A mathematical model for the immunosenescence.

We propose a model for the dynamics of the immune system by considering the subpopulations of virgin and memory T lymphocytes on a time scale corresponding to the human life span. In the deterministic balance equation we introduce a fluctuating term in order to take into account the chronic antigenic stress. Starting from the hypothesis that the depletion of virgin cells with cytotoxic properties (CD8+) is a mortality marker, the model provides survival curves quite similar to the demographic curves.     

A mathematical model for the survivorship curve

A mathematical function describing the various kinds of survivorship curve is formulated with the useful parameter, environmental capacity. Three types of the survivorship curve illlustrated byDeevey can be obtained from changing the value of this function. When a cohort is large and the competition occurs in the scramble type, this function shows the third type ofDeevy 's and this to the first type in the case of low density and the contest type of competition. But the second type would rather be obtained by the action of density independent agencies.     

A mathematical model for toxin accumulation by killer yeasts based on the yeast population growth

The accumulation of toxin by killer yeast populations is modelled starting from a mechanistic approach that explains the toxin production in terms of yeast population growth, and takes into account the environmental inactivation of the toxin. A modified Richard's general equation for limited growth is used to define the function that describes the toxin produced in relation to the yeast biomass increase. The relationship between the rates of cell and toxin production is explicitly shown, and the implications of the resulting proportionality factor are discussed. The model parameters have been adjusted and the model has been validated using experimental data of growth and toxin accumulation from cultures of Pichia membranaefaciens in two different media. The differences between both types of cultures are analysed on the basis of parameter estimates and the predicted rate of toxin production per cell. The results support the hypothesis that biomass production and toxin synthesis are controlled in different ways; they also suggest that the composition of the medium could have a distinct effect on toxin synthesis. Model assumptions are discussed in comparison with a previous model for killer-sensitive interaction of Saccharomyces cerevisiae strains.     

A mathematical model for photoreceptor interactions

The interactions between rods and cones in the retina have been the focus of innumerable experimental and theoretical biological studies in previous decades yet the understanding of these interactions is still incomplete primarily due to the lack of a unified concept of cone photoreceptor organization and its role in retinal diseases. The low abundance of cones in many of the non-primate mammalian models that have been studied make conclusions about the human retina difficult. A more complete knowledge of the human retina is crucial for counteracting the events that lead to certain degenerative diseases, in particular those associated with photoreceptor cell death (e.g., retinitis pigmentosa). In an attempt to gain important insight into the role and interactions of the rods and the cones we develop and analyze a set of mathematical equations that model a system of photoreceptors and incorporate a direct rod-cone interaction. Our results show that the system can exhibit stable oscillations, which correspond to the rhythmic renewal and shedding of the photoreceptors. In addition, our results show the mathematical necessity of this rod-cone direct interaction for survival of both and gives insight into this mechanism.     

A mathematical model for calcium homeostasis

In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.     

A mathematical model for dorsal closure

During embryogenesis, drosophila embryos undergo epithelial folding and unfolding, which leads to a hole in the dorsal epidermis, transiently covered by an extraembryonic tissue called the amnioserosa. Dorsal closure (DC) consists of the migration of lateral epidermis towards the midline, covering the amnioserosa. It has been extensively studied since numerous physical mechanisms and signaling pathways present in DC are conserved in other morphogenetic events and wound healing in many other species (including vertebrates).We present here a simple mathematical model for DC that involves a reduced number of parameters directly linked to the intensity of the forces in the presence and which is applicable to a wide range of geometries of the leading edge (LE). This model is a natural generalization of the very interesting model proposed in Hutson et al. (2003). Being based on an ordinary differential equation (ODE) approach, the previous model had the advantage of being even simpler, but this restricted significantly the variety of geometries that could be considered and thus the number of modified dorsal closures that could be studied.A partial differential equation (PDE) approach, as the one developed here, allows considering much more general situations that show up in genetically or physically perturbed embryos and whose study will be essential for a proper understanding of the different components of the DC process. Even for native embryos, our model has the advantage of being applicable since an early stages of DC when there is no antero-posterior symmetry (approximately verified only in the late phases of DC).We validate our model in a native setting and also test it further in embryos where the zipping force is perturbed through the expression of spastin (a microtubule severing protein). We obtain variations of the force coefficients that are consistent with what was previously described for this setting.     

A mathematical model for radiation-induced myelopoiesis

A model for damage, repair, killing, and repopulation of myelopoietic marrow is presented. Evaluation produces time and dose-rate profiles during and following any complex irradiation. Equations model variable dose rates, multiple exposures, different sources, and arbitrary intervals between treatments. If factors which dominate the control of biological processes can be demonstrated, an option is to set biological rate constants to experimentally determined values. Previously, knowledge did not permit identification of dominating biological processes and their temporal rates. But a unique feature of this study is that unspecified lesions for killing and injury of cells are evaluated from mortality data on the animal species of choice. "Unspecified" is used to indicate a condition of assumption-free modeling of molecular processes, whereby rate constants for cellular effects are simply computed directly from animal mortality data. Coefficients (estimated by maximum-likelihood methods for nonspecific processes) are compared with experimental values for specific processes. The model has many uses, including modeling of the myelopoietic potential as a function of time. Another option is to calculate the whole-body survival curve for cells that control myelopoiesis as a result of the treatment schedule. Also through simple extensions of the model, an extremely complex protocol can be identified with an equivalent prompt dose value--even for partial-body, fractionated exposures.     

A mathematical model for the leukocyte filtration process

Leukocyte filters are applied clinically to remove leukocytes from blood. In order to optimize leukocyte filters, a mathematical model to describe the leukocyte filtration process was developed by modification of a general theoretical model for depth filtration. The model presented here can be used to predict the time-dependent leukocyte filtration as a function of cell-cell interaction in the filter, filter efficiency, filter capacity, filter dimensions, and leukocyte concentration in the suspension applied to the filter. The results of different leukocyte filtration experiments previously reported in the literature could be well described by the present model. (c) 1995 John Wiley & Sons, Inc.     

Infanticide as an alternative male reproductive strategy in langurs: A mathematical model

This paper makes explicit some of the assumptions underlying the view that infanticide is one of two (or more) alternative reproductive strategies utilized by adult male langur monkeys (genus Presbytis). A mathematical model has been developed from these assumptions, and formulae derived which give the equilibrium proportion of infanticidal males expected in langur populations under any given set of reproductive and demographic conditions. Together with estimates of adult male reproductive success obtained from a previous analysis of langur infanticide, these formulae were then used to calculate the precise proportion of infanticidal males expected in natural populations of langurs characterized by specific average male tenures. Further, the number of generations required for such populations to reach their predicted equilibrium was estimated using a simple computer model of langur population dynamics. The present work has thus produced several quantitative predictions which are directly falsifiable with observational data obtainable from wild populations of langur monkeys. With only slight modification, the present model may also be applied to the numerous other primate and non-primate species whose mating systems include many of the same general features as that of langurs.     

A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy

Tumor angiogenesis is the process by which new blood vessels are formed and enhance the oxygenation and growth of tumors. As angiogenesis is recognized as being a critical event in cancer development, considerable efforts have been made to identify inhibitors of this process. Cytostatic treatments that target the molecular events of the angiogenesis process have been developed, and have met with some success. However, it is usually difficult to preclinically assess the effectiveness of targeted therapies, and apparently promising compounds sometimes fail in clinical trials.We have developed a multiscale mathematical model of angiogenesis and tumor growth. At the molecular level, the model focuses on molecular competition between pro- and anti-angiogenic substances modeled on the basis of pharmacological laws. At the tissue scale, the model uses partial differential equations to describe the spatio-temporal changes in cancer cells during three stages of the cell cycle, as well as those of the endothelial cells that constitute the blood vessel walls.This model is used to qualitatively assess how efficient endostatin gene therapy is. Endostatin is an anti-angiogenic endogenous substance. The gene therapy entails overexpressing endostatin in the tumor and in the surrounding tissue. Simulations show that there is a critical treatment dose below which increasing the duration of treatment leads to a loss of efficacy.This theoretical model may be useful to evaluate the efficacy of therapies targeting angiogenesis, and could therefore contribute to designing prospective clinical trials.     

A mathematical model for prokaryotic protein synthesis

A kinetic model for the synthesis of proteins in prokaryotes is presented and analysed. This model is based on a Markov model for the state of the DNA strand encoding the protein. The states that the DNA strand can occupy are: ready, repressed, or having a mRNA chain of length i in the process of being completed. The case i = 0 corresponds to the RNA polymerase attached, but no nucleotides attached to the chain. The Markov model consists of differential equations for the rates of change of the probabilities. The rate of production of the mRNA molecules is equal to the probability that the chain is assembled to the penultimate nucleotide, times the rate at which that nucleotide is attached. Similarly, the mRNA molecules can also be in different states, including: ready and having an amino acid chain of length j attached. The rate of protein synthesis is the rate at which the chain is completed. A Michaelis-Menten type of analysis is done, assuming that the rate of protein degradation determines the ’slow’ time, and that all the other kinetic rates are ‘fast’. In the self-regulated case, this results in a single ordinary differential equation for the protein concentration.     

A mathematical model for indirectly transmitted diseases

We consider a mathematical model for the indirect transmission via a contaminated environment of a microparasite between two spatially distributed host populations having non-coincident spatial domains. The parasite is benign in a first population and lethal in the second one. Global existence results are given for the resulting reaction-diffusion system coupled with an ordinary differential equation. Then, invasion and persistence of the parasite are studied. A simplified model for the transmission of a hantavirus from bank vole to human populations is then analysed.