Sequential computation of elementary modes and minimal cut sets in genome‐scale metabolic networks using alternate integer linear programming

Motivation: Elementary (flux) modes (EMs) have served as a valuable tool for investigating structural and functional properties of metabolic networks. Identification of the full set of EMs in genome‐scale networks remains challenging due to combinatorial explosion of EMs in complex networks. It is often, however, that only a small subset of relevant EMs needs to be known, for which optimization‐based sequential computation is a useful alternative. Most of the currently available methods along this line are based on the iterative use of mixed integer linear programming (MILP), the effectiveness of which significantly deteriorates as the number of iterations builds up. To alleviate the computational burden associated with the MILP implementation, we here present a novel optimization algorithm termed alternate integer linear programming (AILP). Results: Our algorithm was designed to iteratively solve a pair of integer programming (IP) and linear programming (LP) to compute EMs in a sequential manner. In each step, the IP identifies a minimal subset of reactions, the deletion of which disables all previously identified EMs. Thus, a subsequent LP solution subject to this reaction deletion constraint becomes a distinct EM. In cases where no feasible LP solution is available, IP‐derived reaction deletion sets represent minimal cut sets (MCSs). Despite the additional computation of MCSs, AILP achieved significant time reduction in computing EMs by orders of magnitude. The proposed AILP algorithm not only offers a computational advantage in the EM analysis of genome‐scale networks, but also improves the understanding of the linkage between EMs and MCSs. Availability and Implementation: The software is implemented in Matlab, and is provided as supplementary information. Contact: hyunseob.song@pnnl.gov Supplementary information: Supplementary data are available at Bioinformatics online.

[1]  Christian Jungreuthmayer,et al.  Designing optimal cell factories: integer programming couples elementary mode analysis with regulation , 2012, BMC Systems Biology.

[2]  Doraiswami Ramkrishna,et al.  Dynamic models of metabolism: Review of the cybernetic approach , 2012 .

[3]  A. Konopka,et al.  Mathematical Modeling of Microbial Community Dynamics: A Methodological Review , 2014 .

[4]  D. Ramkrishna,et al.  Modeling metabolic systems: the need for dynamics , 2013 .

[5]  Markus J. Herrgård,et al.  Reconstruction and validation of Saccharomyces cerevisiae iND750, a fully compartmentalized genome-scale metabolic model. , 2004, Genome research.

[6]  Stefan Schuster,et al.  Systems biology Metatool 5.0: fast and flexible elementary modes analysis , 2006 .

[7]  Angel Rubio,et al.  Computing the shortest elementary flux modes in genome-scale metabolic networks , 2009, Bioinform..

[8]  Lake-Ee Quek,et al.  A depth-first search algorithm to compute elementary flux modes by linear programming , 2014, BMC Systems Biology.

[9]  Peter Ruhdal Jensen,et al.  Estimating biological elementary flux modes that decompose a flux distribution by the minimal branching property , 2014, Bioinform..

[10]  Bernhard O. Palsson,et al.  BiGG: a Biochemical Genetic and Genomic knowledgebase of large scale metabolic reconstructions , 2010, BMC Bioinformatics.

[11]  Alexander Bockmayr,et al.  Computing Elementary Flux Modes Involving a Set of Target Reactions , 2014, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[12]  Hiroyuki Kurata,et al.  BMC Systems Biology BioMed Central , 2007 .

[13]  Robert Urbanczik,et al.  An improved algorithm for stoichiometric network analysis: theory and applications , 2005, Bioinform..

[14]  Ping Ji,et al.  Decomposing flux distributions into elementary flux modes in genome-scale metabolic networks , 2011, Bioinform..

[15]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[16]  D. Ramkrishna,et al.  A hybrid model of anaerobic E. coli GJT001: Combination of elementary flux modes and cybernetic variables , 2008, Biotechnology progress.

[17]  Y. Schneider,et al.  Metabolic design of macroscopic bioreaction models: application to Chinese hamster ovary cells , 2006, Bioprocess and biosystems engineering.

[18]  H. Raiffa,et al.  3. The Double Description Method , 1953 .

[19]  Steffen Klamt,et al.  Computing complex metabolic intervention strategies using constrained minimal cut sets. , 2011, Metabolic engineering.

[20]  R. Mahadevan,et al.  Using metabolic flux data to further constrain the metabolic solution space and predict internal flux patterns: the Escherichia coli spectrum , 2004, Biotechnology and bioengineering.

[21]  S. Schuster,et al.  Analysis of structural robustness of metabolic networks. , 2004, Systems biology.

[22]  C. Wagner.  Nullspace Approach to Determine the Elementary Modes of Chemical Reaction Systems , 2004 .

[23]  Ryo Tsuboi,et al.  Complementary elementary modes for fast and efficient analysis of metabolic networks , 2014 .

[24]  Egon Balas,et al.  On the Set-Covering Problem , 1972, Oper. Res..

[25]  Eugénio C. Ferreira,et al.  Random sampling of elementary flux modes in large-scale metabolic networks , 2012, Bioinform..

[26]  Christian Jungreuthmayer,et al.  regEfmtool: Speeding up elementary flux mode calculation using transcriptional regulatory rules in the form of three-state logic , 2013, Biosyst..

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

[28]  Steffen Klamt,et al.  Structural and functional analysis of cellular networks with CellNetAnalyzer , 2007, BMC Systems Biology.

[29]  Alexander Bockmayr,et al.  Analysis of Metabolic Subnetworks by Flux Cone Projection , 2011, Algorithms for Molecular Biology.

[30]  Steffen Klamt,et al.  Comparison and improvement of algorithms for computing minimal cut sets , 2013, BMC Bioinformatics.

[31]  Reed L. Taffs,et al.  In silico approaches to study mass and energy flows in microbial consortia: a syntrophic case study , 2009, BMC Systems Biology.

[32]  Steffen Klamt,et al.  CASOP GS: Computing Intervention Strategies Targeted at Production Improvement in Genome-scale Metabolic Networks , 2010, GCB.

[33]  D. Ramkrishna,et al.  Reduction of a set of elementary modes using yield analysis , 2009, Biotechnology and bioengineering.

[34]  Steffen Klamt,et al.  CASOP: a computational approach for strain optimization aiming at high productivity. , 2010, Journal of biotechnology.

[35]  Kalyan Gayen,et al.  Analysis of optimal phenotypic space using elementary modes as applied to Corynebacterium glutamicum , 2006, BMC Bioinformatics.

[36]  Adam M. Feist,et al.  A genome-scale metabolic reconstruction for Escherichia coli K-12 MG1655 that accounts for 1260 ORFs and thermodynamic information , 2007, Molecular systems biology.

[37]  A. Burgard,et al.  Optknock: A bilevel programming framework for identifying gene knockout strategies for microbial strain optimization , 2003, Biotechnology and bioengineering.

[38]  R. Carlson,et al.  Design, construction and performance of the most efficient biomass producing E. coli bacterium. , 2006, Metabolic engineering.

[39]  D. Ramkrishna,et al.  Systematic development of hybrid cybernetic models: Application to recombinant yeast co‐consuming glucose and xylose , 2009, Biotechnology and bioengineering.

[40]  Kristopher A. Hunt,et al.  Complete enumeration of elementary flux modes through scalable demand-based subnetwork definition , 2014, Bioinform..

[41]  Hans C. Bernstein,et al.  Microbial Community Metabolic Modeling: A Community Data‐Driven Network Reconstruction , 2016, Journal of cellular physiology.

[42]  Steffen Klamt,et al.  Two approaches for metabolic pathway analysis? , 2003, Trends in biotechnology.

[43]  D. Ramkrishna,et al.  Cybernetic models based on lumped elementary modes accurately predict strain‐specific metabolic function , 2011, Biotechnology and bioengineering.

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

[45]  Steffen Klamt,et al.  An application programming interface for CellNetAnalyzer , 2011, Biosyst..

[46]  Francisco J. Planes,et al.  Direct calculation of elementary flux modes satisfying several biological constraints in genome-scale metabolic networks , 2014, Bioinform..

[47]  Leonid Khachiyan,et al.  On the Complexity of Dualization of Monotone Disjunctive Normal Forms , 1996, J. Algorithms.

[48]  Mauro Dell'Amico,et al.  Annotated Bibliographies in Combinatorial Optimization , 1997 .

[49]  Christian Jungreuthmayer,et al.  tEFMA: computing thermodynamically feasible elementary flux modes in metabolic networks , 2015, Bioinform..

[50]  J. Stelling,et al.  Combinatorial Complexity of Pathway Analysis in Metabolic Networks , 2004, Molecular Biology Reports.

[51]  R. Mahadevan,et al.  The effects of alternate optimal solutions in constraint-based genome-scale metabolic models. , 2003, Metabolic engineering.

[52]  S. Klamt,et al.  Generalized concept of minimal cut sets in biochemical networks. , 2006, Bio Systems.

[53]  Robert Urbanczik,et al.  Functional stoichiometric analysis of metabolic networks , 2005, Bioinform..

[54]  R. Carlson,et al.  Fundamental Escherichia coli biochemical pathways for biomass and energy production: Identification of reactions , 2004, Biotechnology and bioengineering.

[55]  F. Srienc,et al.  Minimal Escherichia coli Cell for the Most Efficient Production of Ethanol from Hexoses and Pentoses , 2008, Applied and Environmental Microbiology.

[56]  Doraiswami Ramkrishna,et al.  Prediction of metabolic function from limited data: Lumped hybrid cybernetic modeling (L‐HCM) , 2010, Biotechnology and bioengineering.

[57]  Yee Wen Choon,et al.  A hybrid of bees algorithm and flux balance analysis with OptKnock as a platform for in silico optimization of microbial strains , 2013, Bioprocess and Biosystems Engineering.

[58]  A. Kienle,et al.  Experimental and theoretical analysis of poly (β-hydroxybutyrate) formation and consumption in Ralstonia eutropha , 2011 .

[59]  Jörg Stelling,et al.  Parallel Extreme Ray and Pathway Computation , 2009, PPAM.

[60]  Francisco J. Planes,et al.  TreeEFM: calculating elementary flux modes using linear optimization in a tree-based algorithm , 2015, Bioinform..

[61]  S. Schuster,et al.  Structural robustness of metabolic networks with respect to multiple knockouts. , 2008, Journal of theoretical biology.

[62]  Jeffrey D Orth,et al.  What is flux balance analysis? , 2010, Nature Biotechnology.

[63]  Eugénio C. Ferreira,et al.  Selection of Elementary Modes for Bioprocess Control , 2010 .

[64]  Vincent D. Blondel,et al.  Fast computation of minimal elementary decompositions of metabolic flux vectors , 2011, Autom..

[65]  S. Schuster,et al.  Metabolic network structure determines key aspects of functionality and regulation , 2002, Nature.

[66]  B. L. Clarke.  Stoichiometric network analysis , 2008, Cell Biophysics.

[67]  Jörg Stelling,et al.  Large-scale computation of elementary flux modes with bit pattern trees , 2008, Bioinform..

[68]  J. Stelling,et al.  Genome‐scale metabolic networks , 2009, Wiley interdisciplinary reviews. Systems biology and medicine.

[69]  J. Reifman,et al.  Prediction of Metabolic Flux Distribution from Gene Expression Data Based on the Flux Minimization Principle , 2014, PloS one.

[70]  Georges Bastin,et al.  Dynamic metabolic modelling under the balanced growth condition , 2004 .