On the stability of the cell size distribution

A model for the growth of a size-structured cell population reproducing by fission into two identical daughters is formulated and analysed. The model takes the form of a linear first order partial differential equation (balance law) in which one term has a transformed argument. Using semigroup theory and compactness arguments we establish the existence of a stable size distribution under a certain condition on the growth rate of the individuals. An example shows that one cannot dispense with this condition.     

A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators. Supported in part by the National Science Foundation under Grant No. DMS-8722947     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

A model of proliferating cell populations with inherited cycle length

A mathematical model of cell population growth introduced by J. L. Lebowitz and S. I. Rubinow is analyzed. Individual cells are distinguished by age and cell cycle length. The cell cycle length is viewed as an inherited property determined at birth. The density of the population satisfies a first order linear partial differential equation with initial and boundary conditions. The boundary condition models the process of cell division of mother cells and the inheritance of cycle length by daughter cells. The mathematical analysis of the model employs the theory of operator semigroups and the spectral theory of linear operators. It is proved that the solutions exhibit the property of asynchronous exponential growth.     

The effects of cell density and metabolite flux on cellular dynamics

Summary Density-dependent regulation of cell growth in tissue culture is a well-known phenomenon but the mechanism of regulation remains obscure. Here we explore the effects of cell density and metabolite flux on the collective dynamics of a cell population. The intracellular dynamics are modelled by positive feedback kinetic mechanisms of the kind known to apply to yeast cells. Several experimental observations related to glycolytic oscillations are predicted and it is suggested that the general conclusions may be applicable in a broader context.     

Bayesian Inference on the Effect of Density Dependence and Weather on a Guanaco Population from Chile

Understanding the mechanisms that drive population dynamics is fundamental for management of wild populations. The guanaco (Lama guanicoe) is one of two wild camelid species in South America. We evaluated the effects of density dependence and weather variables on population regulation based on a time series of 36 years of population sampling of guanacos in Tierra del Fuego, Chile. The population density varied between 2.7 and 30.7 guanaco/km², with an apparent monotonic growth during the first 25 years; however, in the last 10 years the population has shown large fluctuations, suggesting that it might have reached its carrying capacity. We used a Bayesian state-space framework and model selection to determine the effect of density and environmental variables on guanaco population dynamics. Our results show that the population is under density dependent regulation and that it is currently fluctuating around an average carrying capacity of 45,000 guanacos. We also found a significant positive effect of previous winter temperature while sheep density has a strong negative effect on the guanaco population growth. We conclude that there are significant density dependent processes and that climate as well as competition with domestic species have important effects determining the population size of guanacos, with important implications for management and conservation.     

Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro

We present a three-dimensional individual cell-based, biophysical model to study the effect of normal and malfunctioning growth regulation and control on the spatial-temporal organization of growing cell populations in vitro. The model includes explicit representations of typical epithelial cell growth regulation and control mechanisms, namely 1), a cell-cell contact-mediated form of growth inhibition; 2), a cell-substrate contact-dependent cell-cycle arrest; and 3), a cell-substrate contact-dependent programmed cell death (anoikis). The model cells are characterized by experimentally accessible biomechanical and cell-biological parameters. First, we study by variation of these cell-specific parameters which of them affect the macroscopic morphology and growth kinetics of a cell population within the initial expanding phase. Second, we apply selective knockouts of growth regulation and control mechanisms to investigate how the different mechanisms collectively act together. Thereby our simulation studies cover the growth behavior of epithelial cell populations ranging from undifferentiated stem cell populations via transformed variants up to tumor cell lines in vitro. We find that the cell-specific parameters, and in particular the strength of the cell-substrate anchorage, have a significant impact on the population morphology. Furthermore, they control the efficacy of the growth regulation and control mechanisms, and consequently tune the transition from controlled to uncontrolled growth that is induced by the failures of these mechanisms. Interestingly, however, we find the qualitative and quantitative growth kinetics to be remarkably robust against variations of cell-specific parameters. We compare our simulation results with experimental findings on a number of epithelial and tumor cell populations and suggest in vitro experiments to test our model predictions.     

Cell Growth and Size Homeostasis in Silico

Cell growth in size is a complex process coordinated by intrinsic and environmental signals. In a research work performed by a different group, size distributions of an exponentially growing population of mammalian cells were used to infer cell-growth rate in size. The results suggested that cell growth was neither linear nor exponential, but subject to size-dependent regulation. To explain the observed growth pattern, we built a mathematical model in which growth rate was regulated by the relative amount of mRNA and ribosomes in a cell. Under the growth model and a stochastic division rule, we simulated the evolution of a population of cells. Both the sampled growth rate and size distribution from this in silico population agreed well with experimental data. To explore the model space, alternative growth models and division rules were studied. This work may serve as a starting point to understand the mechanisms behind cell growth and size regulation using predictive models.     

Cell Growth and Size Homeostasis in Silico

Cell growth in size is a complex process coordinated by intrinsic and environmental signals. In a research work performed by a different group, size distributions of an exponentially growing population of mammalian cells were used to infer cell-growth rate in size. The results suggested that cell growth was neither linear nor exponential, but subject to size-dependent regulation. To explain the observed growth pattern, we built a mathematical model in which growth rate was regulated by the relative amount of mRNA and ribosomes in a cell. Under the growth model and a stochastic division rule, we simulated the evolution of a population of cells. Both the sampled growth rate and size distribution from this in silico population agreed well with experimental data. To explore the model space, alternative growth models and division rules were studied. This work may serve as a starting point to understand the mechanisms behind cell growth and size regulation using predictive models.     

Perception and Regulatory Principles of Microbial Growth Control

Fast growth represents an effective strategy for microbial organisms to survive in competitive environments. To accomplish this task, cells must adapt their metabolism to changing nutrient conditions in a way that maximizes their growth rate. However, the regulation of the growth related metabolic pathways can be fundamentally different among microbes. We therefore asked whether growth control by perception of the cell’s intracellular metabolic state can give rise to higher growth than by direct perception of extracellular nutrient availability. To answer this question, we created a simplified dynamical computer model of a cellular metabolic network whose regulation was inferred by an optimization approach. We used this model for a competing species experiment, where a species with extracellular nutrient perception competes against one with intracellular nutrient perception by evaluating their respective average growth rate. We found that the intracellular perception is advantageous under situations where the up and down regulation of pathways cannot follow the fast changing nutrient availability in the environment. In this case, optimal regulation ignores any other nutrients except the most preferential ones, in agreement with the phenomenon of catabolite repression in prokaryotes. The corresponding metabolic pathways remain activated, despite environmental fluctuations. Therefore, the cell can take up preferential nutrients as soon as they are available without any prior regulation. As a result species that rely on intracellular perception gain a relevant fitness advantage in fluctuating nutrient environments, which enables survival by outgrowing competitors.     

Quantifying impacts of plastic debris on marine wildlife identifies ecological breakpoints

Quantifying sublethal effects of plastics ingestion on marine wildlife is difficult, but key to understanding the ontogeny and population dynamics of affected species. We developed a method that overcomes the difficulties by modelling individual ontogeny under reduced energy intake and expenditure caused by debris ingestion. The predicted ontogeny is combined with a population dynamics model to identify ecological breakpoints: cessation of reproduction or negative population growth. Exemplifying this approach on loggerhead turtles, we find that between 3% and 25% of plastics in digestive contents causes a 2.5–20% reduction in perceived food abundance and total available energy, resulting in a 10–15% lower condition index and 10% to 88% lower total seasonal reproductive output compared to unaffected turtles. The reported plastics ingestion is insufficient to impede sexual maturation, but population declines are possible. The method is readily applicable to other species impacted by debris ingestion.     

A mathematical and computational approach for integrating the major sources of cell population heterogeneity

Several approaches have been used in the past to model heterogeneity in bacterial cell populations, with each approach focusing on different source(s) of heterogeneity. However, a holistic approach that integrates all the major sources into a comprehensive framework applicable to cell populations is still lacking.In this work we present the mathematical formulation of a cell population master equation (CPME) that describes cell population dynamics and takes into account the major sources of heterogeneity, namely stochasticity in reaction, DNA-duplication, and division, as well as the random partitioning of species contents into the two daughter cells. The formulation also takes into account cell growth and respects the discrete nature of the molecular contents and cell numbers. We further develop a Monte Carlo algorithm for the simulation of the stochastic processes considered here. To benchmark our new framework, we first use it to quantify the effect of each source of heterogeneity on the intrinsic and the extrinsic phenotypic variability for the well-known two-promoter system used experimentally by Elowitz et al. (2002). We finally apply our framework to a more complicated system and demonstrate how the interplay between noisy gene expression and growth inhibition due to protein accumulation at the single cell level can result in complex behavior at the cell population level.The generality of our framework makes it suitable for studying a vast array of artificial and natural genetic networks. Using our Monte Carlo algorithm, cell population distributions can be predicted for the genetic architecture of interest, thereby quantifying the effect of stochasticity in intracellular reactions or the variability in the rate of physiological processes such as growth and division. Such in silico experiments can give insight into the behavior of cell populations and reveal the major sources contributing to cell population heterogeneity.     

Pattern of variation in avian population growth rates

A central question in population ecology is to understand why population growth rates differ over time. Here, we describe how the long-term growth of populations is not only influenced by parameters affecting the expected dynamics, for example form of density dependence and specific population growth rate, but is also affected by environmental and demographic stochasticity. Using long-term studies of fluctuations of bird populations, we show an interaction between the stochastic and the deterministic components of the population dynamics: high specific growth rates at small densities r(1) are typically positively correlated with the environmental variance sigma(e)(2). Furthermore, theta, a single parameter describing the form of the density regulation in the theta-logistic density-regulation model, is negatively correlated with r(1). These patterns are in turn correlated with interspecific differences in life-history characteristics. Higher specific growth rates, larger stochastic effects on the population dynamics and stronger density regulation at small densities are found in species with large clutch sizes or high adult mortality rates than in long-lived species. Unfortunately, large uncertainties in parameter estimates, as well as strong stochastic effects on the population dynamics, will often make even short-term population projections unreliable. We illustrate that the concept of population prediction interval can be useful in evaluating the consequences of these uncertainties in the population projections for the choice of management actions.     

Competition between two microbial populations in a nonmixed environment: effect of cell random motility

In a nonmixed environment, bacterial population growth can be influenced significantly by cell motility properties as well as by growth kinetic properties. Therefore, in a situation of competition between two bacterial populations for a single chemical nutrient in a nonmixed environment, the outcome may depend upon the respective cell motility properties. In this article, the authors have presented a simple mathematical model for competitive growth of two randomly motile (i.e., possessing no chemotactic behavior) populations in a finite nonmixed region. An understanding of the behavior of this model should provide insight into the behavior of a number of common microbial competition problems. Analysis of this model yields the following results: (1) There may be as many as three possible non-trivial steady-state (or long-time) configurations: when species 1 survives, species 2 dies out; when species 2 survives, species 1 dies out; and species 1 and species 2 coexist. (2) The coexistence state can exist even though one species possesses a smaller intrinsic growth rate constant at all nutrient concentrations, if that same species is sufficiently less motile than the other species. (3) In fact, the species with the smaller maximum specific growth rate may grow to a larger population than the other. (4) The possibility of coexistence can be decided essentially from the results for single population growth.     

Positive feedbacks between bottom-up and top-down controls promote the formation and toxicity of ecosystem disruptive algal blooms: A modeling study

Harmful algal blooms that disrupt and degrade ecosystems (ecosystem disruptive algal blooms, EDABs) are occurring with greater frequency and severity with eutrophication and other adverse anthropogenic alterations of coastal systems. EDAB events have been hypothesized to be caused by positive feedback interactions involving differential growth of competing algal species, low grazing mortality rates on EDAB species, and resulting decreases in nutrient inputs from grazer-mediated nutrient cycling as the EDAB event progresses. Here we develop a stoichiometric nutrient–phytoplankton–zooplankton (NPZ) model to test a conceptual positive feedback mechanism linked to increased cell toxicity and resultant decreases in grazing mortality rates in EDAB species under nutrient limitation of growth rate. As our model EDAB alga, we chose the slow-growing, toxic dinoflagellate Karenia brevis, whose toxin levels have been shown to increase with nutrient (nitrogen) limitation of specific growth rate. This species was competed with two high-nutrient adapted, faster-growing diatoms (Thalassiosira pseudonana and Thalassiosira weissflogii) using recently published data for relationships among nutrient (ammonium) concentration, carbon normalized ammonium uptake rates, cellular nitrogen:carbon (N:C) ratios, and specific growth rate. The model results support the proposed positive feedback mechanism for EDAB formation and toxicity. In all cases the toxic bloom was preceded by one or more pre-blooms of fast-growing diatoms, which drew dissolved nutrients to low growth rate-limiting levels, and stimulated the population growth of zooplankton grazers. Low specific grazing rates on the toxic, nutrient-limited EDAB species then promoted the population growth of this species, which further decreased grazing rates, grazing-linked nutrient recycling, nutrient concentrations, and algal specific growth rates. The nutrient limitation of growth rate further increased toxin concentrations in the EDAB algae, which further decreased grazing-linked nutrient recycling rates and nutrient concentrations, and caused an even greater nutrient limitation of growth rate and even higher toxin levels in the EDAB algae. This chain of interactions represented a positive feedback that resulted in the formation of a high-biomass toxic bloom, with low, nutrient-limited specific growth rates and associated high cellular C:N and toxin:C ratios. Together the elevated C:N and toxin:C ratios in the EDAB algae resulted in very high bloom toxicity. The positive feedbacks and resulting bloom formation and toxicity were increased by long water residence times, which increased the relative importance of grazing-linked nutrient recycling to the overall supply of limiting nutrient (N).     

Identification and characterization of the regulatory elements of the inducible acetamidase operon from Mycobacterium smegmatis

The highly inducible acetamidase promoter from Mycobacterium smegmatis has been used as a tool in the study of mycobacterial genetics. The 4.2 kb acetamidase operon contains four putative open reading frames (ORFs) (amiC, amiA, amiD, and amiS) upstream of the 1.2 kb acetamidase ORF (amiE). In this article, using electrophoretic mobility shift assay and promoter probe analyses with a lacZ reporter system, we show the position of three putative operators within the acetamidase operon in M. smegmatis. Results from these studies reinforce previous findings about the involvement of multiple promoters in the regulation of acetamidase gene expression. Each of the identified operators are positioned upstream of the respective promoter reported in previous studies. We also found that the crude cell lysate of M. smegmatis containing potential regulators, obtained from bacteria grown under inducing or noninducing conditions, binds to specific operators. The binding affinity of each operator with its cognate regulator is significantly different from the other. This supports not only the previous model of acetamidase gene regulation in M. smegmatis but also explains the role of these operators in controlling the expression of respective promoters under different growth conditions.     

Growth characteristics and growth modeling of <Emphasis Type="Italic">Microcystis aeruginosa</Emphasis> and <Emphasis Type="Italic">Planktothrix agardhii</Emphasis> under iron limitation

Although iron is a key nutrient for algal growth just as are nitrogen and phosphorus in aquatic systems, the effects of iron on algal growth are not well understood. The growth characteristics of two species of cyanobacteria, Microcystis aeruginosa and Planktothrix agardhii, in iron-limited continuous cultures were investigated. The relationships between dissolved iron concentration, cell quota of iron, and population growth rate were determined applying two equations, Monod’s and Droop’s equations. Both species produced hydroxamate-type siderophores, but neither species produced catechol-type siderophores. The cell quota of nitrogen for both M. aeruginosa and P. agardhii decreased with decreasing cell quota of iron. The cell quota of phosphorus for M. aeruginosa decreased with decreasing cell quota of iron, whereas those for P. agardhii did not decrease. Iron uptake rate was measured in ironlimited batch cultures under different degrees of iron starvation. The results of the iron uptake experiments suggest that iron uptake rates are independent of the cell quota of iron for M. aeruginosa and highly dependent on the cell quota for P. agardhii. A kinetic model under iron limitation was developed based on the growth characteristics determined in our study, and this model predicted accurately the algal population growth and iron consumption. The model simulation suggested that M. aeruginosa is a superior competitor under iron limitation. The differences in growth characteristics between the species would be important determinants of the dominance of these algal species.