Reaction routes in biochemical reaction systems: Algebraic properties, validated calculation procedure and example from nucleotide metabolism

Abstract. Elementary flux modes (direct reaction routes) are minimal sets of enzymes that can operate at steady state, with all irreversible reactions used in the appropriate direction. They can be interpreted as component pathways of a (bio)chemical reaction network. Here, two different definitions of elementary modes are given and their equivalence is proved. Several algebraic properties of elementary modes are then presented and proved. This concerns, amongst other features, the minimal number of enzymes of the network not used in an elementary mode and the situations where irreversible reactions are replaced by reversible ones. Based on these properties, a refined algorithm is presented, and it is formally proved that this algorithm will exclusively generate all the elementary flux modes of an arbitrary network containing reversible or irreversible reactions or both. The algorithm is illustrated by a biochemical example relevant in nucleotide metabolism. The computer implementation in two different programming languages is discussed.

[1]  F. Young.  Biochemistry , 1955, The Indian Medical Gazette.

[2]  J. Horiuti,et al.  Stoichiometric Number and the Theory of Steady Reaction , 1957 .

[3]  P. C. Milner,et al.  The Possible Mechanisms of Complex Reactions Involving Consecutive Steps , 1964 .

[4]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[5]  F. Nožička.  Theorie der linearen parametrischen Optimierung , 1974 .

[6]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[7]  Bruce L. Clarke,et al.  Complete set of steady states for the general stoichiometric dynamical system , 1981 .

[8]  Peter H. Sellers,et al.  Multiple reaction mechanisms of catalysis , 1982 .

[9]  H G Hers,et al.  Gluconeogenesis and related aspects of glycolysis. , 1983, Annual review of biochemistry.

[10]  Peter H. Sellers,et al.  Combinatorial Classification of Chemical Mechanisms , 1984 .

[11]  C. Salerno,et al.  Hypoxanthine-guanine exchange by intact human erythrocytes. , 1985, Biochemistry.

[12]  Wolfgang Reisig,et al.  Petri Nets , 1985, EATCS Monographs on Theoretical Computer Science.

[13]  D. Fell,et al.  Fat synthesis in adipose tissue. An examination of stoichiometric constraints. , 1986, The Biochemical journal.

[14]  J E Bailey,et al.  MPS: An artificially intelligent software system for the analysis and synthesis of metabolic pathways , 1988, Biotechnology and bioengineering.

[15]  C. W. Groetsch,et al.  Matrix methods and applications , 1988 .

[16]  C Reder,et al.  Metabolic control theory: a structural approach. , 1988, Journal of theoretical biology.

[17]  J J Blum,et al.  Quantitative analysis of intermediary metabolism in rat hepatocytes incubated in the presence and absence of ethanol with a substrate mixture including ketoleucine. , 1989, The Biochemical journal.

[18]  B. Palsson,et al.  Metabolic dynamics in the human red cell. Part I--A comprehensive kinetic model. , 1989, Journal of theoretical biology.

[19]  Manuel Silva Suárez,et al.  Convex geometry and semiflows in P/T nets. A comparative study of algorithms for computation of minimal P-semiflows , 1991, Applications and Theory of Petri Nets.

[20]  Peter H. Sellers,et al.  THE CHARACTERIZATION OF COMPLEX SYSTEMS OF CHEMICAL REACTIONS , 1989 .

[21]  Peter H. Starke,et al.  Analyse von Petri-Netz-Modellen , 1990, Leitfäden und Monographien der Informatik.

[22]  G. Stephanopoulos,et al.  Computer‐aided synthesis of biochemical pathways , 1990, Biotechnology and bioengineering.

[23]  S. Schuster,et al.  Determining all extreme semi-positive conservation relations in chemical reaction systems: a test criterion for conservativity , 1991 .

[24]  Stefan Schuster,et al.  Detecting strictly detailed balanced subnetworks in open chemical reaction networks , 1991 .

[25]  M. Mavrovouniotis.  Synthesis of reaction mechanisms consisting of reversible and irreversible steps. 2. Formalization and analysis of the synthesis algorithm , 1992 .

[26]  Stefan Schuster,et al.  Refined algorithm and computer program for calculating all non-negative fluxes admissible in steady states of biochemical reaction systems with or without some flux rates fixed , 1993, Comput. Appl. Biosci..

[27]  D. Fell.  The Analysis of Flux in Substrate Cycles , 1993 .

[28]  Andrew V. Zeigarnik,et al.  A GRAPH-THEORETICAL MODEL OF COMPLEX REACTION MECHANISMS : BIPARTITE GRAPHS AND THE STOICHIOMETRY OF COMPLEX REACTIONS , 1994 .

[29]  Ralf Hofestädt,et al.  A petri net application to model metabolic processes , 1994 .

[30]  D. Fraenkel,et al.  The effect of increased phosphoglucose isomerase on glucose metabolism in Saccharomyces cerevisiae. , 1994, The Journal of biological chemistry.

[31]  S. Schuster,et al.  ON ELEMENTARY FLUX MODES IN BIOCHEMICAL REACTION SYSTEMS AT STEADY STATE , 1994 .

[32]  V. N. Reddy,et al.  Qualitative analysis of biochemical reaction systems , 1996, Comput. Biol. Medicine.

[33]  Stefan Schuster,et al.  ELEMENTARY MODES OF FUNCTIONING IN BIOCHEMICAL NETWORKS , 1996 .

[34]  R. Heinrich,et al.  The Regulation of Cellular Systems , 1996, Springer US.

[35]  J. Liao,et al.  Pathway analysis, engineering, and physiological considerations for redirecting central metabolism. , 1996, Biotechnology and bioengineering.

[36]  J. C. Nuño,et al.  Network organization of cell metabolism: monosaccharide interconversion. , 1997, The Biochemical journal.

[37]  R. Overbeek,et al.  A reconstruction of the metabolism of Methanococcus jannaschii from sequence data. , 1997, Gene.

[38]  Suzanne M. Paley,et al.  Integrated pathway/genome databases and their role in drug discovery , 1999 .

[39]  Juan Carlos Nuño,et al.  METATOOL: for studying metabolic networks , 1999, Bioinform..

[40]  R. Heinrich,et al.  Metabolic Pathway Analysis: Basic Concepts and Scientific Applications in the Post‐genomic Era , 1999, Biotechnology progress.

[41]  B. Snel,et al.  Pathway alignment: application to the comparative analysis of glycolytic enzymes. , 1999, The Biochemical journal.

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

[43]  L. Wackett.  Metabolic engineering , 2009, Nature biotechnology.

[44]  B. Palsson,et al.  Assessment of the metabolic capabilities of Haemophilus influenzae Rd through a genome-scale pathway analysis. , 2000, Journal of theoretical biology.

[45]  B. Palsson,et al.  Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective. , 2000, Journal of theoretical biology.

[46]  R. Datta,et al.  A thermodynamic approach to the systematic elucidation of unique reaction routes in catalytic reactions , 2000 .

[47]  Steffen Klamt,et al.  Calculability analysis in underdetermined metabolic networks illustrated by a model of the central metabolism in purple nonsulfur bacteria. , 2002, Biotechnology and bioengineering.