A mathematical model for the mechanism of rapid eye movements induced by an anticholinesterase in the decerebrate cat.

The oculomotor pattern which appears in intact preparations during desynchronized sleep is characterized by the irregular occurrence of isolated ocular movements and bursts of rapid eye movements (REM). This complex oculomotor pattern results from the activity of two premotor structures which influence the extraocular motoneurons during this phase of sleep: one is located in the pontine reticular formation, the other in the vestibular nuclei. In the decerebrate preparation the intravenous injection of an anticholinesterase leads to the appearance of a typical pattern of oculomotor activity, which differs from that occurring during physiological sleep in so far as it consists quite exclusively of bursts of REM which appear at very regular intervals. Lesion experiments as well as unit recordings have shown that these bursts of REM depend in particular upon rhythmic discharges of the vestibular nuclear neurons. The underlying anatomical structures responsible for these bursts of REM are therefore the vestibular nuclei, the oculomotor nuclei and the oculo-orbital system. This mechanism is under the influence of cholinergic reticular neurons which generate the oculomotor rhythm. We have postulated the existence of a self-excitatory cholinergic system, located in the pontine reticular formation, whose steady discharge impinges upon an oscillatory neuronal system located in the dorso-lateral pontine tegmentum, which transforms the tonic input into a sinusoidal final output. We have assumed also that the periodic increases in the discharge frequency of this oscillatory system trigger a fast phase generator acting on the different components of the REM system, and that the behavior of each component follows a first-order differential equation. The state of excitation of the components of the system is defined as proportional to frequency of nerve impulses. Assuming ipsilateral and crossed connections, a pattern of oculomotor activity is obtained that simulates the experimental oculomotor output fairly well. The repetition of the eye jerks is described by a Fourier series. The model proposed in this paper may be taken as a first approach in describing the generation mechanism of REM, and as a theoretical guide to new experimental researches and the development of other more realistic models.     

A model of animal movements in a bounded space

Most studies describing animal movements have been developed in the framework of population dispersion or population dynamics, and have mainly focused on movements in open spaces. During their trips, however, animals are likely to encounter physical heterogeneities that guide their movements and, as a result, influence their spatial distribution. In this paper, we develop a statistical model of individual movement in a bounded space. We introduced cockroaches in a circular arena and quantified accurately the behaviors underlying their movement in a finite space. Close to the edges, we considered that the animals exhibit a linear movement mode with a constant probability per unit time to leave the edge and enter the central zone of the arena. Far from the walls cockroaches were assumed to move according to a diffusive random walk which enabled us to overcome the inherent problem of the quantification of the turning angle distribution. A numerical model implementing the behavioral rules derived from our experiments, confirms that the pattern of the spatial distribution of animals observed can be reliably accounted for by wall-following behaviors combined with a diffusive random walk. The approach developed in this study can be applied to model the movements of animals in various environment under consideration of spatial structure.     

A complex mathematical model of the human menstrual cycle

Despite the fact that more than 100 million women worldwide use birth control pills and that half of the world's population is concerned, the menstrual cycle has so far received comparatively little attention in the field of mathematical modeling. The term menstrual cycle comprises the processes of the control system in the female body that, under healthy circumstances, lead to ovulation at regular intervals, thus making reproduction possible. If this is not the case or ovulation is not desired, the question arises how this control system can be influenced, for example, by hormonal treatments. In order to be able to cover a vast range of external manipulations, the mathematical model must comprise the main components where the processes belonging to the menstrual cycle occur, as well as their interrelations. A system of differential equations serves as the mathematical model, describing the dynamics of hormones, enzymes, receptors, and follicular phases. Since the processes take place in different parts of the body and influence each other with a certain delay, passing over to delay differential equations is deemed a reasonable step. The pulsatile release of the gonadotropin-releasing hormone (GnRH) is controlled by a complex neural network. We choose to model the pulse time points of this GnRH pulse generator by a stochastic process. Focus in this paper is on the model development. This rather elaborate mathematical model is the basis for a detailed analysis and could be helpful for possible drug design.     

A mathematical model of ion movements in grey matter during a stroke

The development of cytotoxic oedema during a stroke consists in cell swelling and shrinking of the extracellular space. This phenomenon is triggered by ion movements through voltage-gated channels, exchangers and pumps. During ischaemia, sodium, calcium and chloride enter the neurons whereas potassium and glutamate are expelled out of the cells. A mathematical model is proposed to represent the long-term dynamics of membrane potentials, cell volumes and ionic concentrations in intracellular and extracellular spaces during a stroke and to study the influence of each ionic current on cell swelling. The model relies on electrophysiological mechanisms and takes into account the behaviour of two types of cells: neurons and also astrocytes known to play a key role in the excitotoxic process in grey matter. The results obtained when a severe or a moderate ischaemia is simulated are consistent with those observed in the in vitro and in vivo experiments. As this model appears to be robust, it is used to perform illustrative simulations aimed at studying the effect of some channel blockers on cell swelling. This approach may help to explore new therapeutic strategies in order to reduce stroke damage.     

A mathematical analysis on public goods games in the continuous space

We consider the population dynamics of two competing species sharing the same resource, which is modeled by the carrying capacity term of logistic equation. One species (farmer) increases the carrying capacity in exchange for a decreased survival rate, while the other species (exploiter) does not. As the carrying capacity is shared by both species, farmer is altruistic. The effect of continuous spatial structure on the performance of such strategies is studied using the reaction diffusion equations. Mathematical analysis on the traveling wave solution of the system revealed; (1) Farmers can never expel exploiters in any traveling wave solution. (2) The expanding velocity of the exploiter population invading the farmer population can be analytically determined and it depends only on a cost of altruism and the diffusion coefficients while it is independent of the benefit of altruism. (3) When the effect of altruism is small, the dynamics of the invasion of exploiters obeys the Fisher-KPP equation. Numerical calculations confirm these results.     

A mathematical analysis of a model for tumour angiogenesis

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. A mathematical model is presented which takes into account two of the most important events associated with the endothelial cells as they form capillary sprouts and make their way towards the tumour i.e. cell migration and proliferation. The numerical simulations of the model compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial cell migration and proliferation.     

Three-dimensional mathematical model analysis of the patellofemoral joint

This paper is concerned with a mathematical model analysis of the patellofemoral joint in the human knee, taking into account the articular surface geometry and mechanical properties of the ligament. It was made by the application of a computer-aided design theory (previously studied) and it was possible to express the articular surface geometries in a mathematical formulation and hence elucidate the joint movement mechanics. This method was then applied to a three-dimensional geometrical model of the patellofemoral joint. For the modelling of tendofemoral contact at large angles of knee flexion, the geodestic line theory was adopted. Applying the Newton-Raphson method and the Runge-Kutta Gil method to the model, variables such as patellar attitudes, patellofemoral contact force and tensile force of the patellar ligament for various knee flexion angles were computed. Applying the Hertzian elastic theory, contact stress was also computed. These results showed good agreement with the previously reported experimental results. As an application for the model, some parameter analyses were performed in terms of the contact stress variations and compared with those of the normal knee. The simulation results indicated that both the Q-angle increase and decrease increased contact stress, the patella alta showed undulating variations of stress while the patella infera showed little change of stress, and the tibial tuberositas elevation showed 20-30% reduction of stress.     

A mathematical model for insulin kinetics. III. Sensitivity analysis of the model

A non-linear mathematical model involving four variables and several constants incorporating beta-cell kinetics, a glucose-insulin feedback system and a gastrointestinal absorption term had been applied in earlier papers to various forms of diabetes mellitus. In this paper, we examine the response of the system to variations in the parameters and to initial conditions using sensitivity analysis. It is found that such a method leads to results that are consistent with clinical findings. Further, it is suggested that such an analysis could help in making some predictions regarding future directions in the therapy of diabetes mellitus.     

Problems in taxonomy of rotifers,exemplified by the Filinia longiseta - terminalis complex

The validity of taxonomical categories in parthenogenic groups is discussed. Special problems of rotifer taxonomy are caused by: facultative or obligatory parthenogenic reproduction, high morphological and genetic variability and paucity of morphological characteristics.Examples of these problems are provided by the Filinia terminalis-longiseta group. The different ecological properties of various populations belonging to this group are emphasized. Suggestions concerning the creation of new taxa are made; in particular, the importance of using ecological data is stressed.Biological Station of the Austrian Academy of Science     

A perturbation method for the analysis of impulse propagation in a mathematical neuron model

A perturbation method is presented for the calculation of signal propagation velocities in a mathematical neuron model. Impulse frequencies and waveforms are also determined. The method uses a non-linear conduction model containing a small parameter, such that analytical solutions are possible, yet the inherently non-linear physiological phenomena involved, can be explained. A simple two-variable membrane equation is used, but the method is generally applicable to various more complete systems.     

Interactions in the metabolism of polyunsaturated fatty acids: analysis by a simple mathematical model

Interactions in the metabolism of polyunsaturated fatty acids have been simulated in a simple model system. In the development of this system it was assumed that simple competitive inhibition occurs between parent acids as they are transformed (via dehydrogenation and chain lengthening) to their derivative acids. Numerical solutions of this model system give the composition of the tissue pool of polyunsaturated acids as a function of the proportion of the parent acids in the diet. Experimental data have been analyzed in the light of relations generated by the model system and the parallels observed substantiate the assumptions postulated in the development of the model system.     

Erythrocyte hemolysis by detergents--a mathematical model and analysis of the concentration and kinetic curves

A mathematical model of erythrocyte lysis by detergents is developed which takes into consideration the kinetics of detergent binding to plasma membrane. Experimentally obtained sigmoidal kinetic and concentration curves of hemolysis are well described by the model. A comparative study is carried out in terms of the model of hemolytic action for five detergents: Triton X-100, sodium dodecylsulfate, sodium deoxycholate, cetyltrimethylammonium bromide, and cetylpyridinium chloride. The amount of detergent which should be bound to an erythrocyte membrane to induce lysis was found to be roughly the same for all detergents studied. However, detergents vary in their affinity to the membrane. Cetylpyridinium displays the highest affinity (and consequently the highest hemolytic activity), whereas deoxycholate has the least one.     

Optimization of behavior by minimization of displeasure in a 2-dimensional sensory space

Three healthy young adult males were placed on a treadmill geared to five speeds and five slopes in combination. At minute 16 they gave estimates of displeasure for sensations aroused in their chest and lower limbs. These estimates were compared to the operant choice of the subjects when they were given the task, in other sessions, of climbing up 300 m on the treadmill at varying speeds and slopes. When speed was imposed, they could adjust slope and reciprocally. The were instructed that the session would end when they had reached the 300 m elevation. The results showed that the subjects adjusted slope and speed inversally; time for climbing up 300 m was almost constant, i.e. they tended to work at constant power in various conditions. Such performance could be predicted from the sum of ratings of displeasure aroused in the lower limbs and in the chest. These results suggest that optimalization of behaviour was obtained by minimization of bidimensional displeasure.     

A mathematical model of phytoremediation for petroleum contaminated soil: sensitivity analysis

Phytoremediation is an attractive treatment technology for many contaminated sites due to its cost effectiveness and public acceptance. We present a sensitivity analysis of important parameters from a screening level model for phytoremediation by grass species of weathered petroleum-contaminated sites. The conceptual framework is that root movement through contaminated soil will enhance contaminant biodegradation by providing a local environment more favorable for petroleum degrading microorganisms—the so-called rhizosphere effect. Common questions in phytoremediation are, “What species should be planted?” and “What management practices should be followed?” These choices may affect degradation kinetics, root biomass (and therefore rhizosphere volume), and the root turnover. Important model parameters are the rate constants, rhizosphere volume, and the rate of root turnover. We present a sensitivity analysis with the aim of identifying the most important factors for improving phytoremediation effectiveness. For simulations of the phytoremediation of weathered diesel range organics, our results indicate that annual species, with higher root turnover, are preferred over perennial species with the caveat of equal degradation rate constants, that is, no species-dependent effects. In addition, the results suggest that the management of nonrhizosphere soil could play an important role in the overall effectiveness of phytoremediation. Finally, the effect of increasing root biomass or increasing the rhizosphere thickness is approximately equivalent with respect to the ultimate removal of the contaminants.     

Modification of the immunoenzyme analysis of haptens as exemplified by testosterone determination

The solid-phase enzyme immunoassay for testosterone (TS), permitting the determination of this hormone at concentrations of up to 0.5 ng/ml, has been developed. The method comprises the adsorption of TS conjugated with soya trypsin inhibitor in the wells of a standard polystyrene assay plate, competition between adsorbed TS and TS under test for the binding sites of specific antibodies, and the detection of antibodies bound to the carrier by means of peroxidase-labeled antispecific antibodies. Antisera to TS have been obtained by the immunization of rabbits with TS conjugated with bovine serum albumin of a known composition. These antisera are specific to TS and do not interact with estrogens and progesterone. The study of their cross reactions with eleven TS derivatives has demonstrated that antibodies reveal the presence of structural changes in ring D of the molecule of TS and are insensitive to variations in ring A. The determinant comprising the 17-OH-group essentially contributes to the binding of antibodies.     

A mathematical model of delayed luminescence oscillations of higher plants and analysis of the reaction rates calculated by the model

The oscillatory regime of delayed millisecond luminescence was obtained by the model developed earlier. We compared the rates of changes in the concentrations of some of the metabolites calculated by the model with the rates calculated by other known models. The results of calculations for the changes in metabolite concentrations after switching off the light were compared with experimental data.     

Steady-state analysis of a mathematical model for capillary network formation in the absence of tumor source

This paper extends the work done in [S. Pamuk, Ph.D. Thesis, Iowa State University, 2000; Bull. Math. Biol. 63 (5) (2001) 801] in that we investigate the condition that is needed for the degradation of basement membrane in a mathematical model for capillary network formation. To do this, the steady-state behavior of tumor angiogenesis factor is studied under restricted assumptions, and the tumor angiogenesis factor threshold that activates the transport equations in the capillary is estimated using this steady state. Therefore, once the concentration of the tumor angiogenesis factor in the inner vessel wall reaches this threshold value, endothelial cells begin to move into the extracellular matrix for the start of angiogenesis. Furthermore, we do believe that the result we obtain in this paper provides an underlying insight into mechanisms of cell migration which are crucial for tumor angiogenesis.     

Development of a high-throughput screening assay based on the 3-dimensional pannus model for rheumatoid arthritis

The 3-dimensional (3-D) pannus model for rheumatoid arthritis (RA) is based on the interactive co-culture of cartilage and synovial fibroblasts (SFs). Besides the investigation of the pathogenesis of RA, it can be used to analyze the active profiles of antirheumatic pharmaceuticals and other bioactive substances under in vitro conditions. For a potential application in the industrial drug-screening process as a transitional step between 2-dimensional (2-D) cell-based assays and in vivo animal studies, the pannus model was developed into an in vitro high-throughput screening (HTS) assay. Using the CyBitrade mark-Disk workstation for parallel liquid handling, the main cell culture steps of cell seeding and cultivation were automated. Chondrocytes were isolated from articular cartilage and seeded directly into 96-well microplates in high-density pellets to ensure formation of cartilage-specific extracellular matrix (ECM). Cell seeding was performed automatically and manually to compare both processes regarding accuracy, reproducibility, consistency, and handling time. For automated cultivation of the chondrocyte pellet cultures, a sequential program was developed using the CyBio Control software to minimize shear forces and handling time. After 14 days of cultivation, the pannus model was completed by coating the cartilage pellets with a layer of human SFs. The effects due to automation in comparison to manual handling were analyzed by optical analysis of the pellets, histological and immunohistochemical staining, and real-time PCR. Automation of this in vitro model was successfully achieved and resulted in an improved quality of the generated pannus cultures by enhancing the formation of cartilage-specific ECM. In addition, automated cell seeding and media exchange increased the efficiency due to a reduction of labor intensity and handling time.     

A mathematical model for the synthesis of Gly-Phe by papain in an aqueous-organic system

The synthesis of the protected dipeptide BocGlyPheOMe, has been modellised when working in an aqueousorganic biphasic system, with papain as a catalyst. The mathematical model takes into account that one of the substrates, PheOMe, has parallel hydrolysis reactions and that the reaction only takes place in the aqueous phase while the whole reaction system is biphasic. The reaction system has been modellised when working in batch as well as when working in fed-batch mode, achieving a good prediction of the product evolution for both working strategies. When working in fed-batch mode, the extension of the undesired parallel reactions has been diminished, the model has been used for a computer aided optimisation of the addition sequence of PheOMe. The results obtained led to a process operation strategy with a compromise between yield and productivity.List of Symbols [i] concentration of any component i - [i] _aq concentration of i in the aqueous phase - [i] _bi concentration of i in the biphasic system - [E] ₀ initial concentration of enzyme - k _e, k_q first order kinetic constants - K _A, K_B equilibrium constants - r _m maximum rate of reaction This worked was financed by the Interministerial Commission for Science and Technology (CICYT)from the Spanish Government under projects number BIO/88-370 and SAF92-0261-CO2-02.