Steady-state multiplicity in bioreactors: bifurcation analysis of cybernetic models

Biological systems have an additional level of complexity compared to other chemical systems because of the effects of metabolic regulation, a defining feature of biosystems. Metabolic regulation in the form of control of enzyme synthesis and activity leads to non-linear behavior in bioreactors. Mathematical models that take into account these control mechanisms can be very successful in capturing the peculiarities of bioreactors such as multiple steady states and periodic phenomena. Cybernetic models model the expression and activation of enzymes by the use of cybernetic control variables and have been used to explain multiplicities in hybridoma reactors. In particular Namjoshi et al. (Biotechnol. BioEng. (2002), accepted) have been able to predict the transition from batch and fed batch to continuous culture in hybridoma experiments (Biotechnol. BioEng. 67(1) (2000) 25). The resulting multiple steady states vary widely in cell mass and waste metabolites. The model captures this multiplicity and its bifurcation analysis has revealed additional steady-state branches, three unstable and one stable. The stability of the additional steady state (steady state 4) was confirmed by dynamic simulations using fed-batch strategy prior to initiation of continuous operation. Steady state 4, in view of its close proximity to steady state 3, appears to have little practical significance. The likelihood of additional steady states that may be significantly different can, however not be ignored. Thus, it seems possible to envisage other states of metabolic activity, displaying alternative flux distributions that could lead to steady states notably different from those already determined. Calculation of the most singular point of the system herein is rendered difficult by both its size and possession of non-differentiable variables. The bifurcation analysis reveals the steady-state behavior under a range of operating conditions and can be used to plan optimum bioreactor operation.     

Traveling-wave Phenomena in a Model of Autocrine Signaling Coupled with Dynamics of the MAPK Cascade

Recently we have revealed a minimal reaction subnetwork in the MAPK (mitogen-activated protein kinase) cascade that is responsible for the emergence of bistable and oscillatory behavior. Here we examine a possible mechanism that provides for the propagation of increased MAPK activity in cell populations by interconnecting the intracellular MAPK subnetwork with the ligand-receptor signaling machinery. Such approach allows for significant reduction of the dimensionality of the parameter space on one hand and the conservation of dynamical complexity of the system on the other hand. The coupled model predicts coexistence of one, two or three different stable steady states, or the coexistence of a stable steady state and periodic solution. We found two robust and physiologically relevant characteristics of the proposed model: (i) There is a very large region of coexistence of at least one stable steady state with non-zero MAPK activity and one steady state with zero MAPK activity in the parameter space. (ii) Spontaneous traveling front waves always switching originally inactive cells into ligand releasing cells emerge in adjacent cell populations, e. g. in healthy and injured tissues. Moreover, the formation of composite traveling front waves and spatial oscillatory patterns of MAPK activity are predicted and discussed.     

Dynamics of a cascade of two continuous stirred tank polymerization reactors with a binary initiator mixture

The dynamic behavior of two continuous stirred tank reactors in series has been investigated for free radical solution polymerization of styrene with a binary mixture of two initiators having different thermal decomposition activities. For a wide range of initiator feed composition, both reactors exhibit quite complex nonlinear steady state and transient behavior. When the reactor residence time is used as a bifurcation parameter, the second reactor can have up to five steady states. For certain range of reactor operating conditions, bifurcations to various types of periodic solutions have been observed, such as Hopf bifurcation, isolas, period doubling, period-doubling cascade, and homoclinics. The effects of other reactor variables, such as total initiator concentration, coolant temperature, and reactor volume ratio on the reactor dynamics, are illustrated to show the complex dynamic behavior of the two-reactor system catalyzed by a mixture of t-butyl perbenzoate and benzoyl peroxide.     

Absorption with exothermic reaction in a falling film column

A gaseous reactant is absorbed in a liquid with a positive heat of absorption. It reacts with the liquid by an exothermic, irreversible first order reaction. Analysis of a CSTR for this process has shown that up to 5 steady state solutions may exist even though the system appears to be quite simple with only one reaction. The existence of an extraordinary large number of steady state solutions is explained by the coupling between absorption and reaction: A low temperature gives no reaction, a high temperature prevents absorption of the gas and hence leads to mass transfer control of the reaction, while an intermediate temperature may permit operation at a third (stable) steady state. In between the stable steady states one finds (as us two unstable steady states.We extend the analysis of the CSTR to the distributed system of a falling film reactor, and we treat both the transient situation in the entrance partAn eigenanalysis of the steady states shows that temperature rise across the gas film may lead to 3 stable steady states even if an isothermal model is     

Lag phase model for transient growth of pseudomonas putida on phenol

The successful design of large‐scale bioreactors requires the ability to predict both steady‐state and dynamic operating conditions. At the same time, mathematical models should not be too complex in order to reduce experimental work required to determine kinetic parameters. A simple model which predicts the behaviour of batch and transient continuous culture operations is presented and experimentally verified. The model is based on two regions of metabolic activity: the lag phase and the active phase. Pseudomonas putida growing on phenol as a substrate in a well‐mixed bioreactor was tested in three modes of operation: batch, continuous start‐up and continuous step‐change. The model is demonstrated to predict all the qualitative aspects of the dynamic phases of growth and is quantitatively accurate.     

Numerical and technological properties of bubble column bioreactors for aerobic processes

This paper describes the method of modelling and numerical simulation of bubble column bioreactors in which the process of aerobic biodegradation of carbonaceous substrate occurs. Such bioreactors belong to systems with distributed state variables. Determining steady states of such objects results in solving non-linear boundary-value problems.The bioreactors with axial dispersion and piston flow with biomass recirculation were analysed. The effectiveness of selected algorithms used to determine the steady states of such bioreactors were compared; the numerical properties of mathematical models of analysed bioreactors were specified, the branches of steady states in bioreactors with axial dispersion and piston flow were determined and the application areas of such bioreactors were defined.     

Biodegradation of phenol in the presence of heavy metals

A microbial growth model was presented for estimating the dynamic behavior of cell growth and substrate consumption in the biodegradation of phenol containing a heavy metal, such as zinc or copper ions. The application of the model for experiments in a batch culture and a continuous culture was examined. The values calculated according to the model corresponded satisfactorily with experimental data, such as the optical density of cells and the concentration of phenol. Using the model, the stability of steady states in a continuous culture was analytically evaluated based on the eigenvalues. The steady states were separated into three categories: (i) a stable steady state where phenol was consumed, (ii) an unstable steady state where phenol was consumed, and (iii) a washout state where phenol was not consumed. The number, kind, and stability of steady states varied significantly with operational conditions such as dilution rate, feed concentration of phenol, and feed concentration of heavy metal ions. The operational conditions required to obtain the stable steady state where phenol was consumed are shown graphically. © 2000 Society of Chemical Industry     

Mathematical modelling of fluidized bed reactors—III: Axial dispersion model

An axial dispersion model has been developed for a continuous fluidized bed catalytic reactor with a cocurrent flow of the emulsion phase gas and the catalyst particles. The influence of some parameters on multiplicity of steady states has been reported. Several examples illustrating the transient behavior of the system are presented. In cases where three steady states are possible it appears that the intermediate steady state is unstable, while the lower and the upper steady states are locally stable. It was noted that the initial temperature of the emulsion phase is a predominant factor in determining which steady state will be approached.     

Determination of multiple steady states in oxidation of monophenols by tyrosinase with enzymatic-enzymatic-chemical model

A family of an enzymatically catalyzed reaction network is studied, which involves the oxidation of monophenols by tyrosinase with enzymatic-enzymatic-chemical model in an isothermal continuous flow stirred tank reactor (CFSTR). This system consists of 11 coupled non-linear equations and is determined to have the capacity to exhibit computational multiple steady states. A set of rate constants and two corresponding steady states are computed. The phenomena of bistability, hysteresis and bifurcation are discussed. Moreover, the capacity of steady state multiplicity is extended to its family of reaction networks.     

Multiplicity of steady states in fluidized bed reactors—V: The effect of catalyst decay

A simplified two-phase model is used to investigate the behaviour of non-isothermal fluidized bed reactors experiencing catalyst decay. The investigation shows that for highly exothermic reaction it is almost always desirable to operate the reactor at the middle unstable steady state, since it gives higher accumulative yield than both the high and the low temperature steady states. A simple feedback control scheme with time varying set point is suggested to stabilize the middle steady state. The dynamic behaviour and stability of the system is investigated for the open-loop reactor (uncontrolled) and the closed loop reactor (controlled).     

Optimum CFST bioreactor design: Experimental study using batch growth parameters for saccharomyces cerevisiae producing ethanol

An optimum CFST bioreactor design equation is presented which can be used with linear growth-associated product inhibition models. Batch experiments were performed to determine Saccharomyces cerevisiae growth parameters required for solving the optimum design equation. The three bioreactors were designed in decreasing volumes to achieve theoretical glucose conversions of 99.9% and 99.99%. At steady state, the bioreactors appeared to reach the theoretical conditions as long as the flowrates were held equal or slightly below the optimum conditions. At very low flowrates, biomass death was observed in downstream reactors while at high flowrates, the theoretical conversions were not achieved. The experimental results confirm that optimally designed bioreactors can offer lower processing volumes to achieve high substrate conversions as compared to single CFST bioreactors.     

The effect of IR compensation on stationary and oscillatory patterns in dual-electrode metal dissolution systems

We report an experimental and model based study on the effect of negative coupling, induced by adding IR compensation, on bistability, and synchronization behavior of a dual-electrode metal dissolution electrochemical system. We show that, unlike the case of a single electrode, IR compensation cannot be used to remove bistability; with a large IR compensation the electrodes do not exhibit uniform steady states and patterned surface develops. In the case of oscillatory system, addition of IR compensation produces aperiodic time series that are characterized by switching between oscillations with 1:1, 1:2, and 2:1 entrainment ratios. For higher negative coupling strengths (i.e., larger magnitude of IR compensation) amplitude death occurs and either coexistence of oscillations with steady state or multiple anti-symmetric steady states are observed.     

Review of the important challenges and opportunities related to modeling of mammalian cell bioreactors

Industrialization of mammalian cell culture has been achieved by integrating knowledge from several applying core concepts of chemical engineering, cellular and molecular biology, and biochemistry. Modeling has been applied to biological and physical processes to gain additional insights into such processes. This article covers modeling of the bioreactor and metabolic processes as it applies to bioprocess. Hydrodynamics of a bioreactor is briefly described while additional focus is given to gas‐liquid mass transfer. Biological modeling is presented in the order of increasing complexity. First steady state models are presented followed by dynamic models, cybernetic models, and finally bioreactor integrated models. The closing discussion summarizes challenges of implementation of model‐based approaches in the biopharmaceutical industry. © 2016 American Institute of Chemical Engineers AIChE J, 63: 398–408, 2017     

Design of tubular reactors in recycle systems

The article presents an approach to design tubular reactors in recycle systems, based on non-linear analysis. A pseudo-homogeneous plug-flow reactor model is used. It is assumed that the separation unit delivers product and recycle streams with fixed composition. The stand-alone reactor has a unique stable steady state. The coupled reactor–separation–recycle system shows four types of conversion versus plant Damköhler number bifurcation diagrams. A feasible steady state exists only if the reactor volume exceeds a critical value. For isothermal reactor, the steady state is unique and stable. For non-isothermal reactor, one or two steady states are possible. In the second situation the low-conversion state is unstable. In some parameter regions, the unique state is unstable. The design should ensure state unicity and stability, which are favoured by large heat-transfer capacity, low coolant temperature and high reactor-inlet temperature. A case study demonstrates that these phenomena can be easily found in real plants.     

Two‐Parameter Continuation and Bifurcation Strategies for Oscillatory Behavior Elimination from a Zymomonas mobilis Fermentation System

Two‐parameter continuation and bifurcation analysis strategies were applied to deal with the oscillatory phenomena of a Zymomonas mobilis ethanol fermentation system. A structured verified non‐linear mathematical model considering the physiological limitations of microorganisms for a single continuous fermenter for ethanol production using Z. mobilis was built to identify the Hopf bifurcation (HB) points, which indicate the oscillatory behavior, using the inlet substrate concentration and the dilution rate as bifurcation parameters. The path of the HB points can be determined with different controlling operating parameters. It was found that with the addition of a small amount of cells or ethanol to the feed stream or by increasing the dilution rate, the oscillations could be eliminated and steady‐state behavior was attained. Using a two‐parameter continuation strategy, the Z. mobilis fermentation system could operate at steady state without oscillatory behavior.     

Experimental study of the behaviour of an isothermal continuous stirred tank reactor in the course of an autocatalytic reaction

The dynamic behaviour of an isothermal continuous stirred tank reactor and of a two-element series of continuous stirred tank reactors in the course of an autocatalytic reaction of bistrichlormethyl-trisulphide with aniline, in methanol at 20°C, was studied. It was shown that the regime of the continuous stirred tank reactor and of the series is unstable ia a certain region of the conversion of the initial reaction component (bis-trichlormethyl-trisulphide) and that in a certain interval of the reaction mixture feed rate the reactor can operate at a single value of the feed rate in two (the series in three) stable states. The values of the bis-trichlormethyl-trisulphide conversion in steady states, as well as the dependence of the conversion on the period of stabilization of the reactor regime, determined experimentally in a continuous stirred tank reactor and in the series are in good agreement with the values computed from the kinetic data, determined in a discontinuous stirred tank reactor.     

Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A → B → C reactions

We use a continuation technique for branches of periodic solutions to investigate the oscillatory behavior of a continuously stirred tank reactor with consecutive A → B → C reactions. This continuation technique allows the computation of entire periodic solution branches, including those with limit points and asymptotically unstable solutions. Our computations reveal dynamic phenomena not seen in previous studies of this reactor. The results include response diagrams exhibiting stable and unstable periodic branches that contain multiple limit points. The presence of these points indicates that the reactor may jump from a steady state to a periodic orbit or from one orbit to another. The computations also illustrate interactions of multiple steady state limit points, Hopf bifurcations and infinite periodic bifurcations.     

SUDDEN DESTABILIZATION OF CONTROLLED CHEMICAL PROCESSES

Input multiplicity occurs when more than one set of inputs can produce the same set of outputs. Input multiple steady states are divided into compatible steady states having process gains of similar sign, and opposed steady stales with process gains of opposite sign. For a system controlled with reset action, only the compatible steady states satisfy the necessary condition for stability. Any disturbance which drives the controlled system from the designed steady state to a less stable or unstable compatible steady state can cause sudden destabilization of the process. Several examples are given of the possible types of behavior resulting from this phenomenon.

Multiple steady states also occur for systems with proportional controllers. For single-input-single-output systems with continuous process characteristics, whether or not reset action is used, two steady states positioned next to each other cannot both be stable under closed-loop control. However, under proportional control, opposed steady states for which 1 + K_cK_p is positive can be stable.     

Performance Enhancement by Unsteady‐State Reactor Operation: Theoretical Analysis for Two‐Sites Kinetic Model

Theoretical analysis of the reactor performance under unsteady‐state conditions was carried out. The reactions are described by two kinetic models, which involve the participation in catalytic reaction of two types of active sites. The kinetic model I assumes the blocking of one of the active sites by a reactant, and the kinetic model II suggests a transformation of active sites of one type into another under the influence of the reaction temperature. The unsteady‐state conditions on the catalyst surface are supposed to be created (i) by forced oscillations of temperature and concentration in the reactor inlet (periodic operation of reactor) and (ii) by catalyst circulation between two reactors in a dual‐reactor system (spatial regulation). The influence of various parameters like concentration of reactant, cycle split, length of period of forced oscillations, temperatures and the ratio of catalyst volumes in the dual‐reactor was investigated with respect to the yield of the desired product. It is shown that for both cases of unsteady‐state conditions (periodic reactor operation as well as in a dual‐reactor system), a mean reaction rate predicted by the kinetic model I was up to two times higher than the steady‐state value. The kinetic model II shows a 20 % increase of the selectivity towards the desired product.