Evolution of rotifer population dynamics in a heterogeneous habitat: mathematical modeling

We present the results of mathematical modeling of a rotifer species inhabiting two coupled habitats with different environmental conditions. We use the modified Consensus model and show that the exchange between the habitats can lead to chaotization of originally regular plankton dynamics and synchronization of plankton biomass oscillations. As a result, the invasion of a chaotic regime takes place.     

T cell homeostasis

The pool of mature T cells comprises a heterogeneous mixture of naive and memory CD4(+) and CD8(+) cells. These cells are long lived at a population level but differ markedly in their relative rates of turnover and survival. Here, we review how contact with exogenous stimuli, notably self MHC ligands and various gamma(c) cytokines, plays a decisive role in controlling normal T cell homeostasis.     

Implications of a simple mathematical model to cancer cell population dynamics

Recent research in cancer progression and treatment indicates that many forms of cancer arise from the development of a small subpopulation of abnormal cancer stem cells (CSCs) that promote cancer growth and spread. Many potential treatments preferentially interact with cells at certain stages of the cell cycle by either selective killing or halting the cell cycle, such as intense, nanosecond-duration pulsed electric fields (nsPEFs). Simple mathematical models of unfed cancer cell populations at the plateau of their growth characteristics may estimate the long-term consequences of these treatments on proliferating and quiescent cell populations. Applying such a model with no transition from the quiescent to proliferating state shows that it is possible for the proliferating cell population to fall below 1 if the quiescent cell population obtains a sufficient competitive advantage with respect to nutrient consumption and/or survival rate. Introducing small, realistic transition rates did not appreciably alter short-term or long-term population behaviour, indicating that the predicted small cell population behaviour (< 1 cell) is not an artefact of the simpler model. Experimental observations of nsPEF-induced effects on the cell cycle suggest that such a model may serve as a first step in assessing the viability of a given cancer treatment in vitro prior to clinical application.     

Experimental and mathematical modeling of population dynamics of rhizospheric bacteria under conditions of cadmium stress

The method of membrane filters was used to study the population dynamics of bacteria belonging to the genera Arthrobacter, Flavobacterium, and Klebsiella in barley (Hordeum vulgare) rhizosphere under conditions of cadmium stress (5-15 mg Cd/g soil). Mathematical modeling allowed us to demonstrate that the phytoprotective effect is implemented via the following succession of events: the bacteria synthesize phytohormones (IAA and ethylene)-->root excretory activity increases-->the number of the bacteria in the rhizoplane grows-->the flux of bacteria migrating from the rhizoplane to the rhizosphere increases-->the number of bacteria binding cadmium ions in the rhizosphere grows-->the amount of free ions entering the plant decreases. Among the bacteria studied, K. mobilis 880 displayed the highest migration and immobilization activity and the best survival rate under conditions of cadmium stress. Consequently, K. mobilis 880 is recommended for use in biopreparations for stimulating plant growth under conditions of heavy metal pollution.     

Calcium dynamics and homeostasis in a mathematical model of the principal cell of the cortical collecting tubule

Calcium (Ca) dynamics are incorporated into a mathematical model of the principal cell in the cortical collecting tubule developed earlier in Strieter et al. (1992a. Am. J Physiol. 263:F1063-1075). The Ca components are modeled after the Othmer-Tang model for IP(3)-sensitive calcium channels (1993, in Experimental and Theoretical Advances in Biological Pattern Formation, 295-319). There are IP(3)-sensitive Ca channels and ATP-driven pumps on the membrane of the endoplasmic reticulum. Calcium enters the cell passively down its electrochemical gradient. A Ca pump and Na/Ca exchange in the basolateral membrane are responsible for the extrusion of cytoplasmic calcium. Na/Ca exchange can also operate in reverse mode to transport Ca into the cell. Regulatory effects of cytoplasmic Ca on the apical Na channels are modeled after experimental data that indicate apical Na permeability varies inversely with cytoplasmic Ca concentration. Numerical results on changes in intracellular Ca caused by decreasing NaCl in the bath and the lumen are similar to those from experiments in Bourdeau and Lau (1990. Am. J Physiol. 258:F1497-1503). This match of simulation and experiment requires the synergistic action of the Na/Ca exchanger and the Ca regulated apical Na permeability. In a homogeneous medium, cytoplasmic Ca becomes oscillatory when extracellular Na is severely decreased, as observed in experiments of cultured principal cells (Koster, H., C. van Os and R. Bindels. 1993. Kidney Int.43:828-836). This essentially pathological situation arises because the hyperpolarization of membrane potential caused by Na-free medium increases Ca influx into the cell, while the Na/Ca exchanger is inactivated by the low extracellular Na and can no longer move Ca out of the cell effectively. The raising of the total amount of intracellular Ca induces oscillatory Ca movement between the cytoplasm and the endoplasmic reticulum. Ca homeostasis is investigated under the condition of severe extracellular Ca variations. As extracellular Ca is decreased, Ca regulation is greatly impaired if Ca does not regulate apical ionic transport. The simulations indicate that the Na/Ca exchanger alone has only limited regulatory capacity. The Ca regulated apical sodium or potassium permeability are essential for regulation of cytoplasmic Ca in the principal cell of the cortical collecting tubule.     

Generalized modeling of ecological population dynamics

Over the past 7 years, several authors have used the approach of generalized modeling to study the dynamics of food chains and food webs. Generalized models come close to the efficiency of random matrix models, while being as directly interpretable as conventional differential-equation-based models. Here, we present a pedagogical introduction to the approach of generalized modeling. This introduction places more emphasis on the underlying concepts of generalized modeling than previous publications. Moreover, we propose a shortcut that can significantly accelerate the formulation of generalized models and introduce an iterative procedure that can be used to refine existing generalized models by integrating new biological insights.     

Chancroid transmission dynamics: a mathematical modeling approach

Mathematical models have long been used to better understand disease transmission dynamics and how to effectively control them. Here, a chancroid infection model is presented and analyzed. The disease-free equilibrium is shown to be globally asymptotically stable when the reproduction number is less than unity. High levels of treatment are shown to reduce the reproduction number suggesting that treatment has the potential to control chancroid infections in any given community. This result is also supported by numerical simulations which show a decline in chancroid cases whenever the reproduction number is less than unity.     

Potassium starvation in yeast: mechanisms of homeostasis revealed by mathematical modeling

The intrinsic ability of cells to adapt to a wide range of environmental conditions is a fundamental process required for survival. Potassium is the most abundant cation in living cells and is required for essential cellular processes, including the regulation of cell volume, pH and protein synthesis. Yeast cells can grow from low micromolar to molar potassium concentrations and utilize sophisticated control mechanisms to keep the internal potassium concentration in a viable range. We developed a mathematical model for Saccharomyces cerevisiae to explore the complex interplay between biophysical forces and molecular regulation facilitating potassium homeostasis. By using a novel inference method ("the reverse tracking algorithm") we predicted and then verified experimentally that the main regulators under conditions of potassium starvation are proton fluxes responding to changes of potassium concentrations. In contrast to the prevailing view, we show that regulation of the main potassium transport systems (Trk1,2 and Nha1) in the plasma membrane is not sufficient to achieve homeostasis.     

Individual‐based and stochastic modeling of cell population dynamics considering substrate dependency

In this article, we propose an individual‐based and stochastic modeling approach that is capable of describing the bacterial cell population dynamics during a batch culture. All stochastic nature inherent in intracellular molecular level reactions and cell division processes were considered in a single model framework by embedding a sub‐model describing individual cell's growth kinetics in a discrete event simulation algorithm. The resultant unique feature of the model is that the effects of the stochasticities on the cell population dynamics can be investigated for different substrate‐dependent cell growth kinetics. When Monod kinetics was used as the sub‐model, the stochasticities only slightly affected the cell mass increase and substrate consumption profiles during the batch culture although they were still important in describing the changes of cell population distributions. When Andrews substrate inhibition kinetics was used, however, it was revealed that the overall cell population dynamics could be seriously influenced by the stochasticities. Under a critical initial substrate level, the cell population could proliferate against the substrate inhibition only when the stochasticities were considered. Biotechnol. Bioeng. 2009;103: 891–899. © 2009 Wiley Periodicals, Inc.     

Predicting the effectiveness of population replacement strategy using mathematical modeling

Charles Taylor and John Marshall explain the utility of mathematical modeling for evaluating the effectiveness of population replacement strategy. Insight is given into how computational models can provide information on the population dynamics of mosquitoes and the spread of transposable elements through A. gambiae subspecies. The ethical considerations of releasing genetically modified mosquitoes into the wild are discussed.     

Autologous regulation of naive T cell homeostasis within the T cell compartment

Naive T cells undergo spontaneous slow proliferation on adoptive transfer into syngeneic T cell (T)-deficient hosts. Recent work has shown that such "homeostatic" T cell proliferation is driven by MHC molecules loaded with self-peptides rather than foreign peptides. Because naive T cells in normal T-sufficient hosts remain in interphase despite continuous contact with self-MHC/peptide ligands, T cells apparently inhibit homeostatic proliferation of neighboring T cells. To address this, we have investigated the requirements necessary for "bystander" T cells to inhibit homeostatic proliferation of other T cells. Three key findings are reported. First, homeostatic proliferation of T cells only occurs in specific microenvironments, namely the T cell compartment of the secondary lymphoid tissues. Second, direct entry into T cell compartments is also required for bystander inhibition of homeostatic proliferation. Third, bystander inhibition is mediated largely by naive rather than activated/memory T cells and does not require proliferation or TCR ligation. These findings suggest that homeostasis of naive T cells is unlikely to be regulated through competition for systemic soluble factors or for specific stimulatory self-MHC/peptide ligands. Rather, the data favor mechanisms that involve competition for local non-MHC stimulatory factors or direct cell-to-cell interactions between the T cells themselves within the T cell compartment.     

A mathematical model of population dynamics for Batesian mimicry system

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.     

A mathematical model of population dynamics for Batesian mimicry system

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.     

On mathematical theory of selection: continuous time population dynamics

Mathematical theory of selection is developed within the frameworks of general models of inhomogeneous populations with continuous time. Methods that allow us to study the distribution dynamics under natural selection and to construct explicit solutions of the models are developed. All statistical characteristics of interest, such as the mean values of the fitness or any trait can be computed effectively, and the results depend in a crucial way on the initial distribution. The developed theory provides an effective method for solving selection systems; it reduces the initial complex model to a special system of ordinary differential equations (the escort system). Applications of the method to the Price equations are given; the solutions of some particular inhomogeneous Malthusian, Ricker and logistic-like models used but not solved in the literature are derived in explicit form.     

A mathematical model for the population dynamics of army ants

A stochastic cellular automata model for the population dynamics of the army antEciton burchelli on Barro Colorado Island in Panama is set up. It is simulated on the computer and shown to give good agreement with biological data. It is analysed using two approximations akin to the mean field approximation in statistical mechanics, and good agreement with the simulations is obtained. Finally, the role of distance between successive statary phase bivouacs is discussed with regard to the rate of colony growth. There are two aspects of the biological system studied here that make it of general importance. First, the population is structured, since the size of each colony of army ants is crucial. Second, the spatial behaviour of the population, as in many others, is not diffusion-like, although it is random. This has implications for the kind of model that is chosen.     

Stability of a diverse immunological memory is determined by T cell population dynamics

The correlation between properties of the T cell memory pool and the two regulatory mechanisms of cell death (apoptosis) and memory entry (differentiation) is investigated mathematically. Apoptosis of T cells occurs at the end of an immune response, removing unwanted activated T cells. T cells escaping apoptosis enter the memory pool composed of T cells specific for previously encountered antigens. We find that the relative efficiencies of these two pathways determine the clonal distribution and the long-term stability of the memory pool by regulating the number of new entries. The main result presented in this paper is that immunological memory of previously encountered pathogens cannot be erased by either severe or repeat infections with a particular pathogen (the diversity of the memory pool is ensured) only if apoptosis and/or memory differentiation are regulated by population dependent processes. Furthermore, vaccination properties are improved significantly by population dependent mechanisms and our mathematical analysis reveals that the T cell population must communicate with other parts of the immune system to ensure optimal performance of immunological memory.     

ZFAT plays critical roles in peripheral T cell homeostasis and its T cell receptor-mediated response

ZFAT, originally identified as a candidate susceptibility gene for autoimmune thyroid disease, has been reported to be involved in apoptosis, development and primitive hematopoiesis. Zfat is highly expressed in T- and B-cells in the lymphoid tissues, however, its physiological function in the immune system remains totally unknown. Here, we generated the T cell-specific Zfat-deficient mice and demonstrated that Zfat-deficiency leads to a remarkable reduction in the number of the peripheral T cells. Intriguingly, a reduced expression of IL-7Rα and the impaired responsiveness to IL-7 for the survival were observed in the Zfat-deficient T cells. Furthermore, a severe defect in proliferation and increased apoptosis in the Zfat-deficient T cells following T cell receptor (TCR) stimulation was observed with a reduced IL-2Rα expression as well as a reduced IL-2 production. Thus, our findings reveal that Zfat is a critical regulator in peripheral T cell homeostasis and its TCR-mediated response.