Development of a structured model for batch cultures of lactic acid bacteria

Abstract

A combined stochastic-deterministic model able to predict the growth curve of microorganisms, from inoculation to death, is presented. The proposed model is based on the assumption that microorganisms can experience two different physiological states: non-proliferating and proliferating. The former being the physiological state of the cells right after their inoculation into the new extracellular environment; the latter the state of microorganisms after adaptation to the new medium. To validate the model, a Lactobacillus bulgaricus strain was tested in a medium at pH 4.6 at two different temperatures (42°C and 35°C). Curves representing the bacterial growth cycle were satisfactorily fitted by means of the proposed model. Moreover, due to the mechanistic structure of the proposed model, valuable quantitative information on the following was obtained: rate of conversion of non-proliferating cells into proliferating cells, growth and death rate of proliferating cells, and rate of nutrient consumption.

List of symbols

${\left. {{{{\rm{d}}N{\left( t \right)}} \over {{\rm{d}}t}}} \right|^{{\max }} }$ :

the maximum of ${{{\rm{d}}N\left( t \right)} \over {{\rm{d}}t}}$

E%:

the relative percent difference between experimental and predicted values

F(t):

the limiting nutrient concentration in the extracellular environment (expressed as g/l) at time t

F ₀ :

the initial concentration of limiting nutrient in the extracellular environment (expressed as g/l)

G(t):

the concentration of generated cells (expressed as cfu ml⁻¹) at time t

k _i :

constant(s), to be regarded as fitting parameters

m :

mean value of the probability density function

M(t):

the concentration of death cells (expressed as cfu ml⁻¹) at time t

n _exp :

the number of experimental data points

N(t):

the microorganism concentration (expressed as cfu ml⁻¹) at time t

N ₀ :

the initial concentration of microorganisms (expressed as cfu ml⁻¹)

P(t):

the concentration of proliferating microorganisms (expressed as cfu ml⁻¹) at time t

Q(t):

the concentration of non-proliferating cells (expressed as cfu ml⁻¹) at time t

Q ₀ :

initial concentration of non-proliferating cells

R ₁(t):

probability density function of conversion of non-proliferating cells to proliferating cells (expressed as cfu ml⁻¹ h⁻¹)

R ₂(t):

the proliferation rate (expressed as cfu ml⁻¹ h⁻¹) at time t

$R_2^{\max }$ :

the maximum of R ₂(t) (expressed as cfu ml⁻¹ h⁻¹)

R ₃(t):

the death rate (expressed as cfu ml⁻¹ h⁻¹) at time t

$Y_{\exp }^{\rm{i}}$ :

the experimental value

$Y_{{\rm{pred}}}^{\rm{i}}$ :

the predicted value

σ:

standard deviation of the probability density function