*Corresponding author Tel:+447413541769 Email:myrtozacharof1981@yahoo.com

_____________________________________________________________________________

Modelling and Simulation of Cell growth Dynamics, Substrate

Consumption and Lactic acid production Kinetics of Lactococcus lactis

Myrto-Panagiota Zacharof1* and Robert W. Lovitt2

1*College of Engineering, Multidisciplinary Nanotechnology Centre, Swansea University, Swansea, SA2 8PP, UK

2College of Engineering,Center of Complex Fluids Processing, Multidisciplinary Nanotechnology Centre, Swansea

University, Swansea, SA2 8PP, UK

___________________________________________________________________________________________

Abstract: Lactococcus lactis species have been and still are extensively investigated due to their significant

commercial importance. Current scientific research focuses on strains utilized in food industry, due to their multiple

uses in food and beverages fabrication. Biomass of Lactococcus lactis is of great interest as well as the end products

of its metabolism such as lactic acid and nisin. However their production is constantly challenged due to end

product inhibition occurring during intensive propagation of the coccus in reactor systems. To successfully predict

the behavior of the culture, the approach of combining mathematics with biology, ergo the development of an

unstructured mathematical model, was taken. Although Luedeking and Piret is the model that has been extensively

used to demonstrate growth in end-product inhibition cultures, its applicability is limited due to its dependance on

the specific growth and product coefficients, particularly related to the culturing conditions used. To overcome these

hurdles, a combination of the non competitive single product end inhibition Taylor and Hinselwood models was

used, with the significance of this model laying in the fact that it offers a feasible alternative to the commonly used

model of Luedeking and Piret for describing fermentation kinetics governed by end-product inhibitions. The fitting

with the experimental values, in batch mode, was tested in terms of the coefficient of determination (R²), having

values 0.97 to 0.99 and suggesting a very good fitting with the experimental data. The model was further developed

to achieve theoretical predictions of volumetric cell productivity in continuous and fed-batch mode of substrate feed

in different culturring systems.

Keywords: end product inhibition, growth kinetics, lactic acid, Lactococcoi, mathematical model

INTRODUCTION

[2]

Lactococcus Lactis (L.lactis) belongs to the genera of Lactic Acid Bacteria (LAB), which are microorganisms

whose distinctive ability is to produce lactic acid, as a major end product of their metabolism, through fermentation,

that is the anaerobic catabolism of carbon compounds existing into several types of substrates. L.lactis has been

graded as a non pathogenic bacterium for humans (Generally Regarded As Safe, GRAS) [1-4], enabling its wide use

in the food industry, as a natural acid–production bioreactor. This bacterium is of particular interest to the

alimentary industry, due to its use as a starter culture, it can be inoculated in a variety of food products, assuring the

production of enough lactic acid [5-7], which will lower the pH point acidic point [8], inhibiting the development of

other pathogenic microbial strains and helping the preservation for longer periods of the products. At this point a

secondary starter culture of different microbial strains can be added so as to produce the desirable flavor, taste,

texture and odor [9-11].However, for the evaluation of productivity, optimization, and prediction of behavior and

control of a fermentation process mathematical modeling remains a valuable asset [12-15]. A mathematical model

can be defined as a set of correlations between the parameters of interest of a studied system. This set of relations

can be represented by mathematical equations which evaluate the dependence between the variables (agitation and

aeration, heat transfer and temperature maintenance, viability and preservation of microorganism, product recovery,

feed rate of nutrients, energy and power requirements in order to improve the fermentation process [16-19].

Each of the above parameters can be mathematically represented and be differentiated according to the mode of

operation of the fermentation system [20- 21]. Mathematical models, depending on their complexity, can be divided

into primary, secondary and tertiary, segregated and structured, and unsegregated and unstructured. They can be

evaluated, according to the bias and the accuracy factor, especially if used for performance prediction. Bias factor

can be used to judge the response of the model to the growth of microorganism and the accuracy factor measures the

overall model error [22-24]. The unstructured models are based on the assumption of a fixed cell composition which

is treated as a sole component, irrespective of the individual cellular structure, while biomass growth is balanced.

They are based on experimental observations (empirical models) that so not take into account the theoretical

stoichiometry of reactions [25-27]. On the other hand, segregated structured models use a multicomponent complex

description of cellular population based on its theoretical heterogeneity, including internal kinetics and cellular

regulation [28]. Due to the extensive demand of experimental data for their structure as well as the complexity of

their structure, they demand extended analytical equations they are not easily applicable [29-31].

The simplicity and applicability of unstructured models has led to the majority of fermentation processes being

evaluated according to these models, since a single state of culturing is used throughout the process, with the most

prominent example being the extensive use of the Monod equation to describe microbial kinetics [33-35].

Fermentation procedures are expected to operate under the restriction of a substrate limited growth model. The

relation between substrate concentration and specific growth rate is the basic parameter for the formation of kinetic

models [36]. For the majority of cellular fermentation processes the kinetics can be described by the Monod

[3]

equation, while several other rate expressions (Table 1) have been proposed to cover different conditions

parameters.

Due to the significant industrial importance of Lactococcus lactis, its fermentation on various sources has been

thoroughly investigated [37] and numerous kinetic studies have been published [38]. Regardless the great variety of

growth and production studies, mathematical modelling of the process remains limited to the use of the Luedeking

and Piret model [39], associated with the experimental definition of a growth-associated and of a non growth-

associated constant, limiting its applicability due to its dependence on the specific growth and product coefficients,

particularly related to the culturing conditions used [40-42]. To overcome these hurdles, a combination of the non

competitive single product end inhibition models of Taylor and Hinselwood were used in this study. To further

evaluate the predictive ability and the significance of estimating parameters, the model was developed into three

different culturing conditions: batch, continuous and fed-batch and two growth development systems: CSTR and

MBR.

MATERIALS AND METHODS

Bacterial strains

Lactococcus lactis NCIMB 700185 was provided in a lyophilised form by National Collection of Industrial Food

and Marine bacteria (NCIMB), Aberdeen, Scotland, United Kingdom.

Culturing Conditions

The strain was cultured in modified optimised liquid medium containing 20 g L-1 glucose, yeast extract 10 g L-1,

bacteriological peptone 4 g L-1, sodium acetate 10 g L-1, sodium chloride 5 g L-1, potassium hydrogen phosphate 5 g

L-1. The medium’s pH was set at 6.5.

Continuously Stirring Tank Reactor (CSTR)

A 2L capacity continuously stirring tank (CSTR) pyrex glass reactor (Figure 1) for the procedure. The fermenter

was designed and equipped accordingly to achieve a constant temperature of 30ºC and pH control, as well as

constant anaerobic conditions through constant flow of gaseous nitrogen. It was equipped with a hydrargiric

thermometer (temperature control) a pH probe (Fischer Scientific, UK) connected with a pH controller automation

apparatus (Electrolab FerMac 260, UK), a magnetic stir bar for agitation (350 r/m), a glass aeration port, a sampling

and inoculation port, a gas flow stainless steel port connected with a filter (Polyvent filter, 0.2µm, Whatman Filters,

UK), a port for alkali/acid feed and stainless steel coils for heat emission. The sampling was performed on an hourly

basis on a 10 hours circle via the sampling port with a 10 ml sterile syringe and all the samples were measured for

biomass concentration, via optical density measurements.

Measurement of cellular growth and biomass

[4]

Determination of cell growth was monitored as an increase of turbidity in terms of optical density (O.D.) at 660 nm

wavelength into a spectrophotometer (PU 8625 UV/VIS Philips, France). The light path of the tube was 1.8 cm.

Measuring the O.D. was carried out on an hourly basis until it reached the late stationary phase. The growth curves

were obtained by plotting the O.D. against time. The maximum and specific growth rates ( maxµ , 1h − and µ, 1h − )

of bacteria were calculated from the logarithmic plots of the O.D. versus time during the exponential growth phase

[5], according to the formula:

µ ( 1−h )=

DT2ln

dt)x(lnd

dtdx

x1

== (1)

where DT (h) = x

)1t2t( − (O.D. at 660nm hourly basis) (2)

Determination of End Products through High Performance Liquid Chromatography

L.lactis was reported to produce mainly lactic acid and a very small amount of acetic acid during the exponential

growth phase of its metabolism. In order to quantify the amount of the produced lactic acid and also to confirm the

fact that the strain was homofermentative all the samples were analysed by the High Performance Liquid

Chromatography (HPLC) method. The HPLC system was connected with a conductivity detector (ED 40

Electrochemical Detector, Dionex, UK) and fitted with an ion exchange column (Dionex, Canada). The solvent

(mobile phase) delivery system was constructed by 2 pumps (pumps A and B) (Varian Co. Canada), with a pressure

operation range between 100 to 1900 mbar. Temperature control of the solvents was maintained with a hotplate

(Millipore Co., UK) at 27°C. All the samples were injected in the system via a sterile HPLC plastic syringe at a 20

µl injection loop connected with the HPLC system.

The mobile phase in gradient flow, with a flow rate of 1 ml min-1, was represented by two solution of NaOH of

0.05mM and 100mM concentration. The solutions were delivered to the pumps via rubber tubes and valves with

each run lasting 38 min. The mobile phase was organised as following: NaOH 0.05 mM was poured for 5min., then

the concentration of the solution increased to 5.5mM for 10min. and then to 100mM for 5min. At the final phase the

NaOH concentration was kept at 50.5 mM for 15min and then at 0.5 mM for 3 min. The operation of the system

was controlled automatically using Prostar Workstation Data analysis software package (Varian Co., Canada).

Determination of Carbohydrate consumption rate

In order to investigate the amount of carbohydrate accumulated and to define the carbohydrate consumption rate

during the fermentation performed by the coccus glucose, concentration was measured using GOD-POD enzymical

method. The samples, which were taken from the fermentation, were transferred into 10ml conical plastic tubes

(Fisherbrand, UK) and centrifuged twice (5000 r/ m*10 m.) (Biofuge Stratos Sorall, Kendro Products, Germany) for

complete biomass removal.

[5]

Mathematical Model Development

The mathematical algorithm was based on the Monod model for microbial kinetics and was represented by three

differential equations each one of them representing product formation (dt

dP), substrate consumption (

dt

ds) and

biomass formation (dt

dx). As it was confirmed experimentally L.lactis is a homolactic strain, its growth is inhibited

by constantly augmenting concentration of the metabolic end product. A mathematical term was incorporated in the

model based on Taylor and Hinselwood models so as to describe inhibition effects. Maintenance of the coccus into

stationary phase was represented by a maintenance coefficient (KdX) defined experimental, in addition to the

parameters representing the constants for product formation (KP) and substrate consumption rate (KS), which were

considered to be pH depended constants. All the equations were developed in Excel 2003 Microsoft office software

written on a VBA language spreadsheet, with the use of numerical methods, using the primary estimates of the

coefficients in parallel with the initial values of the parameters to create a series of time course curves that were

compared to the experimental data. Prior to correlating the model formula simultaneously and validating the

goodness of fit of the model, using the Linewear-Burke fitting plot, initial estimates of all constants were performed.

Each model formula was taken per term of development and the respective experimental data were used, to achieve

the best estimates of all the coefficients used. The basic estimates and the standardisation of all the model constants

were performed on every set of experimental data for the analysis of each set of batch of cultures. Several trials

were also done for the calibration of the model using different values for time base, constants, coefficients and

parameters. The best fitting was used in order to avoid any confusion in the results.

Numerical Analysis of the Experimental Data

Each differential parameter was triplicated to obtain the average data. The data were statistically analysed for

accuracy and precision, calculating standard deviation, standard error, experimental error, regression factor and

reading error (Microsoft Excel software Version 2003). All the numerical data were proven to be highly accurate

and reproducible, having standard deviation of mean below 5% and experimental error below 5%, offering highly

significant results.

RESULTS AND DISCUSSION Batch Culture

The model simulation was done with the experimental result of the pH control fermentations, performed on a 2L

CSTR reactor, using optimised medium for growth of L.lactis. The numerical accuracy of the numerical predictions

given by the models were evaluated in terms of standard deviation (σ-2), assuming that the numerical values follow

[6]

a normal distribution. The data were analysed used the chi-test goodness of fit as well as the Shapiro-Wilk test,

indicating that the data had a close to zero skewness and a close to three kurtosis. In the case of batch growth

modelling, the fitting with the experimental values was evaluated in terms of the coefficient of determination (R²).

R² has values 0 to 1, which become larger as the theoretical predictions given from the model fit to the experimental

values [51]. The R² values for the batch growth model were 0.97to 0.99 suggesting a very good fit with the

experimental data.

The estimation of parameters used in the model was allied to the experimental results available. Constants such as

the biomass coefficient (KdX), the substrate consumption coefficient (KS) and the product formation coefficient (KP)

(Table 2) were calculated experimentally and were used to give the fit for the batch culture performed on a pH

control fermentation on a CSTR.

Lactococcus lactis has complex nutritional needs due to its limited biosynthetic capacity [56]. Several researchers

[17,57-58], have highlighted its growth dependence over amino acids and vitamins that can stimulate significantly

biomass formation and cellular growth. A strong correlation has also been proved to exist between the nitrogen

sources within the substrate sources with L.lactis growth, however this dependence has been found being significant

in the case of other metabolites produced during its growth such as nisin [59-61].

Product yields achieved during L.lactis growth at a yield of 1.6 g g-1 in batch culture. This is due to the strong

dependence of the strain on the pH conditions which were kept constant throughout the course of fermentation, but

as well as to the utilisation of the nutrient media, specifically designed and formulated to support intensive biomass

and lactic acid production. Glucose was the principal carbohydrate source provided in the nutrient media is mainly

responsible for lactic acid production (and energy generation) by L.lactis [64]. The carbon –nitrogen sources such as

yeast extract, peptone, beef extracts although stimulate growth influence bacteriocin production such as nisin

[2,38,55].

Theoretically L.lactis converts 1 mole glucose to 2 moles of lactic acid [6]. Neverthelless, during culturing of

L.lactis observed product yields have been well below though of the theoretical prediction [51,54,62]. mainly due to

strain used, the environmental conditions and the fact that some of the carbon must be fixed as biomass. Lactic acid

production is also not only dependent on the carbohydrate sources composition but also of the culturing conditions

where environmental and physical conditions, i.e. oxygen levels, pH, temperature, sugar concentration, other media

components, and end products concentrations all affect the biomass and product yields [62], particularly pH and

oxygen. As a consequence observed yields of lactic acid to glucose have been found in the range 0.4 to 1.6 g g-1

[54,62] with strain dependent yields between 1.0 to 1.6 g g-1 when is cultured in optimum nutritional and

physicochemical conditions [63-68].

L.lactis growth is inhibited by the major end product of its metabolism, lactic acid; the threshold value for complete

growth inhibition due to lactic acid is 366 mM while growth kinetics are affected down to about 30 mM. During

[7]

growth of L.lactis on pH control fermentation, complete growth inhibition does not occur, but there is a strong

effect of lactic acid on biomass formation. A product inhibition term was incorporated in the model following the

Taylor and Hinselwood end product inhibition model, to describe the growth rate. The biomass formation (dt

dx) rate

is described as follows,

XdkX*dt

dx−µ= (3)

where

+

+=

)()(

)(

)*( max

PK

P

SK

S

P

S

µ

µ

(4)

On further integrating the equation µ becomes:

( ) ( )( ) ( )

+

+=

PSK

PKS

S

P

*

**maxµµ (5)

If the biomass formation rate (dt

dx) is on stationary phase, the maintenance coefficient ( dk X), is included the

formula becomes:

( ) ( )

( ) ( )

−

+

+= Xk

PSK

PKS

dt

dxd

S

P

*

**maxµ (6)

As biomass formation kinetics are strongly related to substrate consumption the equation for substrate consumption

rate (dt

ds) becomes:

dt

dx

sxYdt

ds*

1−= (7)

On further integration the equation becomes

( ) ( )( ) ( )

−

+

+−= Xk

PSK

PKS

sxYdt

dsd

S

P

***

*1 maxµ

(8)

The product formation rate equation (dt

dP) proposed is based on the simplified assumption that the rate of product

formation is directly related to the rate of substrate consumption, through a yield coefficient sdpLacticaciY

[8]

For lactic acid formation the proposed equation is the following:

dt

ds*s

)lacticacid(pYdt

)Lacticacid(dP−= (9)

The results of the investigation were developed and compared to the experimental to simulated data in a range of pH

values (5.5 to 7.0) (Figure 2 [a, b, c, d]). When growth reached stationary phase (in the experiment, 6 hours of

culturing the biomass concentration was 4.5 g L-1 and using the model simulation 4.25 g L-1 was achieved 6.25

hours) As it can be seen, the simulated data did fit to the experimental well in terms of predicting product formation

and growth but required a biomass maintenance coefficient term is used to model the growth accurately. In these

cases the product and cell formation developed in relation to the actual data. The end product inhibition term was

required in the mathematical model to describe the growth, as according to the experimental data available, it is

limited most probably, due to end product inhibition.

The theoretical predictions are in much better accordance with the experimental values for biomass formation and

lactic acid production (R2 0.98), especially in pH 5.5, 6.0 and 7.0. However, there was still some poor fitting at the

end of the growth phase of pH 6.5 where some inhibition, most probably due to the pre-existing amount of lactic

acid in the inoculums, could not be described by the model, and the product and substrate were not modelled very

accurately.

In order to investigate the effect of substrate feed on different concentrations, over growth and cell productivity of

L.lactis on a continuous system, the model was tested against different substrate concentrations and theoretical

predictions were made. The efficiency of the system was evaluated in terms of volumetric cell productivity which is

given by the following equation:

Volumetric Cell Productivity (g L-1 h-1) = Final X concentration / Total Fermentation Time (h) (10)

Using the model a series of simulation were made where the substrate concentrations varied between 2.5 g L-1 to 20

g L-1. All the conditions of the system were kept steady on a 10 h fermentation circle. All the numerical values of

the parameters and constants were kept the same (Table 3). The results (Figure 3) show that the volumetric cell

productivity was higher in pH values of 6.5 and 7 and on substrate concentrations of 15 and 20 g L-1. At lower

substrate concentrations, the substrate becomes completely exhausted in a short period of time, with a lower

concentration of lactococcal biomass. At lower pH values the biomass formation rate is also lower with reduced

productivity.

The maximum volumetric cell productivity for batch culturing was 0.47 g h-1 and was achieved on 20 g L-1 substrate

on pH 6.5. While the mechanical construction and operation of a batch system is far less complicated compared to

other systems of culturing, the volumetric cell productivity is proven to be relatively small when compared to the

theoretical predictions made by modelling the intensive propagation of L.lactis on continuous and on the MBR

systems.

[9]

Culturing in bath mode in this time circle was insufficient for high amounts of lactococcal biomass. The batch

system, although is preferable especially in industrial scale practise due to the mechanical simplicity of the

apparatus used, results in simpler operation, also owing to the easier extraction methods of the desired end products.

Continuous Culture

Having investigated batch cultures, an unstructured kinetic mathematical model was developed for theoretical

predictions of growth of L.lactis on a continuous culture system operating on constant pH control. The mathematical

algorithm was based on the previously developed equations over biomass formation, substrate consumption and

lactic acid production for batch growth of L.lactis. Maintenance and an end product inhibition term were also

incorporated. In order to evaluate the end product inhibition effects during growth the model was tested against

different substrate concentrations, between 2.5 to 20 g L-1. The efficiency and feasibility of the system was

evaluated in terms of volumetric cell productivity according to the equation

Volumetric Cell Productivity (g L-1 h-1) = Final X in the system *D (11)

The terms describing the biomass formation rate and the biomass maintained in the system were kept unchanged.

However several new terms were developed especially those needed to demonstrate the changes in the system. The

biomass leaving the system (Xout) was given by

t*F*inXoutX = (12)

while biomass maintained in the system (Xin) is given by the formula:

outXdtdx

0XinX −+= (13)

In the case of substrate, the substrate inflow rate (Sin) in the system is given by the following equation:

t*F*0SinS = (14)

The substrate outflow rate (Sout) in the system is given by the following equation:

Sout = Sin*F*t (15)

The substrate existing initially in the system (Sins) which is accumulated during growth is given by the following

equation:

SinSoutdtds

SoinsS +−+= (16)

On further analysing the above equation, it becomes:

[10]

inoutd

P

Sins SSXk

PK

P

SK

S

sxY

SoS +−+

+

+−+=

)()(

)(

)*(

*1

maxµ

(17)

On further analysing the above equation, it becomes:

inoutd

S

Pins SSXk

PSK

PKS

sxY

SoS +−+

+

+−+=

)(*)(

)(*)*(*

1 maxµ (18)

When describing the product formation rate (dt

dp), the following equations occur:

+

+−−=

)(

)()(

)*(

*1

*)()(max

PK

P

SK

S

sxY

slacticacidp

Ydt

LacticaciddP

P

S

µ

(19)

On further analysis, it becomes:

+−−=

)(*)*(

)(*)*(*

1*)()( max

PSK

PKS

sxYs

lacticacidpY

dt

LacticaciddP

S

Pµ

(20)

The rate on the product maintained in the system (Pin) is given by the formula:

−−= outinin PSS

sp

YP )(* 0 (21)

The rate of product outflow (Pout) is given by the formula:

tFPP inout **= (22)

The dilution rate (D) is given by the equation:

V

FD = (23)

Theoretical predictions were made over the volumetric cell productivity on every pH set using several dilution rates

and substrate concentrations. All the conditions of the system were kept steady and all the numerical values of the

parameters and the constants were kept the same using in the mathematical model for batch growth. Substrate feed

[11]

concentrations varied between 2.5g L-1 to 20 g L-1 and dilution rate between 0.1 to 0.5 L h-1. The efficiency of the

system was evaluated according to the volumetric cell productivity achieved on every test (Figure 4).

The volumetric cell productivity is higher than the one occurring in batch culture of the coccus on pH 6.5 and 7.0

and substrate concentrations of 15 and 20 g L-1 (Figure 5). In addition, the productivity of the system is strongly

correlated with the substrate inflow rate which is dependent on the flow rate. For every set of pH and substrate

concentration the optimum flow rate is different.

A continuous system of intensive culture of L.lactis can offer higher yield of lactococcal biomass, as using a

continuous system for intensive culture ensures higher yields of biomass, due to simultaneous removal of end

products which are inhibitory for growth. However, its operation especially on a large scale is possibly more

complicated as there are numerous mechanical and physicochemical parameters that need to be considered.

Membrane Bioreactor Culture

Based on the unstructured kinetic mathematical algorithm developed for batch and continuous culture of L.lactis, an

unstructured kinetic mathematical model was developed for theoretical predictions over the growth of L.lactis on a

MBR, incorporating a maintenance coefficient term. The terms describing the biomass formation rate and the

biomass maintained in the system were kept unchanged. However several new terms were developed especially

those needed to demonstrate the changes in the system.

The efficiency of the system was evaluated in terms of volumetric cell productivity, using two methods for substrate

feed, continuous substrate feed and fed-batch, in order to compare the theoretical predictions done over the

volumetric cell productivity (Figure 6). The model was tested against different substrate concentrations and flow

rates varying between 2.5 to 20 g L-1 of substrate feed and 1 to 10 L h-1 flow rate in the optimum pH value for

growth selected (pH 6.5).

In order to describe the biomass maintained in the system (Xin) the following equation is formed,

dt

dxXX in += 0 (24)

The substrate inflow rate (Sin) in the system is given by the following equation:

tFSS sin **0= (25)

The substrate outflow rate (Sout) in the system is given by the following equation:

dt

dFSS inout *=

(26)

[12]

After further analysis of the above equation, it becomes

inoutd

P

Sins SSXk

PK

P

SK

S

sxY

SoS +−+

+

+−+=

)()(

)(

)*(

*1

maxµ

(27)

On further analysing the above equation, it becomes:

inoutd

S

ins SSXkPSK

PKpS

sxY

SS +−+

+

+−+=

)(*)(

)(*)*(*

1 max0

µ

(28)

When describing the flow rate of nutrient in the system and the flow of nutrient in the system the following

equations are developed,

The flow rate of nutrient during time (dt

dF) is described by the following formula:

tFdt

dF**µ= (29)

The flow of the substrate Fs, in the system is given by the following equation:

dt

dFFFs += 0 (30)

When describing the rate of product formation, product maintained in the system and product outflow rate most of

the equations are based on the previously developed models, however to describe the additional characteristics of

the system new equations were developed.

The rate on product maintained in the system (Pin) is given by the formula:

−−= outinin PSS

sp

YP )(* 0 (31)

The rate of product outflow (Pout) is given by the formula:

tFPP inout **= (32)

The volumetric cell productivity of the system was evaluated according to the following equation, both for fed-

batch and continuous substrate feed.

[13]

Volumetric Cell productivity (g L-1 h-1) = Final X in the MBR(g) / Total Fermentation Time (h) (33)

The system’s efficiency was evaluated. The volumetric cell productivity for fed-batch supply of substrate was 28 g

L-1 h-1 at a flux rate of 10 L h-1 while for continuous substrate supply being 45 g L-1 h-1 at flux rate of 10 L h-1. When

comparing the volumetric cell productivity between the continuous and the fed-batch substrate support, the in

various substrate concentrations and flow rates continuous substrate support was proven to be superior.

The volumetric cell productivity of L.lactis intensively propagated on a MBR system is much higher when

compared with the ones predicted from the models for batch and continuous mode of culturing (Table 4).

Regardless the MBR system proven to be highly productive, it construction and its handling are far more

complicated, when compared with the handling of a batch system of culturing. Many mechanical parameters such as

heat transfer, flux rates, membrane fouling, membrane surface charge and resistance, sterilisation of the mechanical

apparatus and also of the growth media, cleaning and maintenance of the system have to be considered, resulting to

its limited applicability in the industry.

CONCLUSIONS

For the intensive propagation of L.lactis three different production systems were investigated. The coccus was

primarily grown in simple batch cultures without pH control, determining its nutritional needs. The inhibition

effects of the end products were also studied.

To explore further this system in more detail, the coccus was grown in a CSTR incorporating continuous pH

control. A simple mathematical model was developed for the deeper understanding of the growth kinetics of the

coccus when cultured on a batch system. Through this model, theoretical predictions were made and validated over

the volumetric cell productivity of the system when using different substrate concentrations and inoculum size.

Growth and lactic acid productivity of the strain was considered to be highly dependent on pH, temperature and

substrate composition. As growth conditions were defined experimentally, to optimise the culturing systems and

forecasting the potential productivity of the organism if grown. Using these batch data the computational simulation

models were developed validated and provided theoretical predictions for batch cultures. These were then used to

predict the theoretical behaviour of the continuous and the MBR culture under the assumption that product yields

would not vary significantly. Past work on this organism suggests that providing the environmental conditions and

initial medium composition are specified this should be the case.

Based on the model, the volumetric cell productivity of the batch system was relatively low and a continuous

culture and MBR systems were investigated. The model was tested against several substrate concentrations and flow

rates and theoretical predictions were made over the volumetric cell productivity. In both continuous and MBR

culture systems offered higher biomass yields when compared to batch culture. Comparative studies were conducted

between these systems. The selected parameter for comparison was the volumetric cell productivity achieved from

[14]

every system on different substrate concentrations and for the case of continuous culture and a membrane bioreactor

on different dilution and flow rates. These studies show that the MBR system was proven to be highly productive

offering higher yields of biomass in a shorter period of time, especially when operated with continuous substrate

feed.

The significance of this model lies in the fact that it offers an alternative to the commonly used model of Luedeking

and Piret, for describing fermentation kinetics govern by end-product inhibition. Unlike the Luedeking and Piret

model which is only limited to the only a specialised set of experimental conditions and microorganism used and

substrate source, the proposed model can be expanded to various culturing conditions and to a variety of

homofermentating microorganisms, including the commercially important Lactoccoci and Lactobacilli.

NOMECLATURE

µ Specific growth rate 1h −

µmax Maximum specific growth rate 1h −

td Doubling time h

dt

dP

Product formation rate g L-1

dt

ds

Substrate accumulation rate g L-1

dt

dx

Biomass formation rate g L-1

dt

dF

Flow rate of nutrient during time L 1h −

S Dissolved substrate concentration g L-1

X Cell concentration g L-1

P End product concentration g L-1

KS Constant of substrate consumption dependent on pH maintenance g L-1

KP Constant of end product production g L-1

Xdk

Biomass growth maintenance coefficient g substrate g-1 cells h-1

g g-1 h-1

[15]

sxY Yield of biomass from substrate g g-1

sdpLacticaciY

Yield Productivity coefficient of lactic acid production from substrate

g g-1

spY Yield coefficient of lactic acid production from substrate g g-1

F Flux of the nutrient per hour L 1h −

Sin Substrate inflow rate per hour g L-1

1h −

S0 Initial substrate concentration within the nutrient g L-1

t Time step h

Sout Substrate outflow rate per hour g L-1

1h −

Sins Substrate existing initially in the system g L-1

X0 Biomass inoculated initially in the system g L-1

Xout, Biomass outflow g L-1

Xin Biomass maintained in the system, g L-1

D Dilution rate in the system during time 1h −

V Total Volume of Substrate in the system g L-1

FS Flow of substrate in the system g L-1

0F Initial flow in the system g L-1

[16]

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[21]

Growth Kinetics Models Developed by Comments

sK

S

s += *maxµµ

Monod Monod equation can describe adequately process of slow growth and low population density

SK

S

2*maxµ

µ =

Blackman

Substrate limited equation

)1(*maxSK

e−

−= µµ Tessier Substrate limited equation

n

S

n

s

S

SK

SSK

+=+

−− *)*1(* max1

max

µµ

Moser Substrate limited equation

SXK

S

S+

=*

*maxµµ

Contois Growth inhibition due to toxicity of the organic compounds produced

)(*)(**max

PKSK

KS

PS

P

++=

µµ

Jerusalimsky Growth inhibition due to toxicity of the organic compounds produced

[ ])*1(*)( PKSKSPS

−+=µ Hinselwood Growth inhibition due to toxicity of the organic compounds produced

−+= )

1(*)(**max

P

SK

PSKSµµ

Taylor Growth inhibition due to toxicity of the organic compounds produced

)*1(*)(*max

PKSK

S

PS−+

=µ

µ

Taylor Growth inhibition due to toxicity of the organic compounds produced

Nbdt

dNa

dt

dP** +=

Leudeking and Piret Growth inhibition due to toxicity of the organic compounds produced

Table 1: Models used to describe microbial fermentations [12,13,17,23]

[22]

Table 2: Coefficients and parameters used for mathematical modelling

Parameter Definition Dimension Values on

different pH

µmax Maximum Specific growth rate 1h− 5.5 0.22

6 0.24

6.5 0.66

7 0.72

KS Constant g L-1 5.5 0.40

6 0.435

6.5 0.471

7 0.750

KP Constant g L-1 5.5 0.643

6 0.679

6.5 0.714

7 0.750

sxY Yield coefficient (cell growth) g g-1 5.5 0.11

6 0.13

6.5 0.20

7 0.19

sdpLacticaci

Y Yield Productivity coefficient of lactic acid production from substrate

g g-1 1.6

Xk d Maintenance coefficient g g-1 h-1 0.057

[23]

Table 3: Coefficients and Parameters used for theoretical prediction of volumetric cell productivity on

different substrate concentrations of batch growth of L.lactis on a STR on different pH

Parameter Definition Dimension pH Values

µmax Maximum Specific growth rate 1h−

6.5

0.66

KS Constant g L-1 0.471

KP Constant g L-1 0.714

sxY Yield coefficient (cell growth) g g-1 0.20

sdpLacticaci

Y Yield Productivity coefficient of lactic acid production from substrate

g g-1 1.6

Xk d Maintenance coefficient g g-1 h-1 0.057

[24]

Table 4: Volumetric Cell Productivity of each system and improvement over batch growth

Culturing Systems Volumetric Cell Productivity (g L-1

h-

1)

Improvement over Batch system (times)

Batch 0.4 -

Continuous 1.3 3.25

MBR Fed-Batch feed 28 70

MBR Continuous feed 45 112.5

*Corresponding author Tel:+447413541769 Email:myrtozacharof1981@yahoo.com

Fig. 1: Schematic representation of the CSTR used for mathematical modelling studies from the selected lactobacilli

[26]

Fig. 2. A comparison of the mathematical model predictions with the experimental data where (○) lactic acid production, (□) biomass formation and

carbohydrate (△△△△) on pH 6.5 on a STR including end product inhibition (a) pH 5.5 (b) pH 6.0 (c) pH 6.5 (d) pH 7.0.

[27]

Figure 3. Theoretical predictions of volumetric cell productivity on different substrate concentrations of

batch growth of L.lactis on a STR on different pH 5.5, (◊) pH 6, (□) pH 6.5 (⌂), pH 7(☼)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 5 10 15 20 25

Substrate Concentration (g/L)

Volu

metr

ic C

ell P

roductivity (g/L

/h)

[28]

Figure 4. Theoretical predictions of volumetric cell productivity on different substrate concentrations of

continuous growth of L.lactis on different pH 5.5 (☼) pH 6 (◊), pH 6.5, (⌂) pH 7(□)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25

Substrate Concentration (g/L)

Volu

metr

ic c

ell p

roductivity (g/L

/h)

[29]

Figure 5.Comparison between the theoretical predictions of volumetric cell productivity on different

substrate concentrations of fed-batch and continuous substrate feed on growth of L.lactis on a MBR on

different flux rates. For continuous substrate feed the symbols are the following: 2.5 g/l(◊), 5 g/l(⌂)10 g/l

(□), 15 g/l(+) , 20 g/l(-), for fed-batch substrate feed the symbols are the following: : 2.5(□)g/l, 5 g/l(*)10

g/l(☼),15 g/l(--) , 20 g/l (◊)

0

5

10

15

20

25

30

35

40

45

50

0 2 4 6 8 10 12

Flux (L/h)

Vo

lum

etr

ic c

ell P

rod

uc

tiv

ity

(g

/L/h

)