# Answer Set Programming for Computing Constraints-Based Elementary Flux Modes: Application to Escherichia coli Core Metabolism

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## Abstract

**:**

_{2}gradients spanning fully aerobic to anaerobic, can be further reduced to four optimal EFMs using post-processing and Pareto front analysis.

## 1. Introduction

_{2}and anabolic nitrogen, have been reported using tradeoff analysis and Pareto optimization [9,10,11,12]. The methodology tabulates the resource requirements to realize each EFM; these resources can be anabolic, e.g., nitrogen to assemble metabolic enzymes, which are described here as resource investment costs or catabolic, e.g., O

_{2}which serves as an electron acceptor, which are described here as resource operating costs. Some resources can serve both anabolic and catabolic functions like glucose which is both an energy source and carbon source for enzyme synthesis. Optimal phenotypes for acclimating to environments along gradients of resource scarcity can be identified by plotting the resource costs for each EFM in a tradeoff space where Pareto optimality identifies the most competitive phenotypes [13]. Those EFMs that minimize the resource requirements to achieve a target cellular function are considered most competitive because the phenotypes would permit the most biomass to be made based on a finite supply of a substrate. Tradeoff analysis has accurately predicted and interpreted E. coli acclimation to O

_{2}, carbon, and nitrogen, scarcity based on physiological, proteomics, and fluxomics data from E. coli chemostat cultures [14,15].

## 2. Results

#### 2.1. Application on the E. coli core Model

_{2}, NH${}_{4}^{+}$, inorganic phosphate, H

^{+}, H

_{2}O and O

_{2}. Accordingly, all other transport reactions were inactivated. The biomass-producing EFMs were selected to represent cellular growth. To further reduce computational burden, the solution space was limited to EFMs with a O

_{2}operating cost of less than 0.7 O

_{2}moles per biomass C mole and a glucose operating cost of less than seven glucose C moles per biomass C mole. Since the presence of O

_{2}had a large impact on the regulatory constraints, two separate scenarios were considered: (1) aerobic and (2) anaerobic conditions.

_{2}availability, as described previously in Carlson and Srienc, 2004 [9]. EFMs that permitted optimal acclimation to gradients of O

_{2}scarcity had the lowest substrate operating costs (C moles glucose consumed/C mole biomass produced and moles O

_{2}consumed/C mole biomass produced) defining a Pareto front. Four EFMs defined the Pareto surface with the applied constraints (Figure 1, Appendix A).

#### 2.2. Model Modifications

_{2}. The pyruvate formate lyase (PFL) enzyme, which produces formate, is O

_{2}sensitive, but activity is possible when dissolved O

_{2}concentrations are low, as occurs when cells grow rapidly or in high density cell cultures [14,15,45,46]. In the regulation network, the PFL enzyme was disabled in the presence of O

_{2}by transcriptional regulators ArcA and FNR. Removing this regulation rule for formate metabolism resulted in a ∼10-fold increase in the number of total EFMs (Table 1) and a slightly different Pareto front, which predicted formate production at low O

_{2}availability (Figure 2), consistent with experimental data and previous EFM analyses [9,14,15,46]. Briefly, the Pareto front included the most efficient EFM for producing biomass from glucose, the upper left EFM, which also had a relatively high O

_{2}requirement. As environmental O

_{2}availability decreases, optimal use of the network shifts right along the Pareto front quantifying the increased requirement for glucose as metabolic byproducts are secreted. The first predicted byproduct moving down the Pareto surface was acetate, followed by a combination of acetate and formate and, finally, under anaerobic conditions acetate, formate and ethanol.

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Answer Set Programming

#### 4.2. Problem Formulation of EFMs Computation

#### 4.3. Constraints’ Formulation

_{2}flux must be inferior to 30 times the biomass flux: ${\nu}_{tspO2}<30\phantom{\rule{3.33333pt}{0ex}}{\nu}_{BIOMASS}$. Considering there are about 42.5 C moles in the E. coli core biomass, this results in taking the EFMs, the oxygen operating cost of which is less than 0.7 O

_{2}moles per biomass C mole.

#### 4.4. Pareto Surface of Optimal Functioning

_{2}, and that defined in aggregate, a surface of optimal functioning.

_{2}availability spanning sufficiency to anaerobic conditions.

## 5. Conclusions

_{2}availability spanning sufficiency to anaerobic conditions. The tool greatly expands the size range of metabolic models that can be analyzed for EFMs and, thus, greatly expands the potential for using EFMs to interpret complex biological behaviors.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

EFM | Elementary Flux Mode |

EFMA | Elementary Flux Modes Analysis |

FBA | Flux Balance Analysis |

DD | Double Description |

ASP | Answer Set Programming |

LP | Linear Programming |

SMT | Satisfiability Modulo Theories |

MCFM | Minimal Constrained Flux Mode |

MILP | Mixed Integer Linear Program |

## Appendix A. Pareto Optimal Pathways of E. coli

## Appendix B. E. coli Biomass Modifications

_{2}O coefficients in order for the number of electrons and hydrogen and oxygen moles in the biomass to be closer to typical E. coli values. The calculations are detailed in the Excel file attached in Supplementary File S2.

## Appendix C. Pareto Optimal Pathways of E. coli with the Adjusted Biomass

_{2}operating cost less than 1.4 O

_{2}moles per biomass C mole and a glucose operating cost less than 14 C moles per biomass C mole. In addition, maintenance reaction ATPM from the model was disabled. The results are presented in Table A1.

**Table A1.**Number of EFMs retrieved on the modified E. coli core network depending on culturing conditions for the adjusted biomass. Computation time given within brackets. Disabling the formate regulation returned EFMs for both aerobic and anaerobic conditions in a single execution.

Standard Regulation | No Formate Regulation | ||
---|---|---|---|

Processing | Aerobic conditions | 4273 EFMs [2362 s] | 16,411 EFMs [8005 s] |

Anaerobic conditions | 930 EFMs [469 s] | ||

Post-processing | Filtered out MCFMs | 36 MCFMs | 137 MCFMs |

Pareto optimal in biomass yield | 5 EFMs | 9 EFMs |

_{2}availability decreases, EFMs start producing acetate, acetate, and formate and, finally, acetate, formate, and ethanol under anaerobic conditions.

**Figure A1.**E. coli core EFMs sorted by carbon/biomass uptake rate and oxygen/biomass uptake rate. Biomass was modified to include ATP maintenance. Regulation constraints are as described in Orth et al. 2010.

**Figure A2.**E. coli core EFMs sorted by carbon/biomass uptake rate and oxygen/biomass uptake rate. Biomass was modified to include ATP maintenance. Regulation constraints allow the production of formate in aerobic conditions.

## Appendix D. Additional Results

**Table A2.**Additional results observed for the original biomass. Computation time given within brackets.

Constraints | Filtered out MCFMs | EFMs and MCFMs | ||||
---|---|---|---|---|---|---|

With regulation and environment | O_{2} | No O_{2} | Formate | O_{2} | No O_{2} | Formate |

No additional constraints | 0 | 0 | 0 | 4027 [1314 s] | 1459 [602 s] | 28,256 [5572 s] |

Biomass-producing | 0 | 0 | 0 | 2746 [833 s] | 1355 [436 s] | 24,324 [6281 s] |

Biomass-producing Thermodynamic data | 0 | 0 | 0 | 2746 [901 s] | 1355 [471 s] | 24,324 [6843 s] |

Biomass-producing Yields (O_{2} < 0.7) (C < 7) | 39 | 0 | 119 | 1157 [560 s] | 363 [220 s] | 11,136 [4884 s] |

Biomass-producing Thermo and Yields | 39 | 0 | 119 | 1157 [542 s] | 363 [232 s] | 11,136 [5318 s] |

**Table A3.**Additional results observed for the revised biomass; BP: Biomass-Producing. Computation time given within brackets.

Constraints | Filtered out MCFMs | EFMs and MCFMs | |||||
---|---|---|---|---|---|---|---|

With regulation and environment | O_{2} | No O_{2} | Formate | O_{2} | No O_{2} | Formate | |

ATPM | No additional constraints | 0 | 0 | 0 | 8354 [2518 s] | 1260 [473 s] | 33,499 [6676 s] |

Biomass-producing | 3 | 0 | 3 | 7076 [2939 s] | 1156 [428 s] | 29,570 [8697 s] | |

No ATPM | No additional constraints | 0 | 0 | 0 | 7735 [2337s] | 1228 [428s] | 32,098 [6474s] |

Biomass-producing | 3 | 0 | 3 | 6656 [2948 s] | 1140 [441 s] | 28,795 [8664 s] | |

BP Thermodynamic data | 3 | 0 | 3 | 6656 [3027 s] | 1140 [458 s] | 28,795 [8744 s] | |

BP Yields (O_{2} < 1.4) (C < 14) | 36 | 0 | 137 | 4309 [2369 s] | 930 [473 s] | 16,548 [7904 s] | |

BP Thermo and yields | 36 | 0 | 137 | 4309 [2362 s] | 930 [469 s] | 16,548 [8005 s] |

## Appendix E. ASP Encoding

- $\left\{\mathtt{flux}\right(\mathtt{r}\left)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}r\in R\}$ representing the flux values ${\nu}_{r}$ for every reaction r. These are theory atoms valued during the solving by clingo[LP]. The vector $\nu $ composed of all values contained in the
`flux`atoms of a solution is a flux vector. - $\left\{\mathtt{support}\right(\mathtt{r}\left)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}r\in R\}$ representing active reactions, reactions r such that ${z}_{r}=1$. There is no atom
`support(r)`for reactions r for which ${z}_{r}=0$. In this way, the set of all`support`atoms represents the support $Supp\left(\nu \right)$ of the solution flux vector $\nu $.

## Appendix F. ASP Programs

`solve[LP].lp4`: Program implementing the computation of EFMs under constraints. Works with any network and constraints encoded in ASP as presented in Appendix E.`orth_ecoli_core.lp4`: ASP translation of the network, using the encoding established above.`orth_ecoli_core_atp.lp4`: ASP translation of the network with modified biomass.`ecoli_core_regul.lp4`: Full translation of the E. coli core transcriptional regulation network.`ecoli_core_additional_constraints.lp4`: Additional constraints for the E. coli core network, including environments, thermodynamic constraints and operating costs constraints.

## Appendix G. Additional Python Code

## References

- Schuster, S.; Hilgetag, C. On Elementary Flux Modes in Biochemical Reaction Systems at Steady State. J. Biol. Syst.
**1994**, 2, 165–182. [Google Scholar] [CrossRef] - Schuster, S.; Fell, D.; Dandekar, T. A General Definition of Metabolic Pathways Useful for Systematic Organization and Analysis of Complex Metabolic Networks. Nat. Biotechnol.
**2000**, 18, 326–332. [Google Scholar] [CrossRef] [PubMed] - Behre, J.; Wilhelm, T.; von Kamp, A.; Ruppin, E.; Schuster, S. Structural Robustness of Metabolic Networks with Respect to Multiple Knockouts. J. Theor. Biol.
**2008**, 252, 433–441. [Google Scholar] [CrossRef] [PubMed] - Gerstl, M.P.; Klamt, S.; Jungreuthmayer, C.; Zanghellini, J. Exact Quantification of Cellular Robustness in Genome-Scale Metabolic Networks. Bioinformatics
**2015**, 32, 730–737. [Google Scholar] [CrossRef] [PubMed] - Jungreuthmayer, C.; Nair, G.; Klamt, S.; Zanghellini, J. Comparison and Improvement of Algorithms for Computing Minimal Cut Sets. BMC Bioinform.
**2013**, 14, 318. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jungreuthmayer, C.; Ruckerbauer, D.E.; Gerstl, M.P.; Hanscho, M.; Zanghellini, J. Avoiding the Enumeration of Infeasible Elementary Flux Modes by Including Transcriptional Regulatory Rules in the Enumeration Process Saves Computational Costs. PLoS ONE
**2015**, 10, e0129840. [Google Scholar] [CrossRef][Green Version] - Trinh, C.T.; Unrean, P.; Srienc, F. Minimal Escherichia Coli Cell for the Most Efficient Production of Ethanol from Hexoses and Pentoses. Appl. Environ. Microbiol.
**2008**, 74, 3634–3643. [Google Scholar] [CrossRef][Green Version] - Hädicke, O.; Klamt, S. Computing Complex Metabolic Intervention Strategies Using Constrained Minimal Cut Sets. Metab. Eng.
**2011**, 13, 204–213. [Google Scholar] [CrossRef] - Carlson, R.; Srienc, F. Fundamental Escherichia Coli Biochemical Pathways for Biomass and Energy Production: Identification of Reactions. Biotechnol. Bioeng.
**2004**, 85, 1–19. [Google Scholar] [CrossRef] - Carlson, R.P. Metabolic Systems Cost-Benefit Analysis for Interpreting Network Structure and Regulation. Bioinformatics
**2007**, 23, 1258–1264. [Google Scholar] [CrossRef][Green Version] - Carlson, R.P. Decomposition of Complex Microbial Behaviors into Resource-Based Stress Responses. Bioinformatics
**2009**, 25, 90–97. [Google Scholar] [CrossRef] [PubMed] - Carlson, R.P.; Taffs, R.L. Molecular-Level Tradeoffs and Metabolic Adaptation to Simultaneous Stressors. Curr. Opin. Biotechnol.
**2010**, 21, 670–676. [Google Scholar] [CrossRef] [PubMed][Green Version] - Beck, A.; Hunt, K.; Bernstein, H.; Carlson, R. Chapter 15—Interpreting and Designing Microbial Communities for Bioprocess Applications, from Components to Interactions to Emergent Properties. In Biotechnology for Biofuel Production and Optimization; Eckert, C.A., Trinh, C.T., Eds.; Elsevier: Amsterdam, The Netherlands, 2016; pp. 407–432. [Google Scholar] [CrossRef]
- Folsom, J.P.; Carlson, R.P. Physiological, Biomass Elemental Composition and Proteomic Analyses of Escherichia Coli Ammonium-Limited Chemostat Growth, and Comparison with Iron- and Glucose-Limited Chemostat Growth. Microbiology
**2015**, 161, 1659–1670. [Google Scholar] [CrossRef] [PubMed][Green Version] - Folsom, J.P.; Parker, A.E.; Carlson, R.P. Physiological and Proteomic Analysis of Escherichia coli Iron-Limited Chemostat Growth. J. Bacteriol.
**2014**, 196, 2748–2761. [Google Scholar] [CrossRef] [PubMed][Green Version] - Müller, S.; Regensburger, G.; Steuer, R. Enzyme Allocation Problems in Kinetic Metabolic Networks: Optimal Solutions Are Elementary Flux Modes. J. Theor. Biol.
**2014**, 347, 182–190. [Google Scholar] [CrossRef][Green Version] - Wortel, M.; Noor, E.; Ferris, M.; Bruggeman, F.; Liebermeister, W. Metabolic Enzyme Cost Explains Variable Trade-Offs between Microbial Growth Rate and Yield. PLoS Comput. Biol.
**2018**, 14, 1–21. [Google Scholar] [CrossRef][Green Version] - Provost, A.; Bastin, G. Dynamic Metabolic Modelling under the Balanced Growth Condition. J. Process Control
**2004**, 14, 717–728. [Google Scholar] [CrossRef] - Kim, J.I.; Varner, J.D.; Ramkrishna, D. A Hybrid Model of Anaerobic E. Coli GJT001: Combination of Elementary Flux Modes and Cybernetic Variables. Biotechnol. Prog.
**2008**, 24, 993–1006. [Google Scholar] [CrossRef] - Motzkin, T.; Raiffa, H.; Thompson, G.; Thrall, R. The Double Description Method. In Contributions to Theory of Games, Volume 2; Kuhn, H., Tucker, A., Eds.; Princeton University Press: Princeton, NJ, USA, 1953. [Google Scholar]
- Fukuda, K.; Prodon, A. Double Description Method Revisited. In Proceedings of the Combinatorics and Computer Science; Deza, M., Euler, R., Manoussakis, I., Eds.; Springer: Berlin/Heidelberg, Germany, 1996; pp. 91–111. [Google Scholar] [CrossRef]
- Pfeiffer, T.; Sanchez-Valdenebro, I.; Nuno, J.C.; Montero, F.; Schuster, S. METATOOL: For Studying Metabolic Networks. Bioinformatics
**1999**, 15, 251–257. [Google Scholar] [CrossRef][Green Version] - Terzer, M.; Stelling, J. Large-Scale Computation of Elementary Flux Modes with Bit Pattern Trees. Bioinformatics
**2008**, 24, 2229–2235. [Google Scholar] [CrossRef] - Hunt, K.A.; Folsom, J.P.; Taffs, R.L.; Carlson, R.P. Complete Enumeration of Elementary Flux Modes through Scalable Demand-Based Subnetwork Definition. Bioinformatics
**2014**, 30, 1569–1578. [Google Scholar] [CrossRef] [PubMed] - Jungreuthmayer, C.; Ruckerbauer, D.E.; Zanghellini, J. regEfmtool: Speeding up Elementary Flux Mode Calculation Using Transcriptional Regulatory Rules in the Form of Three-State Logic. Biosystems
**2013**, 113, 37–39. [Google Scholar] [CrossRef] [PubMed] - Peres, S.; Jolicœur, M.; Moulin, C.; Dague, P.; Schuster, S. How Important Is Thermodynamics for Identifying Elementary Flux Modes? PLoS ONE
**2017**, 12, 1–20. [Google Scholar] [CrossRef] [PubMed] - Gerstl, M.P.; Jungreuthmayer, C.; Zanghellini, J. tEFMA: Computing Thermodynamically Feasible Elementary Flux Modes in Metabolic Networks. Bioinformatics
**2015**, 31, 2232–2234. [Google Scholar] [CrossRef][Green Version] - Gerstl, M.P.; Ruckerbauer, D.E.; Mattanovich, D.; Jungreuthmayer, C.; Zanghellini, J. Metabolomics Integrated Elementary Flux Mode Analysis in Large Metabolic Networks. Sci. Rep.
**2015**, 5, 8930. [Google Scholar] [CrossRef][Green Version] - Peres, S.; Schuster, S.; Dague, P. Thermodynamic Constraints for Identifying the Elementary Flux Modes. Biochem. Soc. Trans.
**2018**, 46, 641–647. [Google Scholar] [CrossRef] - Peres, S.; Morterol, M.; Simon, L. SAT-Based Metabolics Pathways Analysis without Compilation. In Lecture Note in Bioinformatics; P. Mendes, J.D., Smallbone, K., Eds.; Springer International Publishing: Berlin/Heidelberg, Germany, 2014; Volume 8859, pp. 20–31. [Google Scholar] [CrossRef]
- Morterol, M.; Dague, P.; Peres, S.; Simon, L. Minimality of Metabolic Flux Modes under Boolean Regulation Constraints. In Proceedings of the Workshop on Constraint-Based Methods for Bioinformatics (WCB), Toulouse, France, 5 September 2016. [Google Scholar]
- de Figueiredo, L.F.; Podhorski, A.; Rubio, A.; Kaleta, C.; Beasley, J.E.; Schuster, S.; Planes, F.J. Computing the Shortest Elementary Flux Modes in Genome-Scale Metabolic Networks. Bioinformatics
**2009**, 25, 3158–3165. [Google Scholar] [CrossRef] - Pey, J.; Planes, F.J. Direct Calculation of Elementary Flux Modes Satisfying Several Biological Constraints in Genome-Scale Metabolic Networks. Bioinformatics
**2014**, 30, 2197–2203. [Google Scholar] [CrossRef][Green Version] - Vieira, V.; Rocha, M. CoBAMP: A Python Framework for Metabolic Pathway Analysis in Constraint-Based Models. Bioinformatics
**2019**, 35, 5361–5362. [Google Scholar] [CrossRef] - Rezola, A.; de Figueiredo, L.F.; Brock, M.; Pey, J.; Podhorski, A.; Wittmann, C.; Schuster, S.; Bockmayr, A.; Planes, F.J. Exploring Metabolic Pathways in Genome-Scale Networks via Generating Flux Modes. Bioinformatics
**2010**, 27, 534–540. [Google Scholar] [CrossRef][Green Version] - von Kamp, A.; Klamt, S. Enumeration of Smallest Intervention Strategies in Genome-Scale Metabolic Networks. PLoS Comput. Biol.
**2014**, 10, 1–13. [Google Scholar] [CrossRef] [PubMed] - David, L.; Bockmayr, A. Computing Elementary Flux Modes Involving a Set of Target Reactions. IEEE/ACM Trans. Comput. Biol. Bioinform.
**2014**, 11, 1099–1107. [Google Scholar] [CrossRef] [PubMed] - Song, H.S.; Goldberg, N.; Mahajan, A.; Ramkrishna, D. Sequential Computation of Elementary Modes and Minimal Cut Sets in Genome-Scale Metabolic Networks Using Alternate Integer Linear Programming. Bioinformatics
**2017**, 33, 2345–2353. [Google Scholar] [CrossRef] [PubMed] - Gebser, M.; Schaub, T.; Thiele, S.; Usadel, B.; Veber, P. Detecting Inconsistencies in Large Biological Networks with Answer Set Programming. In Proceedings of the International Conference on Logic Programming, Udine, Italy, 9–13 December 2008; pp. 130–144. [Google Scholar] [CrossRef][Green Version]
- Razzaq, M.; Paulevé, L.; Siegel, A.; Saez-Rodriguez, J.; Bourdon, J.; Guziolowski, C. Computational Discovery of Dynamic Cell Line Specific Boolean Networks from Multiplex Time-Course Data. PLoS Comput. Biol.
**2018**, 14, 1–23. [Google Scholar] [CrossRef] [PubMed] - Frioux, C.; Schaub, T.; Schellhorn, S.; Siegel, A.; Wanko, P. Hybrid Metabolic Network Completion. In Logic Programming and Nonmonotonic Reasoning; Balduccini, M., Janhunen, T., Eds.; Springer International Publishing: Cham, Switzerland, 2017; pp. 308–321. [Google Scholar] [CrossRef][Green Version]
- Janhunen, T.; Kaminski, R.; Ostrowski, M.; Schaub, T.; Schellhorn, S.; Wanko, P. Clingo Goes Linear Constraints over Reals and Integers. arXiv
**2017**, arXiv:1707.04053. [Google Scholar] [CrossRef][Green Version] - Gebser, M.; Kaminski, R.; Kaufmann, B.; Ostrowski, M.; Schaub, T.; Wanko, P. Theory Solving Made Easy with Clingo 5. In Technical Communications of the 32nd International Conference on Logic Programming (ICLP 2016); Carro, M., King, A., Saeedloei, N., Vos, M.D., Eds.; OpenAccess Series in Informatics (OASIcs); Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik: Dagstuhl, Germany, 2016; Volume 52, pp. 2:1–2:15. [Google Scholar] [CrossRef]
- Orth, J.D.; Fleming, R.M.T.; Palsson, B.Ø. Reconstruction and Use of Microbial Metabolic Networks: The Core Escherichia Coli Metabolic Model as an Educational Guide. EcoSal Plus
**2010**, 4. [Google Scholar] [CrossRef] - de Graef, M.R.; Alexeeva, S.; Snoep, J.L.; Teixeira de Mattos, M.J. The Steady-State Internal Redox State (NADH/NAD) Reflects the External Redox State and Is Correlated with Catabolic Adaptation in Escherichia Coli. J. Bacteriol.
**1999**, 181, 2351–2357. [Google Scholar] [CrossRef][Green Version] - Alexeeva, S.; Hellingwerf, K.J.; Teixeira de Mattos, M.J. Requirement of ArcA for Redox Regulation in Escherichia Coli under Microaerobic but Not Anaerobic or Aerobic Conditions. J. Bacteriol.
**2003**, 185, 204–209. [Google Scholar] [CrossRef][Green Version] - Orth, J.D.; Thiele, I.; Palsson, B.Ã. What Is Flux Balance Analysis? Nat. Biotechnol.
**2010**, 28, 245–248. [Google Scholar] [CrossRef] - Peres, S.; Fromion, V. Thermodynamic Approaches in Flux Analysis. In Methods in Molecular Biology; Analysis, M.F., Ed.; Springer: Berlin/Heidelberg, Germany, 2019; Chapter 17. [Google Scholar]
- Klamt, S.; Gagneur, J.; von Kamp, A. Algorithmic Approaches for Computing Elementary Modes in Large Biochemical Reaction Networks. IEE Proc. Syst. Biol.
**2005**, 152, 249–255. [Google Scholar] [CrossRef] - Olivier, B.G.; Bergmann, F.T. SBML Level 3 Package: Flux Balance Constraints Version 2. J. Integr. Bioinform.
**2018**, 15, 1–39. [Google Scholar] [CrossRef] [PubMed] - King, Z.A.; Lu, J.; Dräger, A.; Miller, P.; Federowicz, S.; Lerman, J.A.; Ebrahim, A.; Palsson, B.O.; Lewis, N.E. BiGG Models: A Platform for Integrating, Standardizing and Sharing Genome-Scale Models. Nucleic Acids Res.
**2016**, 44, D515–D522. [Google Scholar] [CrossRef] [PubMed] - Schuster, S.; Dandekar, T.; Fell, D. Detection of Elementary Modes in Biochemical Networks: A Promising Tool for Pathway Analysis and Metabolic Engineering. Trends Biotechnol.
**1999**, 17, 53–60. [Google Scholar] [CrossRef] - Lifschitz, V. What Is Answer Set Programming? In Proceedings of the AAAI 2008, Chicago, IL, USA, 13–17 July 2008; Volume 8, pp. 1594–1597. [Google Scholar]
- Gebser, M.; Kaufmann, B.; Schaub, T. Conflict-Driven Answer Set Solving: From Theory to Practice. Artif. Intell.
**2012**, 187–188, 52–89. [Google Scholar] [CrossRef][Green Version] - Gebser, M.; Kaminski, R.; Kaufmann, B.; Lindauer, M.; Ostrowski, M.; Romero, J.; Schaub, T.; Thiele, S.; Wanko, P. Potassco User Guide, 2nd ed.; University of Potsdam: Potsdam, Germany, 2019. [Google Scholar]
- Covert, M.W.; Palsson, B.O. Constraints-Based Models: Regulation of Gene Expression Reduces the Steady-State Solution Space. J. Theor. Biol.
**2003**, 221, 309–325. [Google Scholar] [CrossRef][Green Version] - King, Z.A.; Dräger, A.; Ebrahim, A.; Sonnenschein, N.; Lewis, N.E.; Palsson, B.O. Escher: A Web Application for Building, Sharing, and Embedding Data-Rich Visualizations of Biological Pathways. PLoS Comput. Biol.
**2015**, 11, e1004321. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kluyver, T.; Ragan-Kelley, B.; Pérez, F.; Granger, B.; Bussonnier, M.; Frederic, J.; Kelley, K.; Hamrick, J.; Grout, J.; Corlay, S.; et al. Jupyter Notebooks—A Publishing Format for Reproducible Computational Workflows; Loizides, F., Schmidt, B., Eds.; Positioning and Power in Academic Publishing: Players, Agents and Agendas; IOS Press: Amsterdam, The Netherlands, 2016; pp. 87–90. [Google Scholar]

**Figure 1.**E. coli core EFMs sorted by carbon/biomass uptake rate and oxygen/biomass uptake rate. Regulation constraints are as described in Orth et al. 2010.

**Figure 2.**E. coli core EFMs sorted by carbon/biomass uptake rate and oxygen/biomass uptake rate. Regulation constraints allow the production of formate in aerobic conditions.

**Figure 3.**Schematic overview of the workflow for computing EFMs under constraints with ASP. The ASP rules representing the metabolic model and additional biological constraints are given as input into clingo[LP] along with the logic program for computing EFMs. From all these rules, the grounder of clingo[LP] builds instance rules, which are sent to the ASP/LP solver. The resulting answer sets are EFMs consistent with all the constraints. These EFMs can be analyzed in post-processing to select the optimal functioning ones.

**Table 1.**Number of EFMs retrieved from the E. coli core network depending on culturing conditions. The computation time of a single clingo[LP] execution given within brackets. Disabling the formate regulation returned EFMs for both aerobic and anaerobic conditions in a single execution.

Standard Regulation | No Formate Regulation | ||
---|---|---|---|

Processing | Aerobic conditions | 1118 EFMs [542 s] | 11,017 EFMs [5318 s] |

Anaerobic conditions | 363 EFMs [232 s] | ||

Post-processing | Filtered out MCFMs | 39 MCFMs | 119 MCFMs |

Pareto optimal in biomass yield | 4 EFMs | 5 EFMs |

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Mahout, M.; Carlson, R.P.; Peres, S.
Answer Set Programming for Computing Constraints-Based Elementary Flux Modes: Application to *Escherichia coli* Core Metabolism. *Processes* **2020**, *8*, 1649.
https://doi.org/10.3390/pr8121649

**AMA Style**

Mahout M, Carlson RP, Peres S.
Answer Set Programming for Computing Constraints-Based Elementary Flux Modes: Application to *Escherichia coli* Core Metabolism. *Processes*. 2020; 8(12):1649.
https://doi.org/10.3390/pr8121649

**Chicago/Turabian Style**

Mahout, Maxime, Ross P. Carlson, and Sabine Peres.
2020. "Answer Set Programming for Computing Constraints-Based Elementary Flux Modes: Application to *Escherichia coli* Core Metabolism" *Processes* 8, no. 12: 1649.
https://doi.org/10.3390/pr8121649