Conditions for duality between fluxes and concentrations in biochemical networks.

Mathematical and computational modelling of biochemical networks is often done in terms of either the concentrations of molecular species or the fluxes of biochemical reactions. When is mathematical modelling from either perspective equivalent to the other? Mathematical duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one manner. We present a novel stoichiometric condition that is necessary and sufficient for duality between unidirectional fluxes and concentrations. Our numerical experiments, with computational models derived from a range of genome-scale biochemical networks, suggest that this flux-concentration duality is a pervasive property of biochemical networks. We also provide a combinatorial characterisation that is sufficient to ensure flux-concentration duality.The condition prescribes that, for every two disjoint sets of molecular species, there is at least one reaction complex that involves species from only one of the two sets. When unidirectional fluxes and molecular species concentrations are dual vectors, this implies that the behaviour of the corresponding biochemical network can be described entirely in terms of either concentrations or unidirectional fluxes.

[1]  M. Muir.  Physical Chemistry , 1888, Nature.

[2]  Ronan M. T. Fleming,et al.  Genome-Scale Reconstruction of Escherichia coli's Transcriptional and Translational Machinery: A Knowledge Base, Its Mathematical Formulation, and Its Functional Characterization , 2009, PLoS Comput. Biol..

[3]  Richard A. Brualdi,et al.  Matrices of Sign-Solvable Linear Systems , 1995 .

[4]  D. Fell,et al.  A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks , 2000, Nature Biotechnology.

[5]  Steffen Klamt,et al.  Minimal cut sets in a metabolic network are elementary modes in a dual network , 2012, Bioinform..

[6]  Ronan M. T. Fleming,et al.  Multiscale Modeling of Metabolism and Macromolecular Synthesis in E. coli and Its Application to the Evolution of Codon Usage , 2012, PloS one.

[7]  P. Gill,et al.  Maintaining LU factors of a general sparse matrix , 1987 .

[8]  Ronald L. Rivest,et al.  Introduction to Algorithms, 3rd Edition , 2009 .

[9]  Nicole Fruehauf,et al.  Enzyme Kinetics And Mechanism , 2016 .

[10]  Steffen Klamt,et al.  Genome-scale strain designs based on regulatory minimal cut sets , 2015, Bioinform..

[11]  David A. Fell,et al.  Detection of stoichiometric inconsistencies in biomolecular models , 2008, Bioinform..

[12]  H. Schneider,et al.  On the singular graph and the Weyr characteristic of anM-matrix , 1978 .

[13]  Steffen Klamt,et al.  Minimal cut sets in biochemical reaction networks , 2004, Bioinform..

[14]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[15]  Ronan M. T. Fleming,et al.  Quantitative prediction of cellular metabolism with constraint-based models: the COBRA Toolbox v2.0 , 2007, Nature Protocols.

[16]  Ronan M. T. Fleming,et al.  Mass conserved elementary kinetics is sufficient for the existence of a non-equilibrium steady state concentration. , 2011, Journal of theoretical biology.

[17]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[18]  M. Noordewier,et al.  Genome Streamlining in a Cosmopolitan Oceanic Bacterium , 2005, Science.

[19]  Jennifer L. Reed,et al.  iRsp1095: A genome-scale reconstruction of the Rhodobacter sphaeroides metabolic network , 2011, BMC Systems Biology.

[20]  Ines Thiele,et al.  rBioNet: A COBRA toolbox extension for reconstructing high-quality biochemical networks , 2011, Bioinform..

[21]  Alicia Dickenstein,et al.  Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry , 2013, Found. Comput. Math..

[22]  David A. Rosenblueth,et al.  An overview of existing modeling tools making use of model checking in the analysis of biochemical networks , 2012, Front. Plant Sci..

[23]  Ronan M. T. Fleming,et al.  Accelerating the DC algorithm for smooth functions , 2018, Math. Program..

[24]  R Heinrich,et al.  Metabolic regulation and mathematical models. , 1977, Progress in biophysics and molecular biology.

[25]  G N Lewis,et al.  A New Principle of Equilibrium. , 1925, Proceedings of the National Academy of Sciences of the United States of America.

[26]  Michael Hucka,et al.  SBMLToolbox: an SBML toolbox for MATLAB users , 2006, Bioinform..

[27]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[28]  T. Duke,et al.  Conformational spread: the propagation of allosteric states in large multiprotein complexes. , 2004, Annual review of biophysics and biomolecular structure.

[29]  Michael A. Saunders,et al.  Robust flux balance analysis of multiscale biochemical reaction networks , 2013, BMC Bioinformatics.

[30]  Ludwig Wilhelmy,et al.  Ueber das Gesetz, nach welchem die Einwirkung der Säuren auf den Rohrzucker stattfindet , 1850 .

[31]  Daniel Hershkowitz,et al.  Ranks of zero patterns and sign patterns , 1993 .

[32]  B. Palsson,et al.  A protocol for generating a high-quality genome-scale metabolic reconstruction , 2010 .

[33]  Ronan M. T. Fleming,et al.  Consistent Estimation of Gibbs Energy Using Component Contributions , 2013, PLoS Comput. Biol..

[34]  Sylvain Soliman,et al.  Invariants and Other Structural Properties of Biochemical Models as a Constraint Satisfaction Problem , 2012, Algorithms for Molecular Biology.

[35]  Nikos Vlassis,et al.  Fast Reconstruction of Compact Context-Specific Metabolic Network Models , 2013, PLoS Comput. Biol..

[36]  B. Palsson,et al.  Flux-concentration duality in dynamic nonequilibrium biological networks. , 2009, Biophysical journal.

[37]  B. Palsson.  Systems Biology: Properties of Reconstructed Networks , 2006 .

[38]  Matthew N. Benedict,et al.  Genome-Scale Metabolic Reconstruction and Hypothesis Testing in the Methanogenic Archaeon Methanosarcina acetivorans C2A , 2011, Journal of bacteriology.

[39]  E. Hill.  Journal of Theoretical Biology , 1961, Nature.

[40]  Steffen Klamt,et al.  Enumeration of Smallest Intervention Strategies in Genome-Scale Metabolic Networks , 2014, PLoS Comput. Biol..

[41]  Steffen Klamt,et al.  Computing Knock-Out Strategies in Metabolic Networks , 2007, J. Comput. Biol..