Dynamics of Predator‐Prey Interactions in Continuous Cultures

This paper studies the behavior of a general unstructured kinetic model for continuous bioreactors involving interactions between predator, prey and a limiting substrate. The analysis carried out in this paper shows how closed analytical conditions for arbitrary growth rates can be derived that describe the conditions for the existence of the interacting species in an oscillatory behavior. It is also demonstrated how practical diagrams in terms of operating and kinetic parameters can be constructed that classify the different behavior predicted by the model. Applications of these general results to a number of experimentally validated models have revealed that the saturation model always predicts hard oscillations for a certain range of dilution rates, for any values of model parameters. Bifurcation diagrams in the operating parameter space permitted the delineation of regions of hard oscillations, regions of static coexistence, regions of predator washout and regions of total washout. The analysis of the double saturation model has proven its ability to predict two Hopf points. Hard oscillations are therefore expected within the dilution rates corresponding to the two Hopf points. Practical diagrams were also constructed to delineate the boundaries separating hard oscillations from static coexistence and washout conditions.     

A model of glucose kinetics and their control by insulin,compartmental and noncompartmental approaches

Compartmental and noncompartmental models are used to quantify, from multiple steady-state tracer experiments, glucose kinetics and the effect of insulin upon them. Some aspects of experiment design are discussed. A physiological three-compartment model of glucose kinetics is proposed which provides a new quantitative picture of insulin control of glucose metabolism. Noncompartmental modeling is shown to have structural errors which prevent physiological insight. Compartmental models make a better use of the informational content of kinetic data, even if more demanding both in terms of modeling and computational effort and in terms of physiological thinking.     

A new class of biophysical models for predicting the probability of decompression sickness in scuba diving.

Interconnected compartmental models have been used for decades in physiology and medicine to account for the observed multi-exponential washout kinetics of a variety of solutes (including inert gases) both from single tissues and from the body as a whole. They are used here as the basis for a new class of biophysical probabilistic decompression models. These models are characterized by a relatively well-perfused, risk-bearing, central compartment and one or two non-risk-bearing, relatively poorly perfused, peripheral compartment(s). The peripheral compartments affect risk indirectly by diffusive exchange of dissolved inert gas with the central compartment. On the basis of the accuracy of their respective predictions beyond the calibration regime, the three-compartment interconnected models were found to be significantly better than the two-compartment interconnected models. The former, on the basis of a number of criteria, was also better than a two-compartment parallel model used for comparative purposes. In these latter comparisons, the models all had the same number of fitted parameters (four), were based on linear kinetics, had the same risk function, and were calibrated against the same dataset. The interconnected models predict that inert gas washout during decompression is relatively fast, initially, but slows rapidly with time compared with the more uniform washout rate predicted by an independent parallel compartment model. If empirically verified, this may have important implications for diving practice.     

Association between plasma zinc concentration and zinc kinetic parameters in premenopausal women

The objective of this study was to measure relationships between plasma zinc (Zn) concentrations and Zn kinetic parameters and to measure relationships of Zn status with taste acuity, food frequency, and hair Zn in humans. The subjects were 33 premenopausal women not taking oral contraceptives and dietary supplements containing iron and Zn. Main outcomes were plasma Zn concentrations, Zn kinetic parameters based on the three-compartment mammillary model using 67Zn as a tracer, electrical taste detection thresholds, and food frequencies. Lower plasma Zn was significantly (P < 0.01) associated with smaller sizes of the central and the lesser peripheral Zn pools, faster disappearance of tracer from plasma, and higher transfer rate constants from the lesser peripheral pool to the central pool and from the central pool to the greater peripheral pool. The break points in the plasma Zn-Zn kinetics relationship were found between 9.94 and 11.5 micromol/l plasma Zn. Smaller size of the lesser peripheral pool was associated with lower frequency of beef consumption and higher frequency of bran breakfast cereal consumption. Hypozincemic women with plasma Zn <10.7 micromol/l or 700 ng/ml had decreased thresholds of electrical stimulation for gustatory nerves. Our results based on Zn kinetics support the conventional cutoff value of plasma Zn (10.7 micromol/l or 700 ng/ml) between normal and low Zn status.     

Some Formal Approaches to the Analysis of Kinetic Data in Terms of Linear Compartmental Systems

A formal approach to the routine analysis of kinetic data in terms of linear compartmental systems is presented. The methods of analysis are general in that they include much of the theory in common use, such as direct solution of differential equations, integral equations, transfer functions, fitting of data to sums of exponentials, matrix solutions, etc. The key to the formalism presented lies in the fact that a basic operational unit—called “compartment”—has been defined, in terms of which physical and mathematical models as well as input and output functions can be expressed. Additional features for calculating linear combinations of functions and for setting linear dependence relations between parameters add to the versatility of this method. The actual computations for the values of model parameters to yield a least squares fit of the data are performed on a digital computer. A general computer program was developed that permits the routine fitting of data and the evolution of models.     

Impulse-response function of splanchnic circulation with model-independent constraints: theory and experimental validation

Modeling physiological processes using tracer kinetic methods requires knowledge of the time course of the tracer concentration in blood supplying the organ. For liver studies, however, inaccessibility of the portal vein makes direct measurement of the hepatic dual-input function impossible in humans. We want to develop a method to predict the portal venous time-activity curve from measurements of an arterial time-activity curve. An impulse-response function based on a continuous distribution of washout constants is developed and validated for the gut. Experiments with simultaneous blood sampling in aorta and portal vein were made in 13 anesthetized pigs following inhalation of intravascular [15O]CO or injections of diffusible 3-O-[11C]methylglucose (MG). The parameters of the impulse-response function have a physiological interpretation in terms of the distribution of washout constants and are mathematically equivalent to the mean transit time (T) and standard deviation of transit times. The results include estimates of mean transit times from the aorta to the portal vein in pigs: T = 0.35 +/- 0.05 min for CO and 1.7 +/- 0.1 min for MG. The prediction of the portal venous time-activity curve benefits from constraining the regression fits by parameters estimated independently. This is strong evidence for the physiological relevance of the impulse-response function, which includes asymptotically, and thereby justifies kinetically, a useful and simple power law. Similarity between our parameter estimates in pigs and parameter estimates in normal humans suggests that the proposed model can be adapted for use in humans.     

Experimental Approach to Evaluate the 11C Perfusion and Diffusion in Small Animal Tissues for HadronPET Applications

The development of a reliable dose monitoring system in hadron therapy is essential in order to control the treatment plan delivery. Positron Emission Tomography (PET) is the only method used in clinics nowadays for quality assurance. However, the accuracy of this method is limited by the loss of signal due to the biological washout processes. Up to the moment, very few studies measured the washout processes and there is no database of washout data as a function of the tissue and radioisotope. One of the main difficulties is related to the complexity of such measurements, along with the limited time slots available in hadron therapy facilities. Thus, in this work, we proposed an alternative in vivo methodology for the measurement and modeling of the biological washout parameters without any radiative devices. It consists in the implementation of a point-like radioisotope source by direct injection on the tissues of interest and its measurement by means of high-resolution preclinical PET systems. In particular, the washout of ¹¹C carbonate radioisotopes was assessed, considering that ¹¹C is is the most abundant β⁺ emitter produced by carbon beams. ¹¹C washout measurements were performed in several tissues of interest (brain, muscle and 9L tumor xenograf) in rodents (Wistar rat). Results show that the methodology presented is sensitive to the washout variations depending on the selected tissue. Finally, a first qualitative correlation between ¹¹C tumor washout properties and tumor metabolism (via ¹⁸F-FDG tracer uptake) was found.     

Dopamine Uptake by Rat Striatal Synaptosomes: A Compartmental Analysis

Abstract: Dopamine (DA) uptake into synaptosomes from rat corpus striatum was studied in the presence of a monoamine oxidase (MAO) inhibitor and dithiothreitol, by means of a filtration technique. Under these conditions a steady state develops rapidly in which the synaptosomal DA content remains constant while the continuing DA uptake is counterbalanced by DA efflux from the synaptosome. Exchange of synaptosomal [³H]DA and [¹⁴C]DA was measured under these conditions. In timecourse experiments it was found that exchange could be described significantly better by a three-compartment model than by a two-compartment model. However, if synaptosomes from reserpine-pretreated animals were used, analysis according to a three-compartment model did not result in a significantly better fit compared with a two-compartment model. Subsequently, kinetic transfer parameters describing DA fluxes between compartments at different DA concentrations were calculated from the fitted exchange curves. A Michaelis-Menten kinetic analysis indicated that only the in-series three-compartment configuration, in which DA is taken up from the medium into one synaptosomal compartment, from which it can subsequently be transferred to a second compartment without direct access to the medium, gave kinetically acceptable results. Transfer parameters in synaptosomes from reserpine-treated rats were comparable to those parameters describing DA transport between the medium and the first intrasynaptosomal compartment as measured under control conditions. Morover, it was found that potassium depolarization of synaptosomes resulted in a release of DA in a quantity similar to that found in the second intrasynaptosomal compartment. It is suggested that the two intrasynaptosomal compartments found correspond to a cytoplasmatic and vesicular DA pool, respectively. The functional significance of these findings is discussed in terms of the regulation of DA levels within the nerve terminal.     

Comparison of two experimental methods for the determination of Michaelis-Menten kinetics of an immobilized enzyme

For the application of immobilized enzymes, the influence of immobilization on the activity of the enzyme should be Known. This influence can be obtained by determining the intrinsic kinetic parameters of the immobilized enzyme, and by comparing them with the kinetic parameters of the suspended enzyme. This article deals with the determination of the intrinsic kinetic parameters of an agarose-gel bead immobilized oxygen-consuming enzyme: L-lactate 2-monooxygenase. The reaction rate of the enzyme can be described by Michaelis-Menten kinetics. Batch conversion experiments using a biological oxygen monitor, as well as steady-state profile measurements within the biocatalyst particles using an oxygen microsensor, were performed. Two different mathematical methods were used for the batch conversion experiments, both assuming a pseudosteady-state situation with respect to the shape of the profile inside the bead. One of the methods used an approximate relation for the effectiveness factor for Michaelis-Menten kinetics which interpolates between the analytical solutions for zero- and first-order kinetics. The other mathematical method was based on a numerical solution and combined a mass balance over the reactor with a mass balance over the bead. The main difference in the application of the two methods is the computer calculation time; the completely numerical calculation procedure was about 20 times slower than the other calculation procedure.The intrinsic kinetic parameters resulting from both experimental methods were compared to check the reliability of the methods. There was no significant difference in the intrinsic kinetic parameters obtained from the two experimental methods. By comparison of the kinetic parameters for the suspended enzyme with the intrinsic kinetic parameters for the immobilized enzyme, it appeared that immobilization caused a decrease in the value of V(m) by a factor of 2, but there was no significant difference in the values obtained for K(m).     

十三碳二元酸发酵过程产酸期拟均相动力学模型及其应用

分析了十三碳二元酸发酵过程中产酸期的代谢特点,对产酸期四相体系发酵动力学进行了研究。提出了菌体生长、产物形成及底物消耗的动力学模型,对模型参数进行了回归估值,并对产酸期进行了拟合,结果表明,模型的计算值和实测值较为吻合,平均相对偏差为3．6％。利用所建模型对产酸期进行了多种操作条件下的模拟计算,结果表明,提高进入产酸期的菌体浓度、缩短菌体生长期时间及降低发酵液中产物浓度具有提高产物形成速率的有效途径。     

十三碳二元酸发酵过程菌体生长期动力学模型及其应用

介绍了由十三碳烷烃生产十三碳二元酸的发酵过程,对其中的菌体生长期的代谢过程进行了分析。提出了以CO₂释放率判断菌体生长状况的方法,据此可确定进入产酸期的最佳时间．建立了菌体生长期底物消耗及菌体生长的动力学模型,对模型参数进行了回归估值。并对菌体生长期进行了拟合。结果表明,模型的计算值和实测值吻合得较好,平均相对偏差为2．4％。利用所建模型对菌体生长期进行多种操作条件下的模拟计算,结果表明,提高蔗糖浓度及初始菌体浓度均能显著地提高菌体生长期结束时的菌体浓度。     

A methodology for performing global uncertainty and sensitivity analysis in systems biology

Accuracy of results from mathematical and computer models of biological systems is often complicated by the presence of uncertainties in experimental data that are used to estimate parameter values. Current mathematical modeling approaches typically use either single-parameter or local sensitivity analyses. However, these methods do not accurately assess uncertainty and sensitivity in the system as, by default, they hold all other parameters fixed at baseline values. Using techniques described within we demonstrate how a multi-dimensional parameter space can be studied globally so all uncertainties can be identified. Further, uncertainty and sensitivity analysis techniques can help to identify and ultimately control uncertainties. In this work we develop methods for applying existing analytical tools to perform analyses on a variety of mathematical and computer models. We compare two specific types of global sensitivity analysis indexes that have proven to be among the most robust and efficient. Through familiar and new examples of mathematical and computer models, we provide a complete methodology for performing these analyses, in both deterministic and stochastic settings, and propose novel techniques to handle problems encountered during these types of analyses.     

Kinetic model of a multienzyme system of blood prostanoid synthesis. I. Mechanism of stabilization of thromboxane and prostacyclin levels

A kinetic scheme of the prostacyclin-thromboxane system has been evolved on the basis of the authors experimental data and the results described elsewhere. The kinetic behavior of the model has been analysed with the aid of computer technology by varying the following parameters: phospholipase activities, free arachidonic acid exchange rates between platelets and endothelium, PGH-synthetase biosynthesis rates, velocities of arachidonic acid pathways other than the cyclooxygenase ones. It has been demonstrated that the biological system is capable of sustaining prostacyclin and thromboxane concentrations at steady fixed levels within a wide range of kinetic parameters.     

Interpretation of Tracer Washout Curves from a Population of Muscle Fibers

Consideration is given to the role of a population of muscle fibers of distributed diameters in the observed washout of a tracer, with particular reference to radioisotopic K and muscle fibers. It is concluded that if washout of tracer from a single fiber is described as first order, then washout of tracer from a population of fibers is apt to appear as first order.     

Linear n-compartment catenary models: Formulas to describe tracer amount in any compartment and identification of parameters from a concentration-time curve

Resolution of kinetic equations and parameter identification are discussed for n-compartment linear catenary models with elimination allowed from any compartment. For a given input, general formulas are derived to describe the tracer amount in any compartment as a function of the model parameters. Conversely, explicit procedures are given to identify the model parameters when the concentration-time curve is known in one arbitrary compartment, the tracer being injected into the same compartment. In this inverse problem, the solution is not unique: the model transfer rate constants can only be localized in a finite set of intervals.     

Kinetic behavior of the multienzyme system of blood prostanoid synthesis

A kinetic scheme of the prostacyclin-thromboxane system has been evolved on the basis of our own experimental material and the results described elsewhere. The kinetic behavior of the model has been analysed with the aid of computer technology by varying the following parameters: phospholipase activities, free arachidonic acid exchange rates between platelets and endothelium, prostaglandin H (PGH) synthetase biosynthesis rates, velocities of arachidonic acid pathways other than cyclooxygenase ones. It has been demonstrated that the biological system is capable of sustaining prostacyclin and thromboxane concentrations at steady fixed levels within a wide range of kinetic parameters.     

A heterogeneous tissue model for measurement of regional blood perfusion in the myocardium using inert gas isotopes

Inert gas isotopes are finding increasing application in the measurement of blood perfusion in the capillary beds of muscle, especially the myocardium. When measuring blood perfusion of the myocardium, washout curves are first produced by precordial monitoring of isotope activity following intracoronary artery injection of an inert gas isotope dissolved in saline. The washout curve data are then applied to a mathematical model to yield blood perfusion rate. Present models for this purpose either ignore any diffusive effects of gas movement (Kety-Schmidt model), or diffusive effects are accounted for by weighting the calculated perfusion value (Zierler's height-over-area technique). A new model is described here for convective and diffusive movement of an inert, nonpolar gas in myocardial tissue. A digital computer simulation of the model equations is used both to simply the model and to show agreement between the model response and experimental ¹³³Xe washout curves from normal and infracted canine hearts. The model assumes that the tail of the washout curves (portion after roughly 1.5 minutes) is caused by a heterogeneous, diffusion-limited tissue structure. The model provides two parameters which can be adjusted to washout curve data using model-matching techniques. These are perfusion rate, and a parameter which is an index of the diffusive nature of the particular myocardial area under study.     

FitSpace Explorer: An algorithm to evaluate multidimensional parameter space in fitting kinetic data

Fitting several sets of kinetic data directly to a model based on numerical integration provides the best method to extract kinetic parameters without relying on the simplifying assumptions required to achieve analytical solutions of rate equations. However, modern computer programs make it too easy to enter an overly complex model, and standard error analysis grossly underestimates errors when a system is underconstrained and fails to reveal the full degree to which multiple parameters are linked through the complex relationships common in kinetic data. Here we describe the application of confidence contour analysis obtained by measuring the dependence of the sum square error on each pair of parameters while allowing all remaining parameters to be adjusted in seeking the best fit. The confidence contours reveal complex relationships between parameters and clearly outline the space over which parameters can vary (the “FitSpace”). The utility of the method is illustrated by examples of well-constrained fits to published data on tryptophan synthase and the kinetics of oligonucleotide binding to a ribozyme. In contrast, analysis of alanine racemase clearly refutes claims that global analysis of progress curves can be used to extract the free energy profiles of enzyme-catalyzed reactions.     

Analysis and identification of stochastic compartment models in pharmacokinetics: implication for cancer chemotherapy

One of the fundamental problems in pharmacokinetics is determining the parameters of multicompartment models from the measured excretion rate and observations on one of the compartments. Since it is frequently impractical to obtain observations for all physiological compartments, and due to the inherent variations in biological systems, a stochastic approach to multicompartment analysis is suggested. The objective of this article is to consider the identification of multicompartment models from a stochastic point of view, with particular emphasis on two- and three-compartment models.The essential part of this investigation is the statistical simulation of the C-T curves of the observed compartment for different biological conditions. We then present a probabilistic characterization of the multirandom parameter family of the rate constant matrix and evaluate the statistical properties of the solution processes for the unobservable compartments.     

Fuzzy Stochastic Petri Nets for Modeling Biological Systems with Uncertain Kinetic Parameters

Stochastic Petri nets (SPNs) have been widely used to model randomness which is an inherent feature of biological systems. However, for many biological systems, some kinetic parameters may be uncertain due to incomplete, vague or missing kinetic data (often called fuzzy uncertainty), or naturally vary, e.g., between different individuals, experimental conditions, etc. (often called variability), which has prevented a wider application of SPNs that require accurate parameters. Considering the strength of fuzzy sets to deal with uncertain information, we apply a specific type of stochastic Petri nets, fuzzy stochastic Petri nets (FSPNs), to model and analyze biological systems with uncertain kinetic parameters. FSPNs combine SPNs and fuzzy sets, thereby taking into account both randomness and fuzziness of biological systems. For a biological system, SPNs model the randomness, while fuzzy sets model kinetic parameters with fuzzy uncertainty or variability by associating each parameter with a fuzzy number instead of a crisp real value. We introduce a simulation-based analysis method for FSPNs to explore the uncertainties of outputs resulting from the uncertainties associated with input parameters, which works equally well for bounded and unbounded models. We illustrate our approach using a yeast polarization model having an infinite state space, which shows the appropriateness of FSPNs in combination with simulation-based analysis for modeling and analyzing biological systems with uncertain information.