Dynamics of continuous stirred-tank biochemical reactor utilizing inhibitory substrate

The model of a continuous-stirred tank biochemical reactor was developed in which the instant uptake rate of substrate was used. The solutions of the model found for the oxidation of phenol by Pseudomonas putida fitted the experimental data better than the results obtained from the models cited in the literature. The model enables control of the culture parameters so that the unwanted washout of the biomass from the bioreactor can be avoided. A review of the models cited in the literature is also presented.     

Bifurcation analysis of piecewise smooth ecological models

The aim of this paper is the study of the long-term behavior of population communities described by piecewise smooth models (known as Filippov systems). Models of this kind are often used to describe populations with selective switching between alternative habitats or diets or to mimic the evolution of an exploited resource where harvesting is forbidden when the resource is below a prescribed threshold. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations.     

Efficient nonlinear predictive control of a biochemical reactor using neural models

This paper describes the application of artificial neural networks to modelling and control of a continuous fermentor. A computationally efficient nonlinear model predictive control (MPC) algorithm with nonlinear prediction and linearisation (MPC-NPL) which needs solving on-line a quadratic programming problem is developed. It is demonstrated that the algorithm results in closed-loop control performance similar to that obtained in nonlinear MPC, which hinges on full on-line non-convex optimisation. The computational complexity of the MPC-NPL algorithm is shown, control accuracy and robustness are also demonstrated in the case of noisy measurements and disturbances affecting the process.     

Robustness analysis of biochemical network models

Biological systems that have been experimentally verified to be robust to significant changes in their environments require mathematical models that are themselves robust. In this context, a necessary condition for model robustness is that the model dynamics should not be sensitive to small variations in the model's parameters. Robustness analysis problems of this type have been extensively studied in the field of robust control theory and have been found to be very difficult to solve in general. The authors describe how some tools from robust control theory and nonlinear optimisation can be used to analyse the robustness of a recently proposed model of the molecular network underlying adenosine 3',5'-cyclic monophosphate (cAMP) oscillations observed in fields of chemotactic Dictyostelium cells. The network model, which consists of a system of seven coupled nonlinear differential equations, accurately reproduces the spontaneous oscillations in cAMP observed during the early development of D. discoideum. The analysis by the authors reveals, however, that very small variations in the model parameters can effectively destroy the required oscillatory dynamics. A biological interpretation of the analysis results is that correct functioning of a particular positive feedback loop in the proposed model is crucial to maintaining the required oscillatory dynamics.     

Bifurcation analysis of periodic SEIR and SIR epidemic models

The bifurcations of the periodic solutions of SEIR and SIR epidemic models with sinusoidally varying contact rate are investigated. The analysis is carried out with respect to two parameters: the mean value and the degree of seasonality of the contact rate. The corresponding portraits in the two-parameter space are obtained by means of a numerical continuation method. Codimension two bifurcations (degenerate flips and cusps) are detected, and multiple stable modes of behavior are identified in various regions of the parameter space. Finally, it is shown how the parametric portrait of the SEIR model tends to that of the SIR model when the latent period tends to zero.     

Transient state analysis of biochemical reactor using coimmobilized system

The transient state analysis of the consecutive sequence of reactions S P ₁ P ₂ taking place inside a porous spherical coimmobilized biocatalyst is discussed for the case in which each step follows Michaelis Menten type kinetics. The theoretical analysis includes intraparticle diffusional limitations. The model equations are solved by the explicit finite difference method. The effect of various parameters of importance on the batch reactor performance is discussed. Comparison of the model with experimental results has been shown.List of Symbols c _p Dimensionless substrate concentration inside the particle, (s _p/ss _o) - c _{pi, j} Dimensionless substrate concentration inside the particle at i, j - c _s Dimensionless substrate concentration at the surface of the particle, (s _s/s ₀) - d _p cm particle diameter - D _s, D _p cm²/s Diffusion coefficient of the substrate S and intermediate P ₁ inside the particle respectively - h Space step size inside the particle - i Grid point inside the particle - j Grid point along the time coordinate - k Time step size - K _m1, K _m2 g/l Michaelis constants for the first and second reaction respectively - K _I1,K _I2 g/l Substrate inhibition parameters for first and second reaction respectively - P _m g/l Product inhibition parameter for the second reaction - P _1p , P _1s g/l Concentration of the intermediate inside the particle and at the surface of the particle respectively - P _2p , P _2s g/l Concentration of the product P ₂ inside the particle and at the surface of the particle respectively - p _1p Dimensionless intermediate concentration inside the particle, (p _1p/s₀) - p _1s Dimensionless intermediate concentration at the surface of the particle, (p _1s /S ₀) - P _2p Dimensionless product concentration inside the particle, (p _2p /S₀) - p _2s Dimensionless product concentration at the surface of the particle, (p _2s/S₀) - p _{1pi, j} Dimensionless intermediate concentration inside the particle at i, j - P _{2pi, j} Dimensionless product concentration inside the particle at i, j - q Ratio of diffusion coefficients, D _p/D _s - r cm Radial position inside the particle - R cm Radius of the pellet - S ₀ g/l Initial substrate concentration in the bulk liquid - S _p g/l Substrate concentration inside the particle - S _s g/l Substrate concentration at the surface of the particle - t s Time, - V _max1 g/(ls) Maximum reaction velocity for the first reaction - V _max2 g/(ls) Maximum reaction velocity for the second reaction - y Dimensionless radial distance, (r/R) - y _{1, j} Dimensionless radial distance at i, j Greek Letters ₁ Parameter, S ₀/K _m1 - ₂ Parameter, S ₀/K _m2 - _I1 Parameter, S ₀/K _I1 - _I2 Parameter, S ₀/K _I2 - _I3 Parameter, S ₀/P _m - Dimensionless time defined as (D _s t/R ²) - ₁ ² V _max1R ²/K_m1D_s - ₂ ² V _max2R ²/K_m2D_s     

Bifurcational and dynamical analysis of a continuous biofilm reactor

A dynamical model of a continuous biofilm reactor is presented. The reactor consists of a three-phase internal loop airlift operated continuously with respect to the liquid and gaseous phases, and batchwise with respect to the immobilized cells. The model has been applied to the conversion of phenol by means of immobilized cells of Pseudomonas sp. OX1 whose metabolic activity was previously characterized (Viggiani, A., Olivieri, G., Siani, L., Di Donato, A., Marzocchella, A., Salatino, P., Barbieri, P., Galli, E., 2006. An airlift biofilm reactor for the biodegradation of phenol by Pseudomonas stutzeri OX1. Journal of Biotechnology 123, 464-477). The model embodies the key processes relevant to the reactor performance, with a particular emphasis on the role of biofilm detachment promoted by the fluidized state. Results indicate that a finite loading of free cells establishes even under operating conditions that would promote wash out of the suspended biophase. The co-operative/competitive effects of free cells and immobilized biofilm result in rich bifurcational patterns of the steady state solutions of the governing equations, which have been investigated in the phase plane of the process parameters. Direct simulation under selected operating conditions confirms the importance of the dynamical equilibrium establishing between the immobilized and the suspended biophase and highlights the effect of the initial value of the biofilm loading on the dynamical pattern.     

Bifurcation analysis of a product inhibition model of a continuous fermentation process

Summary A product inhibition model of a continuous fermentation process is considered. If the yield term is a variable function of ethanol concentration, oscillation in the cell and ethanol concentrations is shown to be a Hopf bifurcation in the underlying system of nonlinear, ordinary differential equations which comprises the model.     

The logical analysis of continuous, non-linear biochemical control networks

We propose a mapping to study the qualitative properties of continuous biochemical control networks which are invariant to the parameters used to describe the networks but depend only on the logical structure of the networks. For the networks, we are able to place a lower limit on the number of steady states and strong restrictions on the phase relations between components on cycles and transients. The logical structure and the dynamical behavior for a number of simple systems of biological interest, the feedback (predator-prey) oscillator, the bistable switch, the phase dependent switch, are discussed. We discuss the possibility that these techniques may be extended to study the dynamics of large many component systems.     

An overview of data models for the analysis of biochemical pathways

Biochemical pathways such as metabolic, regulatory or signal transduction pathways can be viewed as interconnected processes forming an intricate network of functional and physical interactions between molecular species in the cell. The amount of information available on such pathways for different organisms is increasing very rapidly. This is offering the possibility of performing various analyses on the structure of the full network of pathways for one organism as well as across different organisms, and has therefore generated interest in developing databases for storing and managing this information. Analysing these networks remains far from straightforward owing to the nature of the databases, which are often heterogeneous, incomplete or inconsistent. Pathway analysis is hence a challenging problem in systems biology and in bioinformatics. Various forms of data models have been devised for the analysis of biochemical pathways. This paper presents an overview of the types of models used for this purpose, concentrating on those concerned with the structural aspects of biochemical networks. In particular, the different types of data models found in the literature are classified using a unified framework. In addition, how these models have been used in the analysis of biochemical networks is described. This enables us to underline the strengths and weaknesses of the different approaches, as well as to highlight relevant future research directions.     

Structured segregated models and analysis of self-oscillating yeast continuous cultures

A structured segregated model of budding yeast (Saccharomyces cerevisiae) populations is analysed in order to verify its ability to predict the spontaneous oscillations arising in continuous cultures. To obtain tractable and useful information about the relationships among the metabolic modifications during the cell cycle, the control over division and the occurrence of oscillations, very simple assumptions are considered and added to the model. The cell metabolism has been taken into account by assuming a diversification in the yield coefficient during the cell cycle. Moreover, in the oscillatory range, the cell mass is assumed to be constant at budding and to depend on the limiting substrate concentration at division. For a suitable range of parameter values, sustained oscillations are obtained, which can be compared to the experimental ones.List of Symbols D dilution rate - K _h critical substrate for h - K _s saturation constant - h ratio between size at division and size at budding - h _max maximum h value - h _min minimum h value - s substrate concentration - s _in substrate concentration in the input flow - x budded biomass concentration - y unbudded biomass concentration - Y _x yield coefficient for budded biomass - Y _y yield coefficient for unbudded biomass - specific growth rate - _max maximum specific growth rate     

An immobilized alpha-galactosidase continuous flow reactor

An α-galactosidase which will hydrolyze the oligosaccharides melibiose, raffinose, and stachyose has been immobilized on nylon microfibrils suitable for use in large flow-through reactors. This catalyst system is stable for many months, both under use and storage conditions. The immobilized enzyme behaves similarly to the enzyme in solution, characteristically exhibiting both product and substrate inhibition. The catalyst is prepared in situ and a large, 8-liter reactor has been made. The catalyst has been used to reduce the raffinose concentration in beet sugar molasses.     

Construction of biochemical computer models

This paper described methods for constructing (digital) computer models of biochemical systems when the main object is to investigate the system itself, and not to fit experimental data (e.g., from tracer kinetics) to a set of equations. The author describes model-building as an art which is difficult to communicate, but nevertheless gives valuable tips on the conceptual and practical aspects, from his own considerable experience.     

Semiparametric transformation models for multiple continuous biomarkers in ROC analysis

Recent technological advances continue to provide noninvasive and more accurate biomarkers for evaluating disease status. One standard tool for assessing the accuracy of diagnostic tests is the receiver operating characteristic (ROC) curve. Few statistical methods exist to accommodate multiple continuous‐scale biomarkers in the framework of ROC analysis. In this paper, we propose a method to integrate continuous‐scale biomarkers to optimize classification accuracy. Specifically, we develop semiparametric transformation models for multiple biomarkers. We assume that unknown and marker‐specific transformations of biomarkers follow a multivariate normal distribution. Our models accommodate biomarkers subject to limits of detection and account for the dependence among biomarkers by including a subject‐specific random effect. We also propose a diagnostic measure using an optimal linear combination of the transformed biomarkers. Our diagnostic rule does not depend on any monotone transformation of biomarkers and is not sensitive to extreme biomarker values. Nonparametric maximum likelihood estimation (NPMLE) is used for inference. We show that the parameter estimators are asymptotically normal and efficient. We illustrate our semiparametric approach using data from the Endometriosis, Natural History, Diagnosis, and Outcomes (ENDO) study.