A nonlinear numerical model of percutaneous drug absorption.

A nonlinear mathematical model developed by Chandrasekaran et al. is examined to monitor pharmacokinetic profiles in percutaneous drug absorption and is addressed to several associated problems that could occur in the data analysis of in vitro experiments. The formulation of the model gives rise to a nonlinear partial differential equation (PDE) of parabolic type, and a family of finite-difference methods is developed for the numerical solution of the associated initial/boundary-value problem. The value given to a parameter in this family determines the stability properties of the resulting method and whether the solution is obtained explicitly or implicitly. In the case of implicit members of the family it is seen that the solution of the nonlinear PDE is obtained by solving a linear algebraic system, the coefficient matrix of which is tridiagonal. The behaviors of two methods of the family are examined in a series of numerical experiments. Numerical differentiation and integration procedures are combined to monitor the cumulative amount of drug eliminated into the receptor cell per unit area as time increases. It is found that the use of the equation for the simple membrane model to estimate the permeability coefficient and lag time is warranted even if the system should be described by the dual-sorption model, provided cumulative amount versus time data collected for a sufficiently long time are used. However, being different from the behavior in the simple membrane model, the lag time, which can be estimated in this way, is dose-dependent and decreases with increasing donor cell concentration. On the other hand, the permeability coefficient in the dual-sorption model remains constant irrespective of the donor cell concentrations as in the simple membrane model.     

Time lags in ecological systems.

A model representing ten species in a four trophic level community is constructed by using Volterra's equations including time lags, and is solved numerically for some values of the parameters. Classical discrete and continuous time lags yield similar results; simulating with discrete time lags thus appears useful. Parasitic versus predatory structures are compared by measuring the amplitudes of oscillations and the time taken to settle down to equilibrium, and give the following qualitative conclusions for the model.An increase in the carrying capacity of a trophic level increases the destabilizing influence of time lag in that level, this increase is more marked in predatory structures. Time lags appear to cause more violent oscillations in species strongly linked to a predatory system than in weakly linked species. The oscillatory period increases in predatory systems as the time lag is moved to higher levels, while it decreases in parasitic systems.     

V.A.C. instillation: in vitro model. Part 1.

The reproducibility of a V.A.C. (Vacuum Assisted Closure) instillation system was investigated by means of an in vitro model. The relation between the volume of a delivered solution and its removal from the system was studied in foams of various size. The relationship of instillation time periods and the volume of delivered solution was determined.     

Coexistence and competition in HIV infections.

In this paper we extend our explanation of a model for the dynamics of the interaction between HIV and the cells of the immune system (Nowak et al., 1990). We show that the Simpson index of viral diversity is a Lyapunov function for a simplified version of this model. We also present a more general mathematical characterization of the nature of the diversity threshold exhibited by the model, including for the first time heterogeneity in parameters like virus replication rate, cytopathicity and antigenicity. The more general diversity threshold condition includes the different contributions of strains with higher replication rates and cytopathicities or strains that are only weakly recognized by the immune system. This leads to some new insights and a more detailed understanding of why the viral diversity falls once the diversity threshold is exceeded.     

A continuous,multistage tower fermentor. II. Analysis of reactor performance

A method for analyzing the reactor behavior of a continuous, multistage tower fermentor is described. A model consisting of a system of interconnected, ideal subreactors is set up on the basis of the fermentor's configuration and flow pattern. The residence time distribution curve is used to test the validity of the model and the relative quantities of flow streams and regions in the model are determined. A least-square fitting procedure between measured and calculated distribution curves is used to identify the proper model. The application of this method to real cultivation conditions is also discussed. Using this approach, the multistage tower fermentor is shown to be equivalent to a cascade of four perfectly mixed tanks with a backtracking stream between stages. The extent of backflow under various conditions has also been determined.     

Generalization of the theory of transition times in metabolic pathways: a geometrical approach.

Cell metabolism is able to respond to changes in both internal parameters and boundary constraints. The time any system variable takes to make this response has relevant implications for understanding the evolutionary optimization of metabolism as well as for biotechnological applications. This work is focused on estimating the magnitude of the average time taken by any observable of the system to reach a new state when either a perturbation or a persistent variation occurs. With this aim, a new variable, called characteristic time, based on geometric considerations, is introduced. It is stressed that this new definition is completely general, being useful for evaluating the response time, even in complex transitions involving periodic behavior. It is shown that, in some particular situations, this magnitude coincides with previously defined transition times but differs drastically in others. Finally, to illustrate the applicability of this approach, a model of a reaction mediated by an allosteric enzyme is analyzed.     

Computer simulation of overshoot in saccadic eye movements.

The human horizontal eye movement system produces quick, precise, conjugate eye movements called saccades. These are important in normal vision. For example, reading tasks exclusively utilize saccadic eye movements. The majority of saccades have dynamic overshoot. The amplitude of this overshoot is independent of saccadic amplitude, and is such that it places the image of the stimulus within the retinal region of maximum acuity within a minimum of time. A computer based model of the saccadic mechanisms was used to study the origin of this overshoot. It was discussed that dynamic overshoot cannot be attributed to biomechanism properites of the eye movement mechanism, but must instead be explained by variations in the controlling nervous activity. The form of this neural controller signal is very similar to that required for a time optimal response of an inertial system.     

Optimal Performance of the Tryptophan Operon of E. coli: A stochastic,Dynamical, Mathematical-Modeling Approach

In this work, we develop a detailed, stochastic, dynamical model for the tryptophan operon of E. coli, and estimate all of the model parameters from reported experimental data. We further employ the model to study the system performance, considering the amount of biochemical noise in the trp level, the system rise time after a nutritional shift, and the amount of repressor molecules necessary to maintain an adequate level of repression, as indicators of the system performance regime. We demonstrate that the level of cooperativity between repressor molecules bound to the first two operators in the trp promoter affects all of the above enlisted performance characteristics. Moreover, the cooperativity level found in the wild-type bacterial strain optimizes a cost-benefit function involving low biochemical noise in the tryptophan level, short rise time after a nutritional shift, and low number of regulatory molecules.     

Steady-state and transient behavior in microbial methanification: II. Mathematical modeling and verification

It has been shown that the Upflow Anaerobic Sludge Bed (UASB) system data reported earlier(1) cannot be explained by simple Monod-type substrate consumption patterns. An autoinhibition model was also ruled out because the substrate concentration range over which hysteresis was observed was much larger than such a model would predict. However, propionic and acetic acids were found to inhibit each other's conversion machineries. Since in the UASB system the biocatalyst is flocculated, it was found that a model additionally incorporating this facet of the reactor set-up could explain the steady-state data very well. Using the parameters generated from steady-state data and data from butyric acid step change,(1) i.e., the entire set of parameters (Table I), a very good agreement between predicted and observed data was found. International Mathematical and Statistical Libraries (IMSL) and Upjohn's NONLIN library combined with various root-finding and integrating subroutines were used for parameter estimation. The model thus described was used to predict the response of the UASB system when acetic acid and propionic acid influent concentrations were stepped-up/down. The agreement between the predicted and observed data was found to be excellent in each case during the step-up schedule. During the step-down the data seemed to indicate that the UASB system, like any other chemostat, responded faster than predicted. This could be due to the fact that when the culture has to "gear up" part of the lag time is the time required for the cell to produce the requisite amount of enzymes. In the case of "gearing down" this time is not required and the system responds faster.     

The feasibility of forecasting influenza epidemics in Cuba.

A large influenza epidemic took place in Havana during the winter of 1988. The epidemiologic surveillance unit of the Pedro Kourí Institute of Tropical Medicine detected the beginning of the epidemic wave. The Rvachev-Baroyan mathematical model of the geographic spread of an epidemic was used to forecast this epidemic under routine conditions of the public health system. The expected number of individuals who would attend outpatient services, because of influenza-like illness, was calculated and communicated to the health authorities within enough time to permit the introduction of available control measures. The approximate date of the epidemic peak, the daily expected number of individuals attending medical services, and the approximate time of the end of the epidemic wave were estimated. The prediction error was 12%. The model was sufficiently accurate to warrant its use as a practical forecasting tool in the Cuban public health system.     

Receptor-mediated cell attachment and detachment kinetics. I. Probabilistic model and analysis.

The kinetics of receptor-mediated cell adhesion to a ligand-coated surface play a key role in many physiological and biotechnology-related processes. We present a probabilistic model of receptor-ligand bond formation between a cell and surface to describe the probability of adhesion in a fluid shear field. Our model extends the deterministic model of Hammer and Lauffenburger (Hammer, D.A., and D.A. Lauffenburger. 1987. Biophys. J. 52:475-487) to a probabilistic framework, in which we calculate the probability that a certain number of bonds between a cell and surface exists at any given time. The probabilistic framework is used to account for deviations from ideal, deterministic behavior, inherent in chemical reactions involving relatively small numbers of reacting molecules. Two situations are investigated: first, cell attachment in the absence of fluid stress; and, second, cell detachment in the presence of fluid stress. In the attachment case, we examine the expected variance in bond formation as a function of attachment time; this also provides an initial condition for the detachment case. Focusing then on detachment, we predict transient behavior as a function of key system parameters, such as the distractive fluid force, the receptor-ligand bond affinity and rate constants, and the receptor and ligand densities.(ABSTRACT TRUNCATED AT 250 WORDS)     

A kinetic model of the agglutination process.

We propose a kinetic model of the aggregation process in a system consisting of two different types of particles. Aggregating particles (cells) are polyvalent and bear on the surface a huge number of binding sites for the other type of particles, ligands. The ligand is bivalent and has two identical active sites for binding to cells. The cross-linking of the cells by the ligands causes the aggregation phenomenon called agglutination. We obtained the analytical solution of this model task describing the time dependence of the aggregate mean size versus the composition of the system. The comparison of the analytical solution with the experimental data for the agglutination of bacterial cells by bivalent antibodies shows that the main factors affecting agglutination were correctly taken into account.     

A mathematical model of the food-consumer system I. A case without food replenishment

A simple mathematical model was proposed to describe the dynamics of a food-consumer system. The model was based on the Logistic Theory and consisted of Eqs. (4), (5) and (6). The model was divided into the following three cases for further analyss; i) without food supply except at the initial time, ii) with continuous food supply at a constant rate, and iii) with food supply at varying rates. Only the first model was dealth with in this paper. The assumptions of the model 1 are that a definite amount of food is given only once at the initial time and only the feeding by animals is responsible for the decrease of food, and that the rate of decrease is proportional to the amount of animals. It is also assumed that the growth of animal population is represented by the logistic curve, and that the upper limit of the population is proportional to the amount of food at that time. For simplicity the parameters of basic differential equations are assumed to be constant throughout the time course. Analytical solutions of this non-linear model were given by Eqs. (8), (9), (10) and (11). The properties of time course of the food amount and consumer population were discussed from the mathematical and biological points of view. The method of the estimation of the three constants λ,b, and c from the experimental data was also suggested. Since we had no available data for animal populations, we applied the model, regarding reserve substance as x and new plant body as y, to the data of the initial growth of Azuki bean plant in the dark. This model is very simple, but it may be useful for analyzing the behavior of food-consumer system. And it may give some clue to the analysis of the more complex systems.     

Selection in Finite Populations. III. an Experimental Examination

The theory determining the probability of fixation of new mutants in a population that allows increased adaptation has been developed by Kimura. Two experimental systems were used to examine this theory. First, alleles were introduced at low frequencies into populations and their fate observed. The second approach was to follow populations with closely linked mutants that were introduced on different chromosomes and to follow the fate of favorable recombinants. These experiments allowed the investigation of the appropriateness of the mathematical model for the particular biological system. In the experiments reported here using two X-linked Drosophila mutants, y and w, the predictions of the theory were reasonably fulfilled in the first experimental system (introduction of new alleles), but not in the second system (favorable recombinants). However, the theoretical framework seems quite robust in that it allowed a satisfactory explanation of the experimental results for the second system as well. The probability of, and the time until, production of a favorable recombinant is discussed.     

Allosteric oscillatory enzymes: influence of the number of protomers on metabolic periodicities.

The study of a concerted allosteric model for an enzyme activated by the reaction product shows that this system can generate sustained metabolic oscillations regardless of the number of protomers constituting the enzyme. The analysis extends the results previously obtained in a dimeric model for the phosphofructokinase reaction which produces glycolytic periodicities. When the substrate and product concentrations evolve on comparable time scales, the amplitude of oscillations significantly drops as the number of enzyme subunits evolves from 2 to 8. The width of the domain of substrate injection rates which produce oscillations and the periodic variation in enzyme activity also depend on the number of protomers and on the time scale structure of the system. Theoretical predictions are compared with the experiments on glycolytic oscillations in yeast and muscle, and with the structural characteristics of phosphofructokinase. The results are also discussed in relation with the mechanism of cyclic AMP oscillations in the slime mold Dictyostelium discoideum.     

Kinetics of elementary steps in the cytochrome P-450 reaction sequence. IV. Mechanism of the NADPH reduction reaction of cytochrome P-450LM.

The aerobic NADPH reduction of the Triton N-101 disintegrated cytochrome P-450LM system has been studied. At this organization level--the components being dispersed in solution--a first-order monophasic reaction is exhibited. Neither the complex formation of cytochrome and reductase, respectively, nor preceding diffusion is rate limiting. The initial rate follows the ratio reductase/P-450 (mole/mole) thus indicating a Michaelis-Menten type enzyme mechanism. A model treatment of the reaction fits the systems behaviour as a whole. A multiparameter equilibrium state and different specified time function equations were developed for the determination of individual step rate constants and other system parameters as well.     

Chromosomal destabilization during gene amplification.

Acentric extrachromosomal elements, such as submicroscopic autonomously replicating circular molecules (episomes) and double minute chromosomes, are common early, and in some cases initial, intermediates of gene amplification in many drug-resistant and tumor cell lines. In order to gain a more complete understanding of the amplification process, we investigated the molecular mechanisms by which such extrachromosomal elements are generated and we traced the fate of these amplification intermediates over time. The model system consists of a Chinese hamster cell line (L46) created by gene transfer in which the initial amplification product was shown previously to be an unstable extrachromosomal element containing an inverted duplication spanning more than 160 kilobases (J. C. Ruiz and G. M. Wahl, Mol. Cell. Biol. 8:4302-4313, 1988). In this study, we show that these molecules were formed by a process involving chromosomal deletion. Fluorescence in situ hybridization was performed at multiple time points on cells with amplified sequences. These studies reveal that the extrachromosomal molecules rapidly integrate into chromosomes, often near or at telomeres, and once integrated, the amplified sequences are themselves unstable. These data provide a molecular and cytogenetic chronology for gene amplification in this model system; an early event involves deletion to generate extrachromosomal elements, and subsequent integration of these elements precipitates a cascade of chromosome instability.     

A transient excited state model for sodium permeability changes in excitable membranes.

In this paper we explore the properties of a mathematical model for the passive sodium permeability system of excitable membranes. This model is distinguished by the explicit inclusion of a rate constant which depends not on instantaneous voltage, but on rate of voltage change. Actually, the model is a rather modest modification of the Hodgkin-Huxley model, but displays some behaviors which the H-H model does not. Among these behaviors are a pronounced inactivation shift (for certain parameter values), a difference between inactivation time constant as measured by turning off a sodium current under sustained depolarization and as measured by double pulse experiments, skip runs under sustained current stimulation, and accommodation to slowing rising currents.     

Replication control of autonomously replicating human sequences.

Three autonomously replicating plasmids carrying human genomic DNA and a vector derived from Epstein-Barr virus were studied by density labelling to determine the number of times per cell cycle these plasmids replicate in human cells. Each of the plasmids replicated semi-conservatively once per cell cycle. The results suggest that these human autonomously replicating sequences undergo replication following the same controls as chromosomal DNA and represent a good model system for studying chromosomal replication. We also determined the time within the S phase of the cell cycle that three of the plasmids replicate. Centromeric alpha sequences, which normally replicate late in S phase when in their chromosomal context, were found to replicate earlier when they mediate replication on an extrachromosomal vector. Reproducible patterns of replication within S phase were found for the plasmids, suggesting that the mechanism specifying time of replication may be subject to experimental analysis with this system.