2006), 225. Google Scholar

[29]

A. Varma and B. Ø. Palsson, Metabolic flux balancing: Basic concepts, scientific and practical use,, Nat. Biotechnol., 12 (1994), 994. doi: 10.1038/nbt1094-994. Google Scholar

show all references

References:
[1]

U. Alon, "An Introduction to Systems Biology: Design Principles of Biological Circuits,", Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007). Google Scholar

[2]

A.-L. Barabási and Z. N. Oltvai, Network biology: Understanding the cell's functional organization,, Nat. Rev. Genet., 5 (2004), 101. Google Scholar

[3]

S. D. Becker, et al., Quantitative prediction of cellular metabolism with constraint-based models: The COBRA Toolbox,, Nat. Protoc., 2 (2007), 727. doi: 10.1038/nprot.2007.99. Google Scholar

[4]

M. J. Best and K. Ritter, "Linear Programming: Active Set Analysis and Computer Programs,", Prentice-Hall, (1985). Google Scholar

[5]

L. M. Blank, L. Kuepfer and U. Sauer, Large-scale 13C-flux analysis reveals mechanistic principles of metabolic network robustness to null mutations in yeast,, Genome Biol., 6 (2005). doi: 10.1186/gb-2005-6-6-r49. Google Scholar

[6]

H. P. J. Bonarius, G. Schmid and J. Tramper, Flux analysis of underdetermined metabolic networks: The quest for the missing constraints,, Trends Biotechnol., 15 (1997), 308. doi: 10.1016/S0167-7799(97)01067-6. Google Scholar

[7]

S. P. Cornelius, J. S. Lee and A. E. Motter, Dispensability of Escherichia coli's latent pathways,, Proc. Natl. Acad. Sci. USA, 108 (2011), 3124. doi: 10.1073/pnas.1009772108. Google Scholar

[8]

N. C. Duarte, et al., Global reconstruction of the human metabolic network based on genomic and bibliomic data,, Proc. Natl. Acad. Sci. USA, 104 (2007), 1777. doi: 10.1073/pnas.0610772104. Google Scholar

[9]

S. S. Fong, A. R. Joyce and B. Ø. Palsson, Parallel adaptive evolution cultures of Escherichia coli lead to convergent growth phenotypes with different gene expression states,, Genome. Res., 15 (2005), 1365. doi: 10.1101/gr.3832305. Google Scholar

[10]

S. S. Fong, A. Nanchen, B. Ø. Palsson and U. Sauer, Latent pathway activation and increased pathway capacity enable Escherichia coli adaptation to loss of key metabolic enzymes,, J. Biol. Chem., 281 (2006), 8024. doi: 10.1074/jbc.M510016200. Google Scholar

[11]

, ILOG CPLEX (Version 10.2.0)., Available from: \url{http://www.ilog.com/products/cplex/}., (). Google Scholar

[12]

D. E. Kaufman and R. L. Smith, Direction choice for accelerated convergence in hit-and-run sampling,, Oper. Res., 46 (1998), 84. doi: 10.1287/opre.46.1.84. Google Scholar

[13]

D.-H. Kim and A. E. Motter, Slave nodes and the controllability of metabolic networks,, New J. Phys., 11 (2009). doi: 10.1088/1367-2630/11/11/113047. Google Scholar

[14]

M. V. Kritz, M. T. dos Santos, S. Urrutia and J.-M. Schwartz, Organising metabolic networks: Cycles in flux distributions,, J. Theo. Biol., 265 (2010), 250. doi: 10.1016/j.jtbi.2010.04.026. Google Scholar

[15]

O. L. Mangasarian, Uniqueness of solution in linear programming,, Linear Algebra Appl., 25 (1979), 151. doi: 10.1016/0024-3795(79)90014-4. Google Scholar

[16]

A. E. Motter, Improved network performance via antagonism: From synthetic rescues to multi-drug combinations,, BioEssays, 32 (2010), 236. doi: 10.1002/bies.200900128. Google Scholar

[17]

A. E. Motter, N. Gulbahce, E. Almaas and A.-L. Barabási, Predicting synthetic rescues in metabolic networks,, Mol. Syst. Biol., 4 (2008). doi: 10.1038/msb.2008.1. Google Scholar

[18]

T. Nishikawa, N. Gulbahce and A. E. Motter, Spontaneous reaction silencing in metabolic optimization,, PLoS Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000236. Google Scholar

[19]

B. Papp, C. Pál and L. D. Hurst, Metabolic network analysis of the causes and evolution of enzyme dispensability in yeast,, Nature, 429 (2004), 661. doi: 10.1038/nature02636. Google Scholar

[20]

B. Ø. Palsson, "Systems Biology: Properties of Reconstructed Networks,", Cambridge University Press, (2006). Google Scholar

[21]

E. Ravasz, A. Somera, D. Mongru, Z. Oltvai and A.-L. Barabási, Hierarchical organization of modularity in metabolic networks,, Science, 297 (2002), 1551. doi: 10.1126/science.1073374. Google Scholar

[22]

J. L. Reed, T. D. Vo, C. H. Schilling and B. Ø. Palsson, An expanded genome-scale model of Escherichia coli K-12 (iJR904 GSM/GPR),, Genome Biol., 4 (2003). doi: 10.1186/gb-2003-4-9-r54. Google Scholar

[23]

W. Rudin, "Real and Complex Analysis,'', Third edition, (1987). Google Scholar

[24]

R. Schuetz, L. Kuepfer and U. Sauer, Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli,, Mol. Syst. Biol., 3 (2007). doi: 10.1038/msb4100162. Google Scholar

[25]

T. Shlomi, T. Benyamini, E. Gottlieb, R. Sharan and E. Ruppin, Genome-scale metabolic modeling elucidates the role of proliferative adaptation in causing the Warburg effect,, PLoS Comput. Biol., 7 (2011). doi: 10.1371/journal.pcbi.1002018. Google Scholar

[26]

R. L. Smith, Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions,, Oper. Res., 32 (1984), 1296. doi: 10.1287/opre.32.6.1296. Google Scholar

[27]

V. Spirin and L. A. Mirny, Protein complexes and functional modules in molecular networks,, Proc. Natl. Acad. Sci. USA, 100 (2003), 12123. doi: 10.1073/pnas.2032324100. Google Scholar

[28]

P. Szilágyi, On the uniqueness of the optimal solution in linear programming,, Rev. Anal. Numér. Théor. Approx., 35 (2006), 225. Google Scholar

[29]

A. Varma and B. Ø. Palsson, Metabolic flux balancing: Basic concepts, scientific and practical use,, Nat. Biotechnol., 12 (1994), 994. doi: 10.1038/nbt1094-994. Google Scholar

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