Chemical Papers

, Volume 72, Issue 10, pp 2589–2598 | Cite as

Estimation of plasmid concentration in batch culture of Escherichia coli DH5α via simple state observer

  • Fernando Grijalva-Hernández
  • V. Peña Caballero
  • Pablo A. López-Pérez
  • Ricardo Aguilar-LópezEmail author
Original Paper


The aim of this paper is to present an alternative state observer structure for online estimation purposes of the key dynamical variables in a class of batch culture for plasmid production; the latter has been extremely attractive to be used as DNA vaccines or gene therapy. A mathematical model for culture of Escherichia coli DH5α-harboring plasmid was considered a benchmark system for the application of the proposed estimation methodology. Local observability analysis revealed that the system is partially observable for plasmid concentration considering only the biomass concentration in the batch culture as the measured variable. The proposed observer is designed with a simple proportional–integral feedback of the measured biomass concentration, where under the proposed design, the observer gain´s array can compensate the main nonlinearities of the estimation error dynamics. The convergence of estimated variables to the real ones is mathematically analyzed, reaching an asymptotic behavior. Numerical experiments were performed, where a comparison with a standard extended Luenberger observer was done and the proposed estimation methodology revealed a satisfactory performance.


Plasmid concentration monitoring Batch bioreactor PI-type observer Mathematical analysis 

List of symbols


Biomass concentration (g/L)


Glucose concentration (g/L)


Glycerol concentration (g/L)


Acetate concentration (g/L)


Plasmid concentration (g/L)


Affinity constant for glucose (g/L)


Affinity constant for glycerol (g/L)


Affinity constant for acetate (g/L)


Inhibition constant of growth on glycerol by glucose (g/L)


Inhibition constant of growth on acetate by glucose (g/L)


Inhibition constant of growth on glucose by acetate (g/L)


Inhibition constant of growth on glycerol by acetate (g/L)


Inhibition constant of biomass growth by acetate (g/L)


Inhibition constant of biomass growth by plasmid production (g/L)


Acetate yield on glucose


Acetate yield on glycerol


Biomass yield on glucose


Biomass yield on glycerol


Plasmid yield on biomass growth on glucose


Plasmid yield on biomass growth on glycerol


Plasmid yield on biomass growth on acetate


Time (h)


Gain of the observer (g/L)


Gain of the observer (1/h)

Greek letters

\( \mu_{\text{s}} \)

Specific growth rate on glucose (1/h)

\( \mu_{\text{g}} \)

Specific growth rate on glycerol (1/h)

\( \mu_{\text{a}} \)

Specific growth rate on acetate (1/h)

\( \mu_{{\text{max\,s}}} \)

Maximum specific growth rate on glucose (1/h)

\( \mu_{{\text{max\,g}}} \)

Maximum specific growth rate on glycerol (1/h)

\( \mu_{{\text{max\,a}}} \)

Maximum specific growth rate on acetate (1/h)

\( \xi \)

Estimation error



FGH is grateful with Consejo Nacional de Ciencia y Tecnología (CONACyT, Mexico) for the financial support via a postgraduate scholarship.


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Copyright information

© Institute of Chemistry, Slovak Academy of Sciences 2018

Authors and Affiliations

  1. 1.Departamento de Biotecnología Y BioingenieríaCINVESTAV-IPNMexico City, CDMXMexico
  2. 2.División de Ciencias de la Salud e Ingenierías, Campus Celaya-Salvatierra, Departamento de Ingeniería AgroindustrialUniversidad de GuanajuatoGuanajuatoMexico
  3. 3.Escuela Superior de Apan, Universidad Autónoma del Estado de HidalgoApanMexico

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