A steady-state electrochemical model of vascular smooth muscle cells

A model of the steady-state electrochemical response of vascular smooth muscle cells to external stimuli is presented, which accounts for K, Na, and Ca fluxes. The results of the model are broadly in accordance with experimental data 1), at various transmural pressures; 2), with channel and pump blockade; and 3), under manipulation of external ionic concentrations. The model exhibits dual stable states which sometimes coexist, and abrupt transitions between these states may account for nongraded responses in arteries as external potassium or pressure is varied. The simulations suggest that changes in the intracellular sodium concentration ([Na]i) often accompany smooth muscle responses. For example, [Na]i values vary threefold over the range of pressures from 10 to 100 mmHg.     

A delay differential equation model for tumor growth

We present a competition model of tumor growth that includes the immune system response and a cycle-phase-specific drug. The model considers three populations: Immune system, population of tumor cells during interphase and population of tumor during mitosis. Delay differential equations are used to model the system to take into account the phases of the cell cycle. We analyze the stability of the system and prove a theorem based on the argument principle to determine the stability of a fixed point and show that the stability may depend on the delay. We show theoretically and through numerical simulations that periodic solutions may arise through Hopf Bifurcations.Send offprint requests to:Minaya Villasana     

A synaptic model for the kindling effect

In this work we present a model of the kindling effect based on the Hopf bifurcation for a system of ordinary differential equations. The model shows how quantitative changes in the physiological parameters at the microscopic, synaptic scale, produce the afterdischarge which is a macroscopic effect at the neuronal network scale. The presynaptic mechanisms are based on the vesicular hypothesis, or more generally on the quantal theory of synaptic transmission. The postsynaptic processes rely on Granit's law. This model gives a consistent framework which organizes and explains several experimental observations.     

A generalized Fokker-Planck equation in the case of the Volterra model

Zwanzig and Mori's projection-operator method is used in order to derive a generalized nonlinear Fokker-Planck equation for one “relevant” species in the many species conservative Volterra model. The deterministic, autonomous, Markovian equations of motion, when averaged over a suitable ensemble of initial conditions in general give rise to a non-autonomous, non-Markovian stochastic process for the evolution of this relevant species. Moreover, this relevant species may show irreversible damping, although self-interaction terms are absent in the many species model.     

A synergetic model for the verbal transformation effect

We describe a nonlinear dynamical model based on synergetics for the auditory perceptual illusion known as the verbal transformation effect. The model is an extension of a synergetic model of perceptual oscillations of visual ambiguous figures. The main extension is connected to the number of reported alternative phonemic structures, which is typically much greater than the two or three alternatives usually reported in experiments with visual ambiguous figures. The properties of the model, which are derived using basic psychophysical principles, are presented and evaluated on the basis of the fit to earlier empirical work. It will be shown that there is very good agreement between the empirically observed properties of the verbal transformation effect and the properties detected by the model. Received: 19 February 1997 / Accepted in revised form: 27 May 1997     

A systems model for the pupil size effect

The human pupillary control system is a paradigm for linearized biological control systems. It also exhibits a series of interesting nonlinear behaviors, particularly asymmetry, “pupillary escape”, and “pupillary capture.” We present a nonlinear model in which a signal dependent upon pupil size is fed back internally to cause a change in system parameters related to gains and rates of light adaptation. The model was simulated on a digital computer, a variety of experimental data was well matched, and improvements over previous pupil models demonstrated. A candidate physiological mechanism for adaptive components of the model might have the form of an inverse “Henneman coded” neuronal pool.     

A delay equation model for oviposition habitat selection by mosquitoes

We propose a patch type model for mosquitoes that have aquatic larvae inhabiting ponds. Partial differential equations (PDEs) model the larvae on each of several disconnected patches representing the ponds, with conditions varying in each patch, coupled via the adults in the air. From the PDEs a scalar delay differential equation, with multiple delays, for the total adult mosquito population is derived. The various delays represent the larval development times in the patches. The coefficients contain all the relevant information about the sizes and geometry of the individual patches inhabited by the larvae, the boundary conditions applicable to those patches and the diffusivity of the larvae in each patch. For patches of general shapes and sizes, and without the need to specify the criteria by which an adult mosquito selects an oviposition patch, the modern theory of monotone dynamical systems and persistence theory enables a complete determination of the conditions for the mosquito population to go extinct or to persist. More detailed biological insights are obtained for the case when the patches are squares of various sizes, which allows a detailed discussion of the effects of scale, and for two particular criteria by which mosquitoes might select patches for oviposition, being (i) selection based solely on patch area, and (ii) selection based both on area and expected larval survival probability for each patch. In some parameter regimes, counterintuitive phenomena are predicted.     

A model for studying the effect of shear stress on interactions between vascular endothelial cells and smooth muscle cells

Vascular endothelial cells (ECs) are constantly subjected to blood flow-induced shear stress and the influences of neighboring smooth muscle cells (SMCs). In the present study, a coculture flow system was developed to study the effect of shear stress on EC-SMC interactions. ECs and SMCs were separated by a porous membrane with only the EC side subjected to the flow condition. When ECs were exposed to a shear stress of 12 dynes/cm2 for 24 h, the cocultured SMCs tended to orient perpendicularly to the flow direction. This perpendicular orientation of the cocultured SMCs to flow direction was not observed when ECs were exposed to a shear stress of 2 dynes/cm2. Under the static condition, long and parallel actin bundles were observed in the central regions of the cocultured SMCs, whereas the actin filaments localized mainly at the periphery of the cocultured ECs. After 24 h of flow application, the cocultured ECs displayed very long, well-organized, parallel actin stress fibers aligned with the flow direction in the central regions of the cells. Immunostaining of platelet endothelial cell adhesion molecule-1 confirmed the elongation and alignment of the cocultured ECs with the flow direction. Coculture with SMCs under static condition induced EC gene expressions of growth-related oncogene-alpha and monocyte chemotactic protein-1, and shear stress was found to abolish these SMC-induced gene expressions. Our results suggest that shear stress may serve as a down-regulator for the pathophysiologically relevant gene expression in ECs cocultured with SMCs.     

A model for the crowding effect in the growth of tadpoles

Crowded tadpoles in a limited volume grow to divide into two groups, a normally growing group and a stunted group, even if there are a plenty of food. This phenomenon was found to be interpreted by a model involving a mutual inhibition of the growth among individuals. It was assumed that the growth curve of an individual was modelled by a Gompertz growth equation. Through mathematical analysis and computer simulation of the model, it was shown that the average growth rate of the group was a decreasing function of the population density and that the group segregated into two subgroups, a normally growing group and a stunted group in a certain range of density. A theoretical prospect that the population of normally growing group is proportionate to the volume of the aquarium was obtained.     

A continuum model for coupled cells

A continuum model of diffusion-coupled cells that more accurately reflects the presence of low-permeability gap junctions between cells is analyzed. It is shown by a multi-scale analysis that to lowest order the slow evolution of the mean concentration is described by the usual ordinary differential equations for a discrete model. Furthermore, stable non-uniform steady solutions are shown to exist in the continuum model of a one component system, whereas this is impossible for the standard reaction-diffusion model of this system. It is also shown how to average the equations in this continuum model to obtain a system of reaction-diffusion equations with constant coefficients.     

A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.     

A model system to study the effect of SO2 on plant cells

Fumigation with 0.05 μl/l SO₂ decreased the germination rate of fern spores ofAdiantum capillus-veneris L. and influenced greatly the rhizoid development for abnormities such as apical swelling. Even 10 μM sulfite derived from SO₂ fumigation was influential. These SO₂ effects on germination and following rhizoid development were proved to be independent of each other, and could be induced separately by pulse treatment with K₂SO₃ solution given at different timings. According to the analysis using a time-lapse video recorder, growth retardation and apical swelling of rhizoid occurred just after the application of sulfite. It became clear that the fern gametophytes are very sensitive to SO₂, (that is sulfite) as compared with other plants used for air pollutant experiments so far.     

A model for calculating the Bohr effect in hemoglobin equilibria.

The unusual aspects of the reaction of oxygen with hemoglobin are believed to be due to the free energy of the conformational change in the hemoglobin molecule upon oxygenation. The conformational free energy change due to oxygenation can be estimated in terms of the surface free energy of an emuslion droplet of the same size as the hemoglobin molecule. Calculations on the basis of this model lead to an equilibrium constant that varies with pH as in the acid and alkaline Bohr Effects, and that also varies with the ionic strength. The model used in this paper provides a simple way of estimating the variation of the equilibrium constant of a reaction involving a globular protein where the free energy of conformational changes can be evaluated in terms of surface properties.