A great interest was reserved for the lactic acid production in these last years owing to its wide range of applications (John et al. 2007). Nowadays, the fermentative production of lactic acid is the world’s leading technology (about 90% of world production). To increase the efficiency of the lactic acid fermentation processes, various modes of culture have been investigated (Lin and Wang, 2007; Nandasana and Kumar, 2008). Among them, batch culture remains the most commonly used approach in industrial lactic acid production. However, volumetric productivities are low due to the end-product inhibition (Amrane and Prigent, 1996; Kwon et al. 2001). Continuous processes appear therefore as a useful alternative (Amrane and Prigent, 1996; Lin and Wang, 2007). However, volumetric productivities reported for usual continuous cultures remain low (Lin and Wang, 2007); while the efficiency of continuous two-stage bioreactors was demonstrated (Amrane and Prigent, 1996; Schepers et al. 2006). Several structured and unstructured (Nielsen et al. 1991; Burgos-Rubio et al. 2000; Boonmee et al. 2003; Bâati et al. 2004) models are available. Owing to their simplicity and their accuracy in the description of lactic acid fermentation, simple unstructured models were preferred in this work. Several ones are available in the literature, dealing with batch or continuous cultures of lactic acid bacteria (Luedeking and Piret, 1959; Richter and Nottelmann, 2004; Schepers et al. 2006; Balannec et al. 2007). However, there is a lack of models dealing with two-stage reactors for lactic acid fermentation (Kwon et al. 2001; Lin and Wang, 2007; Bouguettoucha et al. 2008). Some unstructured models were previously
developed in the laboratory. To account for the effect of the undissociated lactic acid (and pH), the main inhibitory specie (Kashket,
1987; Gätje and Gottschalk, 1991), an inhibitory term was introduced in the Luedeking and Piret
expression (Balannec et al. 2007); while in the case of substrate limitations,
an additional term was added to the Luedeking and Piret expression to account for cessation of lactic acid production
when carbon became limiting (Amrane, 2001; Bouguettoucha et al. 2007). Both
expressions were also merged to develop a generalised model involving a unique
expression taking into account both effects (Bouguettoucha
et al. 2007; Bouguettoucha et al. 2008). It was successfully applied to
two-stage continuous cultures, which involves an inhibitory effect for the first stage (seed
culture) and a nutritional limitation effect for the second stage (culture) (Bouguettoucha et al. 2009). The Verlhust model which proved to be
relevant to describe growth kinetics (Pandey et al. 2000; Lan et al. 2006;
Vázquez and Murado, 2008) was considered (Balannec et al. 2007; Bouguettoucha
et al. 2007). These models showed interesting predictive potential to
describe continuous two-stage culture of
Whey based
media contained 48 g L The following
supplementation was added to reconstituted whey for the preparation of seed
culture medium (gr L A schematic description of the system can
be found in a previous paper (Bouguettoucha
et al. 2009). Cultures were
carried out in a 2 L reactor (Set 2M, SGI, Toulouse, France, magnetically
stirred (300 rpm) at 42ºC. pH was controlled at 5.9 by automatic addition of 10
mol L Reaction mixture overflowing the first stage and sterile culture medium were fed to the second stage through a peristaltic pump (Watson-Marlow 502U, Volumax, and PAP, SGI, respectively). The second stage was maintained at constant total mass by means of electronic weighing system (382MP8, Sartorius, Palaiseau, France) acting on a solenoid pinch valve (EG2, Sirai, Bioblock, Illkirch, France) in the bleed pipe. 200 mL of sterile seed culture medium were
fed in the first-stage reactor (250 mL glass reactor) and inoculated with 1% (v
v ^{-1}) with
sterile seed culture medium and operated at a constant volume V = 120 mL. The mean residence time in the first stage was therefore set to
12 hrs (_{i}D = 0.083 h_{i}^{-1}), allowing to avoid large
fluctuations of biomass concentrations, due to seed culture conditions close to
wash out conditions (Amrane and Prigent, 1996). After 4-5 hrs, exponential
growth took place in the second-stage reactor; then it was continuously fed at
constant flow rate with both reaction mixture overflowing the first stage and
sterile culture medium, at constant volume (V=
800 mL). Steady state for the second-stage reactor was achieved when both
turbidity and NaOH addition rate (pH control) remained constant over a period
of at least three mean residence times._{c }As required, the mean residence time in the
second-stage was modified by varying the sole feed flow rate of sterile culture
medium V =
800 mL._{c}Just after inoculation and at steady state sterile
media and reaction mixtures were assayed for total biomass, lactose and lactic
acid as previously described (Amrane, 2005). Four samples aseptically harvested
during each run were also tested in order to check the lactic acid and total
biomass concentration calculated on-line: the observed standard deviations were ± 1 and ± 0.2 gr L The Excel solver was used for the resolution of the considered equations and the parameters optimisation. The following definition has been used for the
determination of the residual standard deviation
Only the second stage was considered in this work, since the dilution rate in the first stage was maintained at a constant value. The following assumptions were assumed: (i)
The fermentation
process was carried out in continuous stirred-tank reactors (CSTR), Mass balance can be expressed as follows: For growth
Where At steady state conditions
For production
At steady state conditions
Where V The above equation (Equation 5) can be rearranged as follows:
The Verlhust model which proved to describe satisfactory growth kinetics (Pandey et al. 2000; Vázquez and Murado, 2008) was considered (Balannec et al. 2007; Bouguettoucha et al. 2007; Bouguettoucha et al. 2008):
Where _{max }was the maximum specific growth rate.## Growth modelIntroduction of the mass balance for biomass in the
second stage (Equation 3) into the Verlhust model (Equation 7)
led to the following implicit equation of
As previously observed (Bouguettoucha et al. 2009),
the growth model (Equation 8) did not account for the important decrease
of the biomass concentration at high dilution rate, namely close to wash out (Amrane
and Prigent, 1996). From this, Equation 8 was
modified by replacing the dilution rate in the second stage _{}. Hence, it came for
growth:
Substrate limitation model (SLM) In order to take into account the carbon limitation recorded during culture at controlled pH, the Luedeking-Piret expression was previously modified (Bouguettoucha et al. 2007):
^{-1}.By introducing the Verlhust expression (Equation 7)
into the above modified Luedeking-Piret relation (Equation 10) and by
considering a constant product on substrate yield
Then, from the mass balance for the product in the
second stage (Equation 7), the lactic acid concentration at steady state
in the second stage
To avoid the use of two expressions to describe production rate, depending on culture conditions, an unique expression taking into account both an inhibitory effect and a nutritional limitation effect was considered (Bouguettoucha et al. 2007 and Bouguettoucha et al. 2008):
Where
Similarly to the substrate limitation model Equation
11 and by considering the Henderson-Hasselbach equation Equation 14, the
specific production rate in the second stage
Introduction of the above specific production rate Equation
15 into the mass balance for the product in the second stage Equation 6 led
to the lactic acid concentration at steady state in the second stage
The calculated data displayed in Figure
1 resulted from the optimization of parameters α and β only, and
the optimized values were 0.0008 and 0.028 for α and β respectively.
Indeed, calculated values for biomass and product concentrations at steady
state in the first stage were taken from a previous work and are given in Table
1, as well as the other parameter values also collected in Table 1 and
deduced from previous batch cultures of ^{-1}) which decreased from 0.43 for the previous
model (Equation 8) (Bouguettoucha et al. 2009) to a negligible value (8 x 10^{-5})
for the modified model (Equation 9).The parametric sensitivity of the two additional
parameters α and β involved in this expression showed that as required
the effect of both parameters increased with the dilution rate. A weak effect
of the parameter α was recorded, since 10% variation of α from its
optimal value led to less than 6% variation of the biomass concentration at
stationary state (Figure 1). Contrarily, biomass concentration at steady
state was significantly affected by the parameter β, involved in the
exponential term, since at the higher dilution rate (0.167 h As previously shown, the Luedeking-Piret Model did not account for the slowing down of production recorded at the end of batch culture (Bouguettoucha et al. 2007) and appeared to fail in the description of experimental data recorded during two stages continuous cultures (Bouguettoucha et al. 2009). From this, only the modified substrate limitation model (SLM) and the generalised model (GM) were considered in this work. The calculated data displayed in Figures 2a and 2b were the result of an optimization of the parameters B, which are summarized in Table 2. Similarly
to growth modelization, the other parameter values were considered as indicated
in Table 1. As observed, both modified SLM and GM models led to nearly
similar values of the parameters _{c}A and _{c}B,
similarly to the behaviour previously recorded (Bouguettoucha et al. 2009).
Indeed, at controlled pH
(5.9), the undissociated lactic acid concentration (approximately 0.3 g L_{c}^{-1})
is below the inhibitory threshold (Gätje and Gottschalk, 1991), namely almost
negligible compared to the inhibitory undissociated lactic acid concentration, 8.5 g L^{-1} (Bouguettoucha et al. 2008). Consequently, the inhibition term of Equation 16 had no effect, since it was close to unit; and the main term was therefore
the substrate limitation term. Both models led therefore to
fairly similar calculated data, which matched experimental data (RSD = 2.6 and
2.5, respectively; Table 2). If compared to the fitting given by the
previous models (Bouguettoucha et al. 2009), and similarly to growth modelization
(Figure 1), the improvement was significant at high dilution rates (0.167
h^{-1}; Figure 2a), illustrated by the lower least square values
(between calculated and experimental data) recorded, 2.5 and 0.02 for SLM (Equation
12) and GM (Equation 16) models, versus 10.2 and 5.2 for the
previous SLM and GM models (Bouguettoucha et al. 2009).Calculated volumetric productivity in the
second stage corresponded to the product of the lactic acid concentration at
steady state A and _{c}B (Table 2). As observed, and similarly to product concentrations at
steady state, both modified SLM and GM models matched experimental data; the
residual standard deviation value was nearly 0.3 for both models, versus about 0.4
given by the previous SLM and GM models (Bouguettoucha et al. 2009). As
expected from the above results, the improvement was significant at high
dilution rate (0.167 h_{c}^{-1}), since the least square values decreased
from 0.29 and 0.15 for the previous SLM and GM models (Bouguettoucha et al.
2009) to 0.07 and 0.0006 for the modified SLM (Equation 12) and GM (Equation
16) models.Since the previous models did not satisfactorily
describe biomass data recorded at steady state in the second stage for high
dilution rates, namely close to wash out (Bouguettoucha et al. 2009), the
dilution rate in the second stage _{}. As expected, the
positive effect of the modified model was especially significant at high
dilution rates; it was observed for all considered culture parameters, namely
biomass concentration, lactic acid production and volumetric productivity,
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