Journal of Mathematical Biology

, Volume 78, Issue 4, pp 985–1032 | Cite as

Optimal control of bacterial growth for the maximization of metabolite production

  • Ivan YegorovEmail author
  • Francis Mairet
  • Hidde de Jong
  • Jean-Luc Gouzé


Microorganisms have evolved complex strategies for controlling the distribution of available resources over cellular functions. Biotechnology aims at interfering with these strategies, so as to optimize the production of metabolites and other compounds of interest, by (re)engineering the underlying regulatory networks of the cell. The resulting reallocation of resources can be described by simple so-called self-replicator models and the maximization of the synthesis of a product of interest formulated as a dynamic optimal control problem. Motivated by recent experimental work, we are specifically interested in the maximization of metabolite production in cases where growth can be switched off through an external control signal. We study various optimal control problems for the corresponding self-replicator models by means of a combination of analytical and computational techniques. We show that the optimal solutions for biomass maximization and product maximization are very similar in the case of unlimited nutrient supply, but diverge when nutrients are limited. Moreover, external growth control overrides natural feedback growth control and leads to an optimal scheme consisting of a first phase of growth maximization followed by a second phase of product maximization. This two-phase scheme agrees with strategies that have been proposed in metabolic engineering. More generally, our work shows the potential of optimal control theory for better understanding and improving biotechnological production processes.


Optimal control Nonlinear dynamical systems Mathematical modelling Bacterial growth Biotechnology 



This work was supported in part by the PIA project Reset (ANR-11-BINF-0005), ANR project Maximic (ANR-17-CE40-0024-01), Inria IPL AlgaeInSilico, and Labex SIGNALIFE (ANR-11-LABX-0028-01). The authors thank Johannes Geiselmann and Eugenio Cinquemani for discussions and comments on the manuscript.


  1. Akhmetzhanov AR, Grognard F, Mailleret L (2011) Optimal life-history strategies in seasonal consumer-resource dynamics. Evolution 65(11):3113–25Google Scholar
  2. Anesiadis N, Cluett WR, Mahadevan R (2008) Dynamic metabolic engineering for increasing bioprocess productivity. Metab Eng 10(5):255–66Google Scholar
  3. Banga JR (2008) Optimization in computational systems biology. BMC Syst Biol 2:47MathSciNetGoogle Scholar
  4. Banga JR, Balsa-Canto E, Moles CG, Alonso AA (2005) Dynamic optimization of bioprocesses: efficient and robust numerical strategies. J Biotechnol 117(4):407–19Google Scholar
  5. Bastin G, Dochain D (1990) On-line estimation and adaptive control of bioreactors. Elsevier, AmsterdamGoogle Scholar
  6. Bonnans F, Martinon P, Giorgi D, Grélard V, Maindrault S, Tissot O, Liu J (2017) Bocop 2.0.5—user guideGoogle Scholar
  7. Bonnard B, Chyba M (2003) Singular trajectories and their role in control theory. Mathématiques & applications, vol 40. Springer, PariszbMATHGoogle Scholar
  8. Bosdriesz E, Molenaar D, Teusink B, Bruggeman FJ (2015) How fast-growing bacteria robustly tune their ribosome concentration to approximate growth-rate maximization. FEBS J 282(10):2029–44Google Scholar
  9. Bremer H, Dennis PP (2013) Modulation of chemical composition and other parameters of the cell at different exponential growth rates. In: Slauch JM (ed) Ecosal plus: cellular and molecular biology of E. coli, salmonella, and the enterobacteriaceae. ASM Press, Washington, DCGoogle Scholar
  10. Ceroni F, Blount BA, Ellis T (2016) Sensing the right time to be productive. Cell Syst 3(2):116–7Google Scholar
  11. Cesari L (1983) Optimization—theory and applications: problems with ordinary differential equations, vol 17. Springer, New YorkzbMATHGoogle Scholar
  12. Chaves M, Gouzé JL (2011) Exact control of genetic networks in a qualitative framework. Automatica 47(6):1105–12MathSciNetzbMATHGoogle Scholar
  13. Chubukov V, Gerosa L, Kochanowski K, Sauer U (2014) Coordination of microbial metabolism. Nat Rev Microbiol 12(5):327–40Google Scholar
  14. Cinar A, Parulekar SJ, Ündey C, Birol G (2003) Batch fermentation: modeling, monitoring, and control. Marcel Dekker, New YorkGoogle Scholar
  15. Clark CW (1990) Mathematical bioeconomics: the optimal management of renewable resources. Wiley, New YorkzbMATHGoogle Scholar
  16. Clarke FH, Ledyaev YS, Stern RJ, Wolenski PR (1998) Nonsmooth analysis and control theory. Springer, New YorkzbMATHGoogle Scholar
  17. de Hijas-Liste GM, Klipp E, Balsa-Canto E, Banga JR (2014) Global dynamic optimization approach to predict activation in metabolic pathways. BMC Syst Biol 8:1Google Scholar
  18. de Jong H, Casagranda S, Giordano N, Cinquemani E, Ropers D, Geiselmann J, Gouzé JL (2017a) Mathematical modelling of microbes: metabolism, gene expression and growth. J R Soc Interface 14:20170502Google Scholar
  19. de Jong H, Geiselmann J, Ropers D (2017b) Resource reallocation in bacteria by reengineering the gene expression machinery. Trends Microbiol 25(6):480–93Google Scholar
  20. Del Vecchio D, Dy AJ, Qian Y (2016) Control theory meets synthetic biology. J R Soc Interface 13(120):20160380Google Scholar
  21. El-Samad H, Kurata H, Doyle JC, Gross CA, Khammash M (2005) Surviving heat shock: control strategies for robustness and performance. Proc Natl Acad Sci USA 102(8):2736–41Google Scholar
  22. Ewald J, Bartl M, Dandekar T, Kaleta C (2017) Optimality principles reveal a complex interplay of intermediate toxicity and kinetic efficiency in the regulation of prokaryotic metabolism. PLoS Comput Biol 13(2):e1005371Google Scholar
  23. Farewell A, Neidhardt FC (1998) Effect of temperature on in vivo protein synthetic capacity in Escherichia coli. J Bacteriol 180(17):4704–10Google Scholar
  24. Fracassi C, Postiglione L, Fiore G, di Bernardo D (2016) Automatic control of gene expression in mammalian cells. ACS Synth Biol 5(4):296–302Google Scholar
  25. Gabasov R, Kirillova FM (1982) Singular optimal control. Plenum Press, New YorkzbMATHGoogle Scholar
  26. Giordano N, Mairet F, Gouzé JL, Geiselmann J, de Jong H (2016) Dynamical allocation of cellular resources as an optimal control problem: novel insights into microbial growth strategies. PLoS Comput Biol 12(3):e1004802Google Scholar
  27. Grigorieva EV, Khailov EN (2007) Optimal control of a nonlinear model of economic growth. Discrete Contin Dyn Syst Ser B Supplement:456–466Google Scholar
  28. Hinshelwood CN (1952) On the chemical kinetics of autosynthetic systems. J Chem Soc (Res) 745–755Google Scholar
  29. Iglesias PA, Ingalls BF (eds) (2010) Control theory and systems biology. MIT Press, CambridgezbMATHGoogle Scholar
  30. Izard J, Gomez Balderas C, Ropers D, Lacour S, Song X, Yang Y, Lindner A, Geiselmann J, de Jong H (2015) A synthetic growth switch based on controlled expression of RNA polymerase. Mol Syst Biol 11(11):840Google Scholar
  31. Johnson FH, Lewin I (1946) The growth rate of E. coli in relation to temperature, quinine and coenzyme. J Cell Physiol 28(1):47–75Google Scholar
  32. Kalisky T, Dekel E, Alon U (2007) Cost-benefit theory and optimal design of gene regulation functions. Phys Biol 4(4):229–45Google Scholar
  33. Khalil AS, Collins JJ (2010) Synthetic biology: applications come of age. Nat Rev Genet 11(5):367–80Google Scholar
  34. Koch AL (1988) Why can’t a cell grow infinitely fast? Can J Microbiol 34(4):421–6Google Scholar
  35. Larrabee KL, Phillips JO, Williams GJ, Larrabee AR (1980) The relative rates of protein synthesis and degradation in a growing culture of Escherichia coli. J Biol Chem 255(9):4125–30Google Scholar
  36. Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman & Hall/CRC Press, Boca RatonzbMATHGoogle Scholar
  37. Lo TM, Chng SH, Teo WS, Cho HS, Chang MW (2016) A two-layer gene circuit for decoupling cell growth from metabolite production. Cell Syst 3(2):133–43Google Scholar
  38. Markley N (2004) Principles of differential equations. Wiley-Interscience, HobokenzbMATHGoogle Scholar
  39. Milias-Argeitis A, Rullan M, Aoki SK, Buchmann P, Khammash M (2016) Automated optogenetic feedback control for precise and robust regulation of gene expression and cell growth. Nat Commun 7:12546Google Scholar
  40. Molenaar D, van Berlo R, de Ridder D, Teusink B (2009) Shifts in growth strategies reflect tradeoffs in cellular economics. Mol Syst Biol 5:323Google Scholar
  41. Mosteller RD, Goldstein RV, Nishimoto KR (1980) Metabolism of individual proteins in exponentially growing Escherichia coli. J Biol Chem 255(6):2524–32Google Scholar
  42. Naumov GV (2003) Construction of the switching curve for optimal control problems with chattering control. Izvestiya RAN: Teoriya i Sistemy Upravleniya 3:46–51 In RussianMathSciNetzbMATHGoogle Scholar
  43. Olson EJ, Tabor JJ (2014) Optogenetic characterization methods overcome key challenges in synthetic and systems biology. Nat Chem Biol 10(7):502–11Google Scholar
  44. Oyarzún DA, Stan GB (2013) Synthetic gene circuits for metabolic control: design trade-offs and constraints. J R Soc Interface 10(78):20120671Google Scholar
  45. Pavlov MY, Ehrenberg M (2013) Optimal control of gene expression for fast proteome adaptation to environmental change. Proc Natl Acad Sci USA 110(51):20527–32Google Scholar
  46. Poelwijk FJ, de Vos MG, Tans SJ (2011) Tradeoffs and optimality in the evolution of gene regulation. Cell 146(3):462–70Google Scholar
  47. Pontryagin LS, Boltyansky VG, Gamkrelidze RV, Mishchenko EF (1964) The mathematical theory of optimal processes. Macmillan, New YorkGoogle Scholar
  48. Sauro HM (2017) Control and regulation of pathways via negative feedback. J R Soc Interface 14(127):20160848Google Scholar
  49. Savageau MA (1983) Escherichia coli habitats, cell types, and molecular mechanisms of gene control. Am Nat 122(6):732–44Google Scholar
  50. Schaechter M, Ingraham JL, Neidhardt FC (2006) Microbe. ASM Press, Washington, DCGoogle Scholar
  51. Schattler H, Ledzewicz U (2012) Geometric optimal control: theory, methods and examples. Interdisciplinary applied mathematics, vol 38. Springer, New YorkzbMATHGoogle Scholar
  52. Schattler H, Ledzewicz U (2015) Optimal control for mathematical models of cancer therapies: an application of geometric methods. Interdisciplinary applied mathematics, vol 42. Springer, New YorkzbMATHGoogle Scholar
  53. Schuetz R, Kuepfer L, Sauer U (2007) Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli. Mol Syst Biol 3:119Google Scholar
  54. Schuster S, Pfeiffer T, Fell D (2008) Is maximization of molar yield in metabolic networks favoured by evolution? J Theor Biol 252(3):497–504MathSciNetzbMATHGoogle Scholar
  55. Scott M, Gunderson CW, Mateescu EM, Zhang Z, Hwa T (2010) Interdependence of cell growth and gene expression: origins and consequences. Science 330(6007):1099–103Google Scholar
  56. Scott M, Klumpp S, Mateescu EM, Hwa T (2014) Emergence of robust growth laws from optimal regulation of ribosome synthesis. Mol Syst Biol 10:747Google Scholar
  57. Sontag ED (2005) Molecular systems biology and control. Eur J Control 11(5):396–435MathSciNetzbMATHGoogle Scholar
  58. Stephanopoulos GN, Aristidou AA, Nielsen J (1998) Metabolic engineering: principles and methodologies. Academic Press, San DiegoGoogle Scholar
  59. Surovstev IV, Morgan JJ, Lindahl PA (2007) Whole-cell modeling framework in which biochemical dynamics impact aspects of cellular geometry. J Theor Biol 244(1):154–66MathSciNetGoogle Scholar
  60. Uhlendorf J, Miermont A, Delaveau T, Charvin G, Fages F, Bottani S, Batt G, Hersen P (2012) Long-term model predictive control of gene expression at the population and single-cell levels. Proc Natl Acad Sci USA 109(35):14271–6Google Scholar
  61. van den Berg HA, Kiselev YN, Kooijman SALM, Orlov MV (1998) Optimal allocation between nutrient uptake and growth in a microbial trichome. J Math Biol 37(1):28–48MathSciNetzbMATHGoogle Scholar
  62. van Elsas JD, Semenov AV, Costa R, Trevors JT (2011) Survival of Escherichia coli in the environment: fundamental and public health aspects. ISME J 5(2):173–83Google Scholar
  63. Venayak N, Anesiadis N, Cluett WR, Mahadevan R (2015) Engineering metabolism through dynamic control. Curr Opin Biotechnol 34:142–52Google Scholar
  64. Venkateswarlu C (2005) Advances in monitoring and state estimation of bioreactors. J Sci Indus Res 63:491–8Google Scholar
  65. Waldherr S, Oyarzún DA, Bockmayr A (2015) Dynamic optimization of metabolic networks coupled with gene expression. J Theor Biol 365:469–85MathSciNetzbMATHGoogle Scholar
  66. Weiße AY, Oyarzún DA, Danos V, Swain PS (2015) Mechanistic links between cellular trade-offs, gene expression, and growth. Proc Natl Acad Sci USA 112(9):1038–47Google Scholar
  67. Yegorov I, Bratus A, Todorov Y (2015) Synthesis of optimal control in a mathematical model of economic growth under R&D investments. Appl Math Sci 9(91):4523–64Google Scholar
  68. Yegorov I, Grognard F, Mailleret L, Halkett F (2017a) Optimal resource allocation for biotrophic plant pathogens. IFAC-PapersOnLine 50(1):3154–9Google Scholar
  69. Yegorov I, Mairet F, Gouzé JL (2017b) Optimal resource allocation for bacterial growth with degradation. IFAC-PapersOnLine 50(1):9858–63Google Scholar
  70. Yegorov I, Mairet F, Gouzé JL (2018) Optimal feedback strategies for bacterial growth with degradation, recycling and effect of temperature. Optim Control Appl Methods 39(2):1084–1109MathSciNetzbMATHGoogle Scholar
  71. Yi T-M, Huang Y, Simon MI, Doyle J (2000) Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc Natl Acad Sci USA 97(9):4649–53Google Scholar
  72. Yong J, Zhou XY (1999) Stochastic controls: Hamiltonian systems and HJB equations. Springer, New YorkzbMATHGoogle Scholar
  73. Zelikin MI, Borisov VF (1994) Theory of chattering control with applications to astronautics, robotics, economics, and engineering. Birkhauser, BostonzbMATHGoogle Scholar
  74. Zelikin MI, Borisov VF (2005) Singular optimal regimes in problems of mathematical economics. J Math Sci 130(1):4409–4570MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.North Dakota State UniversityFargoUSA
  2. 2.Ifremer PBANantesFrance
  3. 3.InriaUniv. Grenoble AlpesGrenobleFrance
  4. 4.BIOCORE team, Inria Sophia-Antipolis Méditerranée, Univ. Côte d’Azur, Inria, INRA, CNRS2004 Route des LuciolesValbonneFrance

Personalised recommendations