General characteristics of optimal feed rate profiles for various fed-batch fermentation processes

General Characteristics of the optimal feed rate profiles have been deduced for various fed-batch fermentation processes by analyzing singular controls and singular arcs. The optimal control sequences depend on the shapes of the specific growth and product formation rates, mu andpi, and the initial conditions. For fed-batch processes described by four mass balance equations, the most general optimal control sequence consists of a period of maximum feed rate, a period of minimum feed rate (a batch period), a period of singular feed rate (variable and intermediate), and a batch period. Degenerate sequences in which one or more periods are missing can result with a particular set of initial conditions. If the fermentation time is not critical, the singular control maximizes the net yield of product and only when the time is also important, it balances a trade off between the yield of product and the specific growth rate which dictates the fermentation time. With the sequence of optimal control known, the optimal feed rate profile determination is reduced to a problem of determining switching times.     

On the optimal control of fed-batch reactors with substrate-inhibited kinetics

The optimal feed rate profiles, for fed-batch fermentation that maximizes the biomass production and accounts for time, are analyzed. The solution can be found only if the final arc of the optimal control is a batch arc, since in this case the final concentrations of substrate and biomass can be determined by ulterior conditions on the mass balance and on the final growth rate of biomass and thus it is possible to solve the resulting time optimal problem by using Green's theorem. This evidences the "turnpike property" of the solution, which tries to spend the maximum time on or at least near the singular arc along which the substrate concentration is maintained constant. The optimality of the final batch arc is related to the time operational cost in the performance index. The sequence of the control depends on the initial conditions for which six different regions, with the respective patterns, have been identified, in case the performance index allows the control sequence to have a final batch.     

Optimization of biphasic growth of Saccharomyces carlsbergensis in fed-batch culture

The optimal glucose feeding policy for the fed-batch culture of Saccharomyces carlsbergensis is presented. The biphasic nature of growth results in a singular feed rate policy that is unique to this organism. When the operating cost is high, the reduction in operating time forces the cells to utilize both glucose and ethanol toward the end of fermentation time and results in a decreasing rate of glucose addition, unlike the normally observed in creasing feed rate. The optimal feeding policy depends heavily on the initial conditions and is highly sensitive to changes in kinetic parameters. A semiempirical scheme for feedback optimization is suggested for the fed-batch yeast culture.     

Cell-mass maximization in fed-batch cultures

Complete solutions are provided for cell-mass maximization for free and fixed final times and constant and variable yields. The optimal feed rate profile is a concatenation of maximum, minimum and singular feed rates. The exact sequence and duration of each feed rate depends primarily on the initial substrate concentration, and degenerate cases arise due to the magnitude constraint on the feed rate and the length of final time t _f. When the final time is free and not in the performance index, it is infinite for constant yield so that any form of feed rate leads to the same amount of cells, while for variable yield the singular feed rate is exponential and maximizes the yield. For fixed final time the singular feed rate for constant yield is exponential and maximizes the specific growth rate by maintaining the substrate concentration constant, while for variable yield, it is semi-exponential and the substrate concentration starts near the maximum specific growth rate and moves toward the maximum yield. A simple sufficient condition for existence of singular feed rate requires an existence of a region bounded by the maxima of specific growth and cellular yield. Otherwise, the optimal feed rate profile is a bang-bang type and the bioreactor operates in batch mode.     

Optimization of feeding profile for a fed-batch bioreactor by an evolutionary algorithm

The optimal feeding profile of a fed batch process was designed by means of an evolutionary algorithm. The algorithm chromosomes include the real-valued parameters of a profile function, defined by previous knowledge. Each chromosome is composed of the parameters that define the feeding profile: the feed rates, the singular arc parameters and the switching times between the profile states. The feed profile design was tested on a fed-batch process simulation. The accepted profiles were smooth and similar to those derived analytically in other studies. Two selection functions, roulette wheel and geometric ranking, were compared. In order to overcome the problem of model mismatches, a novel optimization scheme was carried out. During its operation the process was sampled, the model was updated and the optimization procedure was applied. The on-line optimization showed improvement in the objective function for relatively low sample times. Choosing the sampling frequencies depends on the process dynamics and the time required for the measurements and optimization. Further study on experiments of fed-batch process demonstrated the use of complex, non-differentiable model and produced improved process performances using the optimal feeding profile.     

Run-to-run fed-batch optimization for protein production using recombinant Escherichia coli

A two-phase design approach is introduced to determine the optimal feed rate, fed glucose concentration and fermentation time to maximize protein productivity using recombinant Escherichia coli BL21 (pBAW2) strain. The first phase is applied to determine a primary S-system kinetic model using batch time-series data. Two runs were carried out in the second phase to achieve the maximum protein productivity for the fed-batch fermentation process. The computational results using the S-system kinetic model obtained from the second run are in better agreement with the experiments than those using the kinetic model obtained from batch time-series data. For cross-validation, two extra fed-batch experiments with different feed strategies were carried out for comparison with the optimal fed-batch result. From the experimental results, this approach could improve productivity by at least 3%.     

Simple nonsingular control approach to fed-batch fermentation optimization

Determination of the optimal feed rate for fed-batch fermentation is normally a problem in singular control with a state inequality constraint and as such is, in general, difficult to solve, especially for those described by a large number of dynamic mass balance equations. In this article we use a new set of state variables and the culture volume as the control variable. In this way the problem is converted to one of nonsingular control with the magnitude and rate constraints on the manipulated variable and can be numerically solved by a gradient-based technique, thus avoiding the difficulty associated with singular control problems. Examples are given to illustrate the method.     

Optimization of fed-batch culture of hybridoma cells using dynamic programming: single and multi feed cases

The optimization of fed-batch culture of hybridoma cells is accomplished on a mathematical model using dynamic programming. Optimal feed trajectories are found using a seventh order model for a single feed stream containing both glucose and glutamine and for two separate feed streams of glucose and glutamine. Compared to a constant feed rate, optimal trajectories can improve the final MAb concentration by 11 % for the single feed case and by 20% for the multifeed case. Higher MAb concentrations can be expected for fed-batch optimization with feed enriched in nutrients.     

Enhanced production of alpha-amylase in fed-batch cultures of Bacillus subtilis TN106[pAT5

Growth of Bacillus subtilis TN106[pAT5] and synthesis of plasmid-encoded protein (alpha-amylase) are investigated in batch, continuous, and fed-batch cultures using a defined medium containing glucose and/or starch as the carbohydrate source. The batch culture studies reveal that reduced availability of arginine hampers growth of recombinant cells (which lack an arginine synthesis gene) but promotes production of alpha-amylase and substitution of glucose by starch as the carbohydrate source leads to slower growth of recombinant cells and increased production of alpha-amylase per unit cell mass. Retention of recombinant cells over prolonged periods in continuous cultures is not possible without continuous application of antibiotic selection pressure owing to segregational plasmid instability. Fed-batch experiments with constant volumetric feed rate demonstrate that alpha-amylase production is enhanced at lower feed concentration of starch (sole carbohydrate source) and lower volumetric feed rate. Such slow addition of starch is however not conducive for growth of recombinant cells. The expression of the thermostable alpha-amylase gene carried on the recombinant plasmid pAT5 (derived from a plasmid isolated from a thermophilic bacterium) is promoted at higher temperatures, while growth of recombinant cells is depressed. In all batch and fed-batch experiments, production of alpha-amylase is observed to be inversely related to growth of recombinant cells. The efficacy of two-stage bioreactor operations, with growth of recombinant cells being promoted in the first stage and alpha-amylase production in the second stage, in attaining increased bulk alpha-amylase activity is demonstrated. (c) 1993 John Wiley & Sons, Inc.     

Optimal substrate feeding policy for a fed batch fermentation with substrate and product inhibition kinetics

The optimal substrate feeding policy for the fed batch fermentation which is governed by product and substrate inhibited kinetics is presented. The conjunction point between nonsingular and singular arcs and the feeding policy along the singular arc are derived analytically in terms of the concentrations of substrate and product and the liquid volume. Thus, it is possible to determine the feeding rate by monitoring the state variables (i.e., closed loop control). As a specific example, an optimization study of the fed batch fermentation for ethanol production by Saccharomyces cerevisiae is presented. It is shown that the optimal feeding patterns are heavily dependent upon the initial conditions. The point selectivity provides the guideline for predicting the optimal feeding patterns and explaining the results of rigorous mathematical analysis.     

On-line optimization of recombinant product in a fed-batch bioreactor

In this paper, an efficient scheme for on-line optimization of a recombinant product in a fed-batch bioreactor is presented. This scheme is based on the parametrization of the system states and the elimination of a subset of the dynamic equations in the mathematical model of the fed-batch bioreactor. The fed-batch bioreactor considered here involves the production of chloramphenicol acetyltransferase (CAT) in a genetically modified E. coli. The optimal inducer and the glucose feed rates are obtained using the proposed optimization approach. This approach is compared with the traditional optimization approach, where all the states and the manipulated variables are parametrized. The approach presented in this paper results in a 5-fold improvement in the computational time for the recombinant product optimization. The optimization technique is employed in an on-line optimization scheme, when parametric drift and a disturbance in the manipulated variable is present. Feedback from the process is introduced through resetting the initial conditions of the model and through an observer for estimating the time varying parameter. The simulation results indicated improvement in the amount of product formed, when the optimal profile is regenerated during the course of the batch.     

Minimal time control of fed-batch bioreactor with product inhibition

This paper is devoted to the minimal time control problem for fed-batch bioreactors, in presence of an inhibitory product, which is released by the biomass proportionally to its growth. We first consider a growth rate with substrate saturation and product inhibition, and we prove that the optimal strategy is fill and wait (bang-bang). We then investigate the case of the Jin growth rate which takes into account substrate and product inhibition. For this type of growth function, we can prove the existence of singular arc paths defining singular strategies. Several configurations are addressed depending on the parameter set. For each case, we provide an optimal feedback control of the problem (of type bang-bang or bang-singular-bang). These results are obtained gathering the initial system into a planar one by using conservation laws. Thanks to Pontryagin maximum principle, Green’s theorem, and properties of the switching function, we obtain the optimal synthesis. A methodology is also proposed in order to implement the optimal feeding strategies.     

Model-based optimization of viral capsid protein production in fed-batch culture of recombinant <Emphasis Type="Italic">Escherichia coli</Emphasis>

An optimized fed-batch cultivation process for the production of the polyoma virus capsid protein VP1 in recombinant Escherichia coli BL21 bacteria is presented. The optimization procedure maximizing the amount of desired protein is based on a mathematical model. The model distinguishes an initial cell growth phase from a protein production phase initiated by inducer injection. A new approach to model the target protein formation rate was elaborated, where product formation is primarily dependent on the specific biomass growth rate. Lower growth rates led to higher specific protein concentrations. The model was identified from a series of fed-batch experiments designed for parameter identification purposes and possesses good prediction quality. Then the model was used to determine optimal open-loop control profiles by manipulating the substrate feed rates in both phases as well as the induction time. Feed-rate optimization has been solved using Pontryagin's maximum principle. The solution was validated experimentally. A significant improvement of the process performance index was achieved.     

Genetic algorithms with filters for optimal control problems in fed-batch bioreactors

When using a genetic algorithm (GA) to solve optimal control problems that can arise in a fed-batch bioreactor, the most obvious direct approach is to rely on a finite dimensional discretization of the optimal control problem into a nonlinear programming problem. Usually only the control function is discretized, and the continuous control function is approximated by a series of piecewise constant functions. Even though the piecewise discretized controls that the GA produces for the optimal control problem may give good performances, the control policies often show very high activity and differ considerably from those obtained using a continuous optimization strategy. The present study introduces a few filters into a real-coded genetic algorithm as additional operators and investigates the smoothing capabilities of the filters employed. It is observed that inclusion of a filter significantly smoothens the optimal control profile and often encourages the convergence of the algorithm. The applicability of the technique is illustrated by solving two previously reported optimal control problems in fed-batch bioreactors that are known to have singular arcs.     

Improvement of heterologous protein productivity using recombinant Yarrowia lipolytica and cyclic fed-batch process strategy

A cyclic fed-batch bioprocess is designed and a significant improvement of rice alpha-amylase productivity of recombinant Yarrowia lipolytica is illustrated. A bioprocess control strategy developed and reported here entails use of a genetically stable recombinant cloned for heterologous protein, use of optimized media for cell growth and enzyme production phases, and process control strategy enabling high cell-density culture and high alpha-amylase productivity. This process control can be achieved through maintaining a constant optimal specific cell growth rate at a predetermined value (i.e., 0.1 h-1), controlling medium feed rate commensurate with the cell growth rate, and maintaining a high cell-density culture (i.e., 60-70 g/L) for high productivity of cloned heterologous protein. The volumetric enzyme productivity (1, 960 units/L. h) achieved from the cyclic fed-batch process was about 3-fold higher than that of the fed-batch culture process (630 units/L. h).     

Feedback optimization of fed-batch fermentation

The problem of feedback optimization of the feed rate for fed-batch fermentation processes is formulated in the framework of singular control theory and switching hypersurfaces. Using four differential balance equations that describe a general class of fedbatch processes and a general objective function to be minimized, it is shown that under certain restrictions the feedback optimization of the feed rate can be realized as a nonlinear function of the state variables, such as the concentrations of cell mass, substrate and product, and the fermentor volume. The restrictions on the initial conditions, the fermentation kinetics and the objective function, that are needed for realization of the feedback optimization, are provided. Fed-batch fermentation models of lysine and alcohol are used to construct switching curves and to illustrate the feedback optimization of the feed flow rates.     

Effect of specific production rate of recombinant protein on multimerization of plasmid vector and gene expression level

In fed-batch cultures of recombinant Escherichia coli BL21(DE3)[pT7-G3IL2] at high cell concentration, the post-induction specific growth rate was carefully regulated by controlled medium feed to maximize the synthesis level of recombinant fusion interleukin-2, G3.IL-2. A maximum concentration of G3.IL-2 (11.25 g l(-1)) was achieved in the induced recombinant culture growing at the rate of 0.056 h(-1). A steep decrease in the expression level of G3.IL-2 was observed at the post-induction specific growth rates higher than its optimal value (0.056 h(-1)). In the induced recombinant cultures, plasmid multimerization was observed and highly dependent on specific growth and production rate: a higher post-induction specific growth rate and an increased specific production rate tended to significantly promote it much further. Moreover, plasmid stability was found to decrease rapidly in a faster growing culture.     

On-line optimal control for fed-batch culture of baker's yeast production

A method of on-line optimal control for fed-batch culture of bakers yeast production is proposed. The feed rate is taken as the control variable. The specific growth rate of the yeast is the output variable and is determined from the balance equation of oxygen. A moving model is obtained by using the data from the feed rate and the specific growth rate. Based on the moving model, an optimal feed rate for fed-batch culture is then achieved.