Pontryagin's principle for control problems in age-dependent population dynamics

In this paper, Pontryagin's principle is proved for a fairly general problem of optimal control of populations with continuous time and age variable. As a consequence, maximum principles are developed for an optimal harvesting problem and a problem of optimal birth control.     

An age-dependent stochastic model of population growth

A stochastic model of population growth is treated using the Bellman-Harris theory of agedependent stochastic branching processes. The probability distribution for the population size at any time and the expectation are obtained when it is assumed that there is probability (1−σ), 0≤σ<1, of the organism dividing into two at the end of its lifetime, and probability σ that division will not take place.     

Optimization of harvesting age in an integral age-dependent model of population dynamics

The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of the problem is provided. The model shows the presence of a zero-investment period. A preliminary asymptotic analysis indicates possible turnpike properties of the optimal harvesting age. Biological interpretation of all results is provided.     

A model for the age-dependent dynamics of growth of terrestrial mammals

Two hypothetical biological mechanisms were proposed that largely determine the dynamics of growth of overland mammals. The first mechanism is the maintenance of a spatial similarity of the anatomy of the organism during its growth, and the second is the maintenance of the stability of the internal environment of the organism. On the basis of the advanced hypothesis, a model of age-dependent dynamics of growth was constructed, and a differential equation describing the changes in the body mass with time was derived. According to this model, the dynamics of growth and the mass of an adult individual are determined by two energy constants that characterize the mechanism of food assimilation and the energy expenditures for the movement of the individual in space. The deviation between the solution obtained and the experimental data on age-dependent changes in the mass of the human body was on the average 6%. These insignificant deviations were explained in the framework of the proposed hypothesis, which indicates its validity.     

An integrated model of cardiac mitochondrial energy metabolism and calcium dynamics

We present an integrated thermokinetic model describing control of cardiac mitochondrial bioenergetics. The model describes the tricarboxylic acid (TCA) cycle, oxidative phosphorylation, and mitochondrial Ca(2+) handling. The kinetic component of the model includes effectors of the TCA cycle enzymes regulating production of NADH and FADH(2), which in turn are used by the electron transport chain to establish a proton motive force (Delta mu(H)), driving the F(1)F(0)-ATPase. In addition, mitochondrial matrix Ca(2+), determined by Ca(2+) uniporter and Na(+)/Ca(2+) exchanger activities, regulates activity of the TCA cycle enzymes isocitrate dehydrogenase and alpha-ketoglutarate dehydrogenase. The model is described by twelve ordinary differential equations for the time rate of change of mitochondrial membrane potential (Delta Psi(m)), and matrix concentrations of Ca(2+), NADH, ADP, and TCA cycle intermediates. The model is used to predict the response of mitochondria to changes in substrate delivery, metabolic inhibition, the rate of adenine nucleotide exchange, and Ca(2+). The model is able to reproduce, qualitatively and semiquantitatively, experimental data concerning mitochondrial bioenergetics, Ca(2+) dynamics, and respiratory control. Significant increases in oxygen consumption (V(O(2))), proton efflux, NADH, and ATP synthesis, in response to an increase in cytoplasmic Ca(2+), are obtained when the Ca(2+)-sensitive dehydrogenases are the main rate-controlling steps of respiratory flux. These responses diminished when control is shifted downstream (e.g., the respiratory chain or adenine nucleotide translocator). The time-dependent behavior of the model, under conditions simulating an increase in workload, closely reproduces experimentally observed mitochondrial NADH dynamics in heart trabeculae subjected to changes in pacing frequency. The steady-state and time-dependent behavior of the model support the hypothesis that mitochondrial matrix Ca(2+) plays an important role in matching energy supply with demand in cardiac myocytes.     

An image-based model of calcium waves in differentiated neuroblastoma cells

Calcium waves produced by bradykinin-induced inositol-1,4, 5-trisphosphate (InsP(3))-mediated release from endoplasmic reticulum (ER) have been imaged in N1E-115 neuroblastoma cells. A model of this process was built using the "virtual cell," a general computational system for integrating experimental image, biochemical, and electrophysiological data. The model geometry was based on a cell for which the calcium wave had been experimentally recorded. The distributions of the relevant cellular components [InsP(3) receptor (InsP(3)R)], sarcoplasmic/endoplasmic reticulum calcium ATPase (SERCA) pumps, bradykinin receptors, and ER] were based on 3D confocal immunofluorescence images. Wherever possible, known biochemical and electrophysiological data were used to constrain the model. The simulation closely matched the spatial and temporal characteristics of the experimental calcium wave. Predictions on different patterns of calcium signals after InsP(3) uncaging or for different cell geometries were confirmed experimentally, thus helping to validate the model. Models in which the spatial distributions of key components are altered suggest that initiation of the wave in the center of the neurite derives from an interplay of soma-biased ER distribution and InsP(3) generation biased toward the neurite. Simulations demonstrate that mobile buffers (like the indicator fura-2) significantly delay initiation and lower the amplitude of the wave. Analysis of the role played by calcium diffusion indicated that the speed of the wave is only slightly dependent on the ability of calcium to diffuse to and activate neighboring InsP(3) receptor sites.     

Mathematical modeling of calcium homeostasis in yeast cells

In this study, based on currently available experimental observations on protein level, we constructed a mathematical model to describe calcium homeostasis in normally growing yeast cells (Saccharomyces cerevisiae). Simulation results show that tightly controlled low cytosolic calcium ion level can be a natural result under the general mechanism of gene expression feedback control. The calmodulin (a sensor protein) behavior in our model cell agrees well with relevant observations in real cells. Moreover, our model can qualitatively reproduce the experimentally observed response curve of real yeast cell responding to step-like disturbance in extracellular calcium ion concentration. Further investigations show that the feedback control mechanism in our model is as robust as it is in real cells.     

An age-dependent epidemic model with application to measles

A combined epidemic-demographic model is developed which models the spread of an infectious disease throughout a population of constant size. The model allows for births, deaths, temporary or permanent immunity, and immunization. The relationship of this model to previously studied epidemic and demographic models is illustrated. An advantage of this model is that all epidemic and demographic parameters may be estimated. The stability of the equilibrium point corresponding to the elimination of the disease is studied and a threshold value is found which indicates whether the disease will die out or remain endemic in the population. The application of the model to measles indicates that immunization levels needed to reduce the incidence to near zero may not be as high as previously predicted.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Viable controls in age-dependent population dynamics

The classical age-dependent population model is considered, where the mortality depends on total population and the fertility is age dependent. Conditions are determined under which such a system may be steered to a specific population level and held there.     

Stable controls in age-dependent population dynamics

The classical age-dependent population model is considered, in which mortality depends on total population and fertility is age-dependent. It is shown that in general such systems are not completely controllable with respect to a control variable in the mortality function, but that in certain circumstances a suitable control can be found to hold the population at a specified level.     

Microencapsulation of yeast cells in the calcium alginate membrane

Summary Cells of Saccharomyces cerevisiae (ATCC 24858) were encapsulated in the calcium alginate membrane and cultured. Swelling of the capsule was prevented by adding 0.2 g CaCl₂ to 1 L growth medium. The dry cell concentration based on the inner volume of the capsule reached 309 g/L, which was much higher than could be obtained by cell entrapment. All the cells remained inside the capsule during the cultivation. The flux of CO₂ through the capsule membrane increased approximately twice by adding a nonionic surfactant to the CaCl₂ solution during the step of capsule formation.     

The dynamics of cytosolic calcium in photoreceptor cells

Analysis of the light-induced changes of cytosolic Ca²⁺ ([Ca²⁺]_i) in photoreceptor cells has been taken a step further with two recently published studies^(1,2). In one, changes in [Ca²⁺]_i were measured in single detached rod outer segments from Gecko in response to various light intensities. The advances of the other⁽²⁾ are embodied in its employment of transgenic Drosophila, whose photoreceptors express a visual pigment that is insensitive to the wavelength of light used in the fluorescence imaging of [Ca²⁺]_i. These studies provide a better basis for understanding the regulation of Ca²⁺-mediated events in photoreceptor cells.     

Stability and bifurcation in age-dependent population dynamics

A general form for an integral cohort model reflecting Easterlin's hypothesis is constructed. It is found that the basic dynamics of the model can be determined by a knowledge of the slope of the net reproduction rate at the equilibrium. Several forms for the net reproduction rate are examined numerically, and the resulting bifurcation processes are compared with those of the related first order difference equation modeling a two age group population. The results suggest that the situation usually described as “chaos” is likely to be less prevalent than earlier studies of first and second order difference equations have suggested.     

Cameleon calcium indicator reports cytoplasmic calcium dynamics in Arabidopsis guard cells

Cytoplasmic free calcium ([Ca2+]cyt) acts as a stimulus-induced second messenger in plant cells and multiple signal transduction pathways regulate [Ca2+]cyt in stomatal guard cells. Measuring [Ca2+]cyt in guard cells has previously required loading of calcium-sensitive dyes using invasive and technically difficult micro-injection techniques. To circumvent these problems, we have constitutively expressed the pH-independent, green fluorescent protein-based calcium indicator yellow cameleon 2.1 in Arabidopsis thaliana (Miyawaki et al. 1999; Proc. Natl. Acad. Sci. USA 96, 2135-2140). This yellow cameleon calcium indicator was expressed in guard cells and accumulated predominantly in the cytoplasm. Fluorescence ratio imaging of yellow cameleon 2.1 allowed time-dependent measurements of [Ca2+]cyt in Arabidopsis guard cells. Application of extracellular calcium or the hormone abscisic acid (ABA) induced repetitive [Ca2+]cyt transients in guard cells. [Ca2+]cyt changes could be semi-quantitatively determined following correction of the calibration procedure for chloroplast autofluorescence. Extracellular calcium induced repetitive [Ca2+]cyt transients with peak values of up to approximately 1.5 microM, whereas ABA-induced [Ca2+]cyt transients had peak values up to approximately 0.6 microM. These values are similar to stimulus-induced [Ca2+]cyt changes previously reported in plant cells using ratiometric dyes or aequorin. In some guard cells perfused with low extracellular KCl concentrations, spontaneous calcium transients were observed. As yellow cameleon 2.1 was expressed in all guard cells, [Ca2+]cyt was measured independently in the two guard cells of single stomates for the first time. ABA-induced, calcium-induced or spontaneous [Ca2+]cyt increases were not necessarily synchronized in the two guard cells. Overall, these data demonstrate that that GFP-based cameleon calcium indicators are suitable to measure [Ca2+]cyt changes in guard cells and enable the pattern of [Ca2+]cyt dynamics to be measured with a high level of reproducibility in Arabidopsis cells. This technical advance in combination with cell biological and molecular genetic approaches will become an invaluable tool in the dissection of plant cell signal transduction pathways.     

Measuring calcium dynamics in living cells with genetically encodable calcium indicators

Genetically encoded calcium indicators (GECIs) allow researchers to measure calcium dynamics in specific targeted locations within living cells. Such indicators enable dissection of the spatial and temporal control of calcium signaling processes. Here we review recent progress in the development of GECIs, highlighting which indicators are most appropriate for measuring calcium in specific organelles and localized domains in mammalian tissue culture cells. An overview of recent approaches that have been undertaken to ensure that the GECIs are minimally perturbed by the cellular environment is provided. Additionally, the procedures for introducing GECIs into mammalian cells, conducting calcium imaging experiments, and analyzing data are discussed. Because organelle-targeted indicators often pose an additional challenge, we underscore strategies for calibrating GECIs in these locations.     

Evolution of age-dependent sex-reversal under adaptive dynamics

We investigate the evolution of the age (or size) at sex-reversal in a model of sequential hermaphroditism, by means of the function-valued adaptive dynamics. The trait is the probability law of the age at sex-reversal considered as a random variable. Our analysis starts with the ecological model which was first introduced and analyzed by Calsina and Ripoll (Math Biosci 208(2), 393–418, 2007). The structure of the population is extended to a genotype class and a new model for an invading/mutant population is introduced. The invasion fitness functional is derived from the ecological setting, and it turns out to be controlled by a formula of Shaw–Mohler type. The problem of finding evolutionarily stable strategies is solved by means of infinite-dimensional linear optimization. We have found that these strategies correspond to sex-reversal at a single particular age (or size) even if the set of feasible strategies is considerably broader and allows for a probabilistic sex-reversal. Several examples, including in addition the population-dynamical stability, are illustrated. For a special case, we can show that an unbeatable size at sex-reversal must be larger than 69.3% of the expected size at death.     

A muscle activation model of variable stimulation frequency response and stimulation history, based on positive feedback in calcium dynamics

 Muscle fiber response to a train of variable-frequency pulses includes the potentiation and catch-like effect. For better understanding of these phenomena, we built an activation model with emphasis on the calcium liberation from and re-sequestration into the sarcoplasmic reticulum, including calcium-induced calcium release. The model had two stable equilibrium points in the calcium concentration. Changes from the low to the high equilibrium point could be produced by high-frequency trains of pulses and would account for the potentiation. The model also showed a catch-like effect, as a long-lasting increment of muscle force after the application of a single extra pulse. The increase in force appeared in resting muscle, disappeared when the muscle was potentiated, and reappeared briefly if the stimulation was continued for long periods. Received: 31 January 2000 / Accepted in revised form: 2 August 2000