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Mathematical model for describing oscillations of bacterial biomass

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Abstract

A mathematical model of microbial cultivation is developed that describes the emergence of biomass oscillations under continuous conditions with a period shorter than the culture generation time. The model constants at which the calculation results are in the best agreement with experimental data are found. The phase portrait of the system in the oscillation mode is analyzed. The possibility of using the model for describing the behavior of different cultures in different modes is demonstrated.

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Abbreviations

b:

fraction of the substrate entering the cell that is consumed to form intermediates (during substrate transport)

C e :

mass concentration of transport enzymes E in the σ phase, mg/mg

C es :

mass concentration of complex ES in the σ phase, mg/mg

C l :

mass concentration of intermediates in the cell, mg/mg

\(\bar C_l \) :

steady-state mass concentration of intermediates, mg/mg

\(\tilde C_l \) :

critical mass concentration of intermediates at which the development of bioerrors and cell inactivation are activated, mg/mg

C x :

mass concentration of macromolecular components X in phase II, mg/mg

C y :

mass concentration of complex XL in phase II, mg/mg

D:

dilution rate, h−1

D cr :

critical dilution rate at which biomass oscillations begin, h−1

E:

membrane enzymes catalyzing transport

ES:

intermediate enzyme-substrate complex forming during substrate transport

F:

rate of lysis of pathogenic cells, mg/(1 h)

g:

mass fraction of low-molecular-weight components (substrate and intermediates) in substrate transport and synthesis of macromolecules

g′, g″:

stoichiometric coefficients in reactions of the model of the microlevel

j:

total fictitious flux of substrate supply into the σ phase, mg/(1 h)

k n :

rate constant for the nth reaction in the cell (n = 2−7), h−1

k −n :

rate constant for the intracellular reaction that is reverse to the nth reaction (n = 1, 5), h−1

k s :

specific mass-transfer coefficient of the substrate, l/(mg h)

L:

monomeric intermediates

S:

limiting carbon-containing substrate

t:

time, h

u:

fraction of intermediates consumed by the cell for synthesis of macromolecules

W n :

rate of the nth reaction in the cell (n = 1−7), h−1

W −n :

rate of the intracellular reaction that is reverse to the nth reaction (n = 1, 5), h−1

X:

macromolecular cellular components (enzymes, proteins, DNA, RNA, etc.)

XL:

intermediate complex forming during synthesis of macromolecular components

Z:

biomass, mg/l

\(\bar Z\) :

steady-state biomass concentration, mg/l

Z M :

pathogenic biomass, mg/l

Z V :

viable biomass, mg/l

Z 0 :

biomass concentration at the initial moment of time, g/l

α:

fraction of viable cells

γ:

mass fraction of the products of lysis of the pathogenic biomass that can be consumed by viable cells as a substrate

μ:

specific biomass growth rate, h−1

ρi :

average density of phase i, i.e., the ratio of the total weight of the phase to the volume of the system, mg/l

ρs :

residual substrate concentration, mg/l

ρ 0s :

substrate concentration in the nutrient solution, mg/l

ρɛ :

average CO2 density in phase I, mg/l

τ:

time lag, i.e., time required to prepare the inactivation mechanism, h

υi :

specific rate of growth of phase i, h−1

χ:

specific rate of development of bioerrors on the scale of a cell population, h−1

i:

phase

σ:

σ phase

ϕ:

phase II

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Authors and Affiliations

  1. Mendeleev University of Chemical Technology of Russia, Miusskaya pl. 9, Moscow, 125047, Russia

    A. S. Skichko & E. M. Kol’tsova

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  1. A. S. Skichko
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  2. E. M. Kol’tsova
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Original Russian Text © A. S. Skichko, E. M. Kol’tsova, 2006, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2006, Vol. 40, No. 5, pp. 540–550.

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Skichko, A.S., Kol’tsova, E.M. Mathematical model for describing oscillations of bacterial biomass. Theor Found Chem Eng 40, 503–513 (2006). https://doi.org/10.1134/S0040579506050071

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