Static and dynamic behavior of a class of unstructured models of continuous bioreactors with growth associated product

The static and dynamic behavior of a class of unstructured models of continuous bioprocesses, for which the product is growth associated, are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which both the rates of utilization of limiting substrate and product formation are linearly proportional to the specific cell growth rate. The kinetic expressions are allowed to assume general forms of substrate and nonbiomass product. The steady-state analysis allows the derivation of analytical results and the construction of a useful picture in the models' parameter space delineating the different static behavior these models can predict, including unique steady states and bistability. The analysis of the dynamic behavior allows the derivation of general analytical conditions for the occurrence of periodic behavior in the models. It is also shown that the subclass of these models for which the specific cell growth rate expression is monotonic with respect to the nonbiomass product is unable to predict a stable oscillatory behavior regardless of the expression of the growth rate. These results illustrate the fundamental weakness of this class of unstructured models in predicting transient behavior in continuous cultures. The effect of kinetic and operating parameters on the stability characteristics of these models is also investigated.     

Output regulation of a class of unstructured models of continuous bioreactors: steady-state approaches

This article deals with the output regulation of continuous bioreactors in the face of constant disturbances and inverse dynamics. Nonlinear controllers developed on the basis of approximate equilibrium manifolds can almost attenuate measurable or unmeasurable disturbances on the output. This nonlinear feed-forward/feedback control framework without any tuning parameters can be directly implemented to strictly nonlinear systems. Under dynamic actuator constraints and the availability of only output signals for use in the control law, closed-loop simulations demonstrate that the proposed control techniques are superior to a nonlinear PI control scheme based on the identified Hammerstein model.     

Intelligent modelling of bioprocesses: a comparison of structured and unstructured approaches

This contribution moves in the direction of answering some general questions about the most effective and useful ways of modelling bioprocesses. We investigate the characteristics of models that are good at extrapolating. We trained three fully predictive models with different representational structures (differential equations, differential equations with inheritance of rates and a network of reactions) on Saccharopolyspora erythraea shake flask fermentation data using genetic programming. The models were then tested on unseen data outside the range of the training data and the resulting performances were compared. It was found that constrained models with mathematical forms analogous to internal mass balancing and stoichiometric relations were superior to flexible unconstrained models, even though no a priori knowledge of this fermentation was used.Paper presented at the international conference on trends in monitoring and control of life science applications, 7–8 October 2002, Lyngby, Denmark.     

Chemostat cultures of yeasts, continuous culture fundamentals and simple unstructured mathematical models.

Fundamental aspects of chemostat cultures are reviewed. Using yeast cultures as examples, it is shown that steady states in chemostats may be predicted quantitatively by combining the correct number of unstructured kinetic models with expressions for existing stoichiometric constraints. The necessary number of such kinetic models corresponds to the number of limiting substrates and increases with the number of different metabolic pathways available to the strain. This is demonstrated by an experimental comparison of yeast growth limited by glucose alone for which metabolism is oxidative, and growth doubly limited by both glucose and oxygen, which occurs according to an oxido-reductive metabolism. The steady state data for such experiments can in principle be predicted based on a minimal amount of information by a simple stoichiometric model. It represents the overall stoichiometry of growth by a superposition of a fully oxidative and a fully reductive growth reaction and uses the concept of "aerobicity" to characterize the relative importance of the two reactions.     

Control strategies of continuous bioprocesses based on biological activities

Based on the material balance principle applied to microbial reactions in continuous bioprocesses, the concept of reaction rate control has been developed theoretically. This concept provides a more direct way of controlling biological activities than the control of physical or chemical parameters in practice today. From an analysis of dynamic and steady-state experiments, two control systems for carbon dioxide production rate control during the continuous culture of baker's yeast have been designed and evaluated experimentally. In these control methods, intracellular NADH concentration is used as an immediate indication of the onset of glucose repression. A more sophisticated master controller based on the respiratory quotient can be combined with these control methods. The resulting control system provides a means to indirectly optimize biomass production while preventing ethanol formation in the continuous culture of baker's yeast.     

Fundamental bottlenecks in the application of continuous bioprocesses.

Continuous culture is applied mainly as a research tool and much less as a production process. Fundamental bottlenecks in continuous culture are discussed to help shed light on this apparent contradiction. Based on a discussion of technical, process related, and economic/market bottlenecks it is concluded that the often mentioned productivity argument in favor of continuous processing is much too simple. The optimal choice of a process mode is determined by a full understanding of the equipment and production plant factors and of the economic/market factors. Very often the resulting choice will be the fed batch and/or the cell retention process mode which is characterized by low growth rates. Therefore more research towards product formation at low growth rates (less than 0.05 h-1) is needed.     

A new training method for hybrid models of bioprocesses

Modeling of bioprocesses for engineering applications is a very difficult and time consuming task, due to their complex nonlinear dynamic behavior. In the last years several propositions for hybrid models, and especially serial approaches, were published and discussed, in order to combine analytical prior knowledge with the learning capabilities of Artificial Neural Networks (ANN). These approaches often require synchronous and equidistant sampled training data. However, in practice concentrations are mostly off-line measured, rare, and asynchronous. In this paper a new training method especially suited for very few asynchronously sampled data is presented and applied for modeling animal cell cultures. The achieved model is able to predict the concentrations of the reaction components inside a stirred tank bioreactor.     

Long-time behavior of continuous time models in genetic algebras

In [2] the solutions of Andreoli's differential equation in genetic algebras with genetic realization were shown to converge to equilibria. Here we derive an explicit formula for these limits.     

Estimation of multiple biomass growth rates and biomass concentration in a class of bioprocesses

In this paper, an approach to the estimation of multiple biomass growth rates and biomass concentration is proposed for a class of aerobic bioprocesses characterized by on-line measurements of dissolved oxygen and carbon dioxide concentrations, as well as off-line measurements of biomass concentration. The approach is based on adaptive observer theory and includes two steps. In the first step, an adaptive estimator of two out of three biomass growth rates is designed. In the second step, the third biomass growth rate and the biomass concentration are estimated, using two different adaptive estimators. One of them is based on on-line measurements of dissolved oxygen concentration and off-line measurement of biomass concentrations, while the other needs only on-line measurements of the carbon dioxide concentration. Simulations demonstrated good performance of the proposed estimators under continuous and batch-fed conditions.     

Classification of continuous diffraction patterns: a numerical study

The very intense and short pulses of future X-ray free electron lasers may allow the atomic resolution imaging of small, non-periodic objects. Preliminary estimates show that images obtained from single pulses do not contain statistically enough photons to allow successful reconstruction. Therefore multiple exposures of randomly oriented identical replicas have to be taken and the individual images have to be classified according to the object's orientation. The classification has been analytically treated by Huldt et al. [Huldt, G., Szoke, A., Hajdu, J., 2003. J. Struct. Biol. 144, 219.]. In this paper we extend the analytical results with numerical model calculations. This allows us to simulate realistic situations, which we will face in real experiments. We find significant deviations from the analytical expectations, even in the ideal case of spherical particles with random atomic distributions. We introduce a new norm for the individual scattering patterns and describe a criterion to select images belonging to similar orientation, which makes the classification more reliable in practice. We also discuss the effects of particle shape and size, partial orientational ordering, the measurement's resolution and the charge error caused by the Coulomb explosion.     

Discrete and continuous mathematical models of intraspecific behavior in the pharmacoethology of the mouse

An algorithm and software library were compiled in order to interpret the intraspecies agonistic animal behaviour in terms of discrete or continuous mathematical model. Applied aspects of the use of mathematical models in pharmacoethology were shown on concrete examples. The ways of construction of standard prototypes, and the integrative criteria of psychotropic drugs action were developed. The possibility was shown of identification of unknown substances by comparing with standard drugs by calculating the norm of standardized matrices.     

Kinetic studies and unstructured models of lymphocyte metabolism in fed-batch culture

The growth of two lymphocyte cell lines, a hybridoma cell line and a human cutaneous T cell lymphoma (HuT78), was studied in fed-batch culture, and unstructured models of growth developed. A criteria was established to insure that the growth rate varied by less than a specified tolerance throughout the culture period. Glutamine and serum were growth-limiting nutrients for both cell lines with half-maximal growth rates at 0. 53 mM glutamine and 0. 55%(v/v) serum for the hybridoma cells and 0. 21 mM glutamine and 1. 5% serum for the HuT-78 cells. Over the range of glucose concentrations from 5. 5 mM to 28 mM, the specific growth rate of hybridoma cells was independent of glucose concentration, whereas glucose concentrations above 5. 5 mM inhibited HuT-78 growth. For both cell lines, the growth rate was significantly inhibited by the addition of ammonium, although the hybridoma cell line was more affected by ammonia than was the HuT-78 cell line. Growth of HuT-78 cells increased in the presence of interleukin-2. Unstructured models for the hybridoma cells were similar to other models presented in the literature. Applications of these models to adoptive immunotherapy are discussed.     

Congruent epidemic models for unstructured and structured populations: analytical reconstruction of a 2003 SARS outbreak

Both the threat of bioterrorism and the natural emergence of contagious diseases underscore the importance of quantitatively understanding disease transmission in structured human populations. Over the last few years, researchers have advanced the mathematical theory of scale-free networks and used such theoretical advancements in pilot epidemic models. Scale-free contact networks are particularly interesting in the realm of mathematical epidemiology, primarily because these networks may allow meaningfully structured populations to be incorporated in epidemic models at moderate or intermediate levels of complexity. Moreover, a scale-free contact network with node degree correlation is in accord with the well-known preferred mixing concept. The present author describes a semi-empirical and deterministic epidemic modeling approach that (a) focuses on time-varying rates of disease transmission in both unstructured and structured populations and (b) employs probability density functions to characterize disease progression and outbreak controls. Given an epidemic curve for a historical outbreak, this modeling approach calls for Monte Carlo calculations (that define the average new infection rate) and solutions to integro-differential equations (that describe outbreak dynamics in an aggregate population or across all network connectivity classes). Numerical results are obtained for the 2003 SARS outbreak in Taiwan and the dynamical implications of time-varying transmission rates and scale-free contact networks are discussed in some detail.     

Multilayer adaptive control of continuous bioprocesses using optimising control technique. Case study: Bakers' yeast culture

High costs associated with many fermentation processes in an increasingly competitive industry make any prompt application of modern control techniques to industrial bioprocesses very desirable. However, this is often hampered by the lack of adequate mathematical models, on the one hand, and by the absence of continuous, on-line measurement of the most relevant process variables, on the other hand. This paper addresses these problems and offers a new strategy to control continuous bioprocesses using a hierarchical structure such that neither structured process models nor continuous measurement of all relevant variables have to be available. The control system consists of two layers. The lower layer represents a dynamic adaptive follow-up control of a continuously measured output — in our case dissolved oxygen concentration. This variable is supposed to be strongly correlated with the key output variable — in our case cellular concentration which is not continuously available for measurement. The higher layer is then designed to maintain a desired profile of the process key output using a set-point optimising control technique. The Integrated System Optimisation and Parameter Estimation method used operates on an appropriately chosen steady-state performance criterion. A prerequisite for successful application of the proposed approach is an approximate steady-state model, describing the relationship between the measured output and the process key output variable. Furthermore, occasional in situ, off-line or laboratory measurement values of the key output variable are needed. Promising simulation results of the biomass concentration control, by manipulating the air flow-rate in the continuous bakers' yeast culture are presented.     

LFT models of continuous biotechnological processes

The aim of the paper is to obtain suitable state-space models of continuous biotechnological processes (CBTP) in the framework of Linear Fractional Transformation (LFT). The LFT models are starting point in most of the advanced robust control design and analysis methods. Therefore, a linearized process model in the state-space is used whose elements are supposed to vary within certain bounds to represent the nonlinear behaviour of the real plant. The performance specifications are defined in the frequency domain through weighting functions. Two LFT models of CBTP are obtained ready for controller design aimed to optimize robust stability margins and robust performance, respectively.     

Reactivity of a new class of P450 enzyme models

P450 enzyme models carrying a SO3(-) ligand coordinating to iron have been synthesized and characterized. These complexes show characteristics very similar to those of iron-heme cofactors of P450 enzymes. Their reactivity towards different reactions catalyzed by P450 enzymes such as epoxidation of double bonds, hydroxylation of non-activated C-H bonds, N-dealkylation of amines, and cleavage of diols (C-C-bond cleavage) has been investigated.     

Discrete and continuous state population models in a noisy world

Simple ecological models operate mostly with population densities using continuous variables. However, in reality densities could not change continuously, since the population itself consists of integer numbers of individuals. At first sight this discrepancy appears to be irrelevant, nevertheless, it can cause large deviations between the actual statistical behaviour of biological populations and that predicted by the corresponding models. We investigate the conditions under which simple models, operating with continuous numbers of individuals can be used to approximate the dynamics of populations consisting of integer numbers of individuals. Based on our definition for the (statistical) distance between the two models we show that the continuous approach is acceptable as long as sufficiently high biological noise is present, or, the dynamical behaviour is regular (non-chaotic). The concepts are illustrated with the Ricker model and tested on the Tribolium castaneum data series. Further, we demonstrate with the help of T. castaneum's model that if time series are not much larger than the possible population states (as in this practical case) the noisy discrete and continuous models can behave temporarily differently, almost independently of the noise level. In this case the noisy, discrete model is more accurate [OR has to be applied].     

Stability in a class of cyclic epidemic models with delay

A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.