Abstract
In this work, the growth of Bacillus thuringiensis var. israelensis, a bioinsecticide producer, is investigated. Experiments were carried out in batch mode in order to obtain kinetic model parameters that were further applied to simulate fedbatch processes. The fedbatch mode allows more flexibility in the control of the substrate concentration in the culture medium. Different techniques, such as constant feeding, "bangbang" control and model based control (exponential feeding and singular control), were compared. For the techniques based on a model, combinations of models with and without a substrate inhibition parameter were used to represent the simulated process and the internal model of the feeding controller. Singular control based on the model with an inhibition parameter proved to be the most robust controller.
Bioinsecticide; Bacillus thuringiensis; modelling; fedbatch; simulation
SIMULATION INVESTIGATIONS TOWARDS THE DEVELOPMENT OF A BACTERIAL BIOPESTICIDE FEDBATCH REACTOR
^{1}^{2}^{*}* To whom correspondence should be addressed. To whom correspondence should be addressed.^{1}Departamento de Engenharia Química  Universidade Federal de Pernambuco  UFPE
Av. Prof. Artur de Sá, S/N  Cidade Universitária  CEP. 50740521  Recife  PE, Brazil
Phone/FAX: (081) 2713992/(081) 2710095  Email: mbsj@npd.ufpe.br
^{2}Departamento de Antibióticos  Universidade Federal de Pernambuco  UFPE
Av. Prof. Artur de Sá, S/N  Cidade Universitária  CEP. 50740521  Recife  PE, Brazil
(Received: May 12, 1997; Accepted: November 4, 1997)
Abstract  In this work, the growth of Bacillus thuringiensis var. israelensis, a bioinsecticide producer, is investigated. Experiments were carried out in batch mode in order to obtain kinetic model parameters that were further applied to simulate fedbatch processes. The fedbatch mode allows more flexibility in the control of the substrate concentration in the culture medium. Different techniques, such as constant feeding, "bangbang" control and model based control (exponential feeding and singular control), were compared. For the techniques based on a model, combinations of models with and without a substrate inhibition parameter were used to represent the simulated process and the internal model of the feeding controller. Singular control based on the model with an inhibition parameter proved to be the most robust controller.
Keywords: Bioinsecticide, Bacillus thuringiensis, modelling, fedbatch, simulation.
INTRODUCTION
Control of insect pests is one of mankinds biggest problems. Chemical pesticides have been used for most agricultural production and public health projects. However, they affect the environment, killing plants and animals, and are harmful to humans. In tropical countries, such as Brazil, the problem is intensified by climatic conditions.
The search for more specific and effective insecticides has increased the interest in biological agents. Bacterial entomopathogens have been extensively studied (Stockdale, 1985). Among them, Bacillus thuringiensis is one of the microorganisms most frequently studied as a toxin producer. It produces a parasporal crystal called delta endotoxin. It is usually grown by submerged fermentation and can be cultivated in complex raw materials, such as sugarcane molasses and cornsteep liquor (Moraes et al., 1996). Studies in batch, continuous and fedbatch reactors have been reported in the literature. However, the modelling of these processes has not received any particular attention and literature concerning models and kinetic parameters is scarce.
Delta endotoxin production is related to spore formation. It is assumed here that a large number of cells produces a correspondingly large number of spores, and consequently a large amount of
toxin. The present work aims at the maximization of biomass concentration. Two kinetic growth models (with and without substrate inhibition of cell growth) were studied. Considering that substrate inhibition was possible, the fedbatch mode was studied by simulation, since the substrate concentration in the culture medium can be controlled by this kind of operation. The following different feed techniques without feedback were compared: constant, exponential and optimized feedings.
MATERIALS AND METHODS
Experimental Procedures
The experiments were carried out in a batch fermenter in order to obtain the parameters for the kinetic models of the process studied.
The microorganism used was a strain of Bacillus thuringiensis var. israelensis, which was obtained from the Departamento de Antibióticos  UFPE (Recife, Brazil). The culture medium used was the same as that used by Kang et al. (1992) and consisted of 5.0 kg/m^{3} glucose, 5.0 kg/m^{3} yeast extract, 4.0 kg/m^{3} (NH_{4})_{2}SO_{4}, 1.0 kg/m^{3} K_{2}HPO_{4}, 0.82 kg/m^{3} MgSO_{4}.7H_{2}O, 0.08 kg/m^{3} CaCl_{2}.2H_{2}O and 0.07 kg/m^{3} MnSO_{4}.H_{2}O. The microorganism was grown in a 1.4´ 10^{2} m^{3}New Brunswick Scientific Company bioreactor containing 1.0´ 10^{2} m^{3} of the medium. Temperature was controlled at 30 ^{o}C and pH at 7.0 with 2.0 N KOH; aeration and agitation were kept at 1.0 v.v.m. and 500 rpm, respectively. The inoculum volume used was 10% of the volume of the fermentation medium.
Growth of the microorganism was determined by measuring the dry weight of cells, which were separated by filtration in a Millipore membrane (0.22 mm), washed with distilled water and then dried at 80 ^{o}C for 24 h. Glucose concentration was obtained by the dinitrosalicylic acid (DNSA) method (Bernfeld, 1955). Viable spores assay was carried out using colonies forming units (CFU/mL).
Modelling of the Batch Fermenter
A nonsegregated and nonstructured perspective was adopted. The mass balances for a batch process are shown here.
Cells:
(1)
Substrate:
(2)
where x is the cell concentration, m is the specific growth rate, s is the limiting substrate (glucose) concentration, Y_{x/s} is the yield coefficient and t is the time. It was assumed that the substrate was converted only into biomass and the maintenance rate was neglected.
The specific growth rate, m , can be described by different equations, according to the process features. In this work, two kinetic models were chosen.
1. Monod model  This is one of the most frequently studied models, because it is very simple, with only two parameters, the maximum specific growth rate, m _{max}, and the saturation constant, k_{s}. This model, developed by Monod (1949), does not consider the substrate inhibition of cell growth.
(3)
2. Hump model  this model considers substrate inhibition, like the Andrews model (Andrews, 1968), with the advantage of having only two parameters k_{1} and k_{2}. It was previously used by Agrawal et al. (1982).
(4)
Modelling of the Simulated FedBatch Fermenter
An overall balance of a fedbatch culture is (Dunn and Mor, 1975):
(5)
where V is the culture volume and f is the feed rate.
The cell and substrate mass balance are described by equations (6) and (7):
(6)
(7)
where s_{F} is the limiting substrate concentration in the feeding.
Both models (Monod and Hump) were applied in fedbatch simulations, using constant, exponential and optimized feedings, without feedback.
 Constant feeding: The feed rate is kept constant.
cte (8)
where f_{0} is the initial feed rate.
 Exponential feeding: The feed rate is controlled by a programme, which is described by the equation (Lim et al., 1977):
(9)
where the initial feed rate, f_{0}, is:
(10)
and x_{0}, s_{0} and V_{0} are the initial values for cell concentration, substrate concentration and volume, respectively.
 Optimized feeding: The optimal control theory is applied here to systems which present only one control variable, u, that appears linearly in the dynamic equations, as in the process described below:
(11)
where the process state variable vector, x, is given by Costa (1996):
(12)
The optimization problem is to find the control variable trajectory through time, in order to maximize or minimize a performance criterion or objective function in the final time (Koppel, 1968):
(13)
The Pontryagin maximum principle (Koppel, 1968) can be used to solve this problem. This principle establishes that an adjunct variable vector, l (t), is necessary to maximize the objective function. This vector must satisfy the equations below:
(14)
(15)
where H=H(x,l ,u) is the Hamiltonian function which must be maximized by the control variable, u(t). The Hamiltonian function is defined as:
(16)
and it must satisfy one of the equations below to reach its maximum value:
, u_{min} < u < u_{max} (17)
H = H_{max}, u = u_{min} or u = u_{max} (18)
However, in some situations in a semicontinuous process the Hamiltonian function does not depend on the control variable during a time period called singular arc. In this case, this principle does not work and the application of another theory, the singular control theory, is necessary.
The control equation can be obtained by writing the Hamiltonian function as (Costa, 1996):
(19)
where:
(20)
and
(21)
On the singular arc, the term f (t) = 0 and its derivatives must be equal to zero as well. The control variable equation can be obtained from successive differentiations until an explicit function of u(t) arises.
The dilution rate, D, was used as the control variable. It is defined as the rate between the volumetric feed rate, f, and the culture volume, V, according to the equation below:
(22)
The control variable usually used is the feed rate. But the process analysis can be simplified by the use of the dilution rate, D, as the control variable (Alves, 1993). Equations (5), (6) and (7) become:
Overall balance:
(23)
Cells:
(24)
Substrate:
(25)
In this case, the overall balance equation is not necessary for the solution of the problem since the mass balances of substrate and cells do not depend explicitly on the volume.
In this work, the performance criterion to be maximized is the cell concentration at the final time, according to the objective function:
(26)
The singular arc is described by the equation (Costa, 1996):
(27)
or
(28)
From equation (27) it can be demonstrated that singular control maximizes the specific growth rate and maintains the substrate concentration at a constant value (s_{sing}).
 Monod Model: This equation does not present a defined maximum, where . In this case, it was decided to bring the value of the derivative, y , close to zero:
(29)
Under those conditions, there are only suboptimal values. From equation (29), the limiting substrate concentration in the singular arc, s_{sing}, is:
(30)
And the control equation is:
(31)
where D_{sing} is the dilution rate for the singular arc.
 Hump Model: In this case, the limiting substrate concentration is obtained from equation (27):
(32)
And the singular dilution rate equation is easily determined:
(33)
RESULTS AND DISCUSSION
Experimental Results
The experiments were carried out in batch mode during 32 hours of fermentation. Figure 1 shows the results obtained for cell and glucose concentrations during growth phase which lasted about 5 hours. At the end of this period, glucose concentration was reduced to near zero. The cell concentration reached the maximum value and then decreased because of sporulation. Spores were detected at the beginning of the stationary phase, when the cells stopped growing. After 32 hours of fermentation, the number of spores was about 3  4 ´ 10^{8} CFU/mL.
Only the growth phase was chosen for the development of the kinetic models, as the maximization of the cell production was intended. The kinetic parameters m _{max}, k_{s}, k_{1} and k_{2} were obtained by the application of an optimization technique. A modification of the SIMPLEX method was used by the algorithm NELDER (Kuester and Mize, 1973). Here, the algorithm determines the minimum of a nonlinear function, which relates the experimental (x_{exp}, s_{exp}) and simulated (x_{sim}, s_{sim}) data for growth of cells and substrate consumption, according to the equations below:
Cells:
(34)
Substrate:
(35)
The general objective function, F, is the sum of the two functions above divided by the number of experimental data, N:
(36)
Comparison between the experimental data and batch simulations for cell concentration, using the kinetic parameters obtained from fermentation.
The kinetic parameters obtained from the method were m _{max} = 0.672 h^{1} and k_{s} = 0.331 kg/m^{3} for the Monod model, and k_{1} = 0.398 m^{3}/kg.h and k_{2} = 4.535 kg/m^{3} for the Hump model. Numerical simulations using these parameters were compared with the experimental data. Figure 2 shows that both models can be adjusted to the experimental data.
Simulation Results
The fedbatch mass balances were solved by a software called" Fedbatch", which had been previously developed in order to carry out the simulations (Cunha, 1997). It can be used to simulate the growth of the microorganism in batch and fedbatch modes, using constant, exponential and optimized feed.
The techniques studied do not use feedback, so they should be based on a model that can be applied in as many situations as possible.
The "Fedbatch" programme uses combinations of the Monod and Hump models, where one describes the fermentation and the other is the base for the feeding control. These combinations are shown in Table 1.
The kinetic parameters obtained from the batch experiments were used in all the simulations which used the following conditions: x_{0} = 0.31 kg/m^{3}; s_{0} = 6.05 kg/m^{3}; Y_{x/s} = 0.643 kg(cells)/kg(substrate); V_{0} = 5.0´ 10^{3} m^{3}; V_{max} = 1.0´ 10^{2} m^{3}; f_{max} = 2.5´ 10^{3} m^{3}/h and s_{F} = 20.0 kg/m^{3}.
For all simulations, the limiting substrate feeding was stopped when the fermenter reached the maximum volume of 1.0´ 10^{2} m^{3}. The total fermentation time was the sum of the fedbatch time and a following batch operation, and was the time when the specific growth rate  based on the controller internal model  was reduced to zero.
The constant substrate feeding mode is not controlled by a model and was simulated by the application of the maximum feed rate the feeding pump would provide. The exponential feeding is controlled by an equation that depends exponentially on the specific growth rate, and the kinetic model will influence the process performance. The singular control is also a function of a model. For the Monod model, the differentiation of the specific growth rate must be near zero (equation 32). The value of y = 0.01 m^{3}/kg.h was chosen by simulation.
The beginning of the singular control feeding can be defined by a comparison between the limiting substrate initial concentration and the substrate concentration calculated for the singular arc (Costa, 1996), and can be any one of the three cases below:

s
_{0} > s
_{sing}:

s
_{0} = s
_{sing}:

s
_{0} < s
_{sing}:
For all these cases, the physical limitations of the feeding pump should be considered. If the feed rate reaches the maximum capacity of the pump, it must be kept at this value until the fermenter filling.
"Bangbang" control can be started in the same way explained for singular control, but the feeding is carried out at the maximum feed rate. This technique does not depend on the model, since it does not have to keep the feed rate at the singular arc.
The cases shown in Table 1 were simulated.
Case I: Monod Monod
Both fermentation and feeding are considered to be based on the Monod model. In this case, the techniques were compared, and both achieved the final cell concentration at the same time (Figure 3). Consequently, constant feeding is recommended because of its easier implementation.
Case II: Hump  Hump
The process and the control are based on the Hump model. Since these models are the same, problems will not occur during fermentation.
From Figure 4 we can observe that, when using constant feeding, it takes longer to reach the final cell concentration because of substrate inhibition of cell growth. On the other hand, singular control and exponential feeding were more efficient.
Case III: Monod  Hump
In this situation, the process follows the Monod model and the control is based on the Hump model. The comparisons in this case use the modelbased techniques (singular and exponential) only, because the other techniques are not governed by the kinetic model.
Figure 5 shows comparisons between the two feeding techniques, using three different initial concentrations for glucose and is used to determine whether a controller based on a substrate inhibition model works in different situations. It was observed that both techniques showed a good performance, but singular control was more efficient since it estimated the end of the reaction time as being closer to the end of the growth.
Case IV: Hump  Monod
In this case, the fermentation is described by the Hump model and feeding, by the Monod model. The curves presented in Figure 6 aim to show which controller model presents the best performance.
The Monod model estimates a total fermentation time shorter than the necessary time, and the fermentation is incomplete. From Figure 6 it can be observed that the controller still has a satisfactory performance for smaller initial substrate concentrations. However, a larger concentration of the glucose causes a smaller specific growth rate, and the changes in cell concentration are not pronounced.
It can be concluded that when inappropriately applied, this kind of control harms the process. So, it is recommended that a controller based on the more robust model, which is able to include a greater number of possibilities, be used. The Hump model was more adequate for the situations here.
Comparison among the feeding operations  constant, exponential, singular control and "bangbang" control  for the Monod model for cell concentration.
Comparison of feeding operations  constant, exponential, singular control and "bangbang" control  for the Hump model of cell concentration.
b) s_{0} = 8.0 kg/m^{3}
c) s_{0} = 14.0 kg/m^{3}
Figure 5: Case III simulation of cell concentration and feed rate, using exponential feeding and singular control.
a) s_{0} = 4.0 kg/m^{3}b) s_{0} = 8.0 kg/m^{3}
c) s_{0} = 14.0 kg/m^{3}
Figure 6: Case IV simulation cell concentration and feed rate, using exponential feeding and singular control.
CONCLUSIONS
In this work, the relatively unexplored area of modelling and simulation of Bacillus thuringiensis growth was investigated for the first time. Experiments were carried out in batch mode in order to obtain the modelling kinetic parameters.
Considering the fedbatch simulations in this study, some conclusions related to the feeding techniques and the models studied are possible. When the process is based on the Monod model, all the feeding techniques reach the final cell concentration at the same time. In this case, a sophisticated control is not necessary. The fermenter should be fed at the maximum feed rate, which is a simpler and easier method. If the process is based on the Hump model, both singular control and exponential feeding techniques lead more rapidly to the final cell concentration, where the former showed the best performance in all cases. Therefore, for a substrate inhibition process, a controller based on this model is recommended.
NOMENCLATURE
D Dilution rate, h^{1}
f Feed rate, m^{3}/h
f, g Vectors in equation (11)
F Objective function for the optimization algorithm
H Hamiltonian function
H_{0} Part of the Hamiltonian function that is not multiplied by the control variable
J Performance criterion
k_{1} Hump model parameter, m^{3}/kg.h
k_{2} Hump model parameter, kg/m^{3}
k_{s} Saturation constant, kg/m^{3}
s Substrate concentration, kg/m^{3}
t Time, h
u Control variable
V Volume of culture broth, m^{3}
x Cell concentration, kg/m^{3}
x State variable vector
Y_{x/s} Yield coefficient, kg (cell) / kg (substrate)
Greek letters
f Part of the Hamiltonian function that multiplies the control variable
l Adjunct variable vector
m Specific growth rate, h^{1}
y Monod model approximate value for , m^{3}/kg.h
Subscripts
0 Initial
f Final
F In the feed
max Maximum
min Minimum
sing Singular
Superscripts
T Transposed
ACKNOWLEDGEMENT
The first author received financial support from the Brazilian Research Council  Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). Financial support from FACEPE is also gratefully acknowledged.
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Publication Dates

Publication in this collection
09 Oct 1998 
Date of issue
Mar 1998
History

Accepted
04 Nov 1997 
Received
12 May 1997