Classification of static behavior of a class of unstructured models of continuous bioprocesses

The stability characteristics of a class of unstructured models of continuous bioreactors are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which the non-biomass product formation rate is linearly proportional to the utilization rate of limiting substrate. The kinetics expressions of cell growth and product synthesis are allowed to assume general forms of substrate and product. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of non-trivial steady states with wash-out conditions, and multistability resulting from hysteresis. These general results are applied to specific examples of bioprocesses and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models.     

A study of oxygen transfer in shake flasks using a non-invasive oxygen sensor

We describe a study of oxygen transfer in shake flasks using a non-invasive optical sensor. This study investigates the effect of different plugs, presence of baffles, and the type of media on the dissolved oxygen profiles during Escherichia coli fermentation. We measured the volumetric mass transfer coefficient (k(L)a) under various conditions and also the resistances of the various plugs. Finally, we compared shake flask k(L)a with that from a stirred tank fermentor. By matching k(L)a's we were able to obtain similar growth and recombinant protein product formation kinetics in both a fermentor and a shake flask. These results provide a quantitative comparison of fermentations in a shake flask vs. a bench-scale fermentor and should be valuable in guiding scale-up efforts.     

Performance of a two-stage fermentor with cell recycle for continuous production of lactic acid

The performasnce of a recycle two-stage fermentor with cell separators after each stage is analyzed numerically for continuous production of lactic acid. In this system, the bleed broth withdrawn from the first stage is provided to the second fermentor to reuse viable cells in the bleed. Biological rate expressions and parametric values are taken from the literature. The effects of operating parameters on the concentrations of total and viable cells, substrate and product in each stage, the lactic acid productivity and the substrate conversion are examined and discussed. With respect to overall productivity and conversion, it is found that the present fermentor system is more efficient than a conventional chemostat fermentor with cell recycle.     

Operability of Continuous Bioprocesses: Static Behavior of a Large Class of Unstructured Models

The complete static behavior of a large class of unstructured models of continuous bioprocesses is classified using elementary concepts of the singularity theory and continuation techniques. The class consists of models for which the cell growth rate is proportional to the rate of utilization of limiting substrate while the kinetics of cell growth, utilization of limiting substrate and synthesis of the desired non-biomass product are allowed to assume general forms of substrate and product. This class of models was used extensively in the literature to model fermentation processes. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of wash-out conditions with non-trivial steady states and multistability resulting from hysteresis. These general results are applied to a number of experimentally validated models of fermentation processes, and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models. Practical criteria are also derived for the safe operation of the bioprocesses.     

Measurement of minimum substrate concentration (Smin) in a recycling fermentor and its prediction from the kinetic parameters of Pseudomonas strain B13 from batch and chemostat cultures.

The minimum substrate concentration required for growth, Smin, was measured for Pseudomonas sp. strain B13 with 3-chlorobenzoate (3CB) and acetate in a recycling fermentor. The substrates were provided alone or in a mixture. Smin values predicted with kinetic parameters from resting-cell batches and chemostat cultures differed clearly from the values measured in the recycling fermentor. When 3CB and acetate were fed as single substrates, the measured Smin values were higher than the individual Smin values in the mixture. The Smin in the mixture reflected the relative energy contributions of the two substrates in the fermentor feed. The energy-based maintenance coefficients during zero growth in the recycling fermentor were comparable for all influent compositions (mean +/- standard deviation, 0.34 +/- 0.07 J mg [dry weight]-1 h-1). Maintenance coefficient values for acetate were significantly higher in chemostat experiments than in recycling-fermentor experiments. 3CB maintenance coefficients were comparable in both experimental systems. The parameters for 3CB consumption kinetics varied remarkably with the experimental growth conditions in batch, chemostat, and recycling-fermentor environments. The results demonstrate that the determination of kinetic parameters in the laboratory for prediction of microbial activity in complex natural systems should be done under conditions which best mimic the system under consideration.     

Dialysis continuous process for ammonium lactate fermentation: Simulated and experimental dialysate-feed,immobilized-cell systems

A generalized mathematical model, previously developed and experimentally validated, was modified and used to computer-simulate two dialysate-feed systems for operating a dialysis continuous process for the ammonium lactate fermentation. The simulations predicted that the feeding of substrate into the dialysate circuit and thence into the fermentor circuit via dialysis should greatly improve the production of cell mass and metabolite product. Experiments were conducted to test the system in which the fermentor is operated without an effluent, thus immobilizing the cells. Dried cheese whey ultrafiltrate was rehydrated to contain a normal concentration of lactose (62 mg/ml), supplemented with yeast with an adapted culture of Lactobacillus bulgaricus. The system was operated without interruption for 26 days. Results during steady-state conditions showed that the system is a new and useful way to immobilize living cells for the purpose of producing a metabolite at a high rate for a prolonged time. The substrate consumed by the cells is converted to product via maintenance metabolism only and is sterilized by dialysis.     

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.     

Performance evaluation of ethanol fermentor systems using a vector-valued objective function

It is shown that the performance evaluation using a vector-valued objection function whose components are the product productivity, the product concentration, and the substrate conversion is quite useful in getting deeper insight into the development of new processes and in determining the operating point. Particular attention is focused on the ethanol fermentation using variety of systems such as the conventional chemostat system, multiple fermentor system, cell recycle system, extractive fermentor system, cell recycle system, extractive fermentor system, and immobilized cell system. The contour map and the projection of the noninferior set are used in investigating the performance improvement and the trade-offs among performance indexes.     

Integration of microbial kinetics and fluid dynamics toward model‐driven scale‐up of industrial bioprocesses

Scale‐up of bioprocesses is hampered by open questions, mostly related to poor mixing and mass transfer limitations. Concentration gradients of substrate, carbon dioxide, and oxygen in time and space, especially in large‐scale high‐cell density fed‐batch processes, are likely induced as the mixing time of the fermentor is usually longer than the relevant cellular reaction time. Cells in the fermentor are therefore repeatedly exposed to dynamic environments or perturbations. As a consequence, the heterogeneity in industrial practices often decreases either yield, titer, or productivity, or combinations thereof and increases by‐product formation as compared to well‐mixed small‐scale bioreactors, which is summarized as scale‐up effects. Identification of response mechanisms of the microorganism to various external perturbations is of great importance for pinpointing metabolic bottlenecks and targets for metabolic engineering. In this review, pulse response experimentation is proposed as an ideal way of obtaining kinetic information in combination with scale‐down approaches for in‐depth understanding of dynamic response mechanisms. As an emerging tool, computational fluid dynamics is able to draw a holistic picture of the fluid flow and concentration fields in the fermentor and finds its use in the optimization of fermentor design and process strategy. In the future, directed strain improvement and fermentor redesign are expected to largely depend on models, in which both microbial kinetics and fluid dynamics are thoroughly integrated.     

Fermentation kinetics of spent sulfite liquor by Saccharomyces cerevisiae

The growth kinetics of the yeast Saccharomyces cerevisiae and the production rate of ethanol have been studied in batch fermentation under anaerobic conditions in a 20-L fermentor. Two substrates were used in fermentation trials: a synthetic mixture of three fermentable sugars, D-glucose, D-mannose, and D-galactose, and a low-yield liquor originating from a bisulfite cooking process. The Monod model adequately described the system in relation to the specific growth rate mu(x) and the specific product formation rate mu(P). Different fermentation parameters (growth rate, substrate utilization, and product formation) were determined for the synthetic mixture and the bisulfite liquor. It was observed that the specific growth rate is much lower in spent sulfite liquor than in a synthetic medium. However, the specific product formation rate remains the same in both media.     

Dialysis Continuous Process for Ammonium-Lactate Fermentation: Improved Mathematical Model and Use of Deproteinized Whey

Separate terms for substrate limitation and product inhibition were incorporated into an equation describing the rate of cell growth for the steady-state fermentation of lactose to lactic acid with neutralization to a constant pH by ammonia. The equation was incorporated into a generalized mathematical model of a dialysis continuous process for the fermentation, developed previously, in which the substrate is fed into the fermentor and the fermentor contents are dialyzed through a membrane against water. The improved model was used to simulate the fermentation on a digital computer, and the results agreed with previous experimental tests using whole whey as the substrate. Further simulations were then made to guide experimental tests using deproteinized whey as the substrate. Dried cheese-whey ultrafiltrate was rehydrated with tap water to contain 242 mg of lactose per ml, supplemented with 8 mg of yeast extract per ml, charged into a 5-liter fermentor without sterilization, adjusted in pH (5.5) and temperature (44°C), and inoculated with an adapted culture of Lactobacillus bulgaricus. The fermentor and dialysate circuits were connected, and a series of steady-state conditions was managed nonaseptically for 71 days. The fermentation of deproteinized whey relative to whole whey, with both highly concentrated, resulted in similar extents of product accumulation but at a lesser rate.     

Computer modeling of antibiotic fermentation with on-line product removal

The fermentation of Streptomyces griseus for the production of cycloheximide is similar to other antibiotic fermentations in that product synthesis is subject to feedback regulation and the desired product is degraded in the fermentation broth. The productivity of this fermentation can thus be dramatically increased by removing the antibiotic from the whole broth as it is produced. One means for effecting this on-line product removal is to contact the whole fermentation broth with neutral polymeric resin immobilized in hydrogel beads. The antibiotic adsorbs to the immobilized resin via hydrophobic interactions. In this work, the adsorption of the antibiotic onto the immobilized resin was characterized. A biochemical model of the fermentation was then used to describe the time profiles of biomass, substrate, and antibiotic in a fermentation system in which whole broth is circulated from the fermentor through a continuously stirred extractor containing the adsorbent beads. Various operating conditions were examined to optimize the productivity of the fermentation.     

Mathematical modelling of dynamics and control in metabolic networks. V. Static bifurcations in single biochemical control loops

Here we expand an earlier study of feedback activation in simple linear reaction sequences by searching the parameter space of biologically realistic rate laws for multiple stable steady states. The impetus for this work is to seek the origin of decision making strategies at the metabolic level, with particular emphasis on the switching between the operating conditions needed to meet changing substrate availability and organism requirements. The control loop considered herein is a linear reaction chain in which the end product of the reaction sequence feedback activates the first reaction in the sequence to produce feedback control. It has been found that the criteria for the existence of multiple steady state solutions in such loops involve only the kinetics of the regulatory enzyme controlling the first reaction and that of end product removal. The effects of these kinetics are examined here using two representative models for the regulatory enzyme: the lumped controller, based on Hill-type kinetics, and the symmetry model. The behavior of these two models is qualitatively similar, and both show the characteristics needed for switching between low and high substrate utilization. The removal rate is assumed to be of the Michaelis-Menten type. Judicious scaling of the governing equations permits separation of genetically determined kinetic parameters from concentration dependent ones. This allows us to conclude that, for a fixed set of kinetic parameters, the steady state flux through the loop can be switched between stable steady states by merely varying metabolite or enzyme concentrations. In particular, when the initial substrate exceeds a certain critical level, the loop can be "switched on" (by a discontinuous increase in the flux through the chain), and similarly, when it falls below a critical level, the pathway is shut down. Similar effects can be realized by varying the ratios of enzyme concentrations. It is proposed that by identifying these critical points one can gain significant insight into the objectives of decision making at the metabolic level.     

Evaluation of feeding strategies in carbon-regulated secondary metabolite production through mathematical modeling

There is now growing evidence that the production of many secondary metabolic by microorganisms is subjected to carbon-catabolite regulation. Even though the exact mode of this regulation is not yet clear, an engineering analysis of the production process is still possible based upon a suitable hypothesis. By way of simulation of penicillin fermentation data obtained from the literature, a mechanistic model involving a substrate inhibition kinetics of product formation has been verified in this paper. Such a model has been found successful not only in predicting simple sugar-feeding strategy, but also a complicated computer guided strategy based upon controlling biomass growth rates in the tropo and idiophases. Using this model, for strategies for sugar feeding into penicillin fermentation have been investigated. These results show that similar penicillin productivities can be obtained using any of these strategies provided fermentations are carried out under optimal conditions corresponding to the strategy chosen. Effect of maximum oxygen transfer capacity of the fermentor under the conditions of fungal growth has been incorporated using an upper limit of biomass concentration on achievement of which the fermentations must be stopped due to serious oxygen limitations. Results of model simulations with such limits throw light upon the way in which different fermentors may behave with respect to product formation.     

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.     

Multiplicity and stability analysis of microorganisms in continuous culture: effects of metabolic overflow and growth inhibition

Metabolic overflow (enhanced uptake of substrate and secretion of intermediates) is a phenomenon often observed for cells grown under substrate excess. Growth inhibition by substrate and/or product is also normally found for this kind of culture. An effort is made in this work to analyze the dynamic behavior of a continuous culture subject to metabolic overflow and growth inhibition by substrate and/or product. Analysis of a model system shows that in a certain range of operating conditions three nonwashout steady state solutions are possible. Local stability analysis indicates that only two of them are stable thus leading to multiplicity and hysteresis. Further analysis of the intrinsic effects of different terms describing the metabolic overflow and growth inhibitions reveals that for the model system and the parameters considered, the combined effects of product inhibition and an enhanced formation rate of product under substrate excess cause the multiplicity and hysteresis. Growth inhibition by substrate and/or an enhanced substrate uptake appear not to be necessary conditions. The combined effects of enhanced product formation and product inhibition can also lead to unusual dynamic behavior such as a prolonged time period to reach a steady state, oscillatory transition from one steady state to another, and sustained oscillations. Using the occurrence of multiplicity and oscillation as criteria, the operating regime of a continuous culture can be divided into four domains: one with multiplicity and oscillation, one with unique steady state but possible oscillatory behavior, the other two with unique and stable steady state. The model predictions are in accordance with recent experimental results. The results presented in this work may be used as guidelines for choosing proper operating conditions of similar culture systems to avoid undesired instability and multiplicity. Copyright 1998 John Wiley & Sons, Inc.     

Continuous production of ethanol by yeast "immobilized" in a membrane-contained fermentor

A dialysate-feed, immobilized-cell dialysis continuous fermentation system was investigated as a method of relieving product inhibition in the conversion of glucose to ethanol by cells of Saccharomyces cerevisiae ATCC 4126. The substrate was fed into a continuous dialysate circuit and then into a batch fermentor circuit via diffusion through the microporous membranes of an intermediate dialyzer. Simultaneously, product was withdrawn from the fermentor circuit through the dialyzer membranes into the dialysate circuit and out in the effluent. Since the fermentor was operated without an effluent, the cells essentially were immobilized and converted substrate to product by maintenance metabolism. Contrary to prior results with this novel system for the continuous fermentation of lactose to lactate by lactobacillus cells, a steady state of yeast cells in the fermentor did not occur initially but was obtained by the depletion of medium nitrogen and the prevention of cell breakage, although the substrate and product concentrations then became unsteady. The inherent advantages of the system was offset in the ethanol fermentation by relatively low productivity, which appeared to be limited by membrane permeability.     

Evidence for the occurrence of an oxygen limitation during soil bioremediation by solid-state fermentation

Bioremediation methods are a promising way of dealing with soil and subsoil contamination by organic substances. This biodegradation process is supported by micro-organisms which use the organic carbon from the pollutants as energy source and cells building blocks. The scope of this work is to study the main parameters of the process and the physical limiting steps. Several ground samples from an actual petroleum hydrocarbon contaminated site have been tested. Four fixed bed column reactors and one rotating fermentor are used, enabling the study of the influence of different operating variables on the biodegradation kinetics. The stoichiometric equation for bacteria growth and pollutant degradation has been established, allowing the determination of mass balances. Biodegradation monitoring is achieved by continuously measuring the emissions of carbon dioxide. Biodegradation rates and pollution load decrease in the two kinds of bioreactors are also compared.     

The rotorfermentor. II. Application to ethanol fermentation.

The kinetics of microbial growth and product formation are described as applied to the high cell concentration scheme of the rotorfermentor. A bench scale pilot plant was designed and built in order to demonstrate the operational feasibility of the rotorfermentor. The fermentation of glucose to ethanol by Saccharomyces cerevisiae ATCC 4126 was used. When the rotorfermentor was used with a glucose feed concentration of 104 g/liter almost 100% glucose utilization was obtained and the ethanol productivity rate was 27.3 g ethanol/liter hr which was found to be about 10 times greater than the ethanol productivity obtained from an ordinary continuous stirred tank (CST) fermentor. The ethanol experimental results obtained from the rotorfermentor and an ordinary CST fermentor were used as a basis to assess the economic feasibility of the rotorfermentor. The economics of an industrial scale ordinary CST fermentor with and without cell recycle is compared with a rotorfermentor unit for the same ethanol production throughput. For the process conditions considered in this case, calculations showed that the rotorfermentor may replace both a CST fermentor and cell centrifuge resulting in lower capital equipment costs and lower power consumption requirements.     

Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.