A Heuristic Search for Optimal Parameter Values of Three Biokinetic Growth Models for Describing Batch Cultivations of Streptococcus Pneumoniae in Bioreactors Luciana Montera1, , Antonio C.L. Horta1 , Teresa C. Zangirolami2, Maria do Carmo Nicoletti3 , Talita S. Carmo4 , and Viviane M. Gon¸calves4 1
2
PPG-Biotechnology - UFSCar, SP - Brazil
[email protected] Dept. of Chemical Engineering - UFSCar, SP - Brazil 3 Dept. of Computer Science - UFSCar, SP - Brazil 4 Butantan Institute, SP - Brazil
Abstract. Simulated annealing (SA) is a stochastic search procedure which can lead to a reliable optimization method. This work describes a dynamic mathematical model for Streptococcus pneumoniae batch cultivations containing 8 unknown parameters, which were calibrated by a SA algorithm through the minimization of an evaluation function based on the performance of the model on real experimental data. Three kinetic expressions, the Monod, Moser and Tessier equations, commonly employed to describe microbial growth were tested in the model simulations. SA convergence was achieved after 13810 interactions (about 10 minutes of computing time) and the Tessier equation was identified as the kinetic expression which provided the best fit to the cultivation dataset used for parameter estimation. The model comprising the Tessier equation, estimated parameter values supplied by SA and mass balance equations was further validated by comparing the simulated results to 3 experimental datasets from new cultivations carried out in similar conditions. Keywords: simulated annealing, microbial growth models, biokinetic parameters estimation.
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Introduction
Estimation of model parameters is the most difficult task on the development of mathematical expressions capable of describing chemical and biochemical processes. Levenberg Marquardt is an example of a traditional deterministic global optimization method used for parameter estimation [1]. More recently, the usage of stochastic optimization methods such as genetic algorithm (GA) or simulated annealing (SA) to handle parameter estimation in large, non-linear models has increased [2]. Although some publications address the application of GA for
Corresponding author.
N.T. Nguyen et al. (Eds.): IEA/AIE 2008, LNAI 5027, pp. 359–368, 2008. c Springer-Verlag Berlin Heidelberg 2008
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coupling simulation and parameter calibration in dynamic, non-linear models, typical of biochemical processes ([1], [3]), studies about parameter estimation mediated by SA in dynamic, biological systems have not been reported yet. This paper describes the application of SA for parameter estimation of dynamic, nonstructured models of Streptococcus pneumoniae batch cultivations for conjugated vaccine production. Capsular polysaccharide (PS) is a component of the bacterial cell wall and plays a key role on the protection provided by the vaccine. Large-scale PS production is carried out in submerged, anaerobic cultivations of S. pneumoniae. Although its growth requirements and metabolism have already been well characteryzed [4], it is still a challenge to maintain optimal cultivation conditions during S. pneumoniae growth, mainly due to the huge amount of lactic acid accumulated in the medium, which, in turn, affects cell growth and PS formation. Consequently, the production of low cost, high quality vaccines relies on the modeling, monitoring and control of PS production process. Besides the investigation of SA as a more effective way of determining biokinetic parameter values associated to microbial growth models, three different kinetic models which are particularly very popular for the description of microbial growth in suspension were considered: Monod, Tessier and Moser models [5]. For each model, the search for an optimal set of its parameter values was conducted by simulated annealing, directed by an evaluation function based on the performance of the model on empirical data. The three models, customized by their own optimal set of parameter values, were then used in a comparative analysis of their modeling capability.
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Material and Methods
This section describes the procedures used in the laboratory cultivations as well as the strategy adopted to increase the amount of data, which provided the empirical data used in the experiments described in Section 4. Experimental Data from Batch Cultivations of Streptococcus pneumoniae Microorganism, Cultivation Medium Composition and Analytical Methods: The four experiments were conducted at the Biotechnology Center of Butantan Institute (S. Paulo - Brazil), using Streptococcus pneumoniae serotype 6B strain ST 433/03, which was obtained from Adolfo Lutz Institute, Bacteriology Section, SP, Brazil. A variant of the Hoeprich medium, containing glucose as the main carbon source, dialyzed yeast extract and other aminoacids as nitrogen sources and salts was employed in all experiments. Detailed information about the medium composition analytical methods is given by [4]. Experimental Procedure: The experiments were conducted in 5L BioFlo 2000 bioreactors, monitored by the LabView 7.1 program. Table 1 summarizes the cultivation conditions employed at each experiment, which were very similar. In this paper, the experiments as well as the data collected in each of them are referred to as F erm2 , F erm5 , F erm10 and F erm11 .
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Table 1. Experimental conditions adopted in batch cultivations of S. pneumoniae. Overpressure of the cultivation vessel was kept at 0.1 bar for all experiments. Temperature (o C) N2 flow rate (L/min) pH Agitation speed (rpm) Bioreactor volume (L) Initial glucose concentration (g/L)
F erm2 37 1.0 7.2-7.0 200 5.0 30
F erm5 37 1.0 7.3-7.0 200 8.8 30
F erm10 37 0.5 7.6-7.1 100 2.0 15
F erm11 36 1.5 7.4-7.2 250 8.8 30
Handling Laboratory Experimental Data The data was collected at one-hour interval yielding 6-8 data instances per batch, which were described by the values of four variables, namely: Cell concentration (CX ); Glucose concentration (CS ); PS concentration (CP S ) and Lactic acid concentration (CL ). The data in the original dataset F erm5 , which was used for parameter calibration, went through a few modifications aiming at both, to increase of the number of data instances as well as to remove noisy data by means of interpolation and smoothing, respectively. The dataset F erm5 was input to a smoother/interpolator process (implemented as the perfect smoother, described in [6]) that produced an extended smoothed version of this dataset, also referred as F erm5 . The interpolating function used by the smoother was adjusted for interpolating 29 new data instances between two experimental instances (INT = 2). The number of the original dataset instances, 8, was extended to 211. As the perfect smoother process requires a user-defined value for the parameter (the smoothing parameter), a few different values were tried and the one which suited the best (λ = 8) was chosen.
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S. pneumoniae Growth Modeling
To achieve satisfactory results during a microbial cultivation, some strategies aiming at the process optimization should be considered. Most of the available optimization strategies are based on the appropriate use of mathematical models that describe a particular microbial growth. The models are based on kinetic expressions which relate the specific growth rate of the microorganisms to the limiting substrate concentration and on a set of differential equations describing the substrate consumption, biomass and product formation. There are several simple empirical expressions that are currently used to describe the growth kinetics in bioprocesses [5]. As mentioned earlier in this paper, the search for the optimal kinetic growth parameter values was conducted in relation to three different empirical expressions, namely, Monod, Tessier and Moser models. This section describes the assumptions, the relevant parameters as well the main equations related to each model. Assuming that the cell growth is limited only by the substrate concentration (glucose, in the experiments), the cell growth rate (represented by μ) can be described by the Monod equation (1).
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μ=
μMAX · CS KS + CS
(1)
The Monod equation (1) describes substrate-limited growth only when the growth and population density are low. Two other equations assume that the cell growth is limited only by the substrate concentration, and are known as Tessier and Moser, described by eq. (2) and eq. (3), respectively. The model proposed by Moser is an extension of the Monod equation and it is generally regarded as superior in fitting due to its extra parameter, the exponent n. On the other hand, while the Monod equation was inspired in the enzymatic kinectic rate expression deduced by Michaelis-Menten, the kinetic expression formulated by Tessier is purely empirical [5]. μ = μMAX · (1 − e−KCS ) μMAX · CSn μ= KS + CSn
(2) (3)
In order to grow, microorganisms consume nutrients from the medium (substrate). Part of the substrate is used as energy and carbon source by the microorganisms for the building of new cells and the remaining is converted into other products. In a system, the concentrations of cells, substrate and products through time are described by the mass balance equations. Considering the S. pneumoniae cultivation where glucose is the limiting substrate and lactic acid and capsular polysaccharide are the main products, the equations (4) to (7) express the mass balances for the key variables as a set of ordinary differential equations (ODEs) dependent on the time t. Consider, for example, eq. (4). For specific values of μ, kd and the initial condition CX = CX (t0 ) (at a time t = t0 ), given a set of values tk , t0 ≤ tk ≤ tend , solving the equations means to integrate them at the interval of time defined by tk , t0 ≤ tk ≤ tend , in order to find the corresponding CX(t) value at each time t1 . Table 2 describes each variable and parameter involved in equations (1) to (7). dCX = (μ − kd ) · CX dt μ dCS = + m · CX dt YXS μ dCS = · CX dt YXL μ dCP S = + β · CX dt YXP 1
(4) (5) (6) (7)
In the experiments described in Section 4, to solve the differential equations the ode45 MATLAB function was used, which implements the Runge-Kutta (4,5) method.
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Table 2. Variables and parameter used in equations (1) to (7) CX (g/L) Cell conc. YXS (g/g), YXL (g/g), YXP (g/mg)
Variables CS (g/L) CL (g/L) CP S (mg/L) Glucose conc. Lactic acid conc. Polysaccharide conc. Parameters Yield coefficients for cell formation on glucose consumed (YXS ) or for polysaccharide (YXP ) and lactic acid (YXL ) production per mass of cells
M (h − 1) (g/mg)
Maintenance coefficient
Kd (h − 1)
Death constant
μM AX (h − 1)
Maximum specific growth rate
KS (g/L)
Saturation constant
β(mgP L · gX − 1 · h − 1) Non-growth-related product formation coefficient, Luedeking and Piret model [5] K
Exponential decay constant (Tessier model)
In the specific case of the S. pneumoniae cultivation, its growth kinetics (μ) was assumed to be defined by one of the three cell growth rate equations (i.e., eqs. (1), (2) or (3)). For each growth rate expression tested together with the mass balances to fit experimental data, the main challenge was to establish an appropriate set of parameter values for the chosen model. In most cases the parameter values are not known; they must be determined by computational routines that, based on the comparison between experimental and estimated values, try to establish an optimum set of parameter values. In this context, ”optimum set of parameter values” should be understood as the set of values that enables the best fit of the model to the experimental data.
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The Use of Simulated Annealing for Determining Optimal Parameter Values of Three Microbial Growth Models
This section describes the procedures for parameter estimation and model validation. Initially the three selected growth models had their parameter values adjusted by a heuristic algorithm, using the experimental data described in F erm5 . The best model, i.e., the one among the three that better represented the data, was then validated trying to identify its adequacy for modeling similar (but new) S. pneumoniae cultures represented by datasets F erm2 , F erm10 and F erm11 . Simulated annealing (SA) is a stochastic search procedure suitable for finding global minimum (or maximum) of functions. It is mainly used for finding the best solution to a function in a large search space when deterministic methods are not suitable. The use of SA in the experiments described in this section aims at finding an optimal (or quasi-optimal) set of values for the main parameters used by
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the three non-structured growth models associated to the ODEs, so to adjust each model to the experimental data F erm5 . For each time stamp, the objective function to be optimized by the SA represents the difference between the estimated and the experimental values for variables CX , CS , CL and CP S . As the aim is to adjust the model to best ’fit’ the experimental data, the objective function chosen is the quadratic error sum of the variables, as described by following equation, where Cmodel and Cexp represent, respectively, the 4-uple of variable values predicted by the model and the 4-uple of variable values experimentally measured. f inal exp 2 exp 2 model model ((CX(t) − CX(t) ) + (CS(t) − CS(t) ) + F (Cmodel , Cexp ) = t=initial exp 2 exp model 2 (CL(t) − CL(t) ) + (CPmodel S(t) − CP S(t) ) ) model model model Where CX(t) , CS(t) , CL(t) , CPmodel S(t) : model estimated values for biomass, glucose, lactic acid and capsular polysaccharide concentrations at a time t, reexp exp exp , CS(t) , CL(t) , CPexp spectively and CX(t) S(t) : experimental values for biomass, glucose, lactic acid and capsular polysaccharide concentrations at the same time t, respectively. To assess the adequacy of a model for describing the behavior of a system, it is necessary first to estimate a reasonably ’good’ value for its parameters. The determination of optimized parameter values can be done by analytical methods such as the Simplex [7] or the Newton method. Although these methods can quickly find a minimum (or maximum) of a function, the value can be a local minimum (or maximum). Also, analytical methods can be quite complicated to be implemented and, in general, they are limited when dealing with variable restrictions. As an alternative to analytical methods there are the heuristic methods, which are based on search procedures throughout the space of all possible solutions, aiming at finding an optimal (or near optimal) solution. The SA is one of such heuristic methods and its canonical pseudocode is given in Figure 1. The algorithm resembles the hill climbing search; in its inner loop, however, instead of choosing the best move, it picks up a random move2 . The algorithm starts by choosing a random candidate solution, i.e., a set of values for the model parameters (p1 , p2 , ..., pn as shown Table 3). This solution is referred to as cur param. Any random value is acceptable, provided it is within the restriction interval of the corresponding parameter (see Lower and Upper bounds in Table 3). Next, the cost associated with the cur param solution is calculated. At each step of the algorithm, a new candidate solution, identified as new param is randomly chosen in the neighborhood of the current solution, such that the restriction intervals are respected and, in addition, the following condition on the parameter values is satisfied, for all parameters: cur param[pi ]*0 ≤ new param[pi ] ≤ cur param[pi ]*2. The new solution cost is then calculated and both solutions have their cost compared; the one with the smaller cost becomes the current solution. However, there is a chance of the new param, even with a higher cost, to become the current solution which is dependent on both, 2
The function random() in the SA pseudocode, return a random integer value.
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Fig. 1. The SA pseudocode
the temperature T and the difference in cost between the new and the current solution, represented by E. The decision of accepting a solution with higher cost is an attempt to prevent the SA of finding a good solution that, in fact, is just local (a local maximum or a local minimum) by enlarging the search space covered during the algorithm search. As can be observed in the algorithm pseudocode, as the temperature T decreases, the probability P of accepting ’higher-cost’ solutions also decreases, fact that guarantees the algorithm convergence. The initial values for variables T and decreasing factor must be empirically determined since they control the number of interactions the algorithm performs and, consequently, play an important role in its convergence. The initial values of 10,000 and 0.999 for T and decreasing factor, respectively, used in the implemented SA were established after a reasonable number of executions. Table 3 presents the estimated values found for the parameters belonging to Monod, Tessier and Moser expressions as well as for the ones included in the mass balances. The table shows the lower and the upper bounds of each parameter (i.e., the parameter restrictions) as well. Table 3. Parameter restrictions and parameter values of Monod, Tessier and Moser expressions and mass balance equations found by SA
Lower Bound Upper Bound Monod Tessier Moser
Parameters p1 p2 p3 p4 Ks m Kd YXS 0.01 0.0001 0.0001 0.1 6 0.5 0.5 0.8 0.38 0.0004 0.0005 0.13 0.14 0.001 0.01 0.14 0.01 0.0002 0.002 0.13
p5 p6 YXP μM AX 0.1 0.45 0.5 0.85 0.15 0.55 0.13 0.6 0.17 0.47
p7 p8 p9 YXL β N 0.0001 0.1 0.0001 0.4 50 10 0.19 11 0.18 8.6 0.16 13 0.0014
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Monod 73.2
Tessier 38.2
Moser 71.4
Fig. 2. Simulated (represented by lines) and experimental (represented by symbols) values of the S. pneumoniae cultivation main variables using the Tessier growth rate expression. Data from the cultivation F erm5 was used for parameter calibration.
Based on the value of the objective function (F (Cmodel , Cexp )) associated to each model (Table 4), the Tessier model was identified as the one that best described biomass formation, glucose consumption, PS and lactic acid production of the batch S. pneumoniae cultivation represented by F erm5 . Figure 2 shows the predicted values, using the Tessier expression, for variables cell concentration (CX ), glucose concentration (CS ), lactic acid concentration (CL ) and capsular polysaccharide concentration (CP S ) together with the experimental data. The implemented SA algorithm was very efficient, running 13810 iterations in approximately 10 minutes (in a machine with a Pentium 4 processor with 1GB of memory). The number of 13810 iterations was enough to guarantee the SA convergence, which is not dependent on the initial parameter values, as some estimation algorithms are [8]. Next, Tessier model was validated, by examining its adequacy in describing three other similar experiments (datasets F erm2 , F erm10 and F erm11 ). As the experimental CL values for cultivation F erm2 were not measured due to technical problems, this attribute was not included in the objective function when the adequacy of the Tessier model for describing the experiment F erm2 was examined. Figure 3 (a), (b) and (c) compares experimental data and predicted values for the cultivations F erm11 , F erm10 and F erm2 , respectively. The model predicted values were estimated by solving the set of ODEs for the Tessier model as the kinetic expression, with the parameter values tuned by the SA using the data from F erm5 . Only the values of initial conditions were changed to fit the model to the new data. In Table 5, the accuracy of the model on describing different datasets
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Fig. 3. Validation of the Tessier model using the parameter values estimated by SA for three cultivations of S. pneumoniae performed in similar conditions: (a) F erm11 (b) F erm10 (c) F erm2 . Simulated values represented by lines; experimental values represented by symbols.
generated in similar cultivation conditions can be evaluated by analyzing the error between experimental and simulated data for the different validation tests performed. In Figure 3, it can be observed that the model, together with the estimated parameter values supplied by SA, provides a fairly good description of the time-profiles for all variables. It is also clear that the lack of accuracy of the experimental data, mainly at the beginning of the experiments, can affect the evaluation of the model performance. This difficulty in obtaining more accurate measurements of experimental data is generally attributed to the lack of sensitivity of some analytical procedures, mainly of those employed for determining cell and metabolite at low concentrations.
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Cultivation dataset Objective Function value (excluding lactic acid)
F erm2 1.25
F erm5 1.03
F erm10 8.47
F erm11 1.66
Comparing the objective function values obtained for F erm2 and F erm11 with the one calculated for F erm5 , which was used for parameter calibration (Table 5), it is possible to state that the model validation was successful for datasets F erm2 and F erm11 . However, the objective function value for F erm10 was significantly higher. In fact, Table 1 shows that F erm10 was carried out under fairly different conditions of bioreactor volume, agitation speed and initial glucose concentration, what could explain the poor description provided by the model.
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Conclusion
Simulated annealing is a good choice for a robust optimization procedure. The proposed model (and its set of parameters) was successfully validated and its mass balance equations could be easily modified to simulated more promising bioreactor operation forms, such as fed-batch operation. The simulations could be extremely useful for identifying optima cultivation conditions for achieving higher PS concentration, lower acid lactic formation and, consequently, better process yield and productivity.
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