Mathematical modeling of microbial community dynamics: A methodological review

Processes

Volumn 2, Issue 4, 2014, Pages 711-752

  Song, Hyun Seob ^a   Cannon, William R ^a   Beliaev, Alexander S ^a   Konopka, Allan ^a  

^a PACIFIC NORTHWEST NATIONAL LABORATORY   (United States)

Author keywords

Dynamics; Integrative modeling approaches; Mathematical models; Microbial communities

Indexed keywords

EID: 85054271306     PISSN: None     EISSN: 22279717     Source Type: Journal    
DOI: 10.3390/pr2040711     Document Type: Review

References (136)

1. Whitman, W.B.; Coleman, D.C.; Wiebe, W.J. Prokaryotes: The unseen majority. Proc. Natl. Acad. Sci. USA 1998, 95, 6578-6583
2. Microbiology by numbers. Nat. Rev. Microbiol. 2011, 9, 628, doi:10.1038/nrmicro2644
3. Fukuda, S.; Ohno, H. Gut microbiome and metabolic diseases. Semin. Immunopathol. 2014, 36, 103-114
4. Heintz, C.; Mair, W. You are what you host: Microbiome modulation of the aging process. Cell 2014, 156, 408-411
5. Moloney, R.D.; Desbonnet, L.; Clarke, G.; Dinan, T.G.; Cryan, J.F. The microbiome: Stress, health and disease. Mamm. Genome 2014, 25, 49-74
6. Maukonen, J.; Saarela, M. Microbial communities in industrial environment. Curr. Opin. Microbiol. 2009, 12, 238-243
7. Konopka, A. What is microbial community ecology? ISME J. 2009, 3, 1223-1230
8. Bond, P.L.; Smriga, S.P.; Banfield, J.F. Phylogeny of microorganisms populating a thick, subaerial, predominantly lithotrophic biofilm at an extreme acid mine drainage site. Appl. Environ. Microbiol 2000, 66, 3842-3849
9. Caporaso, J.G.; Lauber, C.L.; Costello, E.K.; Berg-Lyons, D.; Gonzalez, A.; Stombaugh, J.; Knights, D.; Gajer, P.; Ravel, J.; Fierer, N.; et al. Moving pictures of the human microbiome. Genome Biol. 2011, 12, doi:10.1186/gb-2011-12-5-r50
10. Wagg, C.; Bender, S.F.; Widmer, F.; van der Heijden, M.G.A. Soil biodiversity and soil community composition determine ecosystem multifunctionality. Proc. Natl. Acad. Sci. USA 2014, 111, 5266-5270
11. Fierer, N.; Lennon, J.T. The generation and maintenance of diversity in microbial communities. Am. J. Bot. 2011, 98, 439-448
12. Zengler, K.; Palsson, B.O. A road map for the development of community systems (cosy) biology. Nat. Rev. Microbiol. 2012, 10, 366-372
13. Haruta, S.; Yoshida, T.; Aoi, Y.; Kaneko, K.; Futamata, H. Challenges for complex microbial ecosystems: Combination of experimental approaches with mathematical modeling. Microbes Environ. 2013, 28, 285-294
14. Mee, M.T.; Wang, H.H. Engineering ecosystems and synthetic ecologies. Mol. Biosyst. 2012, 8, 2470-2483
15. Larsen, P.; Hamada, Y.; Gilbert, J. Modeling microbial communities: Current, developing, and future technologies for predicting microbial community interaction. J. Biotechnol. 2012, 160, 17-24
16. Larsen, P.E.; Gibbons, S.M.; Gilbert, J.A. Modeling microbial community structure and functional diversity across time and space. FEMS Microbiol. Lett. 2012, 332, 91-98
17. Kissling, W.D.; Dormann, C.F.; Groeneveld, J.; Hickler, T.; Kuhn, I.; McInerny, G.J.; Montoya, J.M.; Romermann, C.; Schiffers, K.; Schurr, F.M.; et al. Towards novel approaches to modelling biotic interactions in multispecies assemblages at large spatial extents. J. Biogeogr. 2012, 39, 2163-2178
18. Roling, W.F.M.; van Bodegom, P.M. Toward quantitative understanding on microbial community structure and functioning: A modeling-centered approach using degradation of marine oil spills as example. Front. Microbiol. 2014, 5, doi:10.3389/fmicb.2014.00125
19. Stegen, J.C.; Lin, X.J.; Konopka, A.E.; Fredrickson, J.K. Stochastic and deterministic assembly processes in subsurface microbial communities. ISME J. 2012, 6, 1653-1664
20. Klapper, I.; Dockery, J. Mathematical description of microbial biofilms. SIAM Rev. 2010, 52, 221-265
21. Tringe, S.G.; von Mering, C.; Kobayashi, A.; Salamov, A.A.; Chen, K.; Chang, H.W.; Podar, M.; Short, J.M.; Mathur, E.J.; Detter, J.C.; et al. Comparative metagenomics of microbial communities. Science 2005, 308, 554-557
22. Lidstrom, M.E.; Konopka, M.C. The role of physiological heterogeneity in microbial population behavior. Nat. Chem. Biol. 2010, 6, 705-712
23. Majed, N.; Chernenko, T.; Diem, M.; Gu, A.Z. Identification of functionally relevant populations in enhanced biological phosphorus removal processes based on intracellular polymers profiles and insights into the metabolic diversity and heterogeneity. Environ. Sci. Technol. 2012, 46, 5010-5017
24. Ramkrishna, D.; Mahoney, A.W. Population balance modeling. Promise for the future. Chem. Eng. Sci. 2002, 57, 595-606
25. Hellweger, F.L.; Bucci, V. A bunch of tiny individuals-individual-based modeling for microbes. Ecol. Model. 2009, 220, 8-22
26. Scheffer, M.; Baveco, J.M.; Deangelis, D.L.; Rose, K.A.; Vannes, E.H. Super-individuals a simple solution for modeling large populations on an individual basis. Ecol. Model. 1995, 80, 161-170
27. Gore, J.; Youk, H.; van Oudenaarden, A. Snowdrift game dynamics and facultative cheating in yeast. Nature 2009, 459, 253-256
28. Borenstein, E. Computational systems biology and in silico modeling of the human microbiome. Brief. Bioinform. 2012, 13, 769-780
29. Orth, J.D.; Thiele, I.; Palsson, B.O. What is flux balance analysis? Nat. Biotechnol. 2010, 28, 245-248
30. Trinh, C.T.; Wlaschin, A.; Srienc, F. Elementary mode analysis: A useful metabolic pathway analysis tool for characterizing cellular metabolism. Appl. Microbiol. Biotechnol 2009, 81, 813-826
31. Song, H.-S.; DeVilbiss, F.; Ramkrishna, D. Modeling metabolic systems: The need for dynamics. Curr. Opin. Chem. Eng. 2013, 2, 373-382
32. Yoo, S.J.; Kim, J.H.; Lee, J.M. Dynamic modelling of mixotrophic microalgal photobioreactor systems with time-varying yield coefficient for the lipid consumption. Biores. Technol. 2014, 162, 228-235
33. Urbanczik, R.; Wagner, C. An improved algorithm for stoichiometric network analysis: Theory and applications. Bioinformatics 2005, 21, 1203-1210
34. 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
35. Terzer, M.; Stelling, J. Large-scale computation of elementary flux modes with bit pattern trees. Bioinformatics 2008, 24, 2229-2235
36. Terzer, M.; Stelling, J. Parallel extreme ray and pathway computation. Lect. Notes Comput. Sci. 2010, 6068, 300-309
37. Song, H.S.; Ramkrishna, D. Reduction of a set of elementary modes using yield analysis. Biotechnol. Bioeng. 2009, 102, 554-568
38. Chan, S.H.J.; Ji, P. Decomposing flux distributions into elementary flux modes in genome-scale metabolic networks. Bioinformatics 2011, 27, 2256-2262
39. Ballerstein, K.; von Kamp, A.; Klamt, S.; Haus, U.U. Minimal cut sets in a metabolic network are elementary modes in a dual network. Bioinformatics 2012, 28, 381-387
40. Greenblum, S.; Turnbaugh, P.J.; Borenstein, E. Metagenomic systems biology of the human gut microbiome reveals topological shifts associated with obesity and inflammatory bowel disease. Proc. Natl. Acad. Sci. USA 2012, 109, 594-599
41. Taffs, R.; Aston, J.E.; Brileya, K.; Jay, Z.; Klatt, C.G.; McGlynn, S.; Mallette, N.; Montross, S.; Gerlach, R.; Inskeep, W.P.; et al. In silico approaches to study mass and energy flows in microbial consortia: A syntrophic case study. BMC Syst. Biol. 2009, 3, doi:10.1186/1752-0509-3-114
42. Reed, D.C.; Algar, C.K.; Huber, J.A.; Dick, G.J. Gene-centric approach to integrating environmental genomics and biogeochemical models. Proc. Natl. Acad. Sci. USA 2014, 111, 1879-1884
43. Ramkrishna, D.; Song, H.S. Dynamic models of metabolism: Review of the cybernetic approach. AIChE J. 2012, 58, 986-997
44. Kim, J.I.; Song, H.S.; Sunkara, S.R.; Lali, A.; Ramkrishna, D. Exacting predictions by cybernetic model confirmed experimentally: Steady state multiplicity in the chemostat. Biotechnol. Prog. 2012, 28, 1160-1166
45. Song, H.S.; Ramkrishna, D. Prediction of metabolic function from limited data: Lumped hybrid cybernetic modeling (l-hcm). Biotechnol. Bioeng. 2010, 106, 271-284
46. Song, H.S.; Ramkrishna, D. Cybernetic models based on lumped elementary modes accurately predict strain-specific metabolic function. Biotechnol. Bioeng. 2011, 108, 127-140
47. Song, H.S.; Ramkrishna, D. Prediction of dynamic behavior of mutant strains from limited wild-type data. Metab. Eng. 2012, 14, 69-80
48. Song, H.S.; Ramkrishna, D.; Pinchuk, G.E.; Beliaev, A.S.; Konopka, A.E.; Fredrickson, J.K. Dynamic modeling of aerobic growth of shewanella oneidensis. Predicting triauxic growth, flux distributions, and energy requirement for growth. Metab. Eng. 2013, 15, 25-33
49. Young, J.D.; Henne, K.L.; Morgan, J.A.; Konopka, A.E.; Ramkrishna, D. Integrating cybernetic modeling with pathway analysis provides a dynamic, systems-level description of metabolic control. Biotechnol. Bioeng. 2008, 100, 542-559
50. Faust, K.; Raes, J. Microbial interactions: From networks to models. Nat. Rev. Microbiol. 2012, 10, 538-550
51. Shuler, M.L.; Kargi, F. Bioprocess Engineering; Prentice Hall: Upper Saddle River, NJ, USA, 2002
52. Burmolle, M.; Webb, J.S.; Rao, D.; Hansen, L.H.; Sorensen, S.J.; Kjelleberg, S. Enhanced biofilm formation and increased resistance to antimicrobial agents and bacterial invasion are caused by synergistic interactions in multispecies biofilms. Appl. Environ. Microb. 2006, 72, 3916-3923
53. Rodriguez-Martinez, J.M.; Pascual, A. Antimicrobial resistance in bacterial biofilms. Rev. Med. Microbiol. 2006, 17, 65-75
54. Pak, K.R.; Bartha, R. Mercury methylation by interspecies hydrogen and acetate transfer between sulfidogens and methanogens. Appl. Environ. Microb. 1998, 64, 1987-1990
55. Gause, G.F. Experimental studies on the struggle for existence i mixed population of two species of yeast. J. Exp. Biol. 1932, 9, 389-402
56. Moon, D.C.; Moon, J.; Keagy, A. Direct and indirect interactions. Available online: http://www.nature.com/scitable/knowledge/library/direct-and-indirect-interactions-15650000 (accessed on 11 October 2014)
57. Berry, D.; Widder, S. Deciphering microbial interactions and detecting keystone species with co-occurrence networks. Front. Microbiol. 2014, 5, doi:10.3389/fmicb.2014.00219
58. Wooley, J.C.; Godzik, A.; Friedberg, I. A primer on metagenomics. PLOS Comput. Biol. 2010, 6, doi:10.1371/journal.pcbi.1000667
59. Mande, S.S.; Mohammed, M.H.; Ghosh, T.S. Classification of metagenomic sequences: Methods and challenges. Brief. Bioinform. 2012, 13, 669-681
60. Fuhrman, J.A. Microbial community structure and its functional implications. Nature 2009, 459, 193-199
61. Barabasi, A.L. Scale-free networks: A decade and beyond. Science 2009, 325, 412-413
62. Barabasi, A.L.; Bonabeau, E. Scale-free networks. Sci. Am. 2003, 288, 60-69
63. Albert, R.; Jeong, H.; Barabasi, A.L. Error and attack tolerance of complex networks. Nature 2000, 406, 378-382
64. Stolyar, S.; van Dien, S.; Hillesland, K.L.; Pinel, N.; Lie, T.J.; Leigh, J.A.; Stahl, D.A. Metabolic modeling of a mutualistic microbial community. Mol. Syst. Biol. 2007, 3, 92, doi:10.1038/msb4100131
65. Freilich, S.; Zarecki, R.; Eilam, O.; Segal, E.S.; Henry, C.S.; Kupiec, M.; Gophna, U.; Sharan, R.; Ruppin, E. Competitive and cooperative metabolic interactions in bacterial communities. Nat. Commun. 2011, 2, doi:10.1038/ncomms1597
66. Klitgord, N.; Segre, D. Environments that induce synthetic microbial ecosystems. PLOS Comput. Biol. 2010, 6, doi:10.1371/journal.pcbi.1001002
67. Wintermute, E.H.; Silver, P.A. Emergent cooperation in microbial metabolism. Mol. Syst. Biol. 2010, 6, doi:10.1038/msb.2010.66
68. Segre, D.; Vitkup, D.; Church, G.M. Analysis of optimality in natural and perturbed metabolic networks. Proc. Natl. Acad. Sci. USA 2002, 99, 15112-15117
69. Bordbar, A.; Feist, A.M.; Usaite-Black, R.; Woodcock, J.; Palsson, B.O.; Famili, I. A multi-tissue type genome-scale metabolic network for analysis of whole-body systems physiology. BMC Syst. Biol. 2011, 5, doi:10.1186/1752-0509-5-180
70. Lewis, N.E.; Schramm, G.; Bordbar, A.; Schellenberger, J.; Andersen, M.P.; Cheng, J.K.; Patel, N.; Yee, A.; Lewis, R.A.; Eils, R.; et al. Large-scale in silico modeling of metabolic interactions between cell types in the human brain. Nat. Biotechnol. 2010, 28, U1279-U1291
71. Khandelwal, R.A.; Olivier, B.G.; Roling, W.F.M.; Teusink, B.; Bruggeman, F.J. Community flux balance analysis for microbial consortia at balanced growth. PLOS ONE 2013, 8, doi:10.1371/ journal.pone.0064567
72. Zomorrodi, A.R.; Maranas, C.D. Optcom: A multi-level optimization framework for the metabolic modeling and analysis of microbial communities. PLOS Comput. Biol. 2012, 8, doi:10.1371/journal.pcbi.1002363
73. Zomorrodi, A.R.; Islam, M.M.; Maranas, C.D. D-optcom: Dynamic multi-level and multi-objective metabolic modeling of microbial communities. ACS Synth. Biol. 2014, 3, 247-257
74. Elith, J.; Leathwick, J.R. Species distribution models: Ecological explanation and prediction across space and time. Annu. Rev. Ecol. Evol. Syst. 2009, 40, 677-697
75. Larsen, P.E.; Field, D.; Gilbert, J.A. Predicting bacterial community assemblages using an artificial neural network approach. Nat. Methods 2012, 9, 621-625
76. Schmidt, M.; Lipson, H. Distilling free-form natural laws from experimental data. Science 2009, 324, 81-85
77. Istok, J.D.; Park, M.; Michalsen, M.; Spain, A.M.; Krumholz, L.R.; Liu, C.; McKinley, J.; Long, P.; Roden, E.; Peacock, A.D.; et al. A thermodynamically-based model for predicting microbial growth and community composition coupled to system geochemistry: Application to uranium bioreduction. J. Contam. Hydrol. 2010, 112, 1-14
78. Larowe, D.E.; Dale, A.W.; Regnier, P. A thermodynamic analysis of the anaerobic oxidation of methane in marine sediments. Geobiology 2008, 6, 436-449
79. Rodriguez, J.; Kleerebezem, R.; Lema, J.M.; van Loosdrecht, M.C.M. Modeling product formation in anaerobic mixed culture fermentations. Biotechnol. Bioeng. 2006, 93, 592-606
80. Beard, D.A.; Qian, H. Relationship between thermodynamic driving force and one-way fluxes in reversible processes. PLOS ONE 2007, 2, e144, doi:10.1371/journal.pone.0000144
81. Noor, E.; Bar-Even, A.; Flamholz, A.; Reznik, E.; Liebermeister, W.; Milo, R. Pathway thermodynamics highlights kinetic obstacles in central metabolism. PLOS Comput. Biol. 2014, 10, e1003483, doi:10.1371/journal.pcbi.1003483
82. Zhu, Y.; Song, J.; Xu, Z.; Sun, J.; Zhang, Y.; Li, Y.; Ma, Y. Development of thermodynamic optimum searching (tos) to improve the prediction accuracy of flux balance analysis. Biotechnol. Bioeng. 2013, 110, 914-923
83. Meysman, F.J.; Bruers, S. Ecosystem functioning and maximum entropy production: A quantitative test of hypotheses. Philos. Trans. R. Soc. B 2010, 365, 1405-1416
84. Schrödinger, E. What is Life? The Physical Aspect of the Living Cell, 1st ed.; Cambridge University Press: New York, NY, USA, 1944
85. Prigogine, I. Time, structure, and fluctuations. Science 1978, 201, 777-785
86. Morowitz, H.J. Energy Flow in Biology: Biological Organization as a Problem in Thermal Physics; Ox Bow Press: Woodbridge, NJ, USA, 1979
87. Dewar, R.C. Maximum entropy production and plant optimization theories. Philos. Trans. R. Soc. B 2010, 365, 1429-1435
88. Dewar, R.C.; Juretic, D.; Zupanovic, P. The functional design of the rotary enzyme atp synthase is consistent with maximum entropy production. Chem. Phys. Lett. 2006, 430, 177-182
89. Unrean, P.; Srienc, F. Metabolic networks evolve towards states of maximum entropy production. Metab. Eng. 2011, 13, 666-673
90. Cannon, W.R. Simulating metabolism with statistical thermodynamics. PLOS ONE 2014, 9, e103582, doi:10.1371/journal.pone
91. Schmiedl, T.; Seifert, U. Stochastic thermodynamics of chemical reaction networks. J. Chem. Phys. 2007, 126, doi:10.1063/1.2428297
92. Gillespie, D.T. Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem. 2007, 58, 35-55
93. Boon, E.; Meehan, C.J.; Whidden, C.; Wong, D.H.J.; Langille, M.G.I.; Beiko, R.G. Interactions in the microbiome: Communities of organisms and communities of genes. FEMS Microbiol. Rev. 2014, 38, 90-118
94. Webb, C.T.; Hoeting, J.A.; Ames, G.M.; Pyne, M.I.; Poff, N.L. A structured and dynamic framework to advance traits-based theory and prediction in ecology. Ecol. Lett. 2010, 13, 267-283
95. Laughlin, D.C.; Laughlin, D.E. Advances in modeling trait-based plant community assembly. Trends Plant Sci. 2013, 18, 584-593
96. Shipley, B.; Vile, D.; Garnier, E. From plant traits to plant communities: A statistical mechanistic approach to biodiversity. Science 2006, 314, 812-814
97. Shipley, B.; Paine, C.E.T.; Baraloto, C. Quantifying the importance of local niche-based and stochastic processes to tropical tree community assembly. Ecology 2012, 93, 760-769
98. Laughlin, D.C.; Joshi, C.; van Bodegom, P.M.; Bastow, Z.A.; Fule, P.Z. A predictive model of community assembly that incorporates intraspecific trait variation. Ecol. Lett. 2012, 15, 1291-1299
99. Jin, Q.S.; Roden, E.E. Microbial physiology-based model of ethanol metabolism in subsurface sediments. J. Contam. Hydrol. 2011, 125, 1-12
100. Bouskill, N.J.; Tang, J.; Riley, W.J.; Brodie, E.L. Trait-based representation of biological nitr fication: Model development testing, and predicted community composition. Front. Microbiol. 2012, 3, doi:10.3389/fmicb.2012.00364
101. Edelstein-Keshet, L. Mathematical Models in Biology; Siam: Philadelphia, PA, USA, 1987; Volume 46
102. Fisher, C.K.; Mehta, P. Identifying keystone species in the human gut microbiome from metagenomic timeseries using sparse linear regression. Available online: http://arxiv.org/pdf/1402.0511v1.pdf (accessed on 11 October 2014)
103. Stein, R.R.; Bucci, V.; Toussaint, N.C.; Buffie, C.G.; Ratsch, G.; Pamer, E.G.; Sander, C.; Xavier, J.B. Ecological modeling from time-series inference: Insight into dynamics and stability of intestinal microbiota. PLOS Comput. Biol. 2013, 9, doi:10.1371/journal.pcbi.1003388
104. Mounier, J.; Monnet, C.; Vallaeys, T.; Arditi, R.; Sarthou, A.S.; Helias, A.; Irlinger, F. Microbial interactions within a cheese microbial community. Appl. Environ. Microb. 2008, 74, 172-181
105. Nowak, M.A. Evolutionary Dynamics; Harvard University Press: Cambridge, MA, USA, 2006
106. Nowak, M.A.; Sigmund, K. Evolutionary dynamics of biological games. Science 2004, 303, 793-799
107. Frey, E. Evolutionary game theory: Theoretical concepts and applications to microbial communities. Phys. A 2010, 389, 4265-4298
108. Minty, J.J.; Singer, M.E.; Scholz, S.A.; Bae, C.H.; Ahn, J.H.; Foster, C.E.; Liao, J.C.; Lin, X.N. Design and characterization of synthetic fungal-bacterial consortia for direct production of isobutanol from cellulosic biomass. Proc. Natl. Acad. Sci. USA 2013, 110, 14592-14597
109. Sousa, M.D. Kinetics of distribution of thymus and marrow cells in peripheral lymphoid organs of mouse-ecotaxis. Clin. Exp. Immunol. 1971, 9, 371-380
110. Kleene, S.J.; Hobson, A.C.; Adler, J. Attractants and repellents influence methylation and demethylation of methyl-accepting chemotaxis proteins in an extract of escherichia-coli. Proc. Natl. Acad. Sci. USA 1979, 76, 6309-6313
111. Tindall, M.J.; Maini, P.K.; Porter, S.L.; Armitage, J.P. Overview of mathematical approaches used to model bacterial chemotaxis ii: Bacterial populations. Bull. Math. Biol. 2008, 70, 1570-1607
112. Ferrer, J.; Prats, C.; Lopez, D. Individual-based modelling: An essential tool for microbiology. J. Biol. Phys. 2008, 34, 19-37
113. Resat, H.; Bailey, V.; McCue, L.A.; Konopka, A. Modeling microbial dynamics in heterogeneous environments: Growth on soil carbon sources. Microb. Ecol. 2012, 63, 883-897
114. Tang, Y.N.; Valocchi, A.J. An improved cellular automaton method to model multispecies biofilms. Water Res. 2013, 47, 5729-5742
115. Kang, S.; Kahan, S.; Momeni, B. Simulating microbial community patterning using biocellion. In Engineering and Analyzing Multicellular Systems; Springer: Berlin, Germany, 2014; pp. 233-253
116. Lardon, L.A.; Merkey, B.V.; Martins, S.; Dotsch, A.; Picioreanu, C.; Kreft, J.U.; Smets, B.F. Idynomics: Next-generation individual-based modelling of biofilms. Environ. Microbiol. 2011, 13, 2416-2434
117. Kreft, J.U.; Booth, G.; Wimpenny, J.W.T. Bacsim, a simulator for individual-based modelling of bacterial colony growth. Microbiol.-UK 1998, 144, 3275-3287
118. Ramkrishna, D. Population Balances: Theory and Applications to Particulate Systems in Engineering; Academic Press: Pittsburgh, PA, USA, 2000
119. Henson, M.A. Dynamic modeling of microbial cell populations. Curr. Opin. Biotechnol. 2003, 14, 460-467
120. Shu, C.C.; Chatterjee, A.; Dunny, G.; Hu, W.S.; Ramkrishna, D. Bistability versus bimodal distributions in gene regulatory processes from population balance. PLOS Comput. Biol. 2011, 7, doi:10.1371/journal.pcbi.1002140
121. Fernandes, R.L.; Nierychlo, M.; Lundin, L.; Pedersen, A.E.; Tellez, P.E.P.; Dutta, A.; Carlquist, M.; Bolic, A.; Schapper, D.; Brunetti, A.C.; et al. Experimental methods and modeling techniques for description of cell population heterogeneity. Biotechnol. Adv. 2011, 29, 575-599
122. Scheibe, T.D.; Mahadevan, R.; Fang, Y.L.; Garg, S.; Long, P.E.; Lovley, D.R. Coupling a genome-scale metabolic model with a reactive transport model to describe in situ uranium bioremediation. Microb. Biotechnol. 2009, 2, 274-286
123. Mahadevan, R.; Edwards, J.S.; Doyle, F.J. Dynamic flux balance analysis of diauxic growth in escherichia coli. Biophys. J. 2002, 83, 1331-1340
124. Hanly, T.J.; Henson, M.A. Dynamic flux balance modeling of microbial co-cultures for efficient batch fermentation of glucose and xylose mixtures. Biotechnol. Bioeng. 2011, 108, 376-385
125. Hanly, T.J.; Henson, M.A. Dynamic metabolic modeling of a microaerobic yeast co-culture: Predicting and optimizing ethanol production from glucose/xylose mixtures. Biotechnol. Biofuels 2013, 6, doi:10.1186/1754-6834-6-44
126. Tzamali, E.; Poirazi, P.; Tollis, I.G.; Reczko, M. A computational exploration of bacterial metabolic diversity identifying metabolic interactions and growth-efficient strain communities. BMC Syst. Biol. 2011, 5, doi:10.1186/1752-0509-5-167
127. 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. Progr. 2008, 24, 993-1006
128. Song, H.S.; Morgan, J.A.; Ramkrishna, D. Systematic development of hybrid cybernetic models: Application to recombinant yeast co-consuming glucose and xylose. Biotechnol. Bioeng. 2009, 103, 984-1002
129. Geng, J.; Song, H.S.; Yuan, J.Q.; Ramkrishna, D. On enhancing productivity of bioethanol with multiple species. Biotechnol. Bioeng. 2012, 109, 1508-1517
130. Fang, Y.L.; Scheibe, T.D.; Mahadevan, R.; Garg, S.; Long, P.E.; Lovley, D.R. Direct coupling of a genome-scale microbial in silico model and a groundwater reactive transport model. J. Contam. Hydrol. 2011, 122, 96-103
131. King, E.L.; Tuncay, K.; Ortoleva, P.; Meile, C. In silico geobacter sulfurreducens metabolism and its representation in reactive transport models. Appl. Environ. Microb. 2009, 75, 83-92
132. Zhuang, K.; Izallalen, M.; Mouser, P.; Richter, H.; Risso, C.; Mahadevan, R.; Lovley, D.R. Genome-scale dynamic modeling of the competition between rhodoferax and geobacter in anoxic subsurface environments. ISME J. 2011, 5, 305-316
133. Harcombe, W.R.; Riehl, W.J.; Dukovski, I.; Granger, B.R.; Betts, A.; Lang, A.H.; Bonilla, G.; Kar, A.; Leiby, N.; Mehta, P.; et al. Metabolic resource allocation in individual microbes determines ecosystem interactions and spatial dynamics. Cell Rep. 2014, 7, 1104-1115
134. Zhuang, K.; Ma, E.; Lovley, D.R.; Mahadevan, R. The design of long-term effective uranium bioremediation strategy using a community metabolic model. Biotechnol. Bioeng. 2012, 109, 2475-2483
135. Akaike, H. Information theory and an extension of the maximum likelihood principle. In Proceedings of the 2nd International Symposium on Information Theory; Akadémiai Kiadó: Budapest, Hungary, 1973; pp. 267-281
136. Schwarz, G. Estimating the dimension of a model. Ann. Stat. 1978, 6, 461-464