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.     

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.     

Tools for improving feeding strategies in a SBR with several species

This paper analyzes feeding strategies in a sequential batch reactor (SBR) with the objective of reaching a given (low) substrate level as quickly as possible for a given volume of water. Inside the SBR, several species compete for a single substrate, which leads to a minimal time control problem in which the control variable is the feeding rate. Following Gajardo et al. (2008) SIAM J Control Optim 47(6):2827–2856, we allow the control variable to be a bounded measurable function of time combined with possible impulses associated with instantaneous dilutions. For this problem, the extremal trajectories of the singular arc type are characterized as the strategies used to maintain the substrate at a constant level. Since this optimization problem is difficult to solve, this characterization provides a valuable tool for investigating the optimality of various feeding strategies. Our aim is thus to illustrate the use of this tool by proposing potential optimal feeding strategies, which may then be compared with other more intuitive strategies. This aim was accomplished via several numerical experiments in which two specific strategies are compared.     

A nonsingular optimization approach to the feed rate profile optimization of fedbatch cultures

A new method to calculate the optimal feed rate profile for fedbatch culture is proposed. Instead of the usual singular control approach of taking the feed rate as the control variable, the substrate concentration profile is used as the transformed control variable to avoid the computational difficulty associated with the singular control. Thus, the problem is converted into a nonsingular optimization problem of determining the optimal substrate concentration profile subject to a constraint. The equivalent feed rate profile to match the optimal substrate concentration profile is then generated. With this method the computational difficulty associated with singular controls for high-order systems is circumvented. The proposed method is illustrated by a number of examples.     

Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy

A mathematical model for the depletion of bone marrow under cancer chemotherapy is analyzed as an optimal control problem. The control represents the drug dosage of a single chemotherapeutic agent and pharmacokinetic equations which model its plasma concentration are included. The drug dosages enter the objective linearly. It is shown that optimal controls are bang-bang, i.e. alternate the drug dosages at full dose with rest-periods in between, and that singular controls which correspond to treatment schedules with varying dosages at less than maximum rate are not optimal. Numerical simulations are given to illustrate the effect of the pharmacokinetic equations on the dosages.     

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.     

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.     

Optimal risk management of human alveolar echinococcosis with vermifuge

In this study, we develop a bioeconomic model of human alveolar echinococcosis (HAE) and formulate the optimal strategies for managing the infection risks in humans by applying optimal control theory. The model has the following novel features: (i) the complex transmission cycle of HAE has been tractably incorporated into the framework of optimal control problems and (ii) the volume of vermifuge spreading to manage the risk is considered a control variable. With this model, we first obtain the stability conditions for the transmission dynamics under the condition of constant control. Second, we explicitly introduce a control variable of vermifuge spreading into the analysis by considering the associated control costs. In this optimal control problem, we have successfully derived a set of conditions for a bang-bang control and singular control, which are mainly characterized by the prevalence of infection in voles and foxes and the remaining time of control. The analytical results are demonstrated by numerical analysis and we discuss the effects of the parameter values on the optimal strategy and the transmission cycle. We find that when the prevalence of infection in foxes is low and the prevalence of infection in voles is sufficiently high, the optimal strategy is to expend no effort in vermifuge spreading.     

Energetics of enzyme catalysis. II. Oversaturation, case diagrams, reversible and irreversible behaviour

Most enzymes react in vivo under reversible conditions where the substrate and product concentrations are not far removed from equilibrium values. Under these conditions when the concentration of substrate is increased, in addition to the usual unsaturated and saturated behaviour we find a third type of kinetic regime at high substrate concentration-oversaturation. In this regime the rate limiting transition state involves interconversion of free enzyme forms. For a one substrate/one product enzyme, case diagrams can be constructed which depict the kinetic behaviour as a function of substrate and product concentrations. Six different cases are found and are discussed with the relevant free energy profiles. A systematic procedure is described for the investigation and construction of the case diagram.     

Modelling the optimal timing in metabolic pathway activation—Use of Pontryagin's Maximum Principle and role of the Golden section

The time course of enzyme concentrations in metabolic pathways can be predicted on the basis of the optimality criterion of minimizing the time period in which an essential product is generated. This criterion is in line with the widely accepted view that high fitness requires high pathway flux. Here, based on Pontryagin's Maximum Principle, a method is developed to solve the corresponding constrained optimal control problem in an almost exclusively analytical way and, thus, to calculate optimal enzyme profiles, when linear, irreversible rate laws are assumed. Three different problem formulations are considered and the corresponding optimization results are derived. Besides the minimization of transition time, we consider an operation time in which 90% of the substrate has been converted into product. In that case, only the enzyme at the lower end of the pathway rather than all enzymes are active in the last phase. In all cases, biphasic or multiphasic time courses are obtained. The biological meaning of the results in terms of a consecutive just-in-time expression of metabolic genes is discussed. For the special case of two-enzyme systems, the role of the Golden section in the solution is outlined.     

Optimum operating mode for a class of fermentation

This paper is concerned with optimization of the operating mode of a fermentor. Combining the various modes of operation—batch, semibatch, and continuous—the operating pattern which maximizes the desired metabolic product in a single fermentor is determined by using Kelley's transformation method with Pontryagin's maximum principle. Kelley's transformation method is a device which avoids the singular situation which occurs when the usual procedure of selecting the optimal control function by the maximum principle breaks down. This is the case in the problem considered in this paper. For lysine fermentation, the best operating mode depends on the fermentor capacity and operating time. The results of this study are summarized thus: (i) when the operating time is “long enough,” optimal conditions require that continuous operation follows either semibatch and/or batch operation, and (ii) when the fermentor capacity becomes “large enough,” semibatch operation becomes important.     

Optimal control of batch processes involving simultaneous enzymatic and microbial reactions

Optimal enzyme feed rate profiles have been calculated, based on a model for a fed-batch simultaneous enzymatic and microbial reaction (SEMR) process. The model parameters corresponded to a relatively slow citric acid fermentation. The profiles were calculated using an iterative algorithm based on the minimum principle. Penalty functions were used to enforce inequality constraints on the enzyme feed rate. Significant improvements in the objective function relative to that for the best constant enzyme feed rate were found. The effect on the optimal profiles of changes in the parameters of the model and the objective function were investigated, as was the effect of introducing the stationary state assumption to eliminate glucose concentration as a state variable. Major differences between bang-bang control variable profiles and singular arcs were found, with the singular arc solution slightly better than the optimal bang-bang control.List of Symbols a N-vector of initial conditions - b ₁–b₁₀ parameters defined in Table 2 - c vector of cost parameters - c ₁–c₆ penalty function parameters - E enzyme concentration (U/l) - f N-vector of functions - F enzyme feed rate (U/l-h) - g N-vector of functions - G glucose concentration (g/l) - H Hamiltonian - J objective function - J ^* modified objective function - L number of integration steps per time interval - L number of control variables - M number of time intervals - n iteration index - N number of state variables - P product concentration (g/l) - r ₁ glucose formation rate (g/l-h) - r ₂ product formation rate (g/l-h) - t time (h) - T final time (h) - u L-vector of control variables - x N-vector of state variables - z N-vector of adjoint variables - Z total enzyme fed (U/l) Greek convergence parameter The support of one of the authors by the National Science Foundation (Grant CBT-84-20552) is gratefully acknowledged.     

Modeling and advanced control of recombinant Zymomonas mobilis fed-batch fermentation

This work presents the development of an unstructured kinetic model incorporating the differing degrees of product, substrate, and pH inhibition on the kinetic rates of ethanol fermentation by recombinant Zymomonas mobilis CP4:pZB5 for growth on two substrates. Product inhibition was observed to start affecting the specific growth rate at an ethanol concentration of 20 g/L and the specific productivity at about 35-40 g/L. Specific growth rate was also shown to be more sensitive to inhibition by lowered pH as well. A model for the inhibition of two competing substrates' cellular uptake via membrane transport is proposed. Inhibition functions and model parameters were determined by fitting experimental data to the model. The model was utilized in a nonlinear model predictive control (NMPC) algorithm to control the product concentration during fed-batch fermentation to offset the inhibitory effects of product inhibition. Using the optimal feeding policy determined online, the volumetric productivity of ethanol was improved 16.6% relative to the equivalent batch operation when the final ethanol concentration was reached.     

Solution of immobilized enzyme problems by collocation methods

A method of solution to the problem of enzyme effectiveness factor has been presented. For a rapid estimation of the same, a graphical procedure is discussed which is sufficiently accurate for many practical situations. Applications to systems with rate dependent on position in the pellet and substrate and product inhibition is discussed. The case of concentration dependent diffusivity (facilitated diffusion) can also be solved by a simple transformation.     

Theoretical analysis of intrinsic reaction kinetics and the behavior of immobilized enzymes system for steady-state conditions

Mathematical modeling of immobilized enzymes under different kinetics mechanism viz. simple Michaelis–Menten, uncompetitive substrate inhibition, total competitive product inhibition, total non-competitive product inhibition and reversible Michaelis–Menten reaction are discussed. These five kinetic models are based on reaction diffusion equations containing non-linear terms related to Michaelis–Menten kinetics of the enzymatic reaction. Modified Adomian decomposition method is employed to derive the general analytical expressions of substrate and product concentration for all these five mechanisms for all possible values of the parameters Φ_S (Thiele modulus for substrate), Φ_P (Thiele modulus for product) and α (dimensionless inhibition degree). Also we have presented the general analytical expressions for the mean integrated effectiveness factor for all values of parameters. Analytical results are compared with the numerical results and also with the limiting case results, which are found to be good in agreement.     

Predictive macroscopic modeling of cell growth,metabolism and monoclonal antibody production: Case study of a CHO fed-batch production

We describe a systematic approach to establish predictive models of CHO cell growth, cell metabolism and monoclonal antibody (mAb) formation during biopharmaceutical production. The prediction is based on a combination of an empirical metabolic model connecting extracellular metabolic fluxes with cellular growth and product formation with mixed Monod-inhibition type kinetics that we generalized to every possible external metabolite. We describe the maximum specific growth rate as a function of the integral viable cell density (IVCD). Moreover, we also take into account the accumulation of metabolites in intracellular pools that can influence cell growth. This is possible even without identification and quantification of these metabolites as illustrated with fed-batch cultures of Chinese Hamster Ovary (CHO) cells producing a mAb. The impact of cysteine and tryptophan on cell growth and cell productivity was assessed, and the resulting macroscopic model was successfully used to predict the impact of new, untested feeding strategies on cell growth and mAb production. This model combining piecewise linear relationships between metabolic rates, growth rate and production rate together with Monod-inhibition type models for cell growth did well in predicting cell culture performance in fed-batch cultures even outside the range of experimental data used for establishing the model. It could therefore also successfully be applied for in silico prediction of optimal operating conditions.     

Setting Single-Period Optimal Capacity Levels and Prices for Substitutable Products

In this paper, we consider how a company that has the flexibility to produce two substitutable products would determine optimal capacity levels and prices for these products in a single-period problem. We first consider the case where the firm is a price taker but can determine optimal capacity levels for both products. We then consider the case where the firm can set the price for one product and the optimal capacity level for the other. Finally, we consider the case where capacity is fixed for both products, but the firm can set prices. For each case, we examine the sensitivity of optimal prices and capacities to the problem parameters. Finally, we consider the case where each product is managed by a product manager trying to maximize individual product profits rather than overall firm profits and analyze how optimal price and capacity decisions are affected.     

Chorismate lyase: kinetics and engineering for stability

By removing the enolpyruvyl group from chorismate, chorismate lyase (CL) produces p-hydroxybenzoate (p-HB) for the ubiquinone biosynthetic pathway. We have analyzed CL by several spectroscopic and chemical techniques and measured its kinetic (kcat=1.7 s(-1), K(m)=29 microM) and product inhibition parameters (K(p)=2.1 microM for p-HB). Protein aggregation, a serious problem with wild type CL, proved to be primarily due to the presence of two surface-active cysteines, whose chemical modification or mutation (to serines) gave greatly improved solution behavior and minor effects on enzyme activity. CL is strongly inhibited by its product p-HB; for this reason activity and inhibition measurements were analyzed by both initial rate and progress curve methods. The results are consistent, but in this case where the stable enzyme-product complex rapidly becomes the predominant form of the enzyme, progress curve methods are more efficient. We also report inhibition measurements with several substrate and product analogs that give information on ligand binding interactions of the active site. The biological function of the unusual product retention remains uncertain, but may involve a mechanism of directed delivery to the membrane-bound enzyme that follows CL in the ubiquinone pathway.     

Modeling and static optimization of the ethanol production in a cascade reactor. II. Static optimization

In Part I(1) of this research a complex model was obtained for describing the ethanol fermentation in a cascade reactor. This complexity is due to both the nonlinearity and the large scale representation. Based on techniques of partitioning and relaxation, a decentralized successive approximation method is developed for static optimization. The influence of the way of fermentation during continuous culture in multistage fermentors is studied in the case of a double inhibition of cell growth and product formation by both substrate and final product. The optimal number of reactors is discussed with respect to the strength of the ethanol inhibition, while the interest of head feeding or distributed feeding is evaluated in relation to the strength of substrate inhibition.