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Fermentation kinetics including product and substrate inhibitions plus biomass death: a mathematical analysis

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Abstract

We propose an analytical solution of the kinetic equations describing fermentations. Equations are solved in phase space, i.e. the biomass concentration is written explicitly as a function of the substrate concentration. These results hold even when cell death and an arbitrary number of substrate/product inhibitions are accounted for. Moreover, constant yield needs not be assumed.

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Abbreviations

K i :

Inhibition coefficient (g/l)

k i :

De-dimensionalized K i , k i  = K i /S 0 (−)

K D :

Cell death coefficient (g/l h)

m :

Maintenance coefficient (h−1)

n D :

Number of biomass death terms (–)

n I :

Number of inhibitions (–)

P i :

Product concentration (g/l)

p i :

De-dimensionalized P i , p i  = P i /(α i S 0 ) (–)

S :

Substrate concentration (g/l)

s :

Normalized substrate, s = 1 − S/S 0 (–)

t :

time (h)

X :

Viable biomass concentration (g/l)

x :

Normalized biomass, x = (X − X 0)(1 + ρ)/(YS 0) (–)

Y :

Biomass yield (–)

α i :

Stoichiometric coefficient for product P i (–)

κ :

De-dimensionalized K D , κ = mYK D /(αS 0) (–)

μ :

Maximum specific growth rate (h−1)

μ′ :

Change in μ due to inhibition (–)

ρ :

Ratio of time constants, ρ = m Y/μ (–)

σ i :

a root of \( \mu '\left( {1 - s} \right) + \rho \,k_1 \prod\nolimits_j {\left[ {1 + \left( {1 - s} \right)/k_j } \right]} \) (–)

φ i :

Defined in Eq. 11 (–)

ψ :

Distance to constant yield, ψ = k 1 ρ/(1 + ρ) (–)

0:

Initial value

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Authors and Affiliations

  1. Institute of Materials Research and Engineering, Singapore, 117602, Singapore

    Mathieu Bouville

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  1. Mathieu Bouville
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Correspondence to Mathieu Bouville.

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Bouville, M. Fermentation kinetics including product and substrate inhibitions plus biomass death: a mathematical analysis. Biotechnol Lett 29, 737–741 (2007). https://doi.org/10.1007/s10529-006-9296-z

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  • DOI: https://doi.org/10.1007/s10529-006-9296-z

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