Input Substrate Concentration Research Articles

Stability and bifurcations in a model of chemostat with two inter‐connected inhibitions and a negative feedback loop

This paper deals with a model of chemostat with two cross‐feeding species and involving two inhibitions. The excess of hydrogen inhibits the growth of acetogenic bacteria which, in turn, inhibit the growth of methanogenic hydrogenotrophic bacteria. The model is described by a system of four ordinary differential equations. We established the conditions of existence and stability of equilibria with respect to the operating parameters which are the dilution rate and the input substrate concentrations of the species of the model, then we illustrate the operating and bifurcation diagrams. This technique makes it easy to interpret different regions of operating diagrams. Inhibitions let some regions of stability rise and others vanish.

Mathematical Study of a Two-Stage Anaerobic Model When the Hydrolysis Is the Limiting Step

A two-step model of the anaerobic digestion process is mathematically and numerically studied. The focus of the paper is put on the hydrolysis and methanogenesis phases when applied to the digestion of waste with a high content of solid matter: existence and stability properties of the equilibrium points are investigated. The hydrolysis step is considered a limiting step in this process using the Contois growth function for the bacteria responsible for the first degradation step. The methanogenesis step being inhibited by the product of the first reaction (which is also the substrate for the second one), and the Haldane growth rate is used for the second reaction step. The operating diagrams with respect to the dilution rate and the input substrate concentrations are established and discussed.

Application of simultaneous dynamic optimization in the productivity of microalgae continuous culture

Microalgae biomass is an up-and-coming option for supplying clean and reliable bio-products and energy for future years. However, its industrial production is not yet accomplished due to its economic unfeasibility. To date, several strategies have been used to improve microalgae productivity. Nonetheless, in most of these strategies, the objective function is minimized by nested procedures that have shown limitations dealing with discontinuities, which is very common in microalgae models. This paper describes the application of simultaneous optimization procedures in GEKKO Python package through the exploration of optimal biomass productivity under continuous operation of the strains Dunaliella tertiolecta and Chlorella vulgaris. The results show the successful implementation with fast convergence times: 7.43s to solve 10800 equations with 900 degrees of freedom in Chlorella vulgaris and 11.45s to solve 6600 equations with 600 degrees of freedom in Dunaliella tertiolecta. Furthermore, in biological terms, simulations show that once an optimal pH>8 level is reached in the Chlorella model, the sensitivity of other variables such as Iin decreases dramatically. Therefore, it is possible to achieve high productivity even without increasing the required light intensity. In addition, in Dunaliella case, the results also infer that larger biomass productivity requires larger input substrate concentration.

On the steady state optimization of the biogas production in a two-stage anaerobic digestion model

In this paper, we study the optimization problem of maximizing biogas production at steady state in a two-stage anaerobic digestion model, which was initially proposed in Bernard etal. (Biotechnol Bioeng 75(4):424-438, 2001). Nominal operating points, consisting of steady states where the involved microorganisms coexist, are usually referred to as desired operational conditions, in particular for maximizing biogas production. Nevertheless, we prove that under some conditions related to input substrate concentrations and microorganism biology, characterized by their growth functions, the optimal steady state can be the extinction of one of the two species. We provide some numerical examples of this situation.

Average break-even concentration in a simple chemostat model with telegraph noise

In this paper, we aim to investigate the effect of telegraph noise on the continuous culture of microorganism in a chemostat. With the help of a finite-state Markov chain, we first construct a regime-switching chemostat model and then establish conditions for extinction and persistence of the microorganism. It is shown that the particular outcome of the chemostat is completely determined by a defined average break-even concentration λA: The microorganism becomes extinct in the chemostat when λA>S0, the input substrate concentration; it will persist when λA<S0. In the case of persistence, we also investigate some further dynamic behaviors of the solution including the recurrent level and eventually existent domain. The theoretical results are illustrated by simulations at the end of the paper.

Steady state analysis of a syntrophic model: the effect of a new input substrate concentration

In this work, we are interested in a reduced and simplified model of the anaerobic digestion process. We focus on the acetogenesis and hydrogenetrophic methanogenesis phases. The model describes a syntrophic relashionship between two microbial species (the acetogenic bacteria and the hydrogenetrophic methanogenic bacteria) with two input substrates (the fatty acids and the hydrogen) including both decay terms and inhibition of the acetogenic bacteria growth by an excess of hydrogen in the system. The existence and stability analysis of the steady states of the model points out the existence of a new equilibrium point which can be stable according to the operating parameters of the system. By means of operating diagrams, we show that, whatever the region of space considered, there exists only one locally exponentially stable steady state.

High biomass density promotes density-dependent microbial growth rate

Describing the interactions between a population and its resources is a research topic in both microbiology and population ecology. When there are fewer resources for the individuals in a large population, the overcrowding can lead to a density-dependent effect which is reflected by a negative feedback of' the organism density on the consumption process. In this paper, we investigate the growth rate of an aerobic microbial ecosystem by two series of experiments performed in continuous agitated cultures. Using a constant dilution rate, but different input substrate concentrations in each experiment, the biomass and substrate concentration were measured at steady state to confront their values with those obtained theoretically from the well-known mathematical model of the chemostat using either resource or density-dependent kinetics. The structures of both flocs and microbial communities were monitored in order to interpret the results. The experiments confirm that density-dependent growth-rate can result either from a high concentration of biomass or from the structuration of this biomass into flocs and we have shown that a new parametrized family of growth functions, that we proposed in this paper, suits better the experimental data than Monod or Contois growth functions.

Time-optimal control of concentration changes in the chemostat with one single species

• Control of bioprocesses with applications in wastewater treatment. • Modeling of a chemostat system including a recirculation parameter for the biomass. • Synthetize optimal feedback control laws based on singular strategies. • Singular strategies represent a trade-off in comparison with bang-bang trajectories. • Characterize robust control laws that can be implemented easily for a practitioner. We consider the problem of driving in minimal time a system describing a chemostat model to a target point. This problem finds applications typically in the case where the input substrate concentration changes yielding in a new steady state. One essential feature is that the system takes into account a recirculation of biomass effect. We depict an optimal synthesis and provide an optimal feedback control of the problem by using Pontryagin’s Principle and geometric control theory for a large class of kinetics.

Global stabilization of the chemostat with elayed and sampled measurements and control

The classical model of the chemostat with one substrate, one species and a Haldane type growth rate function is considered. The input substrate concentration is supposed to be constant and the dilution rate is considered as the control. The problem of globally asymptotically stabilizing a positive equilibrium point of this system in the case where the measured concentrations are delayed and piecewise constant with a piecewise constant control is addressed. The result relies on the introduction of a dynamic extension of a new type.

On a Three Step Model of Anaerobic Digestion Including the Hydrolysis of Particulate Matter

AbstractIn this work, we focus on the mathematical analysis of the model of chemostat with enzymatic degradation of a substrate (organic matter) that can partly be under a solid form Simeonov and Stoyanov (2003). The study of this 3-step model is derived from a smaller order sub-model since some variables can be decoupled from the others. We study the existence and the stability of equilibrium points of the sub-model considering both Monod or Haldane growth rates and distinct dilution rates. In the classical chemostat model with monotonic kinetics, it is well known that only one equilibrium point attracts all solutions and that bistability never occurs Smith and Waltman (1995). In the present study, although (i) only monotonic growth rates are considered and (ii) the concentrations of input substrate concentration is less than the break-even concentration, it is shown that the considered sub-model may exhibit bistability. Hence, the importance of hydrolysis in the appearance of positive equilibrium points and the bistability is pointed out. If a non monotonic growth rate is considered, depending on the input substrate concentration, it is shown that at most four positive equilibrium points exist. Furthermore, for any positive initial condition, the solution converges towards one of the positive equilibrium points for which the washout is unstable. Finally, we study the case where the growth rate is density-dependent, such as the Contois kinetics, which may be of interest if we consider that we work in a non homogeneous environment Lobry and Harmand (2006). Depending on the input substrate concentration, we show that the system can exhibit either a bistability or the global stability of the positive equilibrium point or of the washout.

Biochemical Metabolic Modelling Using Fuzzy Type-2.

In his study a new approach, the use of fuzzy logic type-2 in modeling biochemical reactions is shown. In fact, each enzymatic reaction is modeled by means of a "sigmoid transfer function" relating input and output substrate concentrations. The slant of this function is adjusted using fuzzy type-2. This adjustment is conducted depending on the enzymatic reaction type (having activator/inhibitors or not). The obtained model seems promising in order to permit quantitative results to process data concerning adverse drugs reactions. In this paper it is also proved that by fuzzy type-2 logic, the performance characteristics of the modeling will be improved using the proposed method.

Continuous Selection of the Fastest Growing Species in the Chemostat

This paper proposes control laws for the continuous culture of microorganisms, which make it possible to select species which maximize a criterion. In particular, by controlling the dilution rate and the input substrate concentration, the species with the fastest growth rate in chosen environmental conditions can be selected.In a first step a control is proposed for Monod and Droop models in order to achieve periodic substrate stresses, and a closed loop control is proposed to regulate the total biomass concentration. We show that this biomass regulation causes the selection of the fastest growing species if the system has a periodic behavior, and derive new selection criteria. Finally, the method is simulated using the Droop model for selecting species which maximize these criteria.

Measurements of Kinetic Parameters in a Microfluidic Reactor

Continuous flow microfluidic reactors that use immobilized components of enzymatic reactions present special challenges in interpretation of kinetic data. This study evaluates the difference between mass-transfer effects and reduced efficiencies of an enzyme reaction. The kinetic properties of immobilized alkaline phosphatase (AP) were measured by the dephosphorylation of 6,8-difluoro-4-methylumbelliferyl/phosphate to a fluorescent 6,8-difluoro-4-methylumbelliferone. A glass microfluidic chip with an in-channel weir was created for the capture of solid silica microbeads functionalized with enzyme. The input substrate concentrations and flow rates across the bed were varied to probe the flow-dependent transport and kinetic properties of the reaction in the microreactor bed. Unlike previous reactors, substrate was titrated directly over the fixed enzyme bed by controlling the air pressure over the chip reservoirs. The reactor explored substrate conversions from near zero to 100%. The average bed porosity, residence time, and bed resistance were measured with dye pulses. A simple criterion was derived to evaluate the importance of flow-dependent mass-transfer resistances when using microreactors for calculating kinetic rate constants. In the absence of mass-transfer resistances, the Michaelis-Menten kinetic parameters are shown to be flow independent and are appropriately predicted using low substrate conversion data. A comparison of the kinetic parameters with those obtained using solution-phase enzymatic reactions shows a significant decrease in enzyme activity in the immobilized conformation. The immobilized Km of AP is approximately 6 times greater while the kcat is reduced by approximately 28 times. Contradictions found in literature on the evaluation of Michaelis-Menten kinetic parameters for immobilized enzymes in microfluidic reactors are addressed. When product molecules occupy a significant number of enzymatic sites or modify the enzyme activity, the assumed Michaelis-Menten mechanism can no longer be valid. Under these conditions, the calculations of "apparent" kinetic rate constants, based on Michaelis-Menten kinetics, can superficially show a dependence on flow rate conditions even in the absence of mass-transfer resistances. High substrate conversions are shown to depend on flow rate. A kinetic model based on known mechanisms of the alkaline phosphatase enzyme reaction is tested to predict the measurements for high substrate conversion. The study provides a basis for appropriate use of mass-transfer and reaction arguments in successful application of enzymatic microreactors.

OUTPUT TRACKING OF BIOPROCESSES THROUGH RECIRCULATION WITH UNKNOWN INPUT CONCENTRATION

In a recent work, a new regulator of the output of a continuous autocatalytic bioprocess, by means of recirculation loop, has been presented. It was shown that controlling the recirculation flow rate allows the stabilization of the output in presence of an uncontrolled input flow rate. In the present paper, we extend this result when the input substrate concentration is unknown. For this purpose, we propose the design of an observer of the input concentration which, coupled with a slightly different control law reminiscent of the one used in the case where the input concentration is known, guarantees the regulation of the output.

Optimising activated sludge growth rate by intensifying hydrodynamic forces

AbstractIn this study a Couette device was used to examine the influence of hydrodynamic forces on the overall behaviour of activated sludge. The specific growth rate of activated sludge increased with increasing specific energy dissipation rate, reaching a maximum rate at an energy input of approximately 40 m2 s−3, beyond which any further increases resulted in a reduction in the overall growth. At levels above about 250 m2 s−3 the specific growth rate was negligible. Experimental growth rates in the range 0.04–0.20 h−1 were obtained, depending on the energy input levels and substrate (glucose) concentration. A theoretical model successfully predicted both the decrease in the floc radius and microbial activity with increasing energy input into the system, and when combined was able to provide a reasonable prediction for the optimum specific energy dissipation rate to maximise the specific growth rate.© 2003 Society of Chemical Industry

Properties of the biofilm of Thiobacillus ferrooxidans formed in rotating biological contactor

The physico-chemical properties of the Thiobacillus ferrooxidans biofilm formed on the surface of rotating disks in a rotating biological contactor (RBC) were studied under steady-state conditions. The main independent variable was the input substrate (ferrous iron) concentration in the bioreactor. It has been shown that the biofilm thickness was maximal, the biofilm density was minimal and the specific surface area of pores in the biofilm was maximal when the input ferrous iron concentration was between 1 and 2.1 g/l. The biofilm volume remained nearly the same when input substrate concentrations were between 0.49 and 14.21 g/l, corresponding to oxidation rates between 0.35 and 8.6 g/m 2 h.

Nonlinear recursive feedback control for the regulation of substrate concentration in a fixed bed bioreactor

In this work, a recursive nonlinear feedback control law for the regulation of the outlet substrate concentration for a fixed bed bioreactor with immobilized biomass is designed. The control algorithm is based on a reduced order model related with ordinary differential equations and only made use of a discrete substrate outlet concentration for its implementation. It is shown that the proposed control algorithm is robust against disturbances in the input substrate concentration. The performance of the proposed regulation scheme is illustrated by means of numerical simulations. © 1999 Society of Chemical Industry

Predicting the performance of immobilized enzyme reactors using reversible Michaelis–Menten kinetics

A general mathematical model is developed in the present work for predicting the steady state performance of immobilized enzyme reactor performing reversible Michaelis - Menten kinetics. The model takes into account the effect of external diffusional limitations, the axial dispersion and the equilibrium constant on reactor performance quantified as relative substrate conversion and yield. The performance of reactor is characterized using the dimensionless parameters of Damkohler number, Stanton number, Peclet number, the equilibrium constant and the dimensionless input substrate concentration. The reactor performance is described for the two extreme cases of plug flow reactor (PFR) and continuous stirred tank reactor (CSTR) in addition to the intermediate case of dispersed plug flow reactor (DPFR). The performance of reactor is compared for the two cases of zero order and reversible first order kinetics.

Bioremediation of uranium-bearing wastewater: Biochemical and chemical factors influencing bioprocess application

A biotechnological process for the removal of heavy metals from aqueous solution utilizes enzymatically liberated phosphate ligand which precipitates with heavy metals (M) as cell-bound MHPO(4). The enzyme, a phosphatase, obeys Michaelis-Menten kinetics in resting and immobilized cells; an integrated form of the Michaelis-Menten equation was used to calculate the apparent K(m) (K(m app.)) as operating in immobilized cells in flow-through columns by a ratio method based on the use of two enzyme loadings (E(o1), E(o2)) or two input substrate concentrations (S(o1), S(o2)). The calculated K(m app.) (4.08 mM) was substituted into an equation to describe the removal of metals by immobilized cells. In operation the activity of the bioreactor was in accordance with that predicted mathematically, within 10%. The initial tests were done at neutral pH, whereas the pH of industrial wastewaters is often low; an increase in the K(m app.) at low pH was found in previous studies. Immobilized cells were challenged with acidic mine drainage wastewaters, where the limiting factors were chemical and not biochemical. Bioreactors initially lost activity in this water, but recovered to remove uranyl ion with more than 70% efficiency under steady-state conditions in the presence of competing cations and anions. Possible reasons for the bioreactor recovery are chemical crystallization factors. (c) 1997 John Wiley & Sons, Inc.

Activated sludge treatment of wash waters derived from the purification of virgin olive oil in a new manufacturing process

AbstractEffluent derived from the washing of virgin olive oil in a new manufacturing process was found to be readily treated using a completely mixed activated sludge process. The biokinetic coefficients for the process were derived by operating laboratory‐scale plant at four different input chemical oxygen demand (COD) concentrations and four differing solids retention times. Results showed effluent COD to be correlated with input feed growth and proportional to the product of feed strength and specific growth rate. The proportional to the product of feed strength and specific grown rate. The multiple‐substrate model of Admas et al. fitted the data well and could be used productively to determine effluent COD from a wide range of input substrate concentrations.