The ideal free distribution and bacterial growth on two substrates

A population dynamical model describing growth of bacteria on two substrates is analyzed. The model assumes that bacteria choose substrates in order to maximize their per capita population growth rate. For batch bacterial growth, the model predicts that as the concentration of the preferred substrate decreases there will be a time at which both substrates provide bacteria with the same fitness and both substrates will be used simultaneously thereafter. Preferences for either substrate are computed as a function of substrate concentrations. The predicted time of switching is calculated for some experimental data given in the literature and it is shown that the fit between predicted and observed values is good. For bacterial growth in the chemostat, the model predicts that at low dilution rates bacteria should feed on both substrates while at higher dilution rates bacteria should feed on the preferred substrate only. Adaptive use of substrates permits bacteria to survive in the chemostat at higher dilution rates when compared with non-adaptive bacteria.     

Regulation of cellulase synthesis in batch and continuous cultures of Clostridium thermocellum

Regulation of cell-specific cellulase synthesis (expressed in milligrams of cellulase per gram [dry weight] of cells) by Clostridium thermocellum was investigated using an enzyme-linked immunosorbent assay protocol based on antibody raised against a peptide sequence from the scaffoldin protein of the cellulosome (Zhang and Lynd, Anal. Chem. 75:219-227, 2003). The cellulase synthesis in Avicel-grown batch cultures was ninefold greater than that in cellobiose-grown batch cultures. In substrate-limited continuous cultures, however, the cellulase synthesis with Avicel-grown cultures was 1.3- to 2.4-fold greater than that in cellobiose-grown cultures, depending on the dilution rate. The differences between the cellulase yields observed during carbon-limited growth on cellulose and the cellulase yields observed during carbon-limited growth on cellobiose at the same dilution rate suggest that hydrolysis products other than cellobiose affect cellulase synthesis during growth on cellulose and/or that the presence of insoluble cellulose triggers an increase in cellulase synthesis. Continuous cellobiose-grown cultures maintained either at high dilution rates or with a high feed substrate concentration exhibited decreased cellulase synthesis; there was a large (sevenfold) decrease between 0 and 0.2 g of cellobiose per liter, and there was a much more gradual further decrease for cellobiose concentrations >0.2 g/liter. Several factors suggest that cellulase synthesis in C. thermocellum is regulated by catabolite repression. These factors include: (i) substantially higher cellulase yields observed during batch growth on Avicel than during batch growth on cellobiose, (ii) a strong negative correlation between the cellobiose concentration and the cellulase yield in continuous cultures with varied dilution rates at a constant feed substrate concentration and also with varied feed substrate concentrations at a constant dilution rate, and (iii) the presence of sequences corresponding to key elements of catabolite repression systems in the C. thermocellum genome.     

Cultivation of Escherichia coli with mixtures of 3-phenylpropionic acid and glucose: steady-state growth kinetics.

The fate of pollutants in the environment is affected by the presence of easily degradable carbon sources. As a step towards understanding these complex interactions, a model system was explored: the degradation of mixtures of glucose (i.e., an easily degradable substrate) and 3-phenylpropionic acid (3ppa) (a model pollutant) by Escherichia coli ML 30 was studied systematically in carbon-limited continuous culture. The two substrates were always consumed simultaneously regardless of the dilution rate applied. Even at dilution rates higher than the maximum specific growth rate for 3ppa (0.35 +/- 0.05 h-1), the two carbon substrates were utilized together. When cells were grown at a constant dilution rate with different mixtures of 3ppa and glucose, in which 3ppa contributed between 5 and 90% of carbon substrate in the feed medium, the steady-state concentrations of 3ppa and glucose were approximately proportional to the ratio of the two substrates in the feed medium. When cells were cultivated at different dilution rates with a 1:1 mixture (based on carbon) of glucose and 3ppa, an overall maximum specific growth rate of 0.90 +/- 0.05 h-1 and a Monod substrate saturation constant for 3ppa (Ks) of 600 to 700 micrograms liter-1, similar to that measured during growth with 3ppa alone, fitted the experimentally determined steady-state 3ppa concentrations. However, due to the highly differing substrate affinity constants for 3ppa and glucose (Ks approximately 30 to 70 micrograms liter-1), the total steady-state carbon concentration in the culture at a constant dilution rate was determined mainly by the steady-state 3ppa carbon concentration, and it increased with increasing proportions of 3ppa in the feed medium.     

A biochemically structured model for Saccharomyces cerevisiae.

A biochemically structured model for the aerobic growth of Saccharomyces cerevisiae on glucose and ethanol is presented. The model focuses on the pyruvate and acetaldehyde branch points where overflow metabolism occurs when the growth changes from oxidative to oxido-reductive. The model is designed to describe the onset of aerobic alcoholic fermentation during steady-state as well as under dynamical conditions, by triggering an increase in the glycolytic flux using a key signalling component which is assumed to be closely related to acetaldehyde. An investigation of the modelled process dynamics in a continuous cultivation revealed multiple steady states in a region of dilution rates around the transition between oxidative and oxido-reductive growth. A bifurcation analysis using the two external variables, the dilution rate, D, and the inlet concentration of glucose, S(f), as parameters, showed that a fold bifurcation occurs close to the critical dilution rate resulting in multiple steady-states. The region of dilution rates within which multiple steady states may occur depends strongly on the substrate feed concentration. Consequently a single steady state may prevail at low feed concentrations, whereas multiple steady states may occur over a relatively wide range of dilution rates at higher feed concentrations.     

Fuzzy logic control of an unstable bioreactor

A rule based fuzzy controller (FLC) is developed for stabilization of an unstable continuous stirred tank bioreactor (CSTBR) from various start-up conditions. The output variable is the reactor substrate concentration and the manipulated variable is the dilution rate. The performance of the FLC is evaluated by simulating a mathematical model of an unstable CSTBR. FLC is robust to perturbations in the specific growth rate, specific consumption rate and also to a disturbance in the feed substrate concentration. The performance of the FLC is superior to that of a conventional proportional controller.     

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.     

Continuous conversion of D-xylose to ethanol by immobilized pachysolen tannophilus

The yeast Pachysolen tannophilus was entrapped in calcium alginate beads to ferment D-xylose on a continous basis in the presence of high cell densities. Experimental operating variables included the feed D-xylose concentration, the dilution rate, and the fermentor biomass concentration. Under favorable operating conditions, cultures retained at least 50% of their initial productivity after 26 days of operation. The specific ehanol production rate was dependent on the substrate level in the fermentor, passing through an optimum when the D-xylose concentration was between 28 and 35 g/L. Consequently, reactor productivity increased with dilution rate and feed D-xylose concentration until a maximum was reached. The ethanol content of the effluent always decreased with increasing dilution rate, but excessive dilution rates diminished the ethanol content without increasing productivity. Unlike production rate, ethanol yield declined monotonically from 0.35 g/g as the fermentor substrate concentration increased. The yield was 69% of that theoretically possible when the D-xylose concentration was near zero, as opposed to 42% when it was in the range supporting the optimum specific rate of ethanol production. As long as D-xylose was supplied to cells faster than they could consume it, productivity increased with the mass of cells immobilized. The effectiveness factor associated with the calcium alginte beads used in this system was 0.4, indicating that only 40% of the entrapped biomass was effective in converting D-xylose to ethanol because of diffusion limitations.     

The dynamics of single-substrate continuous cultures: the role of transport enzymes

A chemostat limited by a single growth-limiting substrate displays a rich spectrum of dynamics. Depending on the flow rate and feed concentration, the chemostat settles into a steady state or executes sustained oscillations. The transients in response to abrupt increases in the flow rate or the feed concentration are also quite complex. For example, if the increase in the flow rate is small, there is no perceptible change in the substrate concentration. If the increase in the flow rate is large, there is a large increase in the substrate concentration lasting several hours or days before the culture adjusts to a new steady state. In the latter case, the substrate concentration and cell density frequently undergo damped oscillations during their approach to the steady state. In this work, we formulate a simple structured model containing the inducible transport enzyme as the key intracellular variable. The model displays the foregoing dynamics under conditions similar to those employed in the experiments. The model suggests that long recovery times (on the order of several hours to several days) can occur because the initial transport enzyme level is too small to cope with the increased substrate supply. The substrate concentration, therefore, increases until the enzyme level is built up to a sufficiently high level by the slow process of enzyme induction. Damped and sustained oscillations can occur because transport enzyme synthesis is autocatalytic, and hence, destabilizing. At low dilution rates, the response of stabilizing processes, such as enzyme dilution and substrate consumption, becomes very slow, leading to damped and sustained oscillations.     

Production of 1,3-Propanediol by Clostridium butyricum VPI 3266 in continuous cultures with high yield and productivity

The effects of dilution rate and substrate feed concentration on continuous glycerol fermentation by Clostridium butyricum VPI 3266, a natural 1,3-propanediol producer, were evaluated in this work. A high and constant 1,3-propanediol yield (around 0.65 mol/mol), close to the theoretical value, was obtained irrespective of substrate feed concentration or dilution rate. Improvement of 1,3-propanediol volumetric productivity was achieved by increasing the dilution rate, at a fixed feed substrate concentration of 30, 60 or 70 g l⁻¹. Higher 1,3-propanediol final concentrations and volumetric productivities were also obtained when glycerol feed concentration was increased from 30 to 60 g l⁻¹, at D=0.05–0.3 h⁻¹, and from 60–70 g l⁻¹, at D=0.05 and 0.1 h⁻¹·30 g l⁻¹ of 1,3-propanediol and the highest reported value of productivity, 10.3 g l⁻¹ h⁻¹, was achieved at D=0.30 h⁻¹ and 60 g l⁻¹ of feed glycerol. A switch to an acetate/butyrate ratio higher than one was observed for 60 g l⁻¹ of feed glycerol and a dilution rate higher than 0.10 h⁻¹; moreover, at D=0.30 h⁻¹ 3-hydroxypropionaldehyde accumulation was observed for the first time in the fermentation broth of C. butyricum.     

Kinetic studies of constitutive human granulocyte-macrophage colony stimulating factor (hGM-CSF) expression in continuous culture of <Emphasis Type="Italic">Pichia pastoris</Emphasis>

Extracellular human granulocyte-macrophage colony stimulating factor (hGM-CSF) expression was studied under the control of the GAP promoter in recombinant Pichia pastoris in a series of continuous culture runs (dilution rates from 0.025 to 0.2 h⁻¹). The inlet feed concentration was also varied and the steady state biomass concentration increased proportionally demonstrating efficient substrate utilization and constancy of the biomass yield coefficient (Y_x/s) for a given dilution rate. The specific product formation rate (q_P) showed a strong correlation with dilution rates demonstrating growth associated product formation of hGM-CSF. The volumetric product concentration achieved at the highest feed concentration (4×) and a dilution rate of 0.2 h⁻¹ was 82 mg l⁻¹ which was 5-fold higher compared to the continuous culture run with 1× feed concentration at the lowest dilution rate thus translating to a 40 fold increase in the volumetric productivity. The specific product yield (Y_P/X) increased slightly from 2 to 2.5 mg g⁻¹, with increasing dilution rates, while it remained fairly invariant, for all feed concentrations demonstrating negligible product degradation or feed back inhibition. The robust nature of this expression system would make it easily amenable to scale up for industrial production.     

Kinetics of <Emphasis Type="SmallCaps">l</Emphasis>-lysine fermentation: a continuous culture model incorporating oxygen uptake rate

For process design and optimization, it is essential to have a mathematical model that represents the system well. Many past studies do not go beyond empirically fitting experimental data. In the present study, an unstructured model incorporating oxygen uptake and dissolved oxygen concentration was developed for a continuous culture of L-lysine. Specific rate expressions of cell growth, substrate consumption, product formation, and oxygen uptake were developed and incorporated in the model. The model predicts very well the effects of operational parameters, such as the dilution rate and the feed substrate concentration. It is also able to predict the unsteady-state dynamics of continuous L-lysine fermentation.     

Cell growth and population activity of Saccharomyces cerevisiae in two-stage continuous cultivation

Continuous culture in a cascade of vessels with the addition of supplemental nutrients to any stage permits adjustment of the physiological state of the culture in each stage to best achieve a desired performance goal. The yeast Saccharomyces cerevisiae in two-stage continuous cultivation was selected as a model system. With conditions in the first stage held constant- at a selected glucose concentration in the feed stream, dilution rate for the second stage was varied. Cell numbers, dry weight, glucose concentration, respiration coefficient, and titers of several enzymes were determined. The seed rate was defined as the ratio of glucose concentration in the feeds to stage 1 and to stage 2. At low seed rates, the calculated specific growth rate in the second stage was proportional to dilution rate. At higher seed rates, the specific growth rate based on dry weight behaved differently from that based on cell numbers, and the dependence on dilution rate was not linear.     

Continuous cultures limited by a gaseous substrate: Development of a simple, unstructured mathematical model and experimental verification with Methanobacterium thermoautotrophicum

This article presents a simple, unstructured mathematical model describing microbial growth in continuous culture limited by a gaseous substrate. The model predicts constant gas conversion rates and a decreasing biomass concentration with increasing dilution rate. It has been found that the parameters influencing growth are primarily the gas transfer rate and the dilution rate. Furthermore, it is shown that, for correct simulation of growth, the influence of gaseous substrate consumption on the effective gas flow through the system has to be taken into account.Continuous cultures of Methanobacterium thermoautotrophicum were performed at three different gassing rates. In addition to the measurement of the rates of biomass production, product formation, and substrate consumption, microbial heat dissipation was assessed using a reaction calorimeter. For the on-line measurement of the concentration of the growth-limiting substrate, H(2), a specially developed probe has been used. Experimental data from continuous cultures were in good agreement with the model simulations. An increase in gassing rate enhanced gaseous substrate consumption and methane production rates. However, the biomass yield as well as the specific conversion rates remained constant, irrespective of the gassing rate. It was found that growth performance in continuous culture limited by a gaseous substrate is substantially different from "classic" continuous culture in which the limiting substrate is provided by the liquid feed. In this report, the differences between both continuous culture systems are discussed.     

Cultivation of hybridoma cells in continuous cultures: kinetics of growth and product formation

Mouse hybridoma cells were grown in suspension in continuous stirred bioreactors. Cell growth, substrate utilization, and monoclonal antibody (MAb) production were studied using serum-free medium. Steady-state data were obtained at different dilution rates, between 0.012 and 0.039 h(-1) Viability was profoundly affected by dilution rate, particularly near the lower end of the dilution-rate range investigated. MAb concentration and productivity went through a maximum with respect to dilution rate. Lactate yield on glucose declined with in creasing dilution rate. Experiments were carried out to study the effects of medium glucose concentration on cell growth, product formation, and lactate yield on glucose. Reduction of glucose concentration in the feed medium did not considerably affect cell density and MAb concentration in the culture, but lactate levels dropped sharply; lactate yield on glucose declined substantially, indicating alterations in cell metabolic path ways for energy metabolism. Optimization strategy for continuous cell culture is discussed.     

Chemicals in the Environment

Synthetic chemicals are now being incorporated into the earth''s ecosystems at a rate and in such manners as to alarm environmentalists. These chemicals are the uncontrolled waste products of a technological society. Most prominent among them at the present time are organochlorine, organomercurial and lead compounds. Persistent members of these groups disperse in water, air and animal tissues. Also they have the capacity for concentration in animal food chains, thereby reversing the historical expectation of the dilution and degradation of wastes.Examples of damage from environmental residues to man are at this stage speculative but documentation from effects on wild species is abundant. Already several species of birds seem on their way to extinction. These wild species constitute a gratuitous monitoring system which already has signaled clear warnings for the welfare of man.     

Periodic operation of a continuous culture of Baker's yeast

The possibility of enhancing the biomass productivity of a continuous culture of Saccharomyces cerevisiae growing on a glucose-limited medium is addressed. An unstructured Monod-type model is first identified using steady-state data. The culture is subjected to step changes in dilution rate, and it is seen that the Monod model is unable to predict even qualitatively the dynamic response of the culture. Incorporation of a time delay allows significant improvement in the transient fit. It is found that the culture has a time lag of about 3 h in adapting its growth rate. Cycling the dilution rate with a period of 3 h leads to substantial improvement in the average biomass productivity.     

Batch- and Continuous-Culture Transients for Two Substrate Systems

Batch growth of Escherichia coli in the presence of equal initial concentrations of glucose and a secondary substrate (xylose) is characterized by sequential utilization of the substrates, whereas continuous-culture systems with equal concentrations of the two substrates in the feed are characterized by complete utilization of both substrates at both high and low dilution rates. Such systems at steady state at a low dilution rate, when suddenly shifted to a higher dilution rate, experience a transient drop in population density accompanied by accumulation of the secondary substrate but virtually no accumulation of glucose. Systems at steady state with 200 mg of glucose per liter were found to undergo a transient population decrease and eventual recovery when switched to feed containing 200 mg of a secondary substrate per liter.     

Nonlinear feedforward control of bioreactors with input multiplicities

A steady-state nonlinear feedforward controller (FFC) for measurable disturbances is designed for a continuous bioreactor, which is represented by Hammerstein type nonlinear model wherein the nonlinearity is a polynomial with input multiplicities. The manipulated variable is the feed substrate concentration (S_f) and the disturbance variable is the dilution rate (D). The productivity (Q=DP) is considered as the controlled variable. The desired value of Q=3.73 gives two values of feed substrate concentration. The nonlinearity in the gain is considered for relating output to the manipulated variable and separately for the relation between output to disturbance variable. The FFC is also designed for the overall linearized system. The performance of the FFC is evaluated on the nonlinear differential equation model. The FFC is also designed for the model based on a single nonlinear steady-state equation containing both D and S_f. This nonlinear FFC gives the best performance. The nonlinear FFC is also designed by using only linear gain for the disturbance and nonlinear gain for the manipulated variable. Similarly, nonlinear FFC is also designed by using linear gain for the manipulated variable and the nonlinear gain for the disturbance variable. The performances of these FFC schemes are compared.     

Feedback stabilization of fed-batch bioreactors: non-monotonic growth kinetics

This paper deals with the design of a feedback controller for fed-batch microbial conversion processes that forces the substrate concentration C(S) to a desired setpoint, starting from an arbitrary (initial) substrate concentration when non-monotonic growth kinetics apply. This problem is representative for a lot of industrial fermentation processes, with the baker's yeast fermentation as a well-known example. It is assumed that the specific growth rate mu is function of the substrate concentration only. A first approach exploits the availability of on-line measurements of both the substrate and biomass concentration. A second approach is merely based on on-line measurements of the biomass concentration, which provide an estimate for the specific growth rate. After a reformulation of the substrate concentration setpoint into a specific growth rate setpoint, it is demonstrated that the fed-batch process can still be stabilized around any desired operating point along the non-monotonic kinetics.     

The dynamics of single-substrate continuous cultures: the role of ribosomes

When a chemostat is perturbed from its steady state, it displays complex dynamics. For instance, if the identity of the growth-limiting substrate is switched abruptly, the substrate concentration and cell density undergo a pronounced excursion from the steady state that can last several days. These dynamics occur because certain physiological variables respond slowly. In the literature, several physiological variables have been postulated as potential sources of the slow response. These include transport enzymes, biosynthetic enzymes, and ribosomes. We have been addressing this problem by systematically exploring the role of these variables. In previous work Shoemaker et al. (J. Theor. Biol., 222 (2003) 307-322), we studied the role of transport enzymes, and we showed that transients starting from low transport enzyme levels could be quantitatively captured by a model taking due account of transport enzyme synthesis. However, there is some experimental data indicating that slow responses occur even if the initial enzyme levels are high. Here, we analyse this data to show that in these cases, the sluggish response is most probably due to slow adjustment of the ribosome levels. To test this hypothesis, we extend our previous model by accounting for the evolution of both the transport enzyme and the ribosomes. Based on a kinetic analysis of the data in the literature, we assume that the specific protein synthesis rate is proportional to the ribosome level, and the specific ribosome synthesis rate is autocatalytic. Simulations of the model show remarkable agreement with experimentally observed steady states and the transients. Specifically, the model predictions are in good agreement with (1) the steady-state profiles of the cell density, substrate concentration, RNA, proteins, and transport enzymes, (2) the instantaneous specific substrate uptake, growth, and respiration rates in response to a continuous-to-batch shift, and (3) the transient profiles of the cell density, substrate concentration, and RNA in response to feed switches and dilution rate shifts. Time-scale analysis of the model reveals that every transient response is a combination of two fundamental (and simpler) dynamics, namely, substrate-sufficient batch dynamics and cell-sufficient fed-batch dynamics. We obtain further insight into the transient response by analysing the equations describing these fundamental dynamics. The analysis reveals that in feed switches or dilution rate shift-ups, the transport enzyme reaches a maximum before RNA achieves its maximum, and in dilution rate shift-downs the cell density reaches a maximum before RNA achieves a minimum.