Individual-based modelling of bacterial cultures to study the microscopic causes of the lag phase

The lag phase has been widely studied for years in an effort to contribute to the improvement of food safety. Many analytical models have been built and tested by several authors. The use of Individual-based Modelling (IbM) allows us to probe deeper into the behaviour of individual cells; it is a bridge between theories and experiments when needed. INDividual DIScrete SIMulation (INDISIM) has been developed and coded by our group as an IbM simulator and used to study bacterial growth, including the microscopic causes of the lag phase. First of all, the evolution of cellular masses, specifically the mean mass and biomass distribution, is shown to be a determining factor in the beginning of the exponential phase. Secondly, whenever there is a need for an enzyme synthesis, its rate has a direct effect on the lag duration. The variability of the lag phase with different factors is also studied. The known decrease of the lag phase with an increase in the temperature is also observed in the simulations. An initial study of the relationship between individual and collective lag phases is presented, as a complement to the studies already published. One important result is the variability of the individual lag times and generation times. It has also been found that the mean of the individual lags is greater than the population lag. This is the first in a series of studies of the lag phase that we are carrying out. Therefore, the present work addresses a generic system by making a simple set of assumptions.     

On the lag phase and initial decline of microbial growth curves

The lag phase is generally thought to be a period during which the cells adjust to a new environment before the onset of exponential growth. Characterizing the lag phase in microbial growth curves has importance in food sciences, environmental sciences, bioremediation and in understanding basic cellular processes. The goal of this work is to extend the analysis of cell growth curves and to better estimate the duration of the lag phase. A non-autonomous model is presented that includes actively duplicating cells and two subclasses of non-duplicating cells. The growth curves depend on the growth and death rate of these three subpopulations and on the initial proportion of each. A deterministic and a stochastic model are both developed and give the same results. A notable feature of the model is the decline of cells during the early stage of the growth curve, and the range of parameters when this decline occurs is identified. A limited growth model is also presented that accounts for the lag, exponential growth and stationary phase of microbial growth curves.     

Analysis and computer simulation of accretion patterns in bacterial cultures

Patterned growth of bacteria created by interactions between the cells and moving gradients of nutrients and chemical buffers is observed frequently in laboratory experiments on agar pour plates. This has been investigated by several microbiologists and mathematicians usually focusing on some hysteretic mechanism, such as dependence of cell uptake kinetics on pH. We show here that a simpler mechanism, one based on cell torpor, can explain patterned growth. In particular, we suppose that the cell population comprises two subpopulations —one actively growing and the other inactive. Cells can switch between the two populations depending on the quality of their environment (nutrient availability, pH, etc.) We formulate here a model of this system, derive and analyze numerical schemes for solving it, and present several computer simulations of the system that illustrate various patterns formed. These compare favorably with observed experiments.     

Actin filament branching and protrusion velocity in a simple 1D model of a motile cell

We formulate and analyse a 1D model for the spatial distribution of actin density at the leading edge of a motile cell. The model incorporates nucleation, capping, growth and decay of actin filaments, as well as retrograde flow of the actin meshwork and known parameter values based on the literature. Using a simplified geometry, and reasonable assumptions about the biochemical processes, we derive PDEs for the density of actin filaments and their tips. Analytic travelling wave solutions are used to predict how the speed of the cell depends on rates of nucleation, capping, polymerization and membrane resistance. Analysis and simulations agree with experimental profiles for measured actin distributions. Extended versions of the model are studied numerically. We find that our model produces stable travelling wave solutions with reasonable cell speeds. Increasing the rate of nucleation of filaments (by the actin related protein Arp2/3) or the rate of actin polymerization leads to faster cell speed, whereas increasing the rate of capping or the membrane resistance reduces cell speed. We consider several variants of nucleation (spontaneous, tip, and side branching) and find best agreement with experimentally measured spatial profiles of filament and tip density in the side branching case.     

不同生长季节黑果枸杞的根际细菌群落结构

【目的】黑果枸杞是一种耐盐植物,是我国西北干旱区盐渍土改良的优良植物物种,其根际土壤细菌群落结构在不同生长时期的变化特征尚不清楚。【方法】本研究采用Illumina MiSeq高通量测序研究了黑果枸杞3个生长阶段的根际土壤细菌群落结构的动态变化。【结果】所有样品中共获得317467条序列,对应于7028个细菌/古细菌OTUs。根际土壤细菌群落的α多样性显著高于非根际土壤。衰老期根际细菌的多样性和丰富度明显低于营养生长期和花/果期。变形菌门和酸杆菌门的相对丰度随生长时期的演变而逐渐降低,而蓝细菌门则相反。厚壁菌门的丰度在衰老期明显高于营养生长期和花/果期。优势属的组成也随生长期的演变而改变,营养生长期、花/果期、衰老期的优势属数量分别为17、16、4,且组成也具有差异。相似性分析表明营养生长期和花/果期的根际细菌群落具有很高的相似性,衰老期根际细菌群落组成与生长期和花/果期具有很高差异,然而与非根际土壤的群落结构具有较高的相似性。【结论】根际土壤细菌群落多样性和组成随生长期的改变而表现出明显的动态变异性,表明黑果枸杞生长时期对根际土壤细菌群落结构具有重要的影响。     

Dynamics of novel forests of Castilla elastica in Puerto Rico: from species to ecosystems

Novel forests (NFs)—forests that contain a combination of introduced and native species—are a consequence of intense anthropogenic disturbances and the natural resilience of disturbed ecosystems. The extent to which NFs have similar forest function as comparable native secondary forests is a matter of debate in the scientific community. Little is known about the performance of individual species in those forests. This study focuses on the functional attributes of Castilla elastica NFs in Puerto Rico and on the differences between introduced and native species growing side by side in these forests. Rates of processes measured here were later compared with data from literature about NSFs. I hypothesize that juvenile plants of C. elastica in NFs have higher survival rate than those of native species and that C. elastica trees have faster biomass fluxes than native trees. To test the hypotheses, I measured survival rates of juvenile plants and tree growth and characterized the aboveground litter fluxes and storage. Although juvenile plants of native species displayed higher survival rates than those of C. elastica (53% vs. 28%), the latter was dominant in the understory (96%). Stand biomass growth rate was 2.0 ± 0.4 (average ± one standard deviation) Mg·ha⁻¹·year⁻¹ for the whole forest, and Guarea guidonia, a native species, exhibited the highest tree growth. Total litter fall was 9.6 ± 0.5 Mg·ha⁻¹·year⁻¹, and mean litter standing stock was 4.4 ± 0.1 Mg·ha⁻¹. Castilla elastica litter fall decomposed twice as fast as that of native species (5.8 ± 1.1 vs. 3.03 ± 1 k·year⁻¹). Literature comparisons show that the present NFs differ in some rates of processes from NSFs. This study brings unique and detailed supporting data about the ecological dynamics under mature novel forest stands. Further comprehensive studies about NFs are important to strengthen the body of knowledge about the wide range of variation of emerging tropical ecosystems. Due to the large increase in the area covered by NFs, greater attention is needed to understand their functioning, delivery of ecological services and management requirements.     

Using a stem cell and progeny model to illustrate the relationship between cell cycle times of in vivo human tumour cell tissue populations, in vitro primary cultures and the cell lines derived from them

There is increasing evidence that the growth of human tumours is driven by a small proportion of tumour stem cells with self-renewal properties. Multiplication of these cells leads to loss of self-renewal and after division for a finite number of times the cells undergo programmed cell death. Cell cycle times of human cancers have been measured in vivo and shown to vary in the range from two days to several weeks, depending on the individual. Cells cultured directly from tumours removed at surgery initially grow at a rate comparable to the in vivo rate but continued culture leads to the generation of cell lines that have shorter cycle times (1–3 days). It has been postulated that the more rapidly growing sub-population exhibits some of the properties of tumour stem cells and are the precursors of a slower growing sub-population that comprise the bulk of the tumour. We have previously developed a mathematical model to describe the behaviour of cell lines and we extend this model here to describe the behaviour of a system with two cell populations with different kinetic characteristics and a precursor–product relationship. The aim is to provide a framework for understanding the behaviour of cancer tissue that is sustained by a minor population of proliferating stem cells.     

香溪河库湾浮游藻类种类演替及水华发生趋势分析

三峡水库一期工程蓄水前半年(2003年1月—2003年6月)和蓄水后(2003年7月—2004年12月)期间,香溪河库湾浮游藻类的种类演替和数量变化的调查结果。与蓄水前的数据相比,蓄水仅半年绿藻的种类数就明显增加,约相当于蓄水前的3倍;硅藻的种类数略有减少;其余各门藻类的种类数亦有轻度变化。整个调查期间,藻类细胞密度和生物量的最高峰值均出现在S6采样点,细胞密度达6.93×107cells/L(2004年6月),生物量达87.24mg/L(2004年3月),其余采样点基本显示由北向南依次递减的趋势。本文参考早期资料并比较不同类型水域中藻类的种类演替和垂直分布情况,对三峡湖北库区水域水华的发生趋势进行了分析,初步认为三峡干流江段形成藻类水华的几率较小;流动的支流水域,在阳光充足、水温逐渐升高的春季容易发生藻类水华;在较封闭的静水区则随时都有发生藻类水华的可能性。     

Recent progress in modelling and simulation of three-dimensional tumor growth and treatment

This paper illustrates how to apply methods of systems analysis, control theory and simulation to the field of biology and medicine. For this purpose normal and abnormal cell growth has been modelled at different levels. It was possible to simulate three-dimensional tumor growth and different kinds of treatment. The paper shows how tumor treatment may be optimized in the long run using computer simulation experiments as a powerful new tool prior to clinical therapy.     

Phosphorus supply shifts the quotas of multiple elements in algae and Daphnia: ionomic basis of stoichiometric constraints

The growth rate hypothesis posits that the rate of protein synthesis is constrained by phosphorus (P) supply. P scarcity invokes differential expression of genes involved in processing of most if not all elements encompassing an individual (the ionome). Whether such ionome‐wide adjustments to P supply impact growth and trophic interactions remains unclear. We quantified the ionomes of a resource‐consumer pair in contrasting P supply conditions. Consumer growth penalty was driven by not only P imbalance between trophic levels but also imbalances in other elements, reflecting complex physiological adjustments made by both the resource and the consumer. Mitigating such imbalances requires energy and should impact the efficiency at which assimilated nutrients are converted to biomass. Correlated shifts in the handling of multiple elements, and variation in the supplies of such elements could underlie vast heterogeneity in the rates at which organisms and ecosystems accrue biomass as a function of P supply.     

An evaluation of models for the effect of plasmid copy number on bacterial growth rate

The rates of growth of recombinant bacteria depend on their plasmid content. This is modelled by expressing the specific growth rate in terms of the number of copies of the plasmid per cell. Three models in common use have been tested with different Escherichia coli strains and one strain of Bacillus stearothermophilus containing different plasmids. While no particular model was decisively better than others for all data, that of Bentley & Quiroga (Biotechnol. Bioeng. 1993, 42: 222–234) was the best for specific growth rates which vary inversely with the plasmid copy number, and a modified form of the model of Satyagal & Agarwal (Biotechnol. Bioeng. 1989, 33: 1135–1144) was the best for growth rates which increase with the copy number. The differences appear to be linked to the plasmid replication mechanisms. Contrary to some claims, no model portrayed the experimentally observed inflection points.     

Modeling the role of IL-2 in the interplay between CD4+ helper and regulatory T cells: Assessing general dynamical properties

Mathematical models accounting for well-known evidences relating to the dynamics of interleukin 2, helper and regulatory T cells are presented. These models extend an existent model (the so-called cross-regulation model of immunity), by assuming IL-2 as the growth factor produced by helper cells, but used by both helper and regulatory cells to proliferate and survive. Two model variants, motivated by current literature, are explored. The first variant assumes that regulatory cells suppress helper cells by limiting IL-2 production and consuming the available IL-2; i.e. they just trigger competition for IL-2. The second model variant adds to the latter competitive mechanism the direct inhibition of helper cells activation by regulatory cells. The extended models retain key dynamical features of the cross-regulation model. But such reasonable behavior depends on parameter constraints, which happen to be realistic and lead to interesting biological discussions. Furthermore, the introduction of IL-2 in these models breaks the local/specific character of interactions, providing new properties to them. In the extended models, but not in the cross-regulation model, the response triggered by an antigen affects the response to other antigens in the same lymph node. The first model variant predicts an unrealistic coupling of the immune reactions to all the antigens in the lymph node. In contrast, the second model variant allows the coexistent of concomitant tolerant and immune responses to different antigens. The IL-2 derived from an ongoing immune reaction reinforces tolerance to other antigens in the same lymph node. Overall the models introduced here are useful extensions of the cross-regulation formalism. In particular, they might allow future studies of the effect of different IL-2 modulation therapies on CD4+ T cell dynamics.     

Relationship between milk energy intake and growth rate in suckling mammalian young at peak lactation: an updated meta-analysis

The milk energy intakes and growth rates of suckling young at peak lactation of 62 mammalian species and subspecies, all measured using either the weigh–suckle–weigh, the isotope dilution or the isotope transfer method, were evaluated. The mean daily gross energy intakes (GEI) were as low as 12 kJ in a small rodent (mouse) to as high as 249 MJ in a phocid seal (hooded seal Cystophora cristata ), while the daily growth rates at peak lactation ranged from 0.4 to 5.9 kg. Several allometric equations were calculated to explore the relationships between gross energy intake via milk and body weight (BW), growth rate and BW, as well as between gross energy intake via milk and growth rate. The results suggest that both GEI via milk and growth rates are proportional to BW to the power of 0.82. Accordingly, the metabolic weight of suckling mammalian young should be expressed as BW^0.82. The predictive values of the calculated equations indicate that suckling young at peak lactation consume c . 883 kJ day⁻¹ kg⁻¹ BW^0.82 and have growth rates of 32 g day⁻¹ kg⁻¹ BW^0.82. However, large deviations for some species and few outliers were found. These equations could be used to predict values for species that have not been studied, provided that the BW falls within the range of weights used to derive the equations.     

Gene Regulation in Continuous Cultures: A Unified Theory for Bacteria and Yeasts

During batch growth on mixtures of two growth-limiting substrates, microbes consume the substrates either sequentially (diauxie) or simultaneously. The ubiquity of these growth patterns suggests that they may be driven by a universal mechanism common to all microbial species. Recently, we showed that a minimal model accounting only for enzyme induction and dilution, the two processes that occur in all microbes, explains the phenotypes observed in batch cultures of various wild-type and mutant/recombinant cells (Narang and Pilyugin in J. Theor. Biol. 244:326–348, 2007). Here, we examine the extension of the minimal model to continuous cultures. We show that: (1) Several enzymatic trends, attributed entirely to cross-regulatory mechanisms, such as catabolite repression and inducer exclusion, can be quantitatively explained by enzyme dilution. (2) The bifurcation diagram of the minimal model for continuous cultures, which classifies the substrate consumption pattern at any given dilution rate and feed concentrations, provides a precise explanation for the empirically observed correlations between the growth patterns in batch and continuous cultures. (3) Numerical simulations of the model are in excellent agreement with the data. The model captures the variation of the steady state substrate concentrations, cell densities, and enzyme levels during the single- and mixed-substrate growth of bacteria and yeasts at various dilution rates and feed concentrations. This variation is well approximated by simple analytical expressions that furnish deep physical insights. (4) Since the minimal model describes the behavior of the cells in the absence of cross-regulatory mechanisms, it provides a rigorous framework for quantifying the effect of these mechanisms. We illustrate this by analyzing several data sets from the literature.     

Kinetic analysis of the effect of cell density on hybridoma cell growth in batch culture

The effect of cell density on cell growth was investigated in a suspension batch culture of hybridoma cells. The specific growth rate was found to increase with increasing initial cell density and then to decrease with further increases in initial cell density. In order to quantitatively describe the dependence of specific growth rate on cell density, a kinetic model is proposed, which satisfactorily represents the experimental data.     

Nonlinearity in the Growth of Bacterial Colonies: Conditions and Causes

The universally recognized kinetic model of colony growth, introduced by Pirt, predicts a linear increase of colony size. The linearity follows from the assumption that the colony expands through the growth of only such cells that are located immediately behind the moving colony front, in the so-called peripheral zone of constant width and density. In this work, Pirt's model was tested on two bacteria—Alcaligenes sp. and Pseudomonas fluorescens—having markedly distinct cultural properties and grown on an agarized medium with pyruvate. The colony size dynamics was followed for different densities of the inoculum, ranging from a single cell to a microdroplet of bacterial suspension (10⁵–10⁶ cells), and for different depths of the agar layer, determining the amount of available substrate. A linear growth mode was observed only with P. fluorescens and only in the case of growth from a microdroplet. When originating from a single cell, colonies of both organisms displayed nonlinear growth with a distinct peak of K _r (the rate of colony radius increase) occurring after 2–3 days of growth. The growth of P. fluorescens colonies showed virtually no dependence on the depth of the agarized medium, whereas the rate of colony size increase of Alcaligenes sp. turned out to be directly related to the medium layer thickness. The departure from linearity is consistently explained by a new kinetic scheme stipulating a possible contribution to the colony growth not only of peripheral cells but also (much more distinct in Alcaligenes) of cells at the colony center. The colony growth dynamics is determined not only by the concentration of the limiting substrate but also by the amount of autoinhibitor, the synthesis of which is governed by the age of cells. The distinctions of growth from a single cell and microdroplet could also originate as a result of dissociation into the R- and S-forms and competition between the corresponding subpopulations for oxygen and the common substrate.     

Competition and coexistence in host-parasite systems: the myxomatosis case

Co-circulation of several strains of parasites has been observed in many host-parasite systems. However, simple epidemiological models cannot sustain this coexistence. In this work we study the coexistence of viral strains in the myxomatosis case. Myxomatosis, a highly lethal disease of the European rabbit, has been used in Australia and Europe as a biological control of rabbit populations. A few years after its introduction, the original highly virulent strains were almost completely replaced by field strains covering a wide range of virulence. Here, we study several mechanisms that may explain the field observations. First we considered spatial heterogeneity. The establishment of any strain over regions occupied by host populations may delay the spread of any superior competitive virus strain, producing global coexistence in the long term. On the other hand, sub-populations with different resistance levels in epidemiological contact, as observed in the field, can maintain several different virus strains co-circulating. The second class of mechanism introduces diversity among hosts of a local population sharing a territory. We considered different classes of host resistance to myxomatosis: belonging to a resistance class is a random fact. Host age-dependent resistance is also especially considered. These types of population heterogeneity can sustain local coexistence for many years, although exclusion takes place for long enough periods. The concurrent action of both types of mechanisms could explain why the diversity of virus strains is sustained, and the local coexistence. Finally, we briefly discuss the influence of host genetic dynamics in the coevolution of the system.     

Plasticity in newt metamorphosis: the effect of predation at embryonic and larval stages

1. Some organisms under variable predator pressure show induced antipredator defences, whose development incurs costs and may be associated with changes to later performance. This may be of especial relevance to animals with complex life histories involving metamorphosis. 2. This study examines the effect of predation environment, experienced both during embryonic and larval stages, on palmate newt (Triturus helveticus) metamorphosis. Newt eggs were raised until hatching with or without exposure to chemical cues from brown trout (Salmo trutta), and larval development was monitored in the presence or absence of the cues. 3. Exposure to predator cues during the embryonic stage resulted in higher growth rates at the larval stage, reduced time to metamorphosis and size at metamorphosis. Metamorphs also had narrower heads and shorter forelimbs than those from predator‐free treatments. In contrast, exposure to predator cues during the larval stage did not affect metamorph characteristics. 4. These results indicate that developing embryos are sensitive to predator chemical cues and that the responses can extend to later stages. Reversion of induced defences when predation risk ceased was not detected. We discuss the possible adaptive significance of these responses.