Classification of Static and Dynamic Behavior in Chemostat for Plasmid-Bearing,Plasmid-Free Mixed Recombinant Cultures

The singularity theory is combined with continuation techniques to classify the static and dynamic behavior in a chemostat involving the competition between plasmid-bearing and plasmid-free cell populations. The analysis of the static bifurcation allows the derivation of analytical conditions for the coexistence of the competing populations and for the safe operation of the’bioreactor. The analysis of dynamic bifurcation, on the other hand, shows the ability of the model to predict the coexistence of the two populations in a state of stable limit cycle. Analytical conditions with respect to any growth kinetics are derived for the occurrence of Hopf points in the model. The combination of results of both static and dynamic bifurcation helps to construct a useful picture, in the multidimensional parameter space, of the different behavior predicted by the model.     

Classification of Static and Dynamic Behavior in Chemostat for Plasmid-Bearing, Plasmid-Free Mixed Recombinant Cultures

The singularity theory is combined with continuation techniques to classify the static and dynamic behavior in a chemostat involving the competition between plasmid-bearing and plasmid-free cell populations. The analysis of the static bifurcation allows the derivation of analytical conditions for the coexistence of the competing populations and for the safe operation of the'bioreactor. The analysis of dynamic bifurcation, on the other hand, shows the ability of the model to predict the coexistence of the two populations in a state of stable limit cycle. Analytical conditions with respect to any growth kinetics are derived for the occurrence of Hopf points in the model. The combination of results of both static and dynamic bifurcation helps to construct a useful picture, in the multidimensional parameter space, of the different behavior predicted by the model.     

Dynamics analysis of an age distribution model of oscillating yeast cultures

The ability of an age-population balance model to capture experimentally observed oscillatory dynamics of continuous cultures of budding yeast was investigated through numerical simulations. Experiments with continuous yeast cultures have shown that several oscillatory modes can occur at the same operating condition, and that the mode attained depends on the start-up conditions. Numerical simulations of the model did reveal the existence of several stable periodic solutions. However, each occurred over a different range of dilution rates. Experiments also have shown that the steady state in continuous yeast cultures is stable, even under conditions that allow oscillatory dynamics. The stability of the steady state of the age population balance model under conditions that allow oscillatory dynamics was not resolved. The Jacobian matrix at the steady state is highly ill conditioned, with some eigenvalues very close to the imaginary axis. Using different integration routines to solve the model gave different results with regard to the stability of the steady state, one solver finding the steady state to be stable, another finding the steady state to be unstable.     

Fate of commensalistic cultures in identical coupled bioreactors

A comprehensive analysis of static and dynamic behavior of a mixed culture in two identical coupled bioreactors is presented considering anaerobic digestion involving acidogens (X) and methanogens (Y) as the example bioprocess. A single continuous culture may operate at up to seven steady states, including up to four coexistence steady states, with only one coexistence steady state being locally stable. The one-way interaction between X and Y allows for compartmentalization of the system for a stand-alone bioreactor and two coupled bioreactors into two subsystems, which facilitates the analysis of steady state types and stability characteristics of these and classification of dynamic behavior. The bioreactors in the two-reactor system are identical only in terms of feed composition and reactor space time. A two-reactor system may admit up to forty nine steady states, which are comprised of up to forty coexistence steady states, at least at very low interaction rate (R). The static and dynamic analysis of the two-reactor system is facilitated by appropriate grouping of large number of steady states arising for very low R into nine clusters. Numerical illustrations reveal the rich steady state structure of the bioprocess in coupled bioreactors. While a single bioreactor can operate at only one locally stable coexistence steady state, the coupled bioreactors can operate at up to five locally stable coexistence steady states over certain ranges of R. The two-reactor system is operationally more flexible and more robust vis-a-vis single reactor as concerns maintenance of mixed culture. Emergence of four additional steady state clusters and additional coexistence and partial washout steady states at intermediate R reveals that the coupled bioreactors are an example of a complex system.     

Chemical Waves in Embryonic Cell Cycles

In most metazoans, early embryonic development is characterized by a rapid series of cleavage divisions. At the core of the coordination of these divisions is the oscillatory dynamic of Cyclin-dependent kinase 1 (Cdk1), which arises through the integration of a negative feedback with delay and positive feedback. The regulation of this oscillator in large embryos is emerging as an ideal model for quantitative studies of how spatiotemporal coordination is achieved in large complex tissues. Recent work has demonstrated that the reaction-diffusion dynamics regulating the cell cycle can generate traveling waves of enzymatic activity, which ensure the synchronization of cell division processes. Here, we will review how the mechanisms of cell cycle regulation can give rise to chemical waves, and highlight recent experiments on the coordination of cell division through traveling waves.     

Multiplicity,stability, and oscillatory dynamics of the tubular reactor

Numerical bifurcation techniques were developed for studying the multiplicity, stability, and oscillatory dynamics of the nonadiabatic tubular reactor with a single A → B reaction. The techniques illustrate the existence of one, three, five, or seven steady states and bifurcating periodic solutions. We present numerical procedures for computing the Hopf bifurcation formulas which can determine the stability and location of the oscillation without integrating the parabolic partial differential equations. The combination of our Hopf techniques with steady state bifurcation methods enables us to determine all possible steady and stable oscillatory solutions exhibited by distributed parameter models such as the tubular reactor.     

Dissipative structures of autocatalytic reactions in tubular flow reactors

Dissipative structures of autocatalytic reactions with initially uniform concentrations are studied in tubular flow reactors. A unique steady state exists in a continuous stirred tank reactor. Linear stability analysis predicts either a stable node, a focus or an unstable saddle-focus. Sustained oscillations around the unstable focus can occur for high values of Damköhler number. In distributed parameter systems, travelling, standing or complex oscillatory waves are detected. For low values of Damköhler number, travelling waves with pseudo-constant patterns are observed. With intermediate values of Damköhler number, single or multiple standing waves are obtained. The temporal behavior indicates also the appearance of retriggering or echo waves. For high values of Damköhler number, both single peak and complex multipeak oscillations are found. In the cell model, both regular oscillations near the inlet and chaotic behavior downstream are observed. In the dispersion model, higher Peclet numbers eliminate the oscillations. The spatial profile shows a train of pulsating waves for the discret model and a single pulsating or solitary wave for the continuous model.     

Periodic behaviour of a class of unstructured kinetic models for continuous bioreactors

The periodic behaviour of a large dass of unstructured kinetic models for continuous bioreactors is analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which the utilization rate of the limiting substrate is linearly related to the rates of cell growth and product formation. The model kinetics are allowed, on the other hand, to depend on substrate, biomass and product. The stability analysis allows the derivation of general analytical conditions for the occurrence of periodic behaviour in these models. It is shown that for a number of important cases, the occurrence of oscillatory behaviour is conditioned mainly by the kinetics of product formation. The singularity theory also allows the construction of a useful picture in the multidimensional parameter space delineating the different behaviour these models can predict including bistability and stable oscillatory behaviour.     

Multistablity, bistability and bubbles phenomena in a periodically forced ethanol fermentor

The numerical investigation results of the dynamic behavior of an ethanol fermentor excited by external sinusoidal periodic perturbations are reported. The characteristics of the discriminant of the cubic polynomial of the steady state autonomous model of the unforced fermentor were used to divide the parameter space into regions with different number of steady state solutions. The bioethanol fermentor exhibits interesting complicated dynamic behavior when the center of forcing is close to a static limit point (SLP) i.e. the discriminant=0. Numerical simulations have presented evidence for the existence of multistability, bistability and bubbles phenomena in the forced bioethanol fermentor. This is – to the best of our knowledge – the first study that shows these phenomena in the forced bioethanol fermentor. Multistablity is characterized by the coexistence of several attractors. It has been shown that, the multistability exhibits the coexistence of three attractors, two of them are chaotic and one quasi-periodic, and the bistability exhibits the coexistence of two attractors, one of them is periodic and the other is either periodic, quasi-periodic or chaotic attractor. In the multistabiliy region, we have observed that changing the forcing amplitude with the sequence of events there is an appearance of the bistability and then a transition to multistabilty. It has been shown that, the bubble region acts as a period doubling killer, which has a lethal effect in killing the universal period doubling scenario to chaos by reversing the sequence to period halving. It is also shown that, the unforced (14.55% increase in ethanol average concentration relative to the steady state operation) or forced (8.87% increase in ethanol average concentration relative to the steady state operation) unsteady state operations give better fermentor performance than the steady state operations with respect to the average ethanol concentration and yield. The investigation shows that the nature and the position of the center of forcing have significant effect on the dynamic response of the periodically forced fermentors. It has been found that, the forcing could be beneficial or harmful to the fermentor performance depending on the position of center of forcing. The system shows interesting phase planes at certain forcing amplitudes.     

Multiplicity and stability approach to the design of heat integrated multibed PFR

The non-linear behaviour of the heat-integrated multibed plug flow reactor, consisting of feed-effluent heat exchanger (FEHE), furnace, multibed adiabatic tubular reactor with intermediate cooling, and steam generator is studied. A first order, reversible, exothermic reaction is considered. We calculate the hysteresis, isola, boundary-limit, double-zero and double-Hopf varieties. They divide parameter space into regions with different steady state and dynamic behaviour. State multiplicity, isolated branches and oscillatory behaviour may occur for realistic values of model parameters. Implications on design are discussed.     

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.     

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.     

Multiplex shear stress‐induced nucleation in dynamic microcellular foaming process

The effect of multiplex shear stress on the cell nucleation during microcellular foaming process was investigated using a dynamic foaming experimental apparatus. The multiplex oscillatory shear, which is different from previous one‐dimensional screw shear in a “stable” extrusion foaming process, is applied to the polymer melt through an axially vibrated rotor. The experimental results show that, by superimposing an axial vibration on the rotating rotor, the cell density increases and cell size decreases significantly when the shear rate is low. Both the uniformity of cell size and cell distribution are improved under vibration when compared with that without vibration regardless of how the shear rate changes. In addition, a simplified nucleation model based on shear energy has been carried out to qualitatively investigate the effect of both the simple steady shear and the multiplex oscillatory shear on the cell nucleation. Experiments and theoretical predictions all show that cell nucleation could be greatly improved by superimposing the oscillatory shear when the nucleation driving force induced by the steady shear is insufficient. Finally, the shear heat generated by excessive shear and strong vibration should be considered carefully although the isothermal condition was supposed in the present model. POLYM. ENG. SCI. 46:1728–1738, 2006. © 2006 Society of Plastics Engineers.     

Undamped oscillations in bacterial glycolysis models

The exotic dynamical behaviors exhibited in chemical reaction systems, such as multiple steady states, undamped oscillations, chaos, and so on, often result from unstable steady states. A bacterial glycolysis model is studied, which involves the generation of adenosine triphosphate (ATP) in a flow system and consists of eight species and ten reactions. A minimum subnetwork of the bacterial glycolysis model is determined to exhibit an unstable steady state with a positive real eigenvalue, which gives rise to undamped oscillations for a small perturbation. A set of rate constants and the corresponding unstable steady state are computed by using a positive real eigenvalue condition. The phenomena of oscillations and bifurcation are discussed. These results are extended to the bacterial glycolysis model and several parent networks.     

Dynamics of a CSTR for styrene polymerization initiated by a binary initiator system

The steady state and dynamic behavior of a continuous stirred tank reactor has been analyzed for free radical solution polymerization of styrene initiated by a mixture of two initiators having different thermal stabilities. From the steady state analysis of the reactor model with a mean residence time as a bifurcation parameter, four unique regions of steady state solutions are identified in an operating parameter space for a given initiator feed composition. A variety of complex bifurcation behavior such as multiple steady states, Hopf bifurcation and limit cycles have been observed and their stability characteristics have been analyzed. The effects of feed initiator composition and the concentration of the initiator in the feed stream on the reactor dynamics are also presented.     

Studies in the control of tubular reactors—II Stabilization by modal control

The lumped parameter model that was developed in Part I is linearized to obtain the linear dynamical model of the system near an unstable steady state. When concentration and temperature measurements are possible along the reactor length and their number is the same as the number of collocation points, modal state-feedback controllers are designed to relocate the largest eigenvalues to negative values and thus locally stabilize an unstable steady state. Transient calculations of the non-linear system equations are preformed and the domain of attraction of the stabilized steady state is examined for different locations of the eigenvalues of the closed loop system. It is seen that for both problems I and II the domain of attraction becomes very large when the unstable and one stable eigenvalue are shifted near the third largest one. This is true in spite of the large differences in their dynamical characteristics.     

Systematic performance analysis of continuous processes subject to multiple input cycling

Potential improvement in performance of continuous processes via weak periodic perturbations in one or more process inputs is systematically analyzed using the generalized π-criterion. At very low forcing frequencies, the performance of a periodically forced continuous process is tied to the second total differentials of the objective function at steady state. Periodic forcing of multiple inputs and the role of binary interactions between multiple inputs are analyzed in detail. Where permissible, conditions for properness of periodic control are derived and expressions for optimum phase differences (between input pairs), optimum amplitude ratios, and the maximum performance improvement vis-a-vis steady state operation are obtained. The powerful analytical results are applicable for any continuous process. Two illustrations, one dealing with antibiotic selection in recombinant cell cultures and the other dealing with series-parallel reactions, are considered for numerical study. Portions of the operating parameter space (base values of input variables and forcing frequency) where periodic operations involving weak variations in one or more inputs are superior to optimal steady state operation are identified for each illustration. An increase in the number of inputs perturbed is demonstrated to lead to broadening of the regions in the operating parameter space where forced periodic operations are superior to steady state operation and also to improved process performance. Both analytical and numerical results reveal the significant benefits of binary input interactions.     

Effect of longitudinal mixing on microbial growth in a multi-stage column bioreactor

The performance of a multi-stage column bioreactor for nonlimited microbial growth was studied. The back-flow model was employed to describe the flow pattern of the microbial suspension. Analytical solutions for calculating the microbial cell concentrations under steady-state and unsteady-state operations, with and without cells in feed, were obtained and the cell growth behavior was examined. Only when microbial cells were present in the feed and the reaction number was less than a critical number was a stable steady state achieved: the microbial cell concentration was then significantly affected by the extent of longitudinal mixing. In the absence of microorganisms in the feed, the steady state was not stable with either washout or nonlimited growth resulting.     

Isothermal oscillations in surface reactions with coverage independent parameters

A procedure for obtaining necessary and sufficient conditions for the existence of periodic solutions in surface reactions with constant temperature and coverage independent parameters is described. An analytic method for the analysis of bifurcation to periodic solutions is developed. This formulation, based on the fundamental matrix of a system of ordinary differential equations, provides a simple means of examining the behavior of system in the neighborhood of a singular point. This method is applied into a reaction model which can yield oscillatory solutions with both single and multiple steady states. When a unique steady state becomes unstable, a limit cycle develops. With three steady states, a state may develop into a stable or unstable limit cycle or may exhibit only periodic solutions without a limit cycle.     

Hydrodynamical instability of spatially extended bistable chemical systems

Hydrodynamical density fingering of chemical fronts separating two miscible, stable steady states of different chemical composition, and hence density, can lead to complex spatio-temporal dynamics. The most striking feature of such dynamics is the disconnection of droplets of one stable steady state from fingers invading the other stable steady state. Such disconnected droplets do not exist in pure density fingering and are thus the result of the bistable kinetics. We study such dynamics by direct numerical simulations of Darcy's law for flow in Hele-Shaw cells coupled to the kinetic equation for the concentration of a chemically reacting solute controlling the density of the miscible solutions. The concentration of this solute obeys a simple cubic model leading to bistability. Experimental realization of such dynamics in spatially extended Hele-Shaw cells calls for the use of the concept of spatial bistability which implies construction of new continuously fed open reactors.