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.     

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.     

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.     

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.     

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.     

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.     

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.     

Waning of maternal immunity and the impact of diseases: the example of myxomatosis in natural rabbit populations

Myxomatosis is a leporipoxvirus that infects the european rabbit, inducing a high mortality rate. Observations lead us to hypothesize that a rabbit carrying maternal antibodies (or having recovered) can be infected (or re-infected) upon being exposed (or re-exposed) to the virus. Infection will lead to mild disease, boosting host immune protection. Using a modelling approach we show that this phenomenon may lead to a difference of impact of myxomatosis according to its transmission rate. Young are exposed when they still carry maternal antibodies and develop a mild disease in high transmission populations. Our results show that the impact of myxomatosis is generally higher in epidemic situations compared to populations where the virus circulates all the year. As a consequence, waning of acquired immunity and the continuous supply of newborn along the year may reduce the impact of the disease.     

Mathematical modelling of the Warburg effect in tumour cords

The model proposed here links together two approaches to describe tumours: a continuous medium to describe the movement and the mechanical properties of the tissue, and a population dynamics approach to represent internal genetic inhomogeneity and instability of the tumour. In this way one can build models which cover several stages of tumour progression. In this paper we focus on describing transition from aerobic to purely glycolytic metabolism (the Warburg effect) in tumour cords. From the mathematical point of view this model leads to a free boundary problem where domains in contact are characterized by different sets of equations. Accurate stitching of the solution was possible with a modified ghost fluid method. Growth and death of the cells and uptake of the nutrients are related through ATP production and energy costs of the cellular processes. In the framework of the bi-population model this allowed to keep the number of model parameters relatively small.     

Rational design and optimization of downstream processes of virus particles for biopharmaceutical applications: current advances

The advent of advanced therapies in the pharmaceutical industry has moved the spotlight into virus-like particles and viral vectors produced in cell culture holding great promise in a myriad of clinical targets, including cancer prophylaxis and treatment. Even though a couple of cases have reached the clinic, these products have yet to overcome a number of biological and technological challenges before broad utilization. Concerning the manufacturing processes, there is significant research focusing on the optimization of current cell culture systems and, more recently, on developing scalable downstream processes to generate material for pre-clinical and clinical trials. We review the current options for downstream processing of these complex biopharmaceuticals and underline current advances on knowledge-based toolboxes proposed for rational optimization of their processing. Rational tools developed to increase the yet scarce knowledge on the purification processes of complex biologicals are discussed as alternative to empirical, “black-boxed” based strategies classically used for process development. Innovative methodologies based on surface plasmon resonance, dynamic light scattering, scale-down high-throughput screening and mathematical modeling for supporting ion-exchange chromatography show great potential for a more efficient and cost-effective process design, optimization and equipment prototyping.     

A novel approach for estimating growth phases and parameters of bacterial population in batch culture

Using mathematical analysis, a new method has been developed for studying the growth kinetics of bacterial populations in batch culture. First, sampling data were smoothed with the spline interpolation method. Second, the instantaneous rates were derived by numerical differential techniques and finally, the derived data were fitted with the Gaussian function to obtain growth parameters. We named this the Spline-Numerical-Gaussian or SNG method. This method yielded more accurate estimates of the growth rates of bacterial populations and new parameters. It was possible to divide the growth curve into four different but continuous phases based on changes in the instantaneous rates. The four phases are the accelerating growth phase, the constant growth phase, the decelerating growth phase and the declining phase. Total DNA content was measured by flow cytometry and varied depending on the growth phase. The SNG system provides a very powerful tool for describing the kinetics of bacterial population growth. The SNG method avoids the unrealistic assumptions generally used in the traditional growth equations.     

The segmentation clock in mice: interaction between the Wnt and Notch signalling pathways

In the last few years, the efforts to elucidate the mechanisms underlying the segmentation clock in various vertebrate species have multiplied. Early evidence suggested that oscillations are caused by one of the genes under the Notch signalling pathway (like those of the her or Hes families). Recently, Aulehla et al. [Wnt3a plays a major role in the segmentation clock controlling somitogenesis. Dev. Cell 4, 395-406] discovered that Axin2 (a gene under the Wnt3a signalling pathway) also oscillates in the presomitic mesoderm (PSM) of mice embryos and proposed some mechanisms through which the Notch and Wnt3a pathways may interact. They further suggested that a decreasing concentration of Wnt3a along the PSM may be the gradient the segmentation clock interacts with to form somites. These results were reviewed by Rida et al. [A notch feeling of somite segmentation and beyond. Dev. Biol. 265, 2-22], who introduced a complex clockwork comprising genes Hes1, Lfng (under the Notch pathway), and Axin2, as well as their multiple interactions. In the present work we develop a mathematical model based on the Rida et al. review and use it to tackle some of the questions raided by the Aulehla et al. paper: can the Axin2 feedback loop constitute a clock? Could a decreasing Wnt3a signaling constitute the wavefront, where phase is recorded and the spatial pattern laid down? What is the master oscillator?