Modeling Surface Growth of Escherichia coli on Agar Plates

Surface growth of Escherichia coli cells on a membrane filter placed on a nutrient agar plate under various conditions was studied with a mathematical model. The surface growth of bacterial cells showed a sigmoidal curve with time on a semilogarithmic plot. To describe it, a new logistic model that we presented earlier (H.Fujikawa et al., Food Microbiol. 21:501-509, 2004) was modified. Growth curves at various constant temperatures (10 to 34°C) were successfully described with the modified model (model III). Model III gave better predictions of the rate constant of growth and the lag period than a modified Gompertz model and the Baranyi model. Using the parameter values of model III at the constant temperatures, surface growth at various temperatures was successfully predicted. Surface growth curves at various initial cell numbers were also sigmoidal and converged to the same maximum cell numbers at the stationary phase. Surface growth curves at various nutrient levels were also sigmoidal. The maximum cell number and the rate of growth were lower as the nutrient level decreased. The surface growth curve was the same as that in a liquid, except for the large curvature at the deceleration period. These curves were also well described with model III. The pattern of increase in the ATP content of cells grown on a surface was sigmoidal, similar to that for cell growth. We discovered several characteristics of the surface growth of bacterial cells under various growth conditions and examined the applicability of our model to describe these growth curves.     

A semicontinuous culture model that links cell growth to extracellular nutrient concentration

During semicontinuous culture, a sample of fixed volume is removed at regular time intervals to make measurements and/or harvest culture components, and an equal volume of fresh medium is immediately added to the culture, thereby instantaneously enhancing nutrient concentrations and diluting cell concentration. The resulting cell concentration versus time curve (i.e., the actual cell growth curve) has a saw-toothed appearance because of the periodic dilution of cell concentration. The observed cell concentrations correspond to the peaks of the saw-toothed curve. Cell growth rates are estimated from the locus of observed cell concentrations (i.e., from the apparent growth curve obtained by connecting the peaks of the saw-toothed curve). The sole preexisting model (Fencl's mode) for estimating cell growth rate is valid only when the cells are growing exponentially at a constant rate between samplings. This model has limited validity: despite the periodic enhancement of nutrient concentration, cell growth between samplings eventually causes nutrient depletion, and the cells cease to grow exponentially. Failure to recognize the limits of validity for Fencl' model has resulted in many erroneous applications of the model and, consequently, many incorrect estimates of cell growth rates. To provide a means for correctly estimating cell growth rates, Fencl's exponential model was extended, and a new model that describes the effects of nutrient depletion on cell growth in semi-continuous culture was obtained. The new model shows that exhaustion of a single growth-limiting nutrient in semicontinuous culture causes the locus of cell concentrations observed at time intervals of Deltat to follow a logistic growth curve. The actual cell growth rate was shown to equal the apparent logistic growth rate plus the effective dilution rate -Deltat(-1) In (1 - f), where f is the ratio of sample volume to total culture volume. Moreover, the model predicts that both the apparent logistic growth rate and the apparent steady-state cell concentration should rise linearly with the concentration of growth-limiting nutrient in the input medium, but fall linearly with increases in the effective dilution rate. The new logistic model for nutrient-limited cell growth in semicontinuous culture was successfully tested using published data for Asterionella formosa, Cyclotella meneghiniana, Daucus carota, and strain L mouse cells.     

Dynamic mathematical model to predict microbial growth and inactivation during food processing.

Many sigmoidal functions to describe a bacterial growth curve as an explicit function of time have been reported in the literature. Furthermore, several expressions have been proposed to model the influence of temperature on the main characteristics of this growth curve: maximum specific growth rate, lag time, and asymptotic level. However, as the predictive value of such explicit models is most often guaranteed only at a constant temperature within the temperature range of microbial growth, they are less appropriate in optimization studies of a whole production and distribution chain. In this paper a dynamic mathematical model--a first-order differential equation--has been derived, describing the bacterial population as a function of both time and temperature. Furthermore, the inactivation of the population at temperatures above the maximum temperature for growth has been incorporated. In the special case of a constant temperature, the solution coincides exactly with the corresponding Gompertz model, which has been validated in several recent reports. However, the main advantage of this dynamic model is its ability to deal with time-varying temperatures, over the whole temperature range of growth and inactivation. As such, it is an essential building block in (time-saving) simulation studies to design, e.g., optimal temperature-time profiles with respect to microbial safety of a production and distribution chain of chilled foods.     

Dynamic mathematical model to predict microbial growth and inactivation during food processing.

Many sigmoidal functions to describe a bacterial growth curve as an explicit function of time have been reported in the literature. Furthermore, several expressions have been proposed to model the influence of temperature on the main characteristics of this growth curve: maximum specific growth rate, lag time, and asymptotic level. However, as the predictive value of such explicit models is most often guaranteed only at a constant temperature within the temperature range of microbial growth, they are less appropriate in optimization studies of a whole production and distribution chain. In this paper a dynamic mathematical model--a first-order differential equation--has been derived, describing the bacterial population as a function of both time and temperature. Furthermore, the inactivation of the population at temperatures above the maximum temperature for growth has been incorporated. In the special case of a constant temperature, the solution coincides exactly with the corresponding Gompertz model, which has been validated in several recent reports. However, the main advantage of this dynamic model is its ability to deal with time-varying temperatures, over the whole temperature range of growth and inactivation. As such, it is an essential building block in (time-saving) simulation studies to design, e.g., optimal temperature-time profiles with respect to microbial safety of a production and distribution chain of chilled foods.     

Patterns of bacterial destruction in solutions by microwave irradiation

The patterns of destruction of several kinds of bacterial cells suspended in solutions by microwave irradiation were studied. The survival curves of Escherichia coli and Staphylococcus aureus were similar and approximated a set of three linear phases. The curves of Pseudomonas fluorescens and Bacillus cereus spores shifted to the shorter and longer irradiation periods, respectively. The rate constant and initial time of destruction for each linear phase of the survival curve were compared among these organisms. When irradiated E. coli cells were incubated in an agar with a high salt level, fewer cells were recovered. The curve of E. coli cells in the logarithmic phase of growth shifted to shorter exposure periods. There were no significant differences in the survival curves of E. coli cells grown at temperatures of 22–36°C, whereas the curve of cells grown at 44°C shifted to longer periods.     

Mechanics of edematous lungs.

Using the parenchymal marker technique, we measured pressure (P)-volume (P-V) curves of regions with volumes of approximately 1 cm3 in the dependent caudal lobes of oleic acid-injured dog lungs, during a very slow inflation from P = 0 to P = 30 cmH2O. The regional P-V curves are strongly sigmoidal. Regional volume, as a fraction of volume at total lung capacity, remains constant at 0.4-0.5 for airway P values from 0 to approximately 20 cmH2O and then increases rapidly, but continuously, to 1 at P = approximately 25 cmH2O. A model of parenchymal mechanics was modified to include the effects of elevated surface tension and fluid in the alveolar spaces. P-V curves calculated from the model are similar to the measured P-V curves. At lower lung volumes, P increases rapidly with lung volume as the air-fluid interface penetrates the mouth of the alveolus. At a value of P = approximately 20 cmH2O, the air-fluid interface is inside the alveolus and the lung is compliant, like an air-filled lung with constant surface tension. We conclude that the properties of the P-V curve of edematous lungs, particularly the knee in the P-V curve, are the result of the mechanics of parenchyma with constant surface tension and partially fluid-filled alveoli, not the result of abrupt opening of airways or atelectatic parenchyma.     

Isolation and characterization of mesenchymal stem cells from chicken bone marrow

The bone marrow mesenchymal stem cells (BMSCs) are multipotent stem cells, which can differentiate in vitro into many cell types. However, the vast majority of experimental materials were obtained from human, mouse, rabbit and other mammals, but rarely in poultry. So, in this study, Thirty- to sixty-day old chicken was chosen as experimental animal, to isolate and characterize BMSCs from them. To investigate the biological characteristics of chicken BMSCs, immunofluorescence and RT-PCR were used to detect the characteristic surface markers of BMSCs. Growth curves were drawn in accordance with cell numbers. To assess the differentiation capacity of the BMSCs, cells were induced to differentiate into osteoblasts, adipocytes, and endothelial cells. The surface markers of BMSCs, CD29, CD44, CD31, CD34, CD71 and CD73, were detected by immunofluorescence and RT-PCR assays. The growth curves of different passages were all typically sigmoidal. Karyotype analysis showed that these in vitro cultured cells were genetically stable. In addition, BMSCs were successfully induced to differentiate into osteoblasts, adipocytes, and endothelial cells. The results suggest that the BMSCs isolated from chicken possess similar biological characteristics with those separated from other species, and their multi-lineage differentiation potentiality herald a probable application for cellular transplant therapy in tissue engineering.     

Growth of the surface of Corynebacterium diphtheriae

Surface structure and growth of the surface of Corynebacterium diphtheriae mitis strain were investigated by scanning electron microscopy and the immunofluorescence technique. The surface of the cell revealed by the scanning electron microscope showed a few elevated circular zones which encompassed the cell. The cell diameter increased at this zone and this gave the club-shaped appearance to this species. The cell surface labeled with specific antibodies against the whole bacterial cell and tagged with ferritin remained at a constant length during cell division cycles and the new cell surface emerged from the polar ends of the cell. This new wall surface was completely devoid of the ferritin particles indicating that the cell wall component on the old preexistent wall was completely conserved. A similar finding was obtained by immunofluorescence microscopy. C. diphtheriae, unlike Bacillus spp., showed apical growth as has been observed in fungal cells.     

Two continuum models for the spreading of myxobacteria swarms

We analyze the phenomenon of spreading of a Myxococcus xanthus bacterial colony on plates coated with nutrient. The bacteria spread by gliding on the surface. In the first few hours, cell growth is irrelevant to colony spread. In this case, bacteria spread through peninsular protrusions from the edge of the initial colony. We analyze the diffusion through the narrowing reticulum of cells on the surface mathematically and derive formulae for the spreading rates. On the time scale of tens of hours, effective diffusion of the bacteria, combined with cell division and growth, causes a constant linear increase in the colony's radius. Mathematical analysis and numerical solution of reaction-diffusion equations describing the bacterial and nutrient dynamics demonstrate that, in this regime, the spreading rate is proportional to the square root of both the effective diffusion coefficient and the nutrient concentration. The model predictions agree with the data on spreading rate dependence on the type of gliding motility.     

Development of a structured model for batch cultures of lactic acid bacteria

A combined stochastic-deterministic model able to predict the growth curve of microorganisms, from inoculation to death, is presented. The proposed model is based on the assumption that microorganisms can experience two different physiological states: non-proliferating and proliferating. The former being the physiological state of the cells right after their inoculation into the new extracellular environment; the latter the state of microorganisms after adaptation to the new medium. To validate the model, a Lactobacillus bulgaricus strain was tested in a medium at pH 4.6 at two different temperatures (42°C and 35°C). Curves representing the bacterial growth cycle were satisfactorily fitted by means of the proposed model. Moreover, due to the mechanistic structure of the proposed model, valuable quantitative information on the following was obtained: rate of conversion of non-proliferating cells into proliferating cells, growth and death rate of proliferating cells, and rate of nutrient consumption.     

Establishment and characteristics of white ear lobe chicken embryo fibroblast line and expression of six fluorescent proteins in the cells

A white ear lobe chicken embryo (WELCE) fibroblast cell bank, containing 322 tubes of frozen cells, was successfully established from primary explants of 57 embryo samples. The cells were morphologically consistent with fibroblasts, and the growth curve was sigmoidal with a population doubling time (PDT) of 48 h. Karyotyping and G-banding indicated a total chromosome number of 2n=78; the rate of diploidy in the cell bank was 97.62%. The cells were also free from bacterial, fungal, viral and mycoplasma contamination. Analysis of lactate dehydrogenase (LDH) and malate dehydrogenase (MDH) isoenzymes ruled out cross-contamination between cells. In order to study exogenous gene expression, six fluorescent proteins were transfected into the WELCE cells. The transfection efficiency of these genes was between 10.1 and 41.9%. The corresponding fluorescence was distributed throughout the cytoplasm and nucleus 24h after transfection. The results indicate that the quality of the cell line meet the quality requirements of the ATCC (American Type Culture Collection).     

Characterization of initial events in bacterial surface colonization by two Pseudomonas species using image analysis

The processes leading to bacterial colonization on solidwater interfaces are adsorption, desorption, growth, and erosion. These processes have been measured individually in situ in a flowing system in real time using image analysis. Four different substrata (copper, silicon, 316 stainless-steel and glass) and 2 different bacterial species (Pseudomonas aeruginosa and Pseudomonas fluorescens) were used in the experiments. The flow was laminar (Re = 1.4) and the shear stress was kept constant during all experiments at 0.75 N m(-2). The surface roughness varied among the substrata from 0.002 mum (for silicon) to 0.015 mum (for copper). Surface free energies varied from 25.1 dynes cm(-1) for silicon to 31.2 dynes cm(-1) for copper. Cell curface hydrophobicity, reported as hydrocarbon partitioning values, ranged from 0.67 for Ps. fluorescens to 0.97 for Ps. aeruginosa.The adsorption rate coefficient varried by as much as a factor of 10 among the combinations of bacterial strain and substratum material, and was positively correlated with surface free energy, the surface roughness of the substratum, and the hydrophobicity of the cells. The probability of desorption decreased with increasing surface free energy and surface roughness of the substratum. Cell growth was inhibited on copper, but replication of cells overlying an initial cell layer was observed with increased exposure time to the cell-containing bulk water. A mathematical model describing cell accumulation on a substratum is presented.     

Modeling as a tool to manage ecosystems under multiple stresses: an application to Lake Ontario

Present-day ecosystem management involves understanding of the synergistic effect of multiple stressors on multiple and frequently nebulous management end-points. An example is the simultaneous management of nutrient load reductions and salmon stocking in Lake Ontario. In this study, a simple whole-lake annual time scale model was developed to assess the relationship between these two stressors and various ecosystem responses. The model was used to explore the utility of some possible management end-points for ecosystem health. In historical simulations, production per stocked fish and salmon survival appeared to be good indicators, while nutrient recycling rate and average ecosystem-wide food limitation were found to be fairly unresponsive to the two stressors. The model was further used to predict long term averages of salmon biomass and selected health indicators at various sustained loading and stocking rates. Salmon biomass increased with stocking rate at all stocking rates examined, but the rate of increase declined somewhat at high stocking rates. The response of salmon biomass to nutrient loading appeared to be approximately sigmoidal i.e. there was a nutrient threshold below which fish biomass could not be sustained and another nutrient threshold above which salmon biomass either remained constant or even decreased. The response to either stressor was found to be modified by the value of the other stressor, illustrating the importance of ecosystem-level models for aquatic ecosystem management.     

Glucose-6-phosphate dehydrogenase activity as a marker of resolution in the separation of rabbit erythroblasts according to their stage of maturation

The glucose 6 phosphate dehydrogenase (G6PD) activity of erythroblasts, separated at different advancing stages of development by the velocity sedimentation technique at unit gravity, shows a characteristic sigmoidal curve. The G6PD activity is high and constant in the dividing compartment, steeply declines between the polychromatic and the orthochromatic stage, and returns almost constant during the development from the orthochromatic to the reticulocyte stage. This report focuses on the possibility to use the G6PD activity curve to indicate a loss of resolution in the fractionation process. For this purpose two parameters of the graph were used: (h) which represents the distance between the two constant phases of the curve and (s) the slope of the decreasing part of the graph. In this view we have fractionated on a 400 ml linear gradient of sucrose (1%-2% in PBS), increasing amounts of bone marrow cells from anaemic rabbit. Suspensions of 100, 200, 350 and 700 millions of cells (25 ml in PBS) were separated in different experiments and the obtained G6PD activity curves were compared. We have seen that the two parameters h and s remained constant up to 200 millions of cells, while they declined markedly when the number of cells loaded on the gradient increased to 350 and 700 millions. The separated cells were grouped into three sets of fractions: fraction I with cells belonging to the high and constant phase of the graph, fraction II to the decreasing phase and fraction III to the low and constant activity phase.(ABSTRACT TRUNCATED AT 250 WORDS)     

Multi-parameter flow cytometry and cell sorting reveal extensive physiological heterogeneity in <Emphasis Type="Italic">Bacillus cereus</Emphasis> batch cultures

Based on two staining protocols, DiOC₆(3)/propidium iodide (PI) and RedoxSensor Green (an indicator of bacterial reductase activity)/PI, multi-parameter flow cytometry and cell sorting has identified at least four distinguishable physiological states during batch cultures of Bacillus cereus. Furthermore, dependent on the position in the growth curve, single cells gave rise to varying numbers of colonies when sorted individually onto nutrient agar plates. These growing colonies derived from a single cell had widely different lag phases, inferred from differences in colony size. This further highlights the complex population dynamics of bacterial monocultures and further demonstrates that individual bacterial cells in a culture respond in markedly dissimilar ways to the environment, resulting in a physiologically heterogenous and dynamic population.     

Modeling of the Bacterial Growth Curve

Several sigmoidal functions (logistic, Gompertz, Richards, Schnute, and Stannard) were compared to describe a bacterial growth curve. They were compared statistically by using the model of Schnute, which is a comprehensive model, encompassing all other models. The t test and the F test were used. With the t test, confidence intervals for parameters can be calculated and can be used to distinguish between models. In the F test, the lack of fit of the models is compared with the measuring error. Moreover, the models were compared with respect to their ease of use. All sigmoidal functions were modified so that they contained biologically relevant parameters. The models of Richards, Schnute, and Stannard appeared to be basically the same equation. In the cases tested, the modified Gompertz equation was statistically sufficient to describe the growth data of Lactobacillus plantarum and was easy to use.     

Recovery from nutrient starvation by a marine Vibrio sp.

A marine psychrophilic Vibrio sp., Ant-300, recovered from starvation after the addition of 1 volume of complete nutrient medium to 9 volumes of starvation menstruum. Turbidity (measured by optical density), viable cell counts, cell size (measured from electron micrographs), and cellular concentrations of protein, DNA, and RNA were monitored with recovery time. The usual growth curve of bacterial cultures was observed. On a per viable cell basis, protein, DNA, and RNA increased to maximum values just before cell division and then returned to close to the initial starved-cell value during the stationary phase. Cells under complete starvation conditions or missing only one nutrient in the stationary phase responded with cell division resulting in many smaller cells. The length of the lag phase during recovery was directly proportional to the length of the prior starvation period, even when identical numbers of cells were used for recovery. Cells appeared to pass more deeply into dormancy with starvation time.     

Recovery from nutrient starvation by a marine Vibrio sp

A marine psychrophilic Vibrio sp., Ant-300, recovered from starvation after the addition of 1 volume of complete nutrient medium to 9 volumes of starvation menstruum. Turbidity (measured by optical density), viable cell counts, cell size (measured from electron micrographs), and cellular concentrations of protein, DNA, and RNA were monitored with recovery time. The usual growth curve of bacterial cultures was observed. On a per viable cell basis, protein, DNA, and RNA increased to maximum values just before cell division and then returned to close to the initial starved-cell value during the stationary phase. Cells under complete starvation conditions or missing only one nutrient in the stationary phase responded with cell division resulting in many smaller cells. The length of the lag phase during recovery was directly proportional to the length of the prior starvation period, even when identical numbers of cells were used for recovery. Cells appeared to pass more deeply into dormancy with starvation time.     

Modelling time‐varying low‐temperature‐induced mortality rates for pupae of Tuta absoluta (Gelechiidae,Lepidoptera)

In a series of laboratory experiments, acclimated pupae of Tuta absoluta were exposed to various constant low temperatures in order to estimate their maximum survival times (Kaplan–Meier, Lt_99.99). A Weibull function was fitted to the data points, describing maximum survival time as a function of temperature. In another experiment at ?6°C, the progress of mortality increasing with exposure time was identified. These values were fitted by a sigmoidal function converging asymptotically to 100% mortality for very long exposure times. Analysing mortality data from the maximum survival experiment by a generalized linear model showed a significant common slope parameter (p < .001) that reveals parallelism of the survival curves at each temperature if a log time axis is used. These curves appear stretched (time scaled) if plotted with a nonlogarithmic time axis. By combining these mathematical relations, it was possible to calculate a species‐specific ‘mortality surface’ which exhibits mortalities, depending on temperature and duration of exposure. In order to accumulate hourly mortalities for courses of varying temperatures, an algorithm was developed which yields mortality values from that surface taking into account the attained mortality level. In validation experiments, recorded mortalities were compared against modelled mortalities. Prediction of mortality was partially supported by the model, but pupae experiencing intensely fluctuating temperatures showed decreased mortality, probably caused by rapid cold hardening during exposure. Despite this observation, mortality data converged to distinct levels very close to 100% depending on the intensity of temperature fluctuations that were characteristic for different types of experiments. The highest mortality limit occurred at intensely fluctuating temperatures in laboratory experiments. This constituted a benchmark that was not reached under various field conditions. Thus, it was possible to identify temperature limits for the extinction of field populations of Tuta absoluta pupae.     

Cell population dynamics during the differentiative phase of tissue development

The dynamics of a cell population whose numbers are growing exponentially have been described well by a mathematical model based on the theory of age-dependent branching processes. Such a model, however, does not cover the period following exponential growth when cell differentiation curtails population size. This paper offers an extension to the branching process model to remedy this deficiency. The extended model is ideal for describing embryonic growth; its use is illustrated with data from embryonic retina. The model offers a better computational framework for the interpretation of a variety of data (growth curves of cell numbers, DNA histograms, thymidine labelling indices, FLM curves, BUdR-labelled mitoses curves) because age-distributions can be calculated at any stage of development, not just during exponential growth. Proportions of cells in the various phases of the cell cycle can be computed as growth slows. Such calculations show the gradual transition from a population dominated by cells which are young with respect to cell cycle age to one dominated by those which are old, and the effects such biases have on the proportions of cells in each phase.